Approximation of solutions through the Fibonacci wavelets and measure of noncompactness to nonlinear Volterra-Fredholm fractional integral equations
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This paper consists of two significant aims. The first aim of this paper is to establish the criteria for the existence of solutions to nonlinear Volterra-Fredholm (V-F) fractional integral equations on $[0, L]$, where $0<L<\infty$. The fractional integral is described here in the sense of the Katugampola fractional integral of order $\lambda>0$ and with the parameter $\beta>0$. The concepts of the fixed point theorem and the measure of noncompactness are used as the main tools to prove the existence of solutions. The second aim of this paper is to introduce a computational method to obtain approximate numerical solutions to the considered problem. This method is based on the Fibonacci wavelets with collocation technique. Besides, the results of the error analysis and discussions of the accuracy of the solutions are also presented. To the best knowledge of the authors, this is the first computational method for this generalized problem to obtain approximate solutions. Finally, two examples are discussed with the computational tables and convergence graphs to interpret the efficiency and applicability of the presented method.
Abstract
This paper consists of two significant aims. The first aim of this paper is to establish the criteria for the existence of solutions to nonlinear Volterra-Fredholm (V-F) fractional integral equations on $[0, L]$, where $0<L<\infty$. The fractional integral is described here in the sense of the Katugampola fractional integral of order $\lambda>0$ and with the parameter $\beta>0$. The concepts of the fixed point theorem and the measure of noncompactness are used as the main tools to prove the existence of solutions. The second aim of this paper is to introduce a computational method to obtain approximate numerical solutions to the considered problem. This method is based on the Fibonacci wavelets with collocation technique. Besides, the results of the error analysis and discussions of the accuracy of the solutions are also presented. To the best knowledge of the authors, this is the first computational method for this generalized problem to obtain approximate solutions. Finally, two examples are discussed with the computational tables and convergence graphs to interpret the efficiency and applicability of the presented method.
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