A perturbation-based approach for solving fractional-order Volterra-Fredholm integro differential equations and its convergence analysis

• Published in: International journal of computer mathematics Vol. 97; no. 10; pp. 1994 - 2014
• Main Authors: Das, Pratibhamoy, Rana, Subrata, Ramos, Higinio
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.10.2020; Taylor & Francis Ltd
• Subjects: 1A58; Approximation; approximation theory; Caputo fractional derivative; Convergence; convergence analysis; Differential equations; Exact solutions; experimental evidence; Fractional integro differential equation; Integral equations; perturbation approach; Perturbation methods; Volterra-Fredholm integral equation
• ISSN: 0020-7160; 1029-0265
• DOI: 10.1080/00207160.2019.1673892

The present work considers the approximation of solutions of a type of fractional-order Volterra-Fredholm integro-differential equations, where the fractional derivative is introduced in Caputo sense. In addition, we also present several applications of the fractional-order differential equations and integral equations. Here, we provide a sufficient condition for existence and uniqueness of the solution and also obtain an a priori bound of the solution of the present problem. Then, we discuss about the higher-order model equation which can be written as a system of equations whose orders are less than or equal to one. Next, we present an approximation of the solution of this problem by means of a perturbation approach based on homotopy analysis. Also, we discuss the convergence analysis of the method. It is observed through different examples that the adopted strategy is a very effective one for good approximation of the solution, even for higher-order problems. It is shown that the approximate solutions converge to the exact solution, even for higher-order fractional differential equations. In addition, we show that the present method is highly effective compared to the existed method and produces less error.