Unraveling Forward and Backward Source Problems for a Nonlocal Integrodifferential Equation: A Journey through Operational Calculus for Dzherbashian-Nersesian Operator

TL;DR

This paper develops a new operational calculus for the Dzherbashian-Nersesian operator, enabling exact solutions to forward and backward source problems with nonlocal boundary conditions using Mittag-Leffler functions.

Contribution

It introduces a novel operational calculus of Mikusiński's type for the Dzherbashian-Nersesian operator, facilitating explicit solutions to complex integrodifferential equations.

Findings

Derived exact solutions using Mittag-Leffler functions

Established existence and uniqueness conditions

Provided a framework for solving nonlocal boundary problems

Abstract

This article primarily aims at introducing a novel operational calculus of Mikusi\'nski's type for the Dzherbashian-Nersesian operator. Using this calculus, we are able to derive exact solutions for the forward and backward source problems (BSPs) of a differential equation that features Dzherbashian-Nersesian operator in time and intertwined with nonlocal boundary conditions. The initial condition is expressed in terms of Riemann-Liouville integral (RLI). Solution is presented using Mittag-Leffler type functions (MLTFs). The outcomes related to the existence and uniqueness subject to certain conditions of regularity on the input data are established.