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Öğe An inverse problem of recovering the heat source coefficient in a fourth-order time-fractional pseudo-parabolic equation(Elsevier, 2024) Huntul, M. J.; Tekin, I.; Iqbal, Muhammad Kashif; Abbas, MuhammadIn this paper, we have considered the problem of recovering the time dependent source term for the fourth order time fractional pseudoparabolic equation theoretically and numerically, for the first time, by considering an additional measurement at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of physics and mechanics. The existence of unique solution to the problem has been discussed by means of the contraction principle and Banach Fixed-point theorem on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet illposed (very slight errors in the additional input may cause relatively significant errors in the output force), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the forcing term. The proposed problem is discretized using the quintic B-spline (QnB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine lsqnonlin tool has been used to expedite the numerical computations. Both analytical and perturbed data are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.Öğe Inverse coefficient problem for differential equation in partial derivatives of a fourth order in time with integral over-determination(Karaganda State Univ, 2022) Huntul, M. J.; Tekin, I.Derivatives in time of higher order (more than two) arise in various fields such as acoustics, medical ultrasound, viscoelasticity and thermoelasticity. The inverse problems for higher order derivatives in time equations connected with recovery of the coefficient are scarce and need additional consideration. In this article the inverse problem of determination is considered which depends on time, lowest term coefficient in differential equation in partial derivatives of fourth order in time with initial and boundary conditions from an additional integral observation is considered. Under some conditions regularity, consistency and orthogonality of data by using of the contraction principle the unique solvability of the solution of the coefficient identification problem on a sufficiently small time interval has been proved.Öğe Inverse problem for time dependent coefficients in the higher order pseudo-parabolic equation(Amer Inst Mathematical Sciences-Aims, 2025) Huntul, M. J.; Tekin, I.In this paper, we considered an inverse problem of recovering the time dependent potential and force coefficients in the third order pseudoparabolic equation from nonlocal integral observations. Existence and uniqueness of the solution are proved by means of the contraction principle on a small time interval. The stability results for the inverse problem is presented. The unique solvability theorem for this inverse problem is proved. However, since the governing equation is yet ill-posed (very slight errors in the integral input may cause relatively significant errors in the output potential and heat source terms), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown terms. The third order pseudoparabolic problem is discretized using the cubic B-spline (CB-spline) collocation technique and reshaped as nonlinear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both analytical and perturbed data are inverted. Numerical outcomes for two benchmark test examples are reported and discussed. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.Öğe Simultaneous determination of the time-dependent potential and force terms in a fourth-order Rayleigh-Love equation(Wiley, 2023) Huntul, M. J.; Tekin, IbrahimThis paper considers an inverse coefficient problem of simultaneous determination of solely time-dependent potential and force terms with the unknown longitudinal displacement from a Rayleigh-Love equation from two integral overdetermination measurements. Unique solvability of this theorem is investigated theoretically by using contraction principle. Although, the aforesaid inverse identification problem is ill-posed but has a unique solution. We use the cubic B-spline (CBS) collocation and Tikhonov regularization techniques to discretize the direct problem and to obtain stable as well as accurate solutions, respectively. The stability, for the discretized system of the direct problem, is also carried out by means of the von Neumann method. The acquired results demonstrate that accurate as well as stable solutions for the a(t)$$ a(t) $$ and b(t)$$ b(t) $$ are accessed for gamma 1=gamma 2 is an element of{10-8,10-7,10-6,10-5}$$ {\gamma}_1={\gamma}_2\in \left\{1{0}<^>{-8},1{0}<^>{-7},1{0}<^>{-6},1{0}<^>{-5}\right\} $$, when p is an element of{0.1%,1%}$$ p\in \left\{0.1\%,1\%\right\} $$ for both smooth and discontinuous potential and force terms. The stability analysis shows that the discretized system of the direct problem is unconditionally stable. Since the noisy data are introduced, the investigation and analysis model real circumstances where the practical quantities are naturally infested with noise. The technique establishes that accurate, as well as stable, solutions are obtained.
