Numerical Solution of Fractional-Order Fredholm Integrodifferential Equation in the Sense of Atangana–Baleanu Derivative

Jian Wang,Ayesha Jamal,Xuemei Li,Kamran Kamran,Kamal Shah

doi:10.1155/2021/6662808

Abstract

In the present article, our aim is to approximate the solution of Fredholm-type integrodifferential equation with Atangana–Baleanu fractional derivative in Caputo sense. For this, we propose a method based on Laplace transform and inverse LT. In our numerical scheme, the given equation is transformed to an algebraic equation by employing the Laplace transform. The reduced equation will be solved in complex plane. Finally, the solution of the given problem is obtained via inverse Laplace transform by representing it as a contour integral. Then, the trapezoidal rule is used to approximate the integral to high accuracy. We have considered linear and nonlinear fractional Fredholm integrodifferential equations to validate our method.

Highlights

Fractional calculus is the branch of mathematics which generalizes the concept of derivatives and integrals from integer to any positive real order [1,2,3,4,5]
Fractional derivative of a function geometrically accumulates the function. e corresponding accumulation includes the integer-order derivative as a special case. is shows that fractional calculus describes the global dynamics of realworld problems, whereas the classical calculus describes the local dynamics of the corresponding problems
It is observed that the results obtained for α 1 are almost exact, whereas for other values of α, the approximate solutions are in good agreement with the exact solution

Summary

Introduction

Fractional calculus is the branch of mathematics which generalizes the concept of derivatives and integrals from integer to any positive real order [1,2,3,4,5]. Problems involving arbitrary order operators from engineering and other sciences are growing day by day. Applications of fractional calculus can be found in frequencydependent damping behavior of viscoelastic materials [6, 7], heat diffusion [8, 9], economics [10], control theory [11], robotics, and other problems in engineering sciences, see [12] and references therein. Fractional-order operators can be used for accurate modeling of real-world problems [13]. A great effort has been made by the researchers for the existence and uniqueness of the analytical solutions of equations involving fractional operators, for example, see [13, 16, 20]

Objectives

Results

Conclusion