Asymptotic behaviour of the solutions of inverse problems for pseudo-parabolic equations

Yaman, Metin; Gözükızıl, Ömer Faruk

doi:10.1016/s0096-3003(03)00691-x

Cited by 8 publications

(6 citation statements)

References 5 publications

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“…The problem statement is ∂u ∂t + Au = F (x, t) on Ω × (0, T ) D j v u = 0, j ≤ m − 1 on ∂Ω × (0, T ) u = 0, on Ω × {0}, Ω × {T }, where A is the linear elliptic partial differential operator of order 2m with the bounded measurable coefficients such that (Aφ, φ) ≥ ∥φ∥ 2 for all φ ∈ H 2m (Ω) ∩ H m 0 (Ω), µ = constant > 0. In 2004 in [YG03], Yaman and Gözükızıl studied asymptotic behaviour of the solution of the inverse source problem for the pseudo-parabolic equation (u(x, t) − ∆u(x, t)) t − ∆u(x, t) + αu(x, t) = f (t)g(x, t), Q ∞ = Ω × (0, ∞)…”

Section: Introductionmentioning

confidence: 99%

Direct and inverse problems for time-fractional heat equation generated by Dunkl operator

Bekbolat

Serikbaev

Tokmagambetov

2022

Journal of Inverse and Ill-Posed Problems

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In this paper, we study non–local in time evolution type equations generated by the Dunkl operator. Direct and inverse problems are investigated with the Caputo time-fractional heat equation with the parameter 0 < γ ≤ 1 {0<\gamma\leq 1} . In particular, well-posedness properties are established for the forward problem. To adopt techniques of the harmonic analysis, we solve the problems in the Sobolev type spaces associated with the Dunkl operator. Our special interest is an inverse source problem for the Caputo–Dunkl heat equation. As additional data, the final time measurement is taken. Since our inverse source problem is ill-posed, we also show the stability result. Moreover, as an advantage of our calculus used here, we derive explicit formulas for the solutions of the direct and inverse problems.

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Section: Introductionmentioning

confidence: 99%

Direct and inverse problems for time-fractional heat equation generated by Dunkl operator

Bekbolat

Serikbaev

Tokmagambetov

2022

Journal of Inverse and Ill-Posed Problems

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“…For example, inverse problem of identifying the unknown space-dependent coefficient [1], time-dependent coefficients under the integral overdetermination condition [15], integral boundary data [19,21], space-dependent right-hand side under integral condition [17], velocity field, pressure and right-hand side with additional condition as an integral overdetermination condition [16]. Yaman and Gözükizil [31] studied it for the asymptotic behavior of the solutions of unknown source term with the integral condition. Authors of [2,20] proved the existence and the uniqueness of the solution.…”

Section: Introductionmentioning

confidence: 99%

An inverse problem of identifying the time‐dependent potential in a fourth‐order pseudo‐parabolic equation from additional condition

Huntul

Tamsir

Dhiman

2021

Numerical Methods Partial

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The aim of this work is to identify numerically, for the first time, the time‐dependent potential coefficient in a fourth‐order pseudo‐parabolic equation with nonlocal initial data, nonlocal boundary conditions, and the boundary data as overdetermination condition. This problem emerges significantly in the modeling of various phenomena in physics and engineering. From literature we already know that this inverse problem has a unique solution. However, the problem is still ill‐posed by being unstable to noise in the input data. For the numerical realization, we apply the quintic B‐spline (QB‐spline) collocation method for discretizing the pseudo‐parabolic problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved using the MATLAB subroutine lsqnonlin. Moreover, the von Neumann stability analysis is also discussed.

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“…Existence and uniqueness of solutions to inverse problems for parabolic equations are studied by several authors [4,5,7,8,9]. Asymptotic stability of solutions to such problems are investigated in [2,9,10,11].…”

Section: Introductionmentioning

confidence: 99%

Finite time blow up of solutions to an inverse problem for a quasilinear parabolic equation with power nonlinearity

Gür¹,

Yaman²,

Yilmaz³

2016

J. Nonlinear Sci. Appl.

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We consider an inverse problem for quasilinear parabolic equations with type power nonlinearity. Sufficient conditions on initial data for blow up result are obtained with positive initial energy. Overdetermination condition is given as an integral form. To get the blow up result for considered nonlinear inverse parabolic equation, we use the concavity of a special positive function. The life span of the solution is also computed.

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Asymptotic behaviour of the solutions of inverse problems for pseudo-parabolic equations

Cited by 8 publications

References 5 publications

Direct and inverse problems for time-fractional heat equation generated by Dunkl operator

Direct and inverse problems for time-fractional heat equation generated by Dunkl operator

An inverse problem of identifying the time‐dependent potential in a fourth‐order pseudo‐parabolic equation from additional condition

Finite time blow up of solutions to an inverse problem for a quasilinear parabolic equation with power nonlinearity

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