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

Anwar Ahmad, Muhammad Ali, Salman A. Malik

Abstract

This article primarily aims at introducing a novel operational calculus of Mikusiński'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.

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

Abstract

This article primarily aims at introducing a novel operational calculus of Mikusiński'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.
Paper Structure (8 sections, 14 theorems, 88 equations)

This paper contains 8 sections, 14 theorems, 88 equations.

Table of Contents

  1. Introduction and Problems Statement
  2. Preliminaries and Notations
  3. Operational calculus for Dzherbashian-Nersesian operator
  4. Forward and Backward Source Problems
  5. Spectral Problem
  6. Forward Problem
  7. Backward Problem
  8. Concluding Remarks

Key Result

lemma 1

Podlubny The relation given below is valid subject to certain conditions that $\alpha<2$, $\beta$ is any real number, $\mu$ is a value satisfying $\pi\alpha/2<\mu<min\{\pi,\pi\alpha\}$, $z$ is a complex number in a manner that it satisfies $|z|\geq0$, $\mu\leq |arg(z)|\leq\pi$ and $C_{1}$ is a real

Theorems & Definitions (28)

  • definition 1
  • definition 2
  • definition 3
  • definition 4
  • lemma 1
  • lemma 2
  • lemma 3
  • definition 5
  • theorem 1
  • definition 6
  • ...and 18 more