Numerical Analysis of the Chebyshev Collocation Method for Functional Volterra Integral Equations

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 L 2-norm. We also provide the convergence and stability analysis of the proposed method, which indicates that the numerical errors in the L 2-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.

Abstract (portuguese)

O mÃ©todo de colocaÃ§Ã£o baseado em funÃ§Ãµes de base de Chebyshev, acoplado com processo iterativo de Picard, Ã© proposto para resolver uma equaÃ§Ã£o integral funcional de Volterra do segundo tipo. Usando o Teorema do Ponto Fixo de Banach, provamos teoremas sobre a soluÃ§Ã£o de existÃªncia e unicidade na norma L 2. TambÃ©m fornecemos a anÃ¡lise de convergÃªncia e estabilidade do mÃ©todo proposto, o qual indica que os erros numÃ©ricos na norma L 2 decaem exponencialmente, desde que o kernel seja suficientemente suave. Resultados numÃ©ricos sÃ£o apresentados e confirmam a previsÃ£o teÃ³rica da taxa exponencial de convergÃªncia.

Keywords

functional Volterra integral equation collocation method, Picard iteration, mÃ©todo de colocaÃ§Ã£o de equaÃ§Ã£o integral funcional de Volterra, iteraÃ§Ã£o de Picard