Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

Ali, Muhammad; Aziz, Sara; Malik, Salman A.

doi:10.1002/mma.4776

Cited by 31 publications

(17 citation statements)

References 49 publications

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“…Lemma 32 Let, for the function h 1 , h 2 ∈ C 1 ([ a , b ]), the following relation hold:

\frac{d}{d z} (h_{1} (z) * h_{2} (z)) = h_{1} (z) h_{2} (a) + h_{1} (z) * \frac{d}{d z} h_{2} (z) = h_{2} (z) h_{1} (a) + h_{2} (z) * \frac{d}{d z} h_{1} (z) .

…”

Section: Preliminaries and Some Basic Resultsmentioning

confidence: 99%

An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

Samreen

Malik

2021

Math Methods in App Sciences

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In this work, we investigate unique solvability of inverse source problem (ISP) of determining a time‐dependent source term along with solution for a multi‐term fractional differential equation involving two‐parameter fractional derivative in time usually known as Hilfer fractional derivative. We applied the eigenfunction expansion method, and the corresponding spectral problem consists of Bessel differential equation in space variable. The multi‐term fractional order differential equations are solved by Laplace transform and solution involve multinomial Mittag–Leffler type functions. Banach fixed‐point theorem is used to prove unique existence of time‐dependent source term whenever integral type over‐determination condition is given. Some examples are provided to support our analysis.

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“…Lemma 32 Let, for the function h 1 , h 2 ∈ C 1 ([ a , b ]), the following relation hold:

\frac{d}{d z} (h_{1} (z) * h_{2} (z)) = h_{1} (z) h_{2} (a) + h_{1} (z) * \frac{d}{d z} h_{2} (z) = h_{2} (z) h_{1} (a) + h_{2} (z) * \frac{d}{d z} h_{1} (z) .

…”

Section: Preliminaries and Some Basic Resultsmentioning

confidence: 99%

An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

Samreen

Malik

2021

Math Methods in App Sciences

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“…An inverse problem involving generalized fractional derivative in diffusion and wave equations for time and space dependent source terms are discussed in [15]. In [4], for a Space-Time Fractional Diffusion Equation (STFDE) inverse problem of determining a temporal component in the source term from the total energy of the system is considered and the recovery of a space dependent source term from final data is discussed in [3]. Inverse problems of identifying the space and time dependent source terms for a STFDE are considered in [5].…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Determination of a Source Term and Diffusion Concentration for a Multi-Term Space-Time Fractional Diffusion Equation

Malik

Ilyas

Samreen

2021

Mathematical Modelling and Analysis

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An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

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“…The inverse problem is known as inverse problem of coefficient identification or inverse problem of source identification in accordance with this unknown input respectively. In the last few years, there has been increasing interest in investigating the inverse problems of time fractional differential equations (see [14], [15], [16], [17]).…”

Section: Introductionmentioning

confidence: 99%

Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator

Ahmad,

Ali,

Malik

2021

Preprint

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Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives namely Riemann-Liouville, Caputo and Hilfer derivatives are shown to be special cases of the earlier one. The expression for Laplace transform of fractional Dzherbashian-Nersesian operator is constructed. Inverse problems of recovering space dependent and time dependent source terms of a time fractional diffusion equation with involution and involving fractional Dzherbashian-Nersesian operator are considered. The results on existence and uniqueness for the solutions of inverse problems are established. The results obtained here generalize several known results.

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Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

Cited by 31 publications

References 49 publications

An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

An inverse problem for a multi‐term fractional differential equation with two‐parameter fractional derivatives in time and Bessel operator

Simultaneous Determination of a Source Term and Diffusion Concentration for a Multi-Term Space-Time Fractional Diffusion Equation

Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator

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