Abstract
In this work, we propose a new method for solving Volterra integral equations. The technique is based on the Chebyshev spectral collocation method. The application of the proposed method leads Volterra integral equation to a system of algebraic equations that are easy to solve. Some examples are presented and compared with some methods in the literature to illustrate the ability of this technique. The results demonstrate that the new method is more efficient, convergent, and accurate to the exact solution.
Highlights
Many mathematical formulations of physical phenomena contain integral equations; these equations arise in many fields including biology, economics engineering, and medicine; for more details, see [1,2,3,4]
Implementation of the numerical schemes was performed using the personal computer of 2.5 GHz CPU speed including MATLAB 2017 package to perform the simulation results. e accuracy of the method is demonstrated by presenting infinity error norms uE(x) between exact and approximate solutions computed as uE(x) max|u(x) − u(x)|, (48)
A striking feature of the proposed method is that a high level of accuracy is achieved at the first order of approximation than the Adomian Decomposition Method (ADM), for example, whereas the proposed method gives the exact value at only one iteration for the 14th digits. is is because the proposed method selects all the terms as linear operator
Summary
Many mathematical formulations of physical phenomena contain integral equations; these equations arise in many fields including biology, economics engineering, and medicine; for more details, see [1,2,3,4]. Volterra integral equations are usually difficult to solve analytically, so we resort to finding approximate solutions to the problems using numerical or analytical approximation methods. E main purpose of this work is to introduce a new method for solving Volterra integral equations that uses Chebyshev spectral collocation method. Chebyshev spectral collocation methods have been applied successfully in different areas of sciences and engineering because of their ability to give highly accurate solutions of single or system of boundary value problems [9,10,11,12,13]. We introduce a new integral transformation that is the replacement of an integral term in the Volterra integral equation by Chebyshev spectral differential matrix of known elements.
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