Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

Yuldashev, T. K.; Kadirkulov, B. J.

doi:10.3390/fractalfract5020058

Cited by 17 publications

(7 citation statements)

References 32 publications

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“…It is not hard to verify that the series ( 17) is a formal solution to problem (15) (see, e.g., [20], p. 173, [21]). In order to prove that function ( 17) is actually a solution to the problem, it remains to substantiate this formal statement, i.e., to show that the operators A and D ρ t can be applied term-by-term to series (17). Let S j (t) be the partial sum of series (17).…”

Section: Well-posedness Of Problem (2)mentioning

confidence: 99%

“…Proof. Since f does not depend on t, then we have the following form for the Fourier coefficients of ω (see (17))…”

Section: Furthermore From Equation (2) One Hasmentioning

confidence: 99%

“…For example, when the initial temperature and final temperature for the heat equation are not indicated immediately, and it is not required to find, but information about the difference between the initial and final temperatures is sought. To the best of our knowledge, such an inverse problem was discussed only in the paper [17]. The authors considered this problem for the subdiffusion equation including the Caputo fractional derivative, the elliptical part of which is a two-variable differential expression with constant coefficients.…”

Section: Introductionmentioning

confidence: 99%

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On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Ashurov

Fayziev

2022

Fractal Fract

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The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

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Section: Well-posedness Of Problem (2)mentioning

confidence: 99%

“…Proof. Since f does not depend on t, then we have the following form for the Fourier coefficients of ω (see (17))…”

Section: Furthermore From Equation (2) One Hasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Ashurov

Fayziev

2022

Fractal Fract

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“…Inverse problems are a very important part of all sorts of engineering problems [17]. In [18], the inverse problem is considered for fractional partial differential equation with a nonlocal condition on the integral type. The considered equation is a generalization of the Barenblatt-Zheltov-Kochina differential equation, which simulates the filtration of a viscoelastic fluid in fractured porous media.…”

Section: Introductionmentioning

confidence: 99%

Computational Methods for Parameter Identification in 2D Fractional System with Riemann–Liouville Derivative

Brociek

Wajda

Sciuto

et al. 2022

Sensors

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In recent times, many different types of systems have been based on fractional derivatives. Thanks to this type of derivatives, it is possible to model certain phenomena in a more precise and desirable way. This article presents a system consisting of a two-dimensional fractional differential equation with the Riemann–Liouville derivative with a numerical algorithm for its solution. The presented algorithm uses the alternating direction implicit method (ADIM). Further, the algorithm for solving the inverse problem consisting of the determination of unknown parameters of the model is also described. For this purpose, the objective function was minimized using the ant algorithm and the Hooke–Jeeves method. Inverse problems with fractional derivatives are important in many engineering applications, such as modeling the phenomenon of anomalous diffusion, designing electrical circuits with a supercapacitor, and application of fractional-order control theory. This paper presents a numerical example illustrating the effectiveness and accuracy of the described methods. The introduction of the example made possible a comparison of the methods of searching for the minimum of the objective function. The presented algorithms can be used as a tool for parameter training in artificial neural networks.

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“…To the best of our knowledge, the inverse problem of defining the function ϕ in the non-local condition was discussed only in the paper [15]. The authors considered this problem for the subdiffusion equation with the Caputo fractional derivative, the elliptic part of which is a two-variable differential expression with constant coefficients.…”

Section: Introductionmentioning

confidence: 99%

On the uniqueness of solutions of two inverse problems for the subdiffusion equation

Ashurov¹,

Fayziev²

2022

Preprint

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Let A be an arbitrary positive selfadjoint operator, defined in a separable Hilbert space H. The inverse problems of determining the right-hand side of the equation and the function ϕ in the non-local boundary value problemOperator D t on the left-hand side of the equation expresses the Caputo derivative. For both inverse problems u(ξ 1 ) = V is taken as the over-determination condition. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence and uniqueness of a solution to problems is investigated. An interesting effect was discovered: when solving the forward problem, the uniqueness of the solution u(t) was violated, while when solving the inverse problem for the same values of α, the solution u(t) became unique.

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Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

Cited by 17 publications

References 32 publications

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

Computational Methods for Parameter Identification in 2D Fractional System with Riemann–Liouville Derivative

On the uniqueness of solutions of two inverse problems for the subdiffusion equation

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