Abstract
The collocation method based on Chebyshev basis functions, coupled Picard iterative process, is proposed to solve a functional Volterra integral equation of the second kind. Using the Banach Fixed Point Theorem, we prove theorems on the existence and uniqueness solutions in the L2-norm. We also provide the convergence and stability analysis of the proposed method, which indicates that the numerical errors in the L2-norm decay exponentially, provided that the kernel function is sufficiently smooth. Numerical results are presented and they confirm the theoretical prediction of the exponential rate of convergence.
Highlights
In this work we study both theoretical and numerical aspects of solutions of a functional Volterra integral equations of the form x u(x) = g(x) + f x, κ(x, y)u(y) dy, x ∈ (a, b), (1.1)
The existence and uniqueness of the solution of the functional Volterra integral equation (1.1) is guaranteed by the result, whose proof is based on the Banach Fixed Point Theorem
This paper applies a numerical procedure for a nonlinear Volterra integral equations based on Chebyshev and Legendre bases functions considering the collocation method
Summary
We consider the Chebyshev collocation method [4, 5, 6] and the Picard iterative process [14] for the numerical solution of nonlinear integral equation (1.1) Our tasks in this step are to show that the approximate solution converges, under particular conditions, to an exact solution in L2(a, b), and to analysis the rate of this convergence. There is a rich literature of Chebyshev collocation methods related to Volterra integral equations, see [3,16,17,19], the convergence analysis of the functional Volterra integral equation (1.1) is a new study for this theory Such methodology was implemented based on the Chebyshev polynomials of degree N, which has as main advantage an infinite convergence rate (in terms of the order of accuracy), since it uses high degree polynomials as shape functions.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
