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Masa Husam Saeed Desuqi
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Background: Integro-Differential Equations (IDEs), one of the most important mathematical tools in both pure and applied mathematics, arise in many physical problems such as wind ripple in the desert, nano-hydrodynamics, population growth model, glass-forming process.. They have motivated a huge amount of research in recent years. Many researchers have developed numerical schemes for solving these IDEs. Aim: In this work, the researcher proposed three numerical schemes, namely, the Taylor collocation method, Legendre polynomials method and the Haar wavelets method, to approximate the solution of Volterra Integro-Differential Equations (VIDEs). Material and method: These three numerical methods have been applied in the form of algorithms, and Maple software has been developed/used to solve some numerical examples. Results: The numerical results showed that the convergence and accuracy of the aforementioned methods were in good agreement with the analytical solution. Comparison of numerical results, mentioned in tables and figures, showed clearly that the Haar wavelets method provides more accurate results and is, therefore, more effective than other methods. Keywords: Volterra integro-differential equations; Taylor collocation; Legendre polynomials; Haar wavelets. Introduction Importance of Integral Equations: the subject of IDEs have attracted the attention of many scientists and researchers over the past few years due to their wide range of applications like wind ripple, hydro-dynamics, glass-formation, model of population growth and various models in physics, engineering and medicine, namely the mathematical modelling of epidemics, particular when the models contains age-structure or describe spatial epidemics [5, 6]. Literature Review Many numerical schemes for solving VIDEs have been constructed and implemented by many researches. For example, Karamete et al. [19] used collocation method based on Taylor expansion to solve VIDEs. Khater et al. [20] implemented the Legendre polynomials method to approximate the solution of VIDEs subject to initial conditions. Ali [2] has applied the Haar wavelets approach to obtain an approximate solution to VIDEs. Draidi et al. [12] used the product Nystrom in parallel with the sinc-collocation scheme to solve integral equations with Carleman kernel. Hamaydi et al. [16] solved fuzzy integral equations using variational iteration and Taylor expansion techniques. Moreover, Fazeli et al. [14] suggested several numerical schemes to approximate the solution of VIDEs. Other numerical techniques for solving VIDEs are: variational iteration [30], Walsh expansion series [26], Chebyshev collocation [1], Nystr"o" ̈m method [22], differential transform [11], homotopy [25], power series [3] and finite difference [10]. Burton [7, 8] has investigated in the 1980s some stability results for the VIDEs. Zhang [31] has also presented some stability results for VIDEs. Tunc [28] and Staffans [27] proposed a new stability results based on Lyapunov functional for VIDEs. In this work, we suggest three numerical techniques to solve VIDEs, namely, Taylor collocation method, Legendre polynomials and Haar wavelets method. The VIDE under consideration has the form: Y^((n) ) (x)=r(x)+∫_a^x▒F(x,t)Y(t)dt where Y^((n) ) (x)=(d^n Y)/(dx^n ) ,n∈N subject to the initial conditions: Y^((s))=a_s , s=0,1,2,…,(n-1). The kernel F(x,t) and the function r(x) are given. The unknown function Y(x) is to be determined. A major objective is to compare between these numerical techniques in approximating the solution of the VIDE by solving some numerical examples. We organize this work as follows: chapter one deals with some general aspects of VIDEs together with their solvability. In chapter two, we address all the aforementioned methods, namely, Taylor collocation, Legendre polynomials and Haar wavelets methods. We conclude chapter three by solving some VIDEs with known exact solution by the aforementioned algorithms.