Initial and Boundary Value Problems for Fractional differential equations involving Atangana-Baleanu Derivative

TL;DR

This paper develops explicit solutions for initial value problems involving Atangana-Baleanu fractional derivatives by transforming them into Volterra integral equations and applying Laplace transforms, with applications demonstrated through boundary value problems.

Contribution

It introduces a method to solve Atangana-Baleanu fractional differential equations using integral equations and series expansions, providing a new approach for such problems.

Findings

Explicit solutions derived for initial value problems

Successive approximation method effectively solves Volterra equations

Series solutions expressed using orthogonal basis functions

Abstract

Initial value problem involving Atangana-Baleanu derivative is considered. An Explicit solution of the given problem is obtained by reducing the differential equation to Volterra integral equation of second kind and by using Laplace transform. To find the solution of the Volterra equation, the successive approximation method is used and a lemma simplifying the resolvent kernel has been presented. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form of series expansion using orthogonal basis obtained by separation of variables.