An-Najah National University Faculty of Graduate Studies Computational Methods for Solving Nonlinear Volterra Integro - Differential Equation By Farah Khaled Shehadah Abu Thabit Supervisor Prof. Naji Qatanani This Thesis is Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Computiational Mathmatics, Faculty of Graduate Studies, An-Najah National University, Nablus-Palestine. 2019 ii Computational Methods for Solving Nonlinear Volterra Integro - Differential Equation By Farah Khaled Shehadah Abu Thabit This Thesis was defended successfully on 1/12/2019 and approved by: Defense Committee Members Signature - Prof. Naji Qatanani / Supervisor ………………... - Dr. Mahmoud Almanassra / External examiner ………………… - Dr. Adnan Daraghmeh / Internal examiner …....…………… iii Dedication حجرًا,, ألتعثر وضع كافة العراقيل في دربي حجراً كي ال أكون,, و للمن أحبطني,, واستمات “ “ ... بيا !! ,, فزادتني إصرارًا لمصعود عمييا iv Acknowledgments All thanks to God who gave me the strength and determination to implement this achievement. I would sincerely Thanks to Professor Naji Qatanani for guidance, encouragement and supervision during this study and the preparation of the thesis. also,I would like to record my special thanks to my parents " Khaled & Nisreen " and to my sisters "Mahaba, Nour, Menat-Allah and Layan and for my only brother Mutaz for their support, encouragement and great efforts in all stages of my life . In the end, thanks to all the people who helped me in this work. v اإلقرار أنا الموقع أدناه مقدم الرسالة التي تحمل العنوان: Computational Methods for Solving Nonlinear Volterra Integro - Differential Equation إليو اإلشارةما تمت ءباستثناىي نتاج جيدي الخاص, إنماأقر بأن ما اشتممت عميو ىذه الرسالة لة ككل أو أي جزء منيا لم يقدم من قبل لنيل أي درجة عممية أو بحث ن ىذا الرساأحيثما ورد, و أو بحثية أخرى. تعميميةعممي لدى أي مؤسسة Declaration The work provided in this thesis, unless otherwise referenced, is the researcher’s own work, and has not been submitted elsewhere for any degree or qualification. Student’s Name: :اسم الطالب Signature: التوقيع: Date: التاريخ: vi Table of Contents Page Content No. ii Dedication iv Acknowledgments v Declaration ix List of Tables xi List of Figures xii Abstract 1 Introduction 4 Chapter One: Mathematical Preliminaries 5 Classification of Nonlinear Integro -Differential Equations 1.1 5 Types of nonlinear integro - differential equation 1.1.1 7 Singularity of nonlinear integro - differential equation 1.1.2 9 Systems of nonlinear Volterra Integro - Differential Equations 1.2 10 Systems of nonlinear Fredholm Integro - Differential Equations 1.3 10 Kinds of kernel 1.4 12 Existence of the solution of nonlinear Volterra integro- differential equation 1.5 21 Uniqueness of the solution of nonlinear Volterra integro- differential equation 1.6 25 Laplace Transforms 1.7 26 Properties of the Laplace transforms 1.7.1 27 Convolution theorem 1.7.2 28 Chapter Two: Computational Methods for Solving Nonlinear Volterra Integro-Differential Equation 28 Differential transform method with Adomian polynomials (DTM). 2.1 34 Modified Laplace Adomian Decompostion Method 2.2 37 The Variational iteration method (VIM). 2.3 37 Chapter Three: Numerical Examples and results 41 The numerical realization of equation ( ) ( ) Using Differential Transform Method with Adomian Polynomials (DTM) 3.1 45 The numerical realization of equation ( ) ( ) Using the Modified Laplace Adomian Decompostion Method (LADM) 3.2 vii 55 The numerical realization of equation ( ) ( ) Using The Variational Iteration Method (VIM) 3.3 55 The numerical realization of equation ( ) ( ) Using Differential Transform Method with Adomian Polynomials (DTM) . 3.4 57 The numerical realization of equation ( ) ( ) Using Variational Iteration Method (VIM). 3.5 65 The numerical realization of equation ( ) ( ) Using the Modified Laplace Adomian Decompostion Method (LADM) 3.6 62 The numerical realization of equation ( ) ( ) Using the Variational Iteration Method (VIM). 3.7 65 The numerical realization of equation ( ) ( ) Using the Variational Iteration Method (VIM). 3.8 67 The numerical realization of equation ( ) ( ) Using the Modified Laplace Adomian Decompostion Method (LADM). 3.9 69 The numerical realization of equation ( ) ( ) Using the Differential Transform Method (DTM). 3.10 72 Conclusions 73 References 79 Appendix 79 Matlab Code for Differential Transform Method for Example 3.1 85 Matlab Code for Differential Transform Method for Example 3.2 28 Matlab Code for Differential Transform Method for Example 3.4 58 Matlab Code for Modified Laplace Adomian Decompostion Method for Example 3.1 86 Matlab Code for Modified Laplace Adomian Decompostion Method for Example 3.3 89 Matlab Code for Modified Laplace Adomian Decompostion Method for Example 3.4 92 Matlab Code for Variational Iteration Method for Example 3.1 95 Matlab Code for Variational Iteration Method for Example 3.2 97 Matlab Code for Variational Iteration Method for Example 3.3 viii 155 Matlab Code for Variational Iteration Method for Example 3.4 الملخص ب ix List of Tables No. Title Page 3.1 The exact and numerical solutions of applying Algorithm 3.1 on equation (3.1) 45 3.2 The exact and numerical solutions of applying Algorithm 3.2 on equation (3.1) 45 3.3 The exact and numerical solutions of applying Algorithm 3.3 on equation (3.1) 49 3.4 The exact and numerical solutions of applying Algorithm 3.1 on equation (3.2) 52 3.5 The exact and numerical solutions of applying Algorithm 3.3 on equation (3.2) 54 3.6 The exact and numerical solutions of applying Algorithm 3.2 on equation (3.3) 57 3.7 The exact and numerical solutions of applying Algorithm 3.3 on equation (3.3) 59 3.8 The exact and numerical solutions of applying Algorithm 3.3 on equation (3.4) 62 3.9 The exact and numerical solutions of applying Algorithm 3.2 on equation (3.4) 64 3.10 The exact and numerical solutions of applying Algorithm 3.1 on equation (3.4) 64 x List of Figures No. Title Page (3.1)a The exact and numerical solutions of applying Algorithm 3.1 on equation (3.1) 44 (3.1)b The error resulting of applying Algorithm 3.1on equation (3.1) 44 (3.2)a The exact and numerical solutions of applying Algorithm 3.2 on equation (3.1) 94 (3.2)b The error resulting of applying Algorithm 3.2on equation (3.1) 49 (3.3)a The exact and numerical solutions of applying Algorithm 3.3 on equation (3.1) 53 (3.3)b The error resulting of applying Algorithm 3.3on equation (3.1) 53 (3.4) The exact and numerical solutions of applying all Algorithms on equation (3.1) 54 (3.5)a The exact and numerical solutions of applying Algorithm 3.1 on equation (3.2) 56 (3.5)b The error resulting of applying Algorithm 3.1on equation (3.2) 56 (3.6)a The exact and numerical solutions of applying Algorithm 3.3 on equation (3.2) 58 (3.6)b The error resulting of applying Algorithm 3.3on equation (3.2) 58 (3.7) The exact and numerical solutions of applying all Algorithms on equation (3.3) 59 (3.8)a The exact and numerical solutions of applying Algorithm 3.2 on equation (3.3) 61 (3.8)b The error resulting of applying Algorithm 3.2on equation (3.3) 61 (3.9)a The exact and numerical solutions of applying Algorithm 3.3 on equation (3.3) 63 (3.9)b The error resulting of applying Algorithm 3.3on equation (3.3) 63 (3.10) The exact and numerical solutions of applying all Algorithms on equation (3.2) 64 (3.11)a The exact and numerical solutions of applying Algorithm 3.3 on equation (3.4) 66 xi (3.11)b The error resulting of applying Algorithm 3.3on equation (3.4) 66 (3.12)a The exact and numerical solutions of applying Algorithm 3.2 on equation (3.4) 68 (3.12)b The error resulting of applying Algorithm 3.2onequation (3.4) 68 (3.13)a The exact and numerical solutions of applying Algorithm 3.1 on equation (3.4) 70 (3.13)b The error resulting of applying Algorithm 3.1onequation (3.4) 70 (3.14) The exact and numerical solutions of applying all Algorithms on equation (3.4) 71 xii Computational Methods for Solving Nonlinear Volterra Integro - Differential Equation. By Farah Khaled Shehada Abu Thabit Supervisor Prof. Naji Qatanani Abstract In this thesis we focus on the numerical treatment of the nonlinear Volterra integro - differential equation. This equation has wide range of applications in mathematical physics, engineering, mechanics, chemistry, astronomy, biology, economics and potential theory. After introducing some definitions and important concepts of this equation, we will focus our attention mainly on the numerical methods for solving the nonlinear Volterra-integro differential equation. These methods are: Differential Transform method with Adomian polynomials (DTM), Modified Laplace Adomian Decompostion Method (LADM) and the Variational iteration method (VIM). The mathematical framework of these numerical methods together with their convergence properties will be presented. To demonstrate the efficiency of these numerical methods, we construct some numerical examples. Numerical results show clearly that the Variational iteration method (VIM) is the most efficient numerical technique for solving the xiii nonlinear Volterra integro - differential equation in a comparison with the other numerical techniques. 1 Introduction The Volterra integro-differential equation was initiated by Volterra in 1884. It appears in a variety of applications in many fields including continum mechanics, potential theory, geophysics, characterizing many social and many physical applications such as glass forming Process, nano hydrodynamics, heat transfer and all the diffusion process in general. In addition, this equation has an important role in neutron diffusion and biological species coexisting together with increasing and decreasing rates of generating, and wind ripple in the desert. More details about the applications of these equations can be found in [21, 32]. Nonlinear phenomena has a fundamental role in various fields of science and engineering. The nonlinear models of the real life problems are still difficult to solve either analytically or numerically. There has been much attention devoted to the search for better and more efficient solution methods for determining solution of nonlinear models [3]. There are many several analytical and numerical methods for solving integro - differential equations such as, the Adomian decomposition method, the direct computation method, the series solution method, the successive approximation method and the conversion to equivalent differential equations. However, these analytical methods are not easy to use and require huge calculation [36-37]. Alternatively, integro-differential 2 equations can be solved using many numerical methods such as the Legendre wavelet method [33], the Haar wavelet method [25], the linearization method [34], the finite difference method [39], block-pulse functions [8], the Taylor polynomial method [9, 24] and the differential transform method [1, 2, 4, 5]. In recent years, much work has been concentrated on the solutions of Volterra integro-differential equations. For example in [32] the authors used the decomposition method for solving some nonlinear integro- differential equations that arise as model equations for describing turbulent diffusion. In addition, they gave a comparison between the implicit Runge Kutta method and the decomposition technique. In [40] the authors introduced a multi-grid method for solving the nonlinear Urysohn integral equation. Also, in [37] it was shown that the modified decomposition method for mixed nonlinear Volterra -Fredholm integral equations combined with the noise terms phenomena may provide the exact solution by using just two iterations. Moreover, Wazwaz in [37] implemented the modified decomposition method, where he obtained numerical solutions in a rapidly convergent series with components that are elegantly computed. In addition Sweilam [34] used The variational iteration method (VIM) and nonlinear boundary value problems for 4th order integro - differential equations, where Variational Iteration Method is simple and yet a powerful method for solving integro-differential equations [35]. 3 In this work, numerical simulations with different types of nonlinearities will be treated using some numerical techniques namely, Differential Transform of nonlinear integro-differential equation method with Adomian polynomials, Modified Laplace Adomian Decompostion Method (LADM) And the Variational Iteration Method (VIM). In addition, a comparative study to examine the performance of these methods for solving integro - differential equations will be carried out. This can be realized by solving some numerical examples using Matlab software. This thesis is organized as follows: Chapter one introduces some basic concepts and definitions for the nonlinear integro - differential equations. In chapter two, some numerical methods namely, Differential transform of nonlinear integro-differential equation method with Adomian polynomials, Modified Laplace Adomian Decompostion Method (MLADM) and the Variational Iteration Method (VIM), will be addressed. Some numerical examples and results are presented in chapter three and conclusions have been drawn. 4 Chapter One Mathematical Preliminaries Definition (1.1) [38] An integro - differential equation is the equation in which unknown function ( ) appears under an integral sign and contains ordinary derivative ( )( ) The most standard form of the nonlinear integro-differential equation is given as: ( )( ) ( ) ∫ ( ) ( ( )) ( ) ( ) ( ) Where ( )( ) is the derivative of the unknown function ( ) that will be determined , ( ) is known analytic function, ( ) is a known function of two variables called the kernel of the integro- differential equation, is parameter “complex or real ” ( ) ( ) are limits of integration that may be both constants, variables or mixed, and ( ( )) is a nonlinear function. Since equation (1.1) combines differential operator and integral operator, it is necessary to define some initial conditions ( )( ) for the determination of the particular solution ( ) . 5 Definition )1.2( [38]: Linearity: The integro-differential equation is called linear, if the unknown function ( ) inside the integral sign has exponent equal one. Otherwise if the unknown function ( ) has exponent other than one or contains nonlinear functions of ( ) then the integro-differential equation is called nonlinear, for example, the integro-differential equations ( )( ) ∫ ( ) ( ) and ( )( ) ∫ ( ) are a nonlinear integro-differential equations of the second kind. 1.1 Classification of Nonlinear Integro-Differential Equations 1.1.1 Types of Nonlinear Integro-Differential Equations There are different kinds of nonlinear integro-differential equations: 1. Volterra Nonlinear Integro-Differential Equation The Volterra nonlinear integro-differential equation of the second kind appears in the form: ( )( ) ( ) ∫ ( ) ( ( )) ( ) 6 where the lower limit of integration is constant and the upper limit is variable. 2. Fredholm Nonlinear Integro-Differential Equation The Fredholm nonlinear integro-differential equation of the second kind has the form: ( )( ) ( ) ∫ ( ) ( ( )) ( ) where the lower and the upper limit of integration are constant. 3. Volterra-Fredholm Nonlinear Integro-Differential Equation The Volterra - Fredholm nonlinear integro - differential equation of the second kind takes two forms, namely: ( )( ) ( ) ∫ ( ) ( ( )) ∫ ( ) ( ( )) ( ) and ( )( ) ( ) ∫ ∫ ( ) ( ( )) ( ) , - ( ) 7 where ( ) and ( ) are analytic functions, , - and is closed subset of . It is interesting to note that (1.4) contains disjoint Volterra and Fredholm integrals, however (1.5) contains mixed integrals such that the Fredholm integral is the interior one, and Volterra is the exterior integral. Other derivatives of less order may appear as well. Definition )1.3 ([37] : Analytic function: A function is said to be analytic if and only if its Taylor series about x0 converges to the function in some neighborhood for every x0 in its domain. Definition )1.4( [37] : Homogeneity: If ( ) is identically zero in the nonlinear integro-differential equation of the form: ( )( ) ( ) ∫ ( ) ( ( )) ( ) ( ) then it is called homogeneous, otherwise it is called nonhomogeneous. 1.1.2 Singularity of Nonlinear Integro - Differential Equation When one of the limits ( ) ( ) or both are infinite or when the kernel becomes infinite at one or more points within the integration range then the equation is singular, for example, the integro - differential equations https://en.wikipedia.org/wiki/Taylor_series https://en.wikipedia.org/wiki/Neighborhood_(topology) https://en.wikipedia.org/wiki/Domain_of_a_function 8 ( )( ) ( ) ∫ ( ) ( ( )) ( )( ) ( ) ∫ ( ) ( ( )) are a singular nonlinear integro-differential equation of the second kind. (i) Singular Nonlinear Integro-Differential Equation If the kernel in (1.1) is of the form ( ) ( ) ( ) where ( ) does not equal zero and is a differentiable function at x a t , then the nonlinear integro-differential equation is said to be a singular equation with Cauchy kernel where ( ) ∫ ( ) ( ) is understood in the sence of Cauchy Principle Value (CPV) and the notation P.V ∫ ( ) is usually used to denote this. Thus P.V. ∫ ( ) , ∫ ( ) ∫ ( ) - For example ( )( ) ( ) ∫ ( ) | | 9 (ii) Weakly Singular Nonlinear Integro-Differential Equation The kernel is of the form ( ) ( ) | | ( ) Where ( ) is bounded, “i.e. several times continuously differentiable a x , a t with v ( x , t ) and is a constant such that 0< < For example, the equation: ( )( ) ∫ | | ( ( )) is a singular nonlinear integro-differential equation with a weakly singular kernel. 1.2 System of Nonlinear Volterra Integro-Differential Equations A system of nonlinear Volterra integro - differential equations has the form [38]: ( )( ) ( ) ∫ . ( ) ( ( )) ̃ ( ) ̃ ( ( ))/ ( )( ) ( ) ∫ . ( ) ( ( )) ̃ ( ) ̃ ( ( ))/ 3 ( ) The nonlinear functions ̃ are specified. The kernels ( ) ̃ ( ) and the functions ( ), are given as real - valued function . 15 1.3 System of Nonlinear Fredholm Integro-Differential Equations A system of nonlinear Fredholm integro - differential equations has the form[38]: ( )( ) ( ) ∫ . ( ) ( ( )) ̃ ( ) ̃ ( ( ))/ ( )( ) ( ) ∫ . ( ) ( ( )) ̃ ( ) ̃ ( ( ))/ 3 ( ) The nonlinear functions ̃ are specified. The kernels ( ) ̃ ( ) and the functions ( ), are given as real - valued function . 1.4 Kinds of Kernels 1. Separable Kernel A kernel ( ) is said to be separable or (degenerate) if it can be expressed as the sum of a finite number of terms, that is, ( ) ∑ ( ) ( ) ( ) where the functions ( ) ( ) are linearly independent. 2. Difference Kernel A kernel ( ) is called difference kernel , when ( ) ( ) ( ) 11 3. Symmetric (or Hermitian) Kernel A complex-valued function ( ) is called symmetric (or Hermitian) if ( ) ( ) ( ) where the asterisk denotes the complex symmetric conjugate. For a real kernel, we have ( ) ( ) ( ) 4. Skew-Symmetric Kernel The kernel is of the form ( ) ( ) ( ) 5. Abel's Kernels The kernel ( ) is of the form ( ) ( ) | | ( ) where ( ) and the function ( ) is assumed to be differentiable. 6. Polar Kernel ( ) ( ) ( ) ( ) ( ) where are bounded and ( ) 12 7. Logarithmic Kernel The kernel ( ) called a logarithmic kernel if it has the form ( ) ( ) ( ) ( ) ( ) where satisfy the same conditions as in equation (1.16) 1.5 Existence of the Solution of Nonlinear Volterra Integro- Differential Equation For convenience we consider the following nonlinear Volterra integro- differential equation : ( ) . ( ) ∫ ( ) ( ( )) / ( ) with the initial condition : ( ) ( ) where and are continuous functions and is a given constant; in which , - ( ) and denotes the Euclidean n - space with norm ‖ ‖ and let ( ) be the Banach space of all continuous functions from I into equipped with supremum norm ‖ ‖ * ‖ ( )‖ +. 13 Definition (1.5): Normed vector space ( ‖ ‖): Let the set be a linear vector space, a mapping, ‖ ‖ , is called normed vector space if the following propertis satisfy: 1. ‖ ‖ 2. ‖ ‖ 3. ‖ ‖ ‖ ‖ ‖ ‖ 4. ‖ ‖ ‖ ‖ Definition (1.6): Complete space: A normed vector space ( ‖ ‖) is called complete if every Cauchy saquence in converges to an element . Definition (1.7): Banach space: A Banach space is a complete normed vector space ( ‖ ‖) . The existence theorem is based on the topological transversality theorem given by Tidke [36] and is known as Leray – Schauder alternative. For a normed linear space and a number where ( ), we let * ‖ ‖ + and * ‖ ‖ + 14 Definition (1.8) [26]: We should say that is of the Leray - Schauder type provided for any ball in , either (a) There is an such that , or (b) There exist , and (0,1) such that y . Theorem (1.1) [36] (Leray – Schauder alternative) Let D be a convex subset of a normed linear space and let : be a completely continuous operator and let ( ) ={ q D : q q for some 0 < 1 }, Then either 1. ( ) is unbounded, or 2. The operator F has a fixed point. Proof: Assume is bounded and let be a ball containing ( ) in its interior. Since no can satisfy the second property in Definition (1.8), then the operator has a fixed point and the proof is complete. Theorem (1.2) (Banach fixed point space) A fixed point is a point that does not change upon application of a map, system of differential equations, etc. In particular, a fixed point of a function ( ) is a point such that ( ) http://mathworld.wolfram.com/Map.html http://mathworld.wolfram.com/DifferentialEquation.html http://mathworld.wolfram.com/DifferentialEquation.html 15 Theorem (1.3) [31] (The Arzela – Ascoli Theorem) Let T ( (I ) , ), then the following statements are equivalent: (1) T is compact. (2) T is closed subset of ( (I ) , ) and is both uniformly bounded and equicontinuous over I. For convenience we list the following hypotheses : ( ) There exists a constant 0 such that ‖ ( )‖ for t ( ) There exists a continuous function : I =[ 0, ) such that ‖ ( )‖ ( ) (‖ ‖) for every t and u , where H : (0, ) is a continuous non - decreasing function. ( ) For each ∫| ( ) ( )| is satisfied for . ( ) There exists a continuous function : such that 16 ‖ ( )‖ ( )(‖ ‖ ‖ ‖) for every and . Our main results are given through the following theorem : Theorem (1.4) [33]: Suppose that the hypotheses ( ), ( ) and ( ) are satisfied. Then equations (1.18) and (1.19) have a solution defined on provided satisfies ∫ ( ) ∫ ( ) (1.20) Where ‖ ‖ and M (t) = max { (t) , (t) } for t I. Proof . To prove the exsistence of a solution of equations (1.18) - (1.19), we start by applying Theorem (1.1), first we establish the priori bounds on the solution of the problem ( ) . ( ) ∫ ( ) ( ( )) ) / ( ) under the initial condition equation(1.19) for ( ). Let ( ) be a solution of equation (1.21) with initial condition in equation (1.19), then ( ) satisfies the equivelant integral equation ( ) ∫ . ( ) ∫ ( ) ( ( )) / ( ) upon using the hypotheses ( ), ( ), ( ) and equation (1.22) we obtain 17 ‖ ( )‖ ‖ ‖ ∫ ( ). ‖ ( )‖ ∫ ( ) (‖ ( )‖) / ( ) if we set the right side of equation (1.23) as ( ), then ‖ ( )‖ ( ) ( ) ‖ ‖ and ( ) ( ). ( ) ∫ ( ) ( ( )) / ( ) let ( ) ( ) ∫ ( ) ( ( )) ( ) then we obtain ( ) ( ) z( 0) =L( 0 ) = ‖ ‖ and ́( ) ( ) ( ) ( ) ( ( )) ( )( ( ) ( ( ))) i.e. ́( ) ( ) ( ( )) ( ) ( ) Integrating both sides of equation (1.26) from 0 to t and use the change of variables and the condition in equation (1.20) yields ∫ ( ) ∫ ( ) ( ) ∫ ( ) ∫ ( ) ( ) Consequently, we conclude that there is a constant independent of ( ) such that ( ) and hence ( ) for 18 Thus we have ‖ ( )‖ for and then ‖ ( )‖ * ‖ ( )‖ + We define ( ) and rewrite the initial value problem equations (1.18) - (1.19) as follows: if and ( ) ( ) , then it is easy to see that satisfies ( ) , ( ) ∫ . ( ) ∫ ( ) ( ( ) ) / Then we define , * + by ( ) ∫ . ( ) ∫ ( ) ( ( ) ) / ( ) for , If and only if Satisfies (1.18) - (1.19). Then is clearly continuous. Now we need to prove that is completely continuous. So we let a bound sequence { } in , i.e. ‖ ‖ for all where is positive constant. From (1.28 ) and using the hypotheses ( ), ( ), ( ) and letting * ( ) + we have ‖ ‖ ( ( )) consequently { } is uniformly bounded. Now we shall show that the sequence { } is equicontinuous. Let 19 , from (1.28) and using the hypotheses ( ), ( ), ( ) and letting * ( ) + we have | ( ) ( )| ∫ | ( ( ) ∫ ( ) ( ) ) )| ∫ ( )( ( ) ‖ ‖ ∫ ( ) (‖ ( )‖ ‖ ‖) ) ) ∫ ( ( )) ( ) By equation (1.29) we conclude that { } is equicontinuous and hence by theorem (1.2) the operator is completely countinous. Moreover, the set ( ) = {y : q y for some 0 < < 1}, is bounded, since for every y in ( ) the function is a solution of (1.21), for which we have proved ‖ ‖ and hence ‖ ‖ . Now an application of Theorem (1.1), the operator has a fixed point in . This means that equations (1.18) – (1.19) has a solution. This completes the proof of the theorem. 25 1.6 Uniqueness of the Solution of Nonlinear Volterra Integro - Differential Equation Now to show the uniqueness of the solution of nonlinear Volterra integro -differential equations we need to employ the analysis based on the applications of the Banach fixed point theorem coupled with Bielecki type norm and the integral inequalities with explicit estimates (for more details see [6, 19, 20, 28, 29]). We first build the appropriate metric space for our analysis and we let be a constant and consider the space of continuous functions ( ) such that | ( )| , and denote this special space by ( ). We couple the linear space ( ), with Bielecki’s metric : ( ) ‖ ( ) ( ) ‖ , and with Bielecki’s norm: ‖ ‖ ‖ ( )‖ We are now ready to present the main results for the uniqueness of solution of equations (1.18) - (1.19). Theorem (1.5) [27]: Let , ≥ 0, γ > 1 be constants with = γ. Suppose that the functions in equations (1.18) - (1.19) satisfy 21 the conditions: ‖ ( ) ( ̅ ̅)‖ ,‖ ̅‖ ‖ ̅‖- ‖ ( ) ( ̅)‖ ,‖ ̅‖- ‖ ∫ . ∫ ( ) ( ) ) / ‖ If ( ) , then the equations (1.18) - (1.19) has a unique solution ( ) Proof: To prove Theorem (1.4), we need to consider the following equivalent formulation for the equations (1.18)-(1.19), with let ( ) and define the operator by ( )( ) ∫ . ( ) ∫ ( ) ( ( )) ) / ∫ . ∫ ( ) ( ) ) / ∫ . ∫ ( ) ( ) ) / ( ) Now we shall show that into itself, From (1.30) and using the hypotheses we have ‖ ‖ ‖( )( )‖ 22 ‖ . ( ) ∫ ( ) ( ( )) ) / . ∫ ( ) ( ) ) / ‖ ‖ . ∫ ( ) ( ) ) / ‖ 0‖ ( )‖ ∫ ‖ ( )‖ 1 0 ‖ ( )‖ ∫ ‖ ( )‖ 1 0‖ ‖ ‖ ‖ ∫ 1 ‖ ‖ * ( )+ ‖ ‖ 0 1 ‖ ‖ ( ) This proves that the operator maps ( ) into itself. Now we verify that the operator is a contraction map. Let ( ). From equation (1.30) and using the hypotheses we have ( ) ‖ ( )( ) ( )( ) ‖ 23 ‖ . ( ) ∫ ( ) ( ( )) ) / . ( ) ∫ ( ) ( ( )) ) / ‖ 0‖ ( ) ( )‖ ∫ ‖ ( ) ( )‖ 1 0 ‖ ( ) ( )‖ ∫ ‖ ( ) ( )‖ 1 0 ( ) ( ) ∫ 1 ( ) * ( )+ ( ) 0 1 ( ) ( ) Since ( ) < 1, it follows from the Banach fixed point theorem (see that T has a unique fixed point in ( ). The fixed point of T is however solution of equation (1.1). The proof is complete. 24 1.7 Laplace Transforms Definition (1.5) [36] : Let ( ) be a real valud function defined for Suppose that ( ) is multiplied by and the result is integrated with respect to from , if the integral converges then it is a function of , that is * ( )+ ( ) ∫ ( ) ( ) or * ( )+ ( ) ∫ ( ) ( ) ( ) is called the Laplace transform of ( ). Moreover we have ( ) * ( )+ ( ) called the inverse Laplace transform. 1.7.1 Properties of the Laplace Transforms Laplace transforms has the following properties: 1. Constant Multiple: * ( )+ * ( )+ ( ) ( ) For example: * + * + 25 2. Linearity Property: * ( ) ( )+ * ( )+ * ( )+ ( ) ( ) are constants (1.34) For example: * + * + * + 3. Multiplication by * ( )+ ( ) * ( )+ ( ) ( ) ( ) For example: * + * + ( ) ( ) 1.7.2 Convolution Definition (1.6) [38] : If ( ) and ( ) are picewise continuous functions on [ 0, x ) and both satisfy the conditions needed for the existence of Laplace transform, then the convolution of denoted by ( )( ) is defined by the integral : ( ) ( ) ∫ ( ) ( ) or ( ) ( ) ∫ ( ) ( ) 26 note that ( ) ( ) ( )( ) we can easily show that the Laplace transform of the convolution product ( ) ( ) is *( ) ( )+ ,∫ ( ) ( ) - ( ) ( ) Where ( ) ( ) are the Laplace transform for the functions ( ) ( ) respectively. 27 Chapter Two Computational Methods There are many computational techniques available for solving nonlinear Volterra integro-differental equations. In this chapter we will discuss the following methods:Differential Transform Method with Adomian Polynomials (DTM), Modified Laplace Adomian Decompostion Method and the Variational Iteration Method (VIM). 2.1 Differential Transform Method with Adomian Polynomials (DTM) In this section , the (DTM) was discussed in order to solve nonlinear integro - differential equations in a more comprehensive and effective way; the idea is based on the methodology of [10] where the nonlinear term is replaced by its Adomian polynomials and the dependent variable components is replaced by its corresponding differential transform component of the same index. Differential Transform Method This method is based on the (DTM) presented by Zhou [39] in his study of electric circuits, where basic definitions and fundamental theorems of the differential transformation and its applicability to different types of differential and integral equation are given in [3, 4, 39]. 28 The derivative transformation of a function ( ) is defined as follows: ( ) * ( )+ ( ) where ( ) is the premier analytical function and ( ) is the transformed function. Differential inverse transformation of ( ) is defined as follows: ( ) ∑ ( )( ) ( ) Since ( ) is the analytical function, it is clear that ( ) ( ) By combination of equations (2.1) and (2.2) , with the function ( ) can be written as: ( ) ∑ * ( )+ ( ) Basic mathematical properties of the differential transform can easily be obtained and summarized in the following theory: Theorem (2.1) [7]: If ( ) ( ) and ( ) are differential transforms of the functions ( ) ( ) and ( ) respectively then : 1. If ( ) = ( ) ( ) then ( ) ( ) ( ). 2. If ( ) = ( ) ( ) then ( ) ∑ ( ) ( ) . 3. If ( ) = ( ) then ( ) ( ). 4. If ( ) ( ) then ( ) ( )( ) ( ) ( ). 29 5. ( ) then ( ) ( ) 2 6. If ( ) = ( ) then ( ) . /. 7. If ( ) = ( ) then ( ) . /. 8. If ( ) = then ( ) . Theorem (2.2) [ 7 ]: If ( ) ( ) and ( ) are differential transforms of the functions ( ) ( ) and ( ) respectively, then for we have : 1. If ( ) = ∫ ( ) ( ) then ( ) ∑ ( ) ( ) 2. If ( ) = ( ) ∫ ( ) ( ) then ( ) ∑ ∑ ( ) ( ) ( ) The Modified Differential Transform Method The Adomian decomposition method is utilized to the following general nonlinear equation: ( ) ( ) Where is the unknown function, is the highest - order derivative which assumed to be easily reversible, is the remaining linear operator represents a general nonlinear differential operator , and ( ) is the 35 source term. Applying the inverse operator to both sides of equation (2.4) and using the specific conditions we obtain ( ) ( ) ( ) ( ) where the function ( ) represents the terms arising from integrating the source term ( ). The nonlinear operator ( ) is decomposed as ( ) ∑ ̃ ( ) ( ) where ̃ are the Adomian polynomials determined formally as follows [3,2,10,39] ̃ 0 [ (∑ )]1 ( ) the Adomian polynomials of ( ) are introduced as ̃ ( ) ̃ ́( ) ̃ ́( ) ( )( ) ̃ ́( )( ) ( )( ) ( )( ) ̃ ́( ) ( ) ( )( ) ( )( ) ( )( ) and so on. 31 Lemma (2.1) [4]: If ( ) ( ( )) , then ( ) where are the Adomian polynomials ̃ with replacing by ( ) Proof: ( for more details see [4]). The differential transform components of ( ) are computed by utilizing their properties. Then we can write it in the following formula ( for x = 0 ) ( ) ( ( )) ( ) ( ) ́( ( )) ( ) ( ) ́( ( )) ( ) ( )( ( )) ( ) ( ) ́( ( )) ( ) ( ) ( )( ( )) ( ) ( )( ( )) ( ) ( ) ̀( ( )) ( ( ) ( ) ( )) ( )( ( )) ( ) ( ) ( )( ( )) ( ) ( )( ( )) and so on. To use the differential transform method we need to replace the nonlinear term by its Adomian polynomials of the index , and hence the dependent variable components are replaced by their corresponding differential transforms in the recurrence relation of the same index. 32 2.2 Modified Laplace Adomian Decompostion Method (LADM) In this section, we show how the modified Laplace decomposition method can be used to solve the nonlinear Volterra integro - differential equation (1.1). To solve the nonlinear Volterra integro - differential equation (1.1) using the Laplace transform method, it is essential to use the Laplace transforms of the derivatives of ( ) Applying the Laplace transform to both sides of equation (1.1) yields: * ( )+ ( ) ( )( ) ( )( ) * ( )+ ,∫ ( ) ( ( )) - ( ) or equivalenty * ( )+ ( ) ( )( ) ( )( ) * ( )+ ,∫ ( ) ( ( )) - ( ) The next step in Laplace decomposition method is representing the solution in an infinite series given as ( ) ∑ ( ) ( ) 33 To overcome the nonlinearity of ( ( )) we use the Adomian decomposition method given in the form of an infinite series of the Adomian polynomials ̃ , that is ( ( )) ∑ ̃ ( ) ( ) where for every , ̃ is the Adomian polynomial given as ̃ 0 [ (∑ )]1 ( ) Substituting equation (2.10) and equation (2.11) into equation (2.9) gives {∑ ( ) } ( ) ( )( ) ( )( ) * ( )+ {∫ ( )∑ ̃ ( ) } ( ) The Adomian decomposition method admits the use of the following recursive relation * ( )+ ( ) ( ) * ( )+ ( ) {∫ ( )∑ ̃ ( ) } ( ) * ( )+ {∫ ( )∑ ̃ ( ) } ( ) 34 where ( ) and ( ) are Laplace transforms of ( ) and ( ) respectively. Applying the inverse Laplace transform to equations (2.14)- (2.16) gives the required recursive relation as follows ( ) ( ) ( ) ( ) ( ) 2 {∫ ( )∑ ̃ ( ) }3 ( ) ( ) 2 {∫ ( )∑ ̃ ( ) }3 ( ) 2.3 The Variational Iteration Method (VIM) In this section, we will study the newly developed variational iteration method (VIM) was established by Ji-Huan He [11-12], the method provides rapidly convergent successive approximations of the exact solution only if such a closed form solution exists, and not components as in Adomian decomposition method. In this method the problem considered has the form: ( ) ( ) where is a linear operator, is a nonlinear operator, and ( ) is a given function. According to (VIM) [13 - 16], we can build the following correction functional : ( ) ( ) 35 ∫ ( )* ( ) ̃ ( ) ( )+ ( ) where λ is Lagrange multiplier [18] , which can be determined optimally by the variational theory and may be here constant or function, the subscript denotes the approximation, ̃ is considered as a restricted value that means it behaves as a constant, hence ̃ where is the variational derivative [18]. In particular, the correction functional for the nonlinear integro - differential equation (1.1) is ( ) ( ) ∫ ( ) * ( )( ) ( ) ∫ ( ) ( ̃ ( )) + ( ) The variational iteration method needs applying two basic steps, namely: 1. The determination of the Lagrange multiplier that can be identified optimally via integration by parts and by using a restricted variation. after selecting , an iteration formula, without restricted variation, should be used to determine of the successive approximations of ( ) of the solution ( ) 2. The zeroth approximation ( ) can be any selective function. However, the initial values ( ) ́( ) are preferably used for the selective zeroth approximation . 36 The following is a summary of the Lagrange multipliers as derived in [38], and the selective zeroth approximations: ́ ( ( ) ́( )) ( ) ( ) ( ) ( ( ) ́( ) ( )( )) ( ) ( ) ́( ) ( ) ( ( ) ́( ) ( )( ) ( )( )) ( ) ( ) ( ) ́( ) + ( )( ) and so on. Consequently, the solution is given by ( ) ( ) 37 Chapter Three Numerical Examples and Results In this chapter we implement the aforementioned numerical methods in chapter two to solve some numerical examples in order to test the efficiency and accuracy of these methods. This will be carried out using proper algorithms and Matlab software. Comparison between the exact and the approximate solutions will be tabulated and graphically illustrated. Example 3.1 Consider the Volterra nonlinear integro-differential equation : ( )( ) ( ) ∫ ( ) ( ) ( ) with the initial conditions : ( ) ́( ) ( ) The exact solution of equation ( ) is : ( ) . Now, we apply the aforementioned numerical methods in chapter two to solve equations ( ) ( ) 38 3.1 The Numerical Realization Of Equations (3.1) – (3.2) Using The Differential Transform Method With Adomian Polynomials (DTM) The following Algorithm implements the Differential Transform Method with Adomian Polynomials (DTM) using the Matlab software Algorithm (3.1) This algorithm can be illustrated as follows: 1. Input : a) ( ) , ( ): b) : Is a constant parameter. c) ( ) The function of the integral equation. d) ( ) The kernel function. e) ( ( )) Nonlinear term. f) Set ( ) and ( ) . g) The first known iterations ( ) n 2. Calculate: the Adomian polynomials for the nonlinear term . 3. Calculate: The iterations ( ), from for 39 ( ) ( ) ( ) ( ) ( ) * ( ) ( ) + ( ) ( ) ( ) ( ) ( ) * ( ) ( ) ( ) + end 4. Evaluation sum of all u’s in this form : ( ) ∑ ( ) ( ( )) 5. For any subinterval of discrete points of x : i. Input exact solution and compute exact at all points of x. ii. Find absolute error | |. 45 iii. Plot the error and all values x’s. iv. Output ( , approximate , 𝐸 , error ). v. Print the table. vi. Plot and Stop (The process is complete). 6. Output: Approximations ( ) about interval. We use Algorithm ( ) to solve equations ( ) ( ). Table (3.1) contains both the exact and numerical solutions using the Differential Transform Method (DTM) for equations ( ) ( ). Table (3.1): The exact and numerical solutions using the Differential Transform Method with Adomian Polynomials (DTM) with x Exact solution ( ) Numerical solution ( ) Absolute error | | 0 1.00000000000000 1.00000000000000 0 0.0938 1.00883726004162 1.00883726004118 0.00000000000044 0.2188 1.04903788068848 1.04903787857613 0.00000000211235 0.3125 1.10258370680894 1.10258363157239 0.00000007523655 0.4062 1.17939127885956 1.17939023105735 0.00000104780220 0.5 1.28402541668774 1.28401692708333 0.00000848960440 0.6250 1.47790419541173 1.47782318045695 0.00008101495478 0.7188 1.67644158014955 1.67610624329884 0.00033533685071 0.8125 1.93509466932358 1.93392305604841 0.00117161327516 0.9062 2.27322252652152 2.26962882885807 0.00359369766344 1 2.71828182845904 2.708333333333333 0.009948495125712 It can be observed that the maximum error is 0.009948495125712. The exact and approximate solutions are shown in Figure 3.1 (a) and the resulted error is shown in Figure 3.1 (b). 41 Fig. 3.1 (a): A comparison between the exact and approximate solutions in examples (3.1). Fig. 3.1 (b): Absolute error between exact and numerical solutions in example (3.1). 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 The Solution of Nonlinear Volterra Integro - Differential Equation by (DTM) x - axis q ( x ) Approximation solutions exact solutions 0 0.2 0.4 0.6 0.8 1 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 x - axis Ab so lut e E rro r Error 42 3.2 The Numerical Realization Of Equations (3.1) – (3.2) Using The Modified Laplace Adomian Decopostion Method (LADM) Algorithm 3.2 This algorithm can be illustrated as follows: 1. Input : a) ( ) , ( ): b) : Is a constant parameter. c) ( ) The function of the integral equation. d) ( ) The kernel function. e) ( ( )) Nonlinear term. f) Set ( ) and ( ) . 2. Calculate: The Adomian polynomials for the nonlinear term . 3. Calculate: The iterations ( ) 4. Depending on the following recursive relation * ( )+ ( ) * ( )+ ( ) {∫ ( )∑ ( ) } 43 * ( )+ {∫ ( )∑ ( ) } we will need to find all iteration ( ) using Matlab as follow ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ( ) ( ) ( ) ( ) ( ) 44 ( ( )) End 5. For any subinterval of discrete points of x : i. Input exact solution and compute exact at all points of x. ii. Find absolute error | |. iii. Plot the error and all values x’s. iv. Output ( , approximate , 𝐸 , error ). v. Print the table. vi. Plot and Stop (The process is complete). 2. Output: Approximations ( ) about interval. We use Algorithm ( ) to solve equations ( ) ( ). Table (3.2) contains both the exact and numerical solutions using the Laplace Adomian decompostion method for equations ( ) ( ). 45 Table (3.2): The exact and numerical solutions using the Laplace Adomian decompostion method with x Exact solution ( ) Numerical solution ( ) Absolute error | | 0 1.00000000000000 1.000011200126562 0.000011200126562 0.0938 0.66666684327790 1.008809640126562 0.000027619915065 0.2188 1.04903788068848 1.047884640126562 0.001153240561927 0.3125 1.10258370680894 1097667450126562 0.004916256682381 0.4062 1.17939127885956 1.165009640126562 0.014381638732999 0.5 1.28402541668774 1.250012200126562 0.034014216561180 0.6250 1.47790419541173 1.390636200126562 0.087267995285177 0.7188 1.67644158014955 1.516684640126562 0.159756940022989 0.8125 1.93509466932358 1.660167450126562 0.274927219197021 0.9062 2.27322252652152 1.821209640126562 0.452012886394959 1 2.71828182845904 2.000011200126561 0.718270628332484 It can be observed that the maximum error is = 0.718270628332484. The exact and approximate solutions are shown in Figure 3.2 (a) and the resulted error is shown in Figure 3.2 (b). Fig. 3.2 (a): A comparison between the exact and approximate solutions in examples (3.1). 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 x - axis q ( x ) The Solution of Nonlinear Volterra Integro - Differential Eqation by (LADM) exact solutions Approximation solutions 46 Fig. 3.2 (b): Absolute error between exact and numerical solutions in example (3.1). 3.3 The Numerical Realization Of Equation (3.3 )– (3.3 ) Using The Variational Iteration Method (VIM) Algorithm 3.3 This Algorithm can be illustrated as follows: 1. Input : a) ( ) , ( ): b) : Is a constant parameter. c) ( ) The function of the integral equation. d) ( ) The kernel function. e) ( ( )) Nonlinear term. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x -axis Ab so lu te E rro r Error 47 f) Set ( ) and ( ) . 2. Calculate: The iterations ( ) ∫ ( ) ∫ ( ) ( ) ( ) ( ) 3. Set and solve more iterations. 4. For any subinterval of discrete points of x : i. Input exact solution and compute exact at all points of x. ii. Find absolute error | |. iii. Plot the error and all values x’s. iv. Output ( , approximate , 𝐸 , error ). v. Print the table. vi. Plot and Stop (The process is complete). 48 5. Output: Approximations ( ) about interval. We use Algorithm ( ) to solve equations ( ) ( ). Table (3.3) contains both the exact and numerical solutions using the Variational Iteration Method ( ) ( ). Table (3.3): The exact and numerical solutions using the Variational Iteration Method with Adomian Polynomials (DTM) with x Exact solution ( ) Numerical solution ( ) Absolute error | | 0 1.00000000000000 1.00000000000000 0 0.0938 1.008837260041627 1.008837265541627 5 0.2188 1.049037880688489 1.049537885688489 5 0.3125 1.102583706808942 1.02583756858942 0 0.4062 1.179391278859565 1.179391278859565 0 0.5 1.284025416687741 1.284525416687741 0 0.6250 1.477904195411739 1.477954195411739 0.055555555555527 0.7188 1.676441580149551 1.676441585149551 5.5555555555555787 0.8125 1.935094669323582 1.935594669323582 0. 055555555515565 0.9062 2.273222526521525 2.273222526521525 0. 055555555259527 1 2.718281828459046 2.718281828459546 0. 055555552265553 It can be observed that the maximum error is 5.555555552265553 The exact and approximate solutions are shown in Figure 3.3 (a) and the resulted error is shown in Figure 3.3 (b). 49 Fig. 3.3 (a): A comparison between the exact and approximate solutions in examples (3.1). Fig. 3.3 (b): Absolute error between exact and numerical solutions in example (3.1) . 0 0.2 0.4 0.6 0.8 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 x - axis q ( x ) The Solution of Nonlinear Volterra Integro - Differential Equation by (VIM) Approximation solutions exact solutions 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 x 10 -9 x - axis Ab so lut e Er ro r Erorr 55 The exact and approximate solutions of all methods are shown in Figure 3.4. Fig. 3.4: A comparison between the exact and approximate solutions in example (3.1). Example 3.2 Consider the Volterra nonlinear integro-differential equation : ( ) ( ) ( ) ∫ ( ) ( ) with the initial condition: ( ) ( ) The exact solution of equation ( ) is : ( ) = 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 x - axis q ( x ) The solutin of Nonlinear Integro-Differential equation by (DTM),(LADM) and (VIM) exact solutions (LADM) solutions (DTM) solutions (VIM) solutions 51 3.4 The Numerical Realization Of Equations (3.3) – (3.4) Using The Differential Transform Method with Adomian Polynomials (DTM) Using Algorithm (3.1) for equations (3.3) – (3.4) . Table (3.5) contains both the exact and numerical solutions using the Differential Transform Method with Adomian Polynomials (DTM) for equations (3.3) – (3.4). Table (3.6): The exact and numerical solutions using the Differential Transform Method with Adomian Polynomials (DTM) with Exact solution ( ) Numerical solution ( ) Absolute error | | 0 0 0 0 0.0938 0.09352634530384 0.09352634524183 0.00000000006201 0.2188 0.21540541588183 0.21540529299132 0.00000012289051 0.3125 0.30288486837497 0.30288194350543 0.00000292486953 0.4062 0.38583975163390 0.38581026419894 0.00002948743495 0.5 0.46364760900080 0.46346726190476 0.00018034709604 0.6250 0.55859931534356 0.55737143471127 0.00122788063228 0.7188 0.62323229760101 0.61921887947618 0.00401341812483 0.8125 0.68231655487474 0.67113242450924 0.01118413036550 0.9062 0.73622997521529 0.70867363217887 0.02755634303641 1 0.78539816339744 0.72380952380952 0.06158863958792 It can be observed that the maximum error is 0.06158863958792. The exact and approximate results of ( ) are shown in Figure 3.4 (a) and the resulted error is shown in Figure 3.5 (b). 52 Fig. 3.5 (a): A comparison between the exact and approximate solutions in examples (3.2). Fig. 3.5 (b): Absolute error between exact and numerical solutions in example (3.2) . 3.5 The Numerical Realization Of Equations (3.3) – (3.4) Using The Variational Iteration Method (VIM) Using Algorithm (3.3) for equations (3.3) – (3.4) . Table (3.5) contains both the exact and numerical solutions using the Variational Iteration Method (VIM) for equations (3.3) – (3.4) . 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 The Solution of Nonlinear Integro - Differential Equation by (DTM) x - axis q ( x ) Approximation solutions exact solutions 0 0.2 0.4 0.6 0.8 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 x - axis Ab so lut e E rro r Error 53 Table (3.5): The exact and numerical solutions using the Variational Iterationwith It can be observed that the maximum error is 0.035536457537173. The exact and approximate solutions are shown in Figure 3.6 (a) and the resulted error is shown in Figure 3.6 (b). Fig. 3.6 (a): A comparison between the exact and approximate solutions in examples (3.2). 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x - axis q ( x ) The Solution of The Nonlinear Integro - Differential Equation by (VIM) exact solutions Approximation solutions Exact solution ( ) ( ) Numerical solution Absolute error | | 0 0 0 0 0.0938 0.09352634530384 0.093526345303846 0.000000000008066 0.2188 0.21540541588183 0.215405415889905 0.000000001624805 0.3125 0.30288486837497 0.302884869999776 0.000000078719100 0.4062 0.38583975163390 0.385839830353001 0.00000166761338 0.5 0.46364760900080 0.463649276613144 0.000043044026501 0.6250 0.55859931534356 0.558642359370063 0.000043044026501 0.7188 0.62323229760101 0.623556289841569 0.000323992240555 0.8125 0.68231655487474 0.684190697399316 0.001874142524568 0.9062 0.73622997521529 0.745076036939866 0.008846061724573 1 0.78539816339744 0.820934620934621 0.035536457537173 54 Fig. 3.6 (b): Absolute error between exact and numerical solutions in example (3.2). The exact and approximate solutions of all methods are shown in Figure 3.7. Fig. 3.7: A comparison between the exact and approximate solutions in examples (3.2). 0 0.2 0.4 0.6 0.8 1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 x - axis Ab so lute Er ror Error 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x - axis q ( x ) The Solutions of Nonlinear Integro-Differential Equation by (DTM) and (VIM) exact solution (DTM) solution (VIM) solution 55 Example 3.3 Consider the Volterra nonlinear integro-differential equation : ( ) ∫ ( ( )) ( ) with the initial condition: ( ) ( ) The exact solution of equation ( ) is : ( ) = . 3.6 The Numerical Realization of Equations (3.5) – (3.6) Using The Modified Laplace Adomian Decopostion Method (LADM). Using Algorithm (3.2) for equations (3.5) – (3.6), Table (3.6) contains both the exact and numerical solutions using the Laplace Adomian Decopostion Method for equations (3.5) – (3.6) . Table (3.6): The exact and numerical solutions using the Laplace Adomian Decopostion Method (LDTM) with x Exact solution ( ) Numerical solution ( ) Absolute error | | -1 0.367879441171442 0.567667641618306 0.199788200446864 -0.8 0.449328964117222 0.600948258997328 0.151619294880106 -0.6 0.548811636094027 0.650597105956101 0.101785469862075 -0.4 0.670320046035639 0.724664482058611 0.054344436022971 -0.2 0.818730753077982 0.835160023017820 0.016429269939838 0 1.000000000000000 1.000000000000000 0 0.2 1.221402758160170 1.245912348820635 0.024509590660465 0.4 1.491824697641270 1.612770464246234 0.120945766604964 0.6 1.822118800390509 2.160058461368274 0.337939660977765 0.8 2.225540928492468 2.976516212197558 0.750975283705090 56 It can be observed that the maximum error is 0.750975283705090. The exact and approximate solutions are shown in Figure 3.8 (a) and the resulted error is shown in Figure 3.8 (b). Fig. 3.8 (a): A comparison between the exact and approximate solutions in examples (3.3) . Fig. 3.8 (b): Absolute error between exact and numerical solutions in example (3.3). -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 x - axis q ( x ) The Solution of Nonlinear Volterra Integro - Differential Equation by (LADM) Approximation solutions exact solution -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0.5 1 1.5 2 2.5 3 3.5 x - axis Ab so lut e E rro r Error 57 3.7 The Numerical Realization Of Equations (3.5) – (3.6) Using The Variational Iteration Method (VIM) Table (3.7): The exact and numerical solutions using the variational iteriation method (VIM) with x Exact solution ( ) Numerical solution ( ) Absolute error | | -1 0.367879441171442 5.367879441321282 5.555555555149839 -0.8 0.449328964117222 5.449328964125571 5.555555555558355 -0.6 0.548811636094027 5.548811636594228 5.555555555555251 -0.4 0.670320046035639 5.675325546535645 5.555555555555551 -0.2 0.818730753077982 5.818735753577982 5 0 1.000000000000000 1.000000000000000 0 0.2 1.221402758160170 1.221452758165175 5 0.4 1.491824697641270 1.491824697641269 5.555555555555551 0.6 1.822118800390509 1.822118855395290 5.555555555555219 0.8 2.225540928492468 2.225545928483157 5.555555555559361 It can be observed that the maximum error is 5.555555555559361. The exact and approximate solutions are shown in Figure 3.9 (a) and the resulted error is shown in Figure 3.9 (b). Fig. 3.9 (a): A comparison between the exact and approximate solutions in examples (3.3). -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 x - axis q ( x ) The Solution of Nonlinear Volterra Integro - Differential Equation by (VIM) Exact solutions Approximation solutions 58 Fig. 3.9 (b): Absolute error between exact and numerical solutions in example (3.3). The exact and approximate solutions of all methods are shown in Figure 3.7. Fig. 3.10: A comparison between the exact and approximate solutions in examples (3.3). -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 x 10 -10 x - axis Ab so lut e E rro r Error -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 x - axis q ( x ) The Solution of Nonlinear Volterra Integro-Differential Equation by (LADM) AND (VIM) exact solutions (DTM) solutions (VIM) solutions 59 Example 3.4 Consider the Volterra nonlinear integro-differential equation : ( ) ∫ ( ) ( ) with the initial condition: ( ) ( ) The exact solution of equation ( ) is : ( ) 1 3.8 The numerical Realization of equations ( ) ( ) Using The Variational Iteration Method (VIM) Using Algorithm (3.3) for equations (3.7) – (3.8) . Table (3.6) contains both the exact and the numerical results using the Variational Iteration Method for equations (3.7) – (3.8). Table (3.8): The exact and numerical solutions using Variational Iteration Method (VIM) with Exact solution ( ) Numerical solution ( ) Absolute error | | 0 2.000000000000000 2.000000000000000 0 0.0938 2.098340055937721 2.098340054991735 0.000000555925155 0.2188 2.244582335327159 2.24458182938934 0.000005427229555 0.3125 2.366837941173796 2.366836647675151 0.000005427229550 0.4062 2.501102742986564 2.501096554599779 0.000038359154292 0.5 2.648721270700128 2.648699569298986 0.000185507539221 0.6250 2.868245957432222 2.868163173060590 5.001042650006852 0.7188 3.051969369391527 3.051777803836352 0.003142377825653 0.8125 3.253534787213209 3.253135200988730 0.008390931887913 0.9062 3.474900021868255 3.474130861914352 0.020388681771895 1 3.718281828459046 3.716892914345171 0.045913559787484 65 It can be observed that the maximum error is 0.045913559787484. The exact and approximate solutions are shown in Figure 3.11 (a) and the resulted error is shown in Figure 3.11(b). Fig. 3.11 (a): A comparison between the exact and approximate solutions in examples (3.4). Fig. 3.11 (b): Absolute error between exact and numerical solutions in examples (3.4). -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 x- axis q ( x ) The Solution of Nonlinear Volterra Integro - Differential Equation by (VIM) exact solutions Approximatian solutions 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10 -3 x - axis Ab so lut e Er ro r Error 61 3.9 The Numerical Realization Of Equations (3.7) – (3.8) Using The Modified Laplace Adomian Decopostion Method (LADM). Using Algorithm (3.2) for equations (3.7) – (3.8) . Table (3.6) contains both the exact and the numerical results using the Variational Iteration Method for equations (3.7) – (3.8) . Table (3.9): The exact and numerical solutions using Modified Laplace Adomian Decompostion Method (LADM) with Exact solution ( ) Numerical solution ( ) Absolute error | | 0 2.000000000000000 2.000000000000000 0 0.0938 2.098340055937721 2.598345555937721 0.000000555000000 0.2188 2.244582335327159 2.244582335327165 0.000000555000000 0.3125 2.366837941173796 2.366837941173797 0.000000555000000 0.4062 2.501102742986564 2.551152742986564 0 0.5 2.648721270700128 2.648721275755128 0 0.6250 2.868245957432222 2.868245957432221 0.000000555000001 0.7188 3.051969369391527 3.551969369391522 0.000000555000006 0.8125 3.253534787213209 3.253534787213173 0.000000555000036 0.9062 3.474900021868255 3.474955521868575 0.000000555000185 1 3.718281828459046 3.718281828458235 0.000000555000816 It can be observed that the maximum error is = 0.000000555000816. The exact and approximate solutions are shown in Figure 3.12 (a) and the resulted error is shown in Figure 3.12 (b). 62 Fig. 3.12(a): A comparison between the exact and approximate solutions in examples (3.4). Fig 3.12 (b): Absolute error between exact and numerical solutions in examples (3.4). 0 0.2 0.4 0.6 0.8 1 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 The Solution of Nonlinear Volterra Integro - Differantial Equation by (LADM) x - axis q ( x ) exact solutions Approximation solutions 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 7 8 9 x 10 -13 x - axis Ab so lut e E rro r Error 63 3.10 The Numerical Realization Of Equations (3.7) – (3.8) Using The Differential Transform Method With Adomian Polynomials (DTM) Using Algorithm (3.1) for equations (3.7) – (3.8) . Table (3.6) contains both the exact and the numerical results using the Variational Iteration Method for equations (3.7) – (3.8). Table (3.8): The exact and numerical solutions using Differential Transform Method with Adomian Polynomials (DTM) with Exact solution ( ) Numerical solution ( ) Absolute error | | 0 2.000000000000000 2.000000000000000 0 0.0938 2.098340055937721 2.598345555937721 5 0.2188 2.244582335327159 2.244582335327165 0.000000555000001 0.3125 2.366837941173796 2.366837941173725 0.000000555000071 0.4062 2.501102742986564 2.501102742985276 0.000000555001288 0.5 2.648721270700128 2.648721270687366 0.000000000012763 0.6250 2.868245957432222 2.868245957282028 0.000000000150195 0.7188 3.051969369391527 3.051969368686468 0.000000000705060 0.8125 3.253534787213209 3.253534784476825 0.000000002736384 0.9062 3.474900021868255 3.474900012701995 0.000000009166260 1 3.718281828459046 3.718281801146385 0.000000027312661 It can be observed that the maximum error is 0.000000027312661. The exact and approximate solutions are shown in Figure 3.13 (a) and the resulted error is shown in Figure 3.13 (b). 64 Fig. 3.13 (a): A comparison between the exact and approximate solutions in examples (3.4). Fig. 3.13 (b): Absolute error between exact and numerical solutions in examples (3.4). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 x - axis q ( x ) The solution of Nonlinear Volterra Integro - Differential Equation by (DTM) Approximation solutions exact solution 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 2.5 3 3.5 x 10 -8 x - axis Ab so lut e E rro r Erorr 65 The exact and approximate solutions of all methods are shown in Figure 3.14. Fig. 3.14: A comparison between the exact and approximate solutions in examples (3.4). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 x - axis q ( x ) The solutions of Nonlinear Integro-Differential equation by (LADM),(DTM) and (VIM) exact solutions (LADM) solutions (DTM) solutions (VIM) solutions 66 Conclusions Computional methods have been used to solve nonlinear Volterra integro - differential equation . The numerical methods are implemented in a form of algorithms to solve some numerical examples using Matlab software. Based on our numerical results, one sees clearly that the Variational Iteration Method (VIM) is the most effective numerical technique for solving nonlinear Volterra integro - differential equation. 67 References [1] I. Arikoglu, Solution of boundary value problems for integro - differential equations by using differential transform method, Appl. Math. Comput. 168 (2005), pp 1145-1158. [2] I. 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Comput. 127 (2002) 195-206. 73 Appendix Matlab Code for Differential Transform Method (DTM) for Example 3.1  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Laplace Method'))  %% in put data  u(1)=1;  u(2)=0;  u(3)=1;  u(4)=0;  %%A0  G(1)=0; 74  G(2)=0;  G(3)=1;  G(4)=0;  G(5)=1;  %%nonlinear adomian polynomial  %%for i=4  for k=2:6  u(k+3)=(1/((k+1)*(k+2)))*(6*u(k+1)+(8/k)*G(k-1));  end  %%solution of problem  syms x  yapp=0;  for i=1:9  yapp=yapp+(u(i)*x^(i-1));  end  yapp  %%Exact solution 75  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M1=x.^8./24 + x.^6./6 + x.^4./2 + x.^2 + 1;  exact=exp(x.^2);  error=abs(M1-exact);  plot(x,M1,x,exact,'m-.')  %%print  disp(sprintf(' TABLE(1) :) '))  [x' exact' M1' error'] ###################################################### Matlab Code for Differential Transform Method (DTM) for Example 3.2  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display 76  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Transform'))  %% in put data  u(1)=0;  u(2)=1;  %%A0  G(1)=1;  G(2)=1;  G(3)=1;  G(4)=-1/6;  G(5)=1/24;  %% nonlinear adomian polynomial  for k=2:3  u(k+1)=(-1*(k-2)*u(k-1)/(k))+(1/(6*(k+1))*[((-1^k)*(6-11*k+6*k^2- k^3)/factorial(k))]);  End  u(5)=-2*u(3)/4; 77  u(6)=-3*u(4)/5+1/30*((-6/factorial(4))+G(1)/4);  u(7)=-4*u(5)/6+1/36*((24/factorial(5))+G(2)/5);  u(8)=-5*u(6)/7+1/42*((-60/factorial(6))+G(3)/6);  u(9)=-6*u(7)/8+1/48*((120/factorial(7))+G(4)/7);  u(10)=-7*u(8)/9+1/54*((-210/factorial(8))+G(5)/8);  %%solution of problem  syms x  yapp=0;  for i=1:7  yapp=yapp+(u(i)*x^(i-1));  end  yapp  %%Exact solution  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M1= x.^5./5 - x.^3./3 - x.^7/7 + x;  exact=atan(x); 78  error=abs(M1-exact);  plot(x,M1,x,exact,'m-.')  %%print  disp(sprintf(' TABLE(1) :) '))  [x' exact' M1' error'] ####################################################### Matlab Code for Differential Transform Method (DTM) for Example 3.4  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Transform'))  %% in put data  u(1)=0; 79  u(2)=1;  %%A0  G(1)=0;  G(2)=0;  G(3)=1;  G(4)=1;  G(5)=7/12;  G(6)=1/2;  %%non linear adomian polynomial  u(3)=(1/factorial(2))+(1/factorial(2))*(G(1)/factorial(0));  u(4)=(1/factorial(3))+(1/factorial(3))*((G(2)/factorial(1))- (G(1)/(factorial(1)*factorial(0))));  u(5)=(1/factorial(4))+(1/factorial(4))*((G(4)/factorial(3))- (G(3)/(factorial(2)*factorial(2)))+(G(2)/(factorial(3)*factorial(1))));  u(6)=(1/factorial(5))+(1/factorial(5))*((G(5)/factorial(4))- (G(4)/(factorial(2)*factorial(3)))+(G(3)/(factorial(3)*factorial(2)))); 85  u(7)=(1/factorial(6))+(1/factorial(6))*((G(6)/factorial(5))- (G(5)/(factorial(2)*factorial(4)))+(G(4)/(factorial(3)*factorial(3)))- (G(3)/(factorial(2)*factorial(4))));  %%solution of problem  syms x  yapp=0;  for i=1:7  for j=1:8  yapp=yapp+(x^2+3)+(taylor(1+exp(x), x, 'Order', 11));  end  end  yapp;  %%Exact solution  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M1=x.^10/3628800 + x.^9/362880 + x.^8/40320 + x.^7/5040 + x.^6/720 + x.^5/120 + x.^4/24 + x.^3/6 + x.^2/2 + x + 2;  exact=1+exp(x); 81  error=abs(M1-exact);  plot(x,M1,'*',x,exact,'r')  %%print  disp(sprintf(' TABLE(1) :) '))  [x' exact' M1' error'] ####################################################### Matlab Code for Modified Laplace Adomian Decompostion method (MLAD) for Example 3.1  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Laplace Method'))  %In put data  syms t x s 82  %%for i=1  A=2/3;  %%for i=2  U=A*log(A)*t;  F=int(U,t,0,x);  A1=laplace(F);  B=(8*A1/(s^2-6));  C=ilaplace(B);  D=((1/3)*cosh(sqrt(6)*x))+(subs(C,t,x));  %%for i=2  U1=t*(D*log(A)+D);  F1=int(U1,t,0,x);  A2=laplace(F1);  B1=(8*A2/(s^3-6*s));  C1=ilaplace(B1);  D1=(subs(C1,t,x));  %% approximation(1) 83  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M1= (22152124404789197.*cosh(6.^(1/2).*x))./81064793292668928 + (4869473359433779.*x.^2)./27021597764222976 + 4869473359433779/81064793292668928;  exact=exp(x.^2);  error=abs(M1-exact);  plot(x,M1,x,exact,'m')  %%print  disp(sprintf(' TABLE(1) :) '))  [x' exact' M1' error'] ###################################################### Matlab Code for Modified Laplace Adomian Decompostion method (MLAD) for Example 3.2  clc  clear all  format long 84  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Laplace Method'))  %In put data  syms t x s  %%for i=1  A=0.5+0.5*exp(2*x);  %%for i=2  F=(A)^2;  A1=laplace(F);  B=(A1/(s*s-s));  C=ilaplace(B);  D=(subs(C,t,x))*-1;  %%for i=3  K=((A)*(D)*2); 85  A2=laplace(K);  B1=(A2/(s*(s-1)));  C1=ilaplace(B1);  D1=-1*subs(C1,t,x);  %% approximation(1)  x=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ];  M1=0.5+0.5.*exp(2.*x);  exact=exp(x);  error1=abs(M1-exact);  plot(x,M1,x,exact,'m')  %%print  disp(sprintf(' TABLE(1) :) '))  [x' exact' M1' error']  %% approximation(2)  x=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ];  M2= x./4 - exp(2.*x)./4 - exp(4.*x)./48 + exp(x)./3 - 1/16;  exact=exp(x); 86  error=abs(M2-exact);  plot(x,M2,x,exact,'m')  %%print  disp(sprintf(' TABLE(2) :) '))  [x' exact' M2' error']  %% approximation(3)  x=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ]  M3= (3.*x)./16 + (11.*exp(2.*x))./32 - exp(3.*x)./18 + (13.*exp(4.*x))./576 + exp(6.*x)./1440 - (31.*exp(x))./90 - (x.*exp(2.*x))./8 - (x.*exp(x))./3 + x.^2./8 + 19/576;  error=abs(M3-exact);  plot(x,M3,x,exact)  %%print  disp(sprintf(' TABLE(3) :) '))  [x' exact' M3' error'] ####################################################### 87 Matlab Code for Modified Laplace Adomian Decompostion method (MLAD) for Example 3.4  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Laplace Method'))  %In put data  syms t x s  %%for i=1  A=3;  %%for i=2  F=(A)^2;  A1=9/s;  B=(A1/(s*s-s)); 88  C=ilaplace(B);  D=(subs(C,t,x));  %%for i=3  K=((A)*(D)*2);  A2=laplace(K);  B1=(A2/(s*(s-1)));  C1=ilaplace(B1);  D1=-exp(2*x)-exp(x)*((2*x)-1)+subs(C1,t,x);  %%for i=4  K1=((A)*(D1)*2)+(D^2);  A3=laplace(K1);  B2=(A3/(s*(s-1)));  C2=ilaplace(B2);  D2=subs(C2,t,x);  %%for i=5  K2=((D)*(D1)*2)+((A)*2);  A4=laplace(K2); 89  B3=(A4/(s*(s-1)));  C3=ilaplace(B3);  D3=subs(C3,t,x);  %% approximation(1) a) x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1]; (a) M1=x.^4/24 + x.^3/6 + x.^2/2 + x + 2; (b) exact=1+exp(x); (c) error=abs(M1-exact); (d) plot(x,M1,'*',x,exact,'r') (e) %%print (f) disp(sprintf(' TABLE(1) :) ')) (g) [x' exact' M1' error'] b) %% approximation(2) i) x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M2= x.^7/5040 + x.^6/720 + x.^5/120 + x.^4/24 + x.^3/6 + x.^2/2 + x + 2; 95  exact=1+exp(x); (a) error=abs(M2-exact); (b) plot(x,M2,x,exact,'m') (c) %%print (d) disp(sprintf(' TABLE(2) :) ')) (i) [x' exact' M2' error'] 1. %% approximation(3) ii) x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M3= x.^9/362880 + x.^8/40320 + x.^7/5040 + x.^6/720 + x.^5/120 + x.^4/24 + x.^3/6 + x.^2/2 + x + 2;  exact=1+exp(x); (a) error=abs(M3-exact); (b) plot(x,M3,x,exact) (c) %%print (d) disp(sprintf(' TABLE(3) :) ')) (i) [x' exact' M3' error'] (ii) %% approximation(4) 91 ii) x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M4=x.^11/39916800 + x.^10/3628800 + x.^9/362880 + x.^8/40320 + x.^7/5040 + x.^6/720 + x.^5/120 + x.^4/24 + x.^3/6 + x.^2/2 + x + 2;  exact=1+exp(x); (a) error=abs(M4-exact); (b) plot(x,M4,x,exact) (c) %%print (d) disp(sprintf(' TABLE(4) :) ')) (i) [x' exact' M4' error'] (ii) %% approximation(5) ii) x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M5=x.^14/87178291200 + x.^13/6227020800 + x.^12/479001600 + x.^11/39916800 + x.^10/3628800 + x.^9/362880 + x.^8/40320 + x.^7/5040 + x.^6/720 + x.^5/120 + x.^4/24 + x.^3/6 + x.^2/2 + x + 2;  exact=1+exp(x); 92 (a) error1=abs(M5-exact); (b) plot(x,M5,x,exact) (c) %%print (d) disp(sprintf(' TABLE(5) :) ')) (i) [x' exact' M5' error'] ii) %%Plot All Itteration with Exact Solution  plot(x,M1,'*',x,M2,'B',x,M3,'R',x,M4,'Y',x,M5,'G',x,exact,'P') ####################################################### Matlab Code for Variational Iteration Method (VIM) for Example 3.1  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Variation Iterative Method')) 93  %In put data  N=4;  syms x  syms r  syms t  y(1)=1;  %%for i=1  A=int(8*r*1*log(1),r,0,t);  B=int(((t-x)*(0-6*1+4)-A),t,0,x);  U1=y(1)+(B);  B=expand(U1);  %%for i=2  A2=int(((subs(U1,x,r)))*8*r*log((subs(U1,x,r))),r,0,t);  B2=int((t-x)*(diff(diff(subs(U1,x,t)))-6*(subs(U1,x,t))+4)-A2,t,0,x);  U2=U1-B2;  C=expand(U2);  %% approximation(1) 94  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M1=x.^2 + 1;  exact=exp(x.^2);  error=abs(M1-exact);  plot(x,M1,x,exact)  %%print  disp(sprintf(' TABLE(1) :) '))  [x' exact' M1' error']  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%  %% approximation(2)  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M2=(16*pi)/15 - (32.*x)/15 - (log(x - i)*16*i)/15 + (log(x + i)*16*i)/15 + 2.*x.*log(x - i) + 2.*x.*log(x + i) + x.^2 - (58.*x.^3)/45 - x.^4/2 - (9.*x.^5)/25 + (4.*x.^3.*log(x.^2 + 1))/3 + (2.*x.^5.*log(x.^2 + 1))/5 + 1;  exact=exp(x.^2); 95  error=abs(M2-exact);  plot(x,M2,x,exact)  %%print  disp(sprintf(' TABLE(2) :) '))  [x' exact' M2' error']  %%all plot  plot(x,M1,x,exact,'m') ####################################################### Matlab Code for Variational Iteration Method (VIM) for Example 3.2  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Variation Iterative Method')) 96  %In put data  N=4;  syms x  syms r  syms t  y(1)=0;  %%for i=1 a) A=int(r^3,r,0,t); b) B=int(((-1)*((0-(t^3+3*t^2+6*t+6)*exp(-1*t)))+A),t,0,x); c) U1=y(1)+(B); d) B=expand(U1);  %%for i=2 a) A2=int((r^3*exp(-1*(subs(U1,x,r)))),r,0,t); b) B2=int((-1)*((6*(t^2+1)*(diff(subs(U1,x,t))- (t^3+3*t^2+6*t+6)*exp(-1*t)))+A2),t,0,x); i) U2=U1-B2; c) C=expand(U2); 97 d) %%for i=3 e) A3=int((r^3*exp(-1*(subs(U2,x,r)))),r,0,t); f) B3=int((-1)*((6*(t^2+1)*(diff(subs(U2,x,t))- (t^3+3*t^2+6*t+6)*exp(-1*t)))+A3),t,0,x); i) U3=U2-B3; g) L=expand(U3); i) %% approximation(1)  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M1=-1*x.^3/3+x; i) exact= atan(x); ii) error=abs(M1-exact); iii) plot(x,M1,x,exact) iv) %%print v) disp(sprintf(' TABLE(1) :) ')) vi) [x' exact' M1' error'] %% approximation(2) 98  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M2= x.^9/9 - x.^7/7 + x.^5/5 - x.^3/3 + x;  exact= atan(x);  error=abs(M2-exact); i) plot(x,M2,x,exact) ii) %%print iii) disp(sprintf(' TABLE(2) :) ')) iv) [x' exact' M2' error'] %% approximation(3)  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1]; i) M3=x.^13/13 - x.^11/11 + x.^9/9 - x.^7/7 + x.^5/5 - x.^3/3 + x;  exact= atan(x); i) error1=abs(M3-exact); ii) plot(x,M3,x,exact) iii) %%print iv) disp(sprintf(' TABLE(3) :) ')) 99 v) [x' exact' M3' error']  %%all plot  plot(x,M1,'R',x,M2,'Y',x,exact,'B',x,exact,'M') ####################################################### Matlab Code for Variational Iteration Method (VIM) for Example 3.3  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Variation Iterative Method'))  %In put data  N=4;  syms x  syms r 155  syms t  y(1)=1;  %%for i=1 i) A=int(0,r,0,t); ii) B=int((0-1+exp(t)-exp(2*t)-A),t,0,x); iii) U1=y(1)-(B); iv) B=expand(U1);  %%for i=2  A2=int((exp(t-r)*(1-(subs(U1,x,r))^2)),r,0,t); (1) B2=int(diff(subs(U1,x,t)-1+exp(t)-exp(2*t)-A2),t,0,x); (2) U2=U1-B2; (3) C=expand(U2); (4) %%for i=3 (5) A3=int((exp(t-r)*(1-(subs(U2,x,r))^2)),r,0,t); (6) B3=int(diff(subs(U2,x,t)-1+exp(t)-exp(2*t)-A3),t,0,x); (7) U3=U2-B3; (8) L=expand(U3); 151  %% approximation(1) a) x=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ]; (a) M1=x.^4/24 + x.^3/6 + x.^2/2 + x + 0.9999; (b) exact=exp(x); (c) error=abs(M1-exact); (d) plot(x,M1,x,exact) (e) %%print (f) disp(sprintf(' TABLE(1) :) ')) (g) [x' exact' M1' error'] (h) %% approximation(2) b) x=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ]; (a) M2=x.^9/362880 + x.^8/40320 + x.^7/5040 + x.^6/720 + x.^5/120 + x.^4/24 + x.^3/6 + x.^2/2 + x + 1; (b) exact=exp(x); (c) error=abs(M2-exact); (d) plot(x,M2,x,exact) (e) %%print 152 (f) disp(sprintf(' TABLE(2) :) ')) (g) [x' exact' M2' error'] ii) %% approximation(3) c) x=[-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 ]; (a) M3=x.^12/479001600 + x.^11/39916800 + x.^10/3628800 + x.^9/362880 + x.^8/40320 + x.^7/5040 + x.^6/720 + x.^5/120 + x.^4/24 + x.^3/6 + x.^2/2 + x + 1; (b) exact=exp(x); (c) error=abs(M3-exact); (d) plot(x,M3,x,exact) (e) %%print (f) disp(sprintf(' TABLE(3) :) ')) (g) [x' exact' M3' error']  %%all plot  plot(x,M1,'R',x,M2,'Y',x,exact,'B',x,exact,'M') ####################################################### 153 Matlab Code for Variational Iteration Method (VIM) for Example 3.4  clc  clear all  format long  %The general form of nonlinear vollterra integro differential equation  %Display  disp(sprintf('Farah Abu-Thabet'))  disp(sprintf('Variation Iterative Method'))  %In put data  N=4;  syms x  syms r  syms t  y(1)=2;  %%for i=1  A=int(exp(t-r)*((y(1))^2),r,0,t); 154  B=int((0-1-exp(t)+2*t*exp(t)+exp(2*t)-A),t,0,x);  U1=y(1)-(B);  A=expand(U1);  %% for 2  A2=int(((subs(U1,x,r))^2)*(exp(t-r)),r,0,t);  B2=int((subs(diff(U1),x,t))-1-exp(t)+2*t*exp(t)+exp(2*t)-A2,t,0,x);  U2=U1-B2;  B=expand(U2);  %% for 3  A3=int(((subs(U2,x,r))^2)*(exp(t-r)),r,0,t);  B3=int((subs(diff(U2),x,t))-1-exp(t)+2*t*exp(t)+exp(2*t)-A3,t,0,x);  U3=U2-B3;  C= expand(U3);  %% approximation(1)  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M1=7*exp(x) - exp(2.*x)/2 - 3.*x - 2.*x.*exp(x) - 9/2; 155  exact=1+exp(x);  error1=abs(M1-exact);  plot(x,M1,x,exact)  %%print  disp(sprintf(' TABLE(1) :) '))  [x' exact' M1' error']  %% approximation(2)  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M2=52*exp(2.*x) - (257.*x)/4 - (13*exp(3.*x))/9 + exp(4.*x)/48 + (2*exp(x))/3 - (37.*x.*exp(2.*x))/2 + (x.*exp(3.*x))/3 - 24.*x.^2.*exp(1.*x) + 4.*x.^3.*exp(x) + 2.*x.^2.*exp(2.*x) - 17.*x.*exp(x) - (45.*x.^2)/2 - 3.*x.^3 - 7091/144;  exact=1+exp(x);  error=abs(M2-exact);  plot(x,M2,x,exact)  %%print  disp(sprintf(' TABLE(2) :) '))  [x' exact' M2' error'] 156  %% approximation(3)  x=[0 0.0938 0.2188 0.3125 0.4062 0.5 0.6250 0.7188 0.8125 0.9062 1];  M3=(23854975*exp(2.*x))/576 - (1755133633.*x)/20736 - (4810751*exp(3.*x))/7776 + (86570981*exp(4.*x))/248832 - (4698473*exp(5.*x))/480000 + (319343*exp(6.*x))/1944000 - (65*exp(7.*x))/42336 + exp(8.*x)/129024 + (2933236952771*exp(1.*x))/54432000 - (12187981.*x.*exp(2.*x))/288 + (533779.*x.*exp(3.*x))/432 - (4766885.*x.*exp(4.*x))/20736 + (381511.*x.*exp(5.*x))/72000 - (1351.*x.*exp(6.*x))/21600 + (x.*exp(7.*x))/3024 + (733435.*x.^2.*exp(x))/144 - 1433.*x.^3.*exp(x) + (52903.*x.^4.*exp(x))/72 + (128.*x.^5.*exp(x))/5 + 18.*x.^6.*exp(x) - (24.*x.^7.*exp(x))/7 + (1404235.*x.^2.*exp(2.*x))/72 - (60869.*x.^2.*exp(3.*x))/54 - (74527.*x.^3.*exp(2.*x))/12 + (232073.*x.^2.*exp(4.*x))/3456 + (19967.*x.^3.*exp(3.*x))/54 + (2831.*x.^4.*exp(2.*x))/2 –  (7609.*x.^2.*exp(5.*x))/7200 - (8441.*x.^3.*exp(4.*x))/864 - (469.*x.^4.*exp(3.*x))/9 - 174.*x.^5.*exp(2.*x) + (7.*x.^2.*exp(6.*x))/1080 + (3.*x.^3.*exp(5.*x))/40 + (5.*x.^4.*exp(4.*x))/9 + (8.*x.^5.*exp(3.*x))/3 + 8.*x.^6.*exp(2.*x) - (2214919.*x.*exp(x))/216 - (23678779.*x.^2)/576 - (101187.*x.^3)/8 - (252809.*x.^4)/96 - (7347.*x.^5)/20 - (63.*x.^6)/2 - (9.*x.^7)/7 - 482735066144893/5080320000; 157  exact=1+exp(x);  error=abs(M3-exact);  plot(x,M3,x,exact)  %%print  disp(sprintf(' TABLE(3) :) '))  [x' exact' M3' error']  %%all plot  plot(x,M1,'R',x,M2,'Y',x,M3,'G',x,exact,'B') جامعة النجاح الوطنية كمية الدراسات العميا الطرق العددية لحل معادلة فولتيرا التكاممية التفاضمية الغير خطية إعداد فرح خالد شحادة أبو ثابت إشراف اجي قطنانيد. ن أ. قدمت هذه األطروحة استكمااَل لمتطمبات الحصول عمى درجة الماجستير في الرياضيات .فمسطين -بكمية الدراسات العميا، في جامعة النجاح الوطنية، نابمس ،المحوسبة 3139 ب الطرق العددية لحل معادلة فولتيرا التكاممية التفاضمية الغير خطية إعداد و ثابتفرح خالد شحادة أب إشراف .د. ناجي قطنانيأ الممخص تحتوي في ىذه األطروحة ركزنا عمى حل معادلة فولتيرا التكاممية التفاضمية الغير خطية ألنيا عمى مج