Solvability of mixed problems for the wave equation with reflection of the argument

Mussirepova, Elmira; Sarsenbi, A. M.

doi:10.1002/mma.8448

Cited by 5 publications

(4 citation statements)

References 33 publications

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“…This proves the uniform convergence of the series u xxxx (x, t) in the open domain Ω. Thus, we have shown that series (17), (18) satisfy equation (4).…”

Section: Resultssupporting

confidence: 69%

“…The existence and uniqueness of the solution of mixed problems for the heat equation with a complex-valued coefficient was established in [3]. The solvability of mixed problems for a perturbed wave equation with involution and with a variable complex-valued coefficient was studied in [4,5]. The solvability of inverse problems for the perturbed heat equation with involution and with a variable complex-valued coefficient was considered in [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

“…. We enumerate ω 1 , ω 2 , ω 3 , ω 4 different roots of the number 4 √ −1 so that for ρ ∈ S ν , Re (ρω 1 ) ≤ Re (ρω 2 ) ≤ Re (ρω 3 ) ≤ Re (ρω 4 ).…”

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confidence: 99%

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On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

Imanbetova,

Sarsenbi,

Seilbekov

2024

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In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

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“…This proves the uniform convergence of the series u xxxx (x, t) in the open domain Ω. Thus, we have shown that series (17), (18) satisfy equation (4).…”

Section: Resultssupporting

confidence: 69%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

Imanbetova,

Sarsenbi,

Seilbekov

2024

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“…This is particularly true for numerical approaches that deal with this category of problems. Recent research has focused on inverse problems involving perturbations of the involution type, as seen in the literature [8,9,[14][15][16].…”

Section: Direct Problem For Time-fractional Heat Equation With Involu...mentioning

confidence: 99%

The modified fractional‐order quasi‐reversibility method for a class of direct and inverse problems governed by time‐fractional heat equations with involution perturbation

Benabbes,

Boussetila,

Lakhdari

2024

Math Methods in App Sciences

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This study aims to explore two classes of ill‐posed problems governed by a nonclassical fractional heat equation with an involution perturbation. To achieve a stable solution, we introduce a modified pseudo‐parabolic regularization method that involves a correction term through a mixed fractional derivation operator, resulting in a sequence of well‐posed problems that depend on a regularization parameter . We rigorously demonstrate that the resulting approximate problems are well‐posed and present convergence results for this regularization approach.

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Solvability of Mixed Problems for a Fourth-Order Equation with Involution and Fractional Derivative

Kirane

Sarsenbi

2023

Fractal Fract

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In the present work, two-dimensional mixed problems with the Caputo fractional order differential operator are studied using the Fourier method of separation of variables. The equation contains a linear transformation of involution in the second derivative. The considered problem generalizes some previous problems formulated for some fourth-order parabolic-type equations. The basic properties of the eigenfunctions of the corresponding spectral problems, when they are defined as the products of two systems of eigenfunctions, are studied. The existence and uniqueness of the solution to the formulated problem is proved.

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Solvability of mixed problems for the wave equation with reflection of the argument

Abstract: The paper considers mixed problems for the wave equation with reflection of the argument, with a complex-valued variable coefficient. The existence and uniqueness of solution of the studied problems is established by the method of separation of variables.

Cited by 5 publications

References 33 publications

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

The modified fractional‐order quasi‐reversibility method for a class of direct and inverse problems governed by time‐fractional heat equations with involution perturbation

Solvability of Mixed Problems for a Fourth-Order Equation with Involution and Fractional Derivative

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