Direct and Inverse Problems for a Samarskii-Ionkin Type Problem for a Two Dimensional Fractional Parabolic Equation

Kerbal, Sebti; Kadirkulov, B. J.; Kirane, Mokhtar

doi:10.18576/pfda/040301

Cited by 11 publications

(9 citation statements)

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“…We study the solvability of problem (1)-( 5) for various values of the spectral parameter. This work is a further development of the results of [35,[38][39][40][42][43][44][45].…”

Section: Problem Statementmentioning

confidence: 78%

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Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

Yuldashev

Kadirkulov

2020

Axioms

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In this paper, we consider a boundary value problem for a nonlinear partial differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. With respect to the first variable, this equation is a nonlinear fractional differential equation in the positive part of the considering segment and is a second-order nonlinear differential equation with spectral parameter in the negative part of this segment. Using the Fourier series method, the solutions of nonlinear boundary value problems are constructed in the form of a Fourier series. Theorems on the existence and uniqueness of the classical solution of the problem are proved for regular values of the spectral parameter. For irregular values of the spectral parameter, an infinite number of solutions of the mixed equation in the form of a Fourier series are constructed.

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“…We study the solvability of problem (1)-( 5) for various values of the spectral parameter. This work is a further development of the results of [35,[38][39][40][42][43][44][45].…”

Section: Problem Statementmentioning

confidence: 78%

“…When t < 0, by virtue of the conditions of the theorem and applying the Cauchy-Schwarz inequality and Bessel inequality to (45) we similarly obtain the following estimates:…”

Section: Solvability Of Scsnie (30) and (31)mentioning

confidence: 90%

Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

Yuldashev

Kadirkulov

2020

Axioms

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“…More detailed information on nonlocal problems can be found in the monograph [18]. We would like to note some works [14,[30][31][32], where nonlocal problems for partial differential and integro-differential equations with derivatives of integer or fractional orders were studied.…”

Section: Problem Statementmentioning

confidence: 99%

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

Yuldashev

Kadirkulovich

2020

UMJ

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In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

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“…The problems close to our studies were considered in the work of S. Kerbal et al [25]. In this work, for a differential equation of a fractional order with a differential operator of the fourth order in two spatial variables, an initial-boundary value problem is studied, where the boundary conditions in the first variable are specified as Dirichlet-type conditions, and those in the second variable are considered as nonlocal Samarsky-Ionkin-type conditions.…”

Section: Introductionmentioning

confidence: 98%

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

et al. 2023

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The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems are solved using the Fourier method. The properties of eigenfunctions and associated functions of the corresponding spectral problems are studied. Theorems on the existence and uniqueness of solutions to the studied problems are proved.

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Direct and Inverse Problems for a Samarskii-Ionkin Type Problem for a Two Dimensional Fractional Parabolic Equation

Cited by 11 publications

References 0 publications

Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

Boundary Value Problem for Weak Nonlinear Partial Differential Equations of Mixed Type with Fractional Hilfer Operator

Nonlocal Problem for a Mixed Type Fourth-Order Differential Equation With Hilfer Fractional Operator

On Solvability of Some Inverse Problems for a Fractional Parabolic Equation with a Nonlocal Biharmonic Operator

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