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Öğe An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation(Hacettepe Univ, Fac Sci, 2023) Huntul, Mousa J.; Tekin, IbrahimIn this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed (very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms), we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit (ADE) method and reshaped as non-linear least squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem (IP) unique.Öğe An Inverse Problem of Reconstructing the Unknown Coefficient in a Third Order Time Fractional Pseudoparabolic Equation(Univ Craiova, 2024) Huntul, Mousa J.; Tekin, Ibrahim; Iqbal, Muhammad K.; Abbas, MuhammadIn this paper, we have considered the problem of reconstructing the time dependent potential term for the third order time fractional pseudoparabolic equation from an additional data at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of engineering, mechanics and physics. The existence of unique solution to the problem has been discussed by means of the contraction principle 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 ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential), 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 potential term. The proposed problem is discretized using the cubic B-spline (CB-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 perturbed data and analytical are inverted and the numerical outcomes for two benchmark test examples are reported and discussed.Öğe Identification of the Solely Time-Dependent Zero-Order Coefficient in a Linear Bi-Flux Diffusion Equation from an Integral Measurement(2023) Tekin, Ibrahim; Çetın, Mehmet AkıfBi-flux diffusion equation, can be easily affected by the existence of external factors, is known as an anomalous diffusion. In this paper, the inverse problem (IP) of determining the solely time-dependent zero-order coefficient in a linear Bi-flux diffusion equation with initial and homogeneous boundary conditions from an integral additional specification of the energy is considered. The unique solvability of the inverse problem is demonstrated by using the contraction principle for sufficiently small times.Öğe Identification of the time-dependent lowest term in a fourth order in time partial differential equation(2023) Tekin, IbrahimIn this article, identification of the time-dependent lowest term in a fourth order in time partial differential equation (PDE) from knowledge of a boundary measurement is studied by means of contraction mapping.Öğe IDENTIFICATION OF THE TIME-DEPENDENT LOWEST TERM IN A FOURTH ORDER IN TIME PARTIAL DIFFERENTIAL EQUATION(Ankara Univ, Fac Sci, 2023) Tekin, IbrahimIn this article, identification of the time-dependent lowest term in a fourth order in time partial differential equation (PDE) from knowledge of a boundary measurement is studied by means of contraction mapping.Öğe Inverse problem for a nonlinear third order in time partial differential equation(Wiley, 2021) Tekin, IbrahimIn this article, we study the inverse problem of recovering a time-dependent coefficient of a nonlinear third order in time partial differential equation (PDE), which is usually referred to as Moore-Gibson-Thompson equation, from knowledge of a one boundary measurement.Öğe Inverse scattering problem for linear system of four-wave interaction problem with equal number of incident and scattered waves(Springer Basel Ag, 2021) Ismailov, Mansur I.; Tekin, IbrahimThe first order semi-strict hyperbolic system on the semi-axis in the case of equal number of incident and scattered waves are considered. The uniqueness of the inverse scattering problem (the problem of finding the potential with respect to scattering operator) is studied by utilizing it to Gelfand-Levitan-Marchenko type linear integral equations. It is determined the sufficient quantity of scattering problems (on the semi-axis for the same hyperbolic system) for ensuring the uniqueness of the inverse scattering problem.Öğe On an inverse problem for a nonlinear third order in time partial differential equation(Elsevier B.V., 2022) Huntul, Mousa J.; Tekin, IbrahimIn this article, first we convert an inverse problem of determining the unknown timewise terms of nonlinear third order in time partial differential equation (PDE) from knowledge of two boundary measurements to the auxiliary system of integral equations. Then, existence and uniqueness of the solution of this system is proved by means of the contraction principle on a small time interval. Also uniqueness of the solution of the inverse problem is given. However, since the governing equation is yet ill-posed (very slight errors in the temperature input may cause relatively significant errors in the output potential and 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 in time PDE problem is discretized using the FDM and reshaped as non-linear 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. The proposed numerical approach has also been discussed. © 2022 The Author(s)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.Öğe Simultaneous identification of the solely time-dependent potential and source control terms from known moisture moments(Wiley, 2024) Alosaimi, Moataz; Tekin, Ibrahim; Cetin, Mehmet AkifPseudo-parabolic equations are commonly used as mathematical models in mechanics, biology, and physics to address various applied problems. One particular application involves describing moisture transfer dynamics in subsoil layers using pseudo-parabolic equations. This manuscript examines the inverse problem (IP) of identifying the moisture transfer function, along with the time-varying potential and source control terms, in a linear pseudo-parabolic equation from known moisture moments. We prove the existence of a unique solution to the inverse problem for sufficiently small times by employing the contraction principle. The inverse problem is reformulated as a nonlinear least-squares minimization problem, with the unknowns subjected to simple bounds. To guarantee stability, the Tikhonov regularization technique is utilized. For the numerical discretization, we develop the Crank-Nicolson finite-difference method to solve the direct problem. To solve the nonlinear least-squares minimization problem, we utilize the built-in subroutine lsqnonlin from the MATLAB optimization toolbox. We present and thoroughly discuss numerical outcomes for a benchmark test example, providing insights into the performance and effectiveness of the proposed methodology.
