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

Abstract

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.

Keywords:

• Fractional integro differential equation
• Caputo fractional derivative
• Volterra–Fredholm integral equation
• approximation theory
• convergence analysis
• perturbation approach
• experimental evidence

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 26A33
• 1A58
• 34K05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The author, Pratibhamoy Das, wants to thank the Science and Engineering Research Board (SERB), Govt. of India, for providing the research support for the present work under the Early Career Research Award with the grant ECR/2017/001973.