Inverse problems for a multi-term time fractional evolution equation with an involution

Ilyas, Asim; Malik, Salman A.; Saif, Summaya

doi:10.1080/17415977.2021.2000606

Cited by 14 publications

(12 citation statements)

References 25 publications

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“…Kinash et al [12] presented two IPs for a generalized subdiffusion equation with final over-determination condition. Asim et al [8] studied two IPs for a multiterm time-fractional evolution equation with an involution term, interpolating the heat and wave equations.…”

Section: Inverse Source Problem-i (Isp-i)mentioning

confidence: 99%

On the Inverse Problems for a Family of Integro-Differential Equations

Suhaib

Ilyas

Malik

2023

Mathematical Modelling and Analysis

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Section: Inverse Source Problem-i (Isp-i)mentioning

confidence: 99%

On the Inverse Problems for a Family of Integro-Differential Equations

Suhaib

Ilyas

Malik

2023

Mathematical Modelling and Analysis

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“…Malik et al [18] considered an IP for the determination of source term and diffusion concentration for a multi-term fractional diffusion equation with integral type over-specified condition. Asim et al [19] studied two IPs for a multi-term time fractional evolution equation with an involution with appropriate over-specified conditions. Spectral problems with nonlocal boundary conditions have been discussed in [20][21][22].…”

Section: Problem Formulation and Introductionmentioning

confidence: 99%

“…A number of researchers investigate inverse problems for fractional diffusion equations involving the Caputo fractional derivative with respect to time [23, 24]. Multi‐term fractional derivatives enable to model accelerating and retarding sub(super) diffusion, since different powers of

t

dominate as

t\to 0&amp;amp;#x0002B;

and

t\to \infty

in the kernel [19, 25]. Tempered fractional derivatives are used to describe slow transition of anomalous diffusion to a normal one.…”

Section: Problem Formulation and Introductionmentioning

confidence: 99%

An inverse problem for a two‐dimensional diffusion equation with arbitrary memory kernel

Saif

Malik

2023

Math Methods in App Sciences

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An inverse source problem (ISP) of recovering space‐dependent source term along with diffusion concentration for a two‐dimensional diffusion equation involving integral convolution of arbitrary memory kernel in time is considered. The unique existence of the solution is proved using a bi‐orthogonal system of functions obtained from the associated non‐self‐adjoint spectral problem and its adjoint problem which form Riesz basis in L2false[false(0,1false)×false(0,1false)false]$$ {L}&#x0005E;2\left[\left(0,1\right)\times \left(0,1\right)\right] $$. In addition, some particular cases of ISP are described as special cases of our results.

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“…Among the inverse problems for the fractional diffusion equation with Riemann–Liouville and Caputo type derivatives, the most common are inverse source problems with different overdetermination conditions (see, e.g., [20–26] and the literature in them). In the work [24], there also were only obtained the uniqueness theorem for inverse problem of determining the various time‐independent smooth coefficients appearing in time‐fractional diffusion equations, from measurements of the solution on a certain subset at fixed time.…”

Section: Introduction: Formulation Of Problemmentioning

confidence: 99%

Inverse coefficient problem for the time‐fractional diffusion equation with Hilfer operator

Durdiev

2023

Math Methods in App Sciences

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In this paper, the inverse problem of determining the time‐dependent reaction–diffusion coefficient in the Cauchy problem for the time‐fractional diffusion equation with the Hilfer operator by a single observation at the point of the diffusion process is studied. To represent the solution of the direct problem, the fundamental solution of this equation is constructed and properties of this solution are investigated. The fundamental solution contains Fox's ‐functions widely used in fractional calculus. The fundamental solution for the fractional diffusion operator with the Hilfer derivative is also obtained in terms of the Yang ‐function, and it is shown that they coincide. Using estimates of the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown coefficient, which will be used in investigation of inverse problem. The inverse problem is reduced to the equivalent integral equation. For solving this equation, the contracted mapping principle is applied. The local existence and global uniqueness results and the conditional stability estimate of solution are proven.

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Inverse problems for a multi-term time fractional evolution equation with an involution

Cited by 14 publications

References 25 publications

On the Inverse Problems for a Family of Integro-Differential Equations

On the Inverse Problems for a Family of Integro-Differential Equations

An inverse problem for a two‐dimensional diffusion equation with arbitrary memory kernel

Inverse coefficient problem for the time‐fractional diffusion equation with Hilfer operator

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