Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

Torebek, Berikbol T.; Tapdıgoglu, Ramiz

doi:10.1002/mma.4468

Cited by 50 publications

(42 citation statements)

References 21 publications

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“…The nonlocal functional-differential operator (6.4)-(6.5) is self-adjoint [KST17,TT17]. For |ε| < 1, the operator (6.4)-(6.5) has the following eigenvalues λ 2k = 4(1 + ε)k 2 , k ∈ N and λ 2k+1 = (1 − ε)(2k + 1) 2 , k ∈ N ∪ {0} and corresponding eigenfunctions…”

Section: Examplesmentioning

confidence: 99%

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Ruzhansky

Tokmagambetov

Torebek

2019

Journal of Inverse and Ill-Posed Problems

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A class of inverse problems for restoring the right-hand side of a parabolic equation for a large class of positive operators with discrete spectrum is considered. The results on existence and uniqueness of solutions of these problems as well as on the fractional time diffusion (subdiffusion) equations are presented. Consequently, the obtained results are applied for the similar inverse problems for a large class of subelliptic diffusion and subdiffusion equations (with continuous spectrum). Such problems are modelled by using general homogeneous leftinvariant hypoelliptic operators on general graded Lie groups. A list of examples is discussed, including Sturm-Liouville problems, differential models with involution, fractional Sturm-Liouville operators, harmonic and anharmonic oscillators, Landau Hamiltonians, fractional Laplacians, and harmonic and anharmonic operators on the Heisenberg group. The rod cooling problem for the diffusion with involution is modelled numerically, showing how to find a "cooling function", and how the involution normally slows down the cooling speed of the rod.

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Section: Examplesmentioning

confidence: 99%

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Ruzhansky

Tokmagambetov

Torebek

2019

Journal of Inverse and Ill-Posed Problems

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“…In Ahmad et al, there are good references to publications on related issues. We note from recent papers close to the theme of our article. In these papers, different variants of direct and inverse initial‐boundary value problems for evolutionary equations are considered, including problems with nonlocal boundary conditions and problems for equations with fractional derivatives.…”

Section: Reduction To a Mathematical Problemmentioning

confidence: 85%

“…The mathematical problem to for a = 0 was considered in Torebek and Tapdigoglu, and for a = β = 0 in Kirane et al…”

Section: Reduction To a Mathematical Problemmentioning

confidence: 99%

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On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Kirane

Sadybekov

Sarsenbi

2019

Math Methods in App Sciences

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We consider a problem of modeling the thermal diffusion process in a closed metal wire wrapped around a thin sheet of insulation material. The layer of insulation is assumed to be slightly permeable. Therefore, the temperature value from one side affects the diffusion process on the other side. For this reason, the standard heat equation is modified, and a third term with an involution is added. Modeling of this process leads to the consideration of an inverse problem for a one‐dimensional fractional evolution equation with involution and with periodic boundary conditions with respect to a space variable. This equation interpolates heat equation. Such equations are also called nonlocal subdiffusion equations or nonlocal heat equations. The inverse problem consists in the restoration (simultaneously with the solution) of the unknown right‐hand side of the equation, which depends only on the spatial variable. The conditions for overdefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

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“…Here, we prove well‐posedness results for Equation in the homogeneous and non‐homogeneous cases. For more mathematical justifications of anomalous diffusion processes, we refer to other studies and references therein.…”

Section: Introductionmentioning

confidence: 99%

Anomalous diffusion phenomena with conservation law for the fractional kinetic process

Tokmagambetov

Torebek

2018

Math Methods in App Sciences

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Some inverse problems for the nonlocal heat equation with Caputo fractional derivative

Cited by 50 publications

References 21 publications

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations

On an inverse problem of reconstructing a subdiffusion process from nonlocal data

Anomalous diffusion phenomena with conservation law for the fractional kinetic process

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