Salman A. Malik

We considered the direct problem (DP) and inverse problem (IP) of determining a reaction coefficient of the diffusion equation involving Dzherbashian‐Nersesian integro‐differential operator (DNIDO). The solutions of direct and IPs are written in the form of Fox H‐function. The reaction coefficient is supposed to depend only on the first two components of the spatial variable and on the time variable . The over‐specified condition for the IP is the projection of diffusion concentration in the ‐plane. The IP is reduced to an equivalent integral equation (IE), and existence and uniqueness for the solution of the IP is proved by applying the Banach fixed point theorem (BFPT). The stability of the solution of the direct and IPs, when diffusion coefficient, source term, and initial data are perturbed, is proved and some examples are provided. We have generalized our main problem from to . Some numerical examples with an exponentially decaying source and periodic reaction coefficient term are presented.

A system of nonlinear fractional differential equations involving nth level fractional derivative is considered. We prove integration by parts formula for nth level fractional derivative. The result about non-existence of global solution in time has been proved. Our method of proof is based on the weak formulation and judicious choice of test function. Some examples are provided to verify the mathematical analysis.

This article focuses on inverse problem for Hallaire-Luikov moisture transfer equation involving Hilfer fractional derivative in time. Hallaire-Luikov equation is used to study heat and mass transfer in capillary-porous bodies. Spectral expansion method is used to find the solution of the inverse problem. By imposing certain conditions on the functions involved and utilizing certain properties of multinomial Mittag-Leffler function, it is shown that the solution to the equation, known as the inverse problem, is regular and unique. Moreover, the inverse problem exhibits ill-posedness in the sense of Hadamard. The article ends with an example to demonstrate these theoretical findings.

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 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. Solutions are presented using Mittag‐Leffler (ML) type functions. The outcomes related to the existence and uniqueness subject to certain conditions of regularity on the input data are developed.

A diffusion-wave equation with multi-term Hilfer fractional derivatives (HFDs) in time and ultra-hyperbolic operator (UHO) in space has been considered. Fundamental solution of the fractional diffusion-wave equation is obtained by using Laplace and Fourier transform with Mellin-Barnes integral representation. The solution obtained involved the Fox H-function. In addition, we provide some special cases of diffusion-wave equation.

A diffusion equation involving integral convolution in time variable with arbitrary kernel and nonlocal boundary conditions is considered. The existence and uniqueness results for two inverse problems of determining source terms (space‐ and time‐dependent sources) along with diffusion concentration from appropriate over‐specified conditions are presented. A bi‐orthogonal system of functions is used to have series representation of the solutions of the inverse problems. Several special cases such as standard diffusion, multi‐term diffusion, and tempered diffusion equations are discussed, and some examples are provided.

This paper delves into both direct and two inverse source problems associated with a diffusion equation featuring integral convolution over time, while considering non-classical boundary conditions. The inverse source problems are shown to exhibit ill-posed characteristics in accordance with Hadamard’s definition. A bi-orthogonal function system is employed to express series solutions for the inverse source problems. By imposing specific conditions on the provided data, we establish the existence of unique series solutions. Several special cases of the diffusion equation are presented, depending on the nature of the memory kernel. Furthermore, to illustrate the findings regarding inverse source problems, we provide specific examples.