The aim of this paper is to find the time-dependent term numerically in a two-dimensional heat equation using initial and Neumann boundary conditions and nonlocal integrals as over-determination conditions.This is a v...The aim of this paper is to find the time-dependent term numerically in a two-dimensional heat equation using initial and Neumann boundary conditions and nonlocal integrals as over-determination conditions.This is a very interesting and challenging nonlinear inverse coefficient problem with important applications in various fields ranging from radioactive decay,melting or cooling processes,electronic chips,acoustics and geophysics to medicine.Unique solvability theo-rems of these inverse problems are supplied.However,since the problems are still ill-posed(a small modification in the input data can lead to bigger impact on the ultimate result in the output solution)the solution needs to be regularized.Therefore,in order to obtain a stable solution,a regularized objective function is minimized in order to retrieve the unknown coefficient.The two-dimensional inverse problem is discretized using the forward time central space(FTCS)finite-difference method(FDM),which is conditionally stable and recast as a non-linear least-squares minimization of the Tikhonov regularization function.Numerically,this is effectively solved using the MATLAB subroutine lsqnonlin.Both exact and noisy data are inverted.Numerical results for a few benchmark test examples are presented,discussed and assessed with respect to the FTCS-FDM mesh size discretisation,the level of noise with which the input data is contaminated,and the choice of the regularization parameter is discussed based on the trial and error technique.展开更多
The inverse problem of reconstructing the time-dependent thermal conductivity and free boundary coefcients along with the temperature in a two-dimensional parabolic equation with initial and boundary conditions and ad...The inverse problem of reconstructing the time-dependent thermal conductivity and free boundary coefcients along with the temperature in a two-dimensional parabolic equation with initial and boundary conditions and additional measurements is,for the rst time,numerically investigated.This inverse problem appears extensively in the modelling of various phenomena in engineering and physics.For instance,steel annealing,vacuum-arc welding,fusion welding,continuous casting,metallurgy,aircraft,oil and gas production during drilling and operation of wells.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 alternating direction explicit method along with the Tikhonov regularization to nd a stable and accurate numerical solution of nite differences.The root mean square error(rmse)values for various noise levels p for both smooth and non-smooth continuous time-dependent coef-cients Examples are compared.The resulting nonlinear minimization problem is solved numerically using the MATLAB subroutine lsqnonlin.Both exact and numerically simulated noisy input data are inverted.Numerical results presented for two examples show the efciency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.展开更多
In this paper,we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermi...In this paper,we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition.We obtain sufficient conditions for the unique solvability of the inverse problem.The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem.This inverse problem appears extensively in the modelling of various phenomena in engineering and physics.For example,seismology,medicine,fusion welding,continuous casting,metallurgy,aircraft,oil and gas production during drilling and operation of wells.In addition,the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences.The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin.Both analytical and numerically simulated noisy input data are inverted.The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared.Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.Furthermore,the choice of the regularization parameter is also discussed based on the trial and error technique.展开更多
In this paper,we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation,which vanishes at the initial m...In this paper,we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation,which vanishes at the initial moment of time,and/or the convection coefficient along with the temperature for a one-dimensional parabolic equation,from some additional information about the process(the so-called over-determination conditions).Although uniquely solvable these inverse problems are still ill-posed since small changes in the input data can result in enormous changes in the output solution.The finite difference method with the Crank-Nicolson scheme combined with the nonlinear Tikhonov regularization are employed.The resulting minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin.For both exact and noisy input data,accurate and stable numerical results are obtained.展开更多
文摘The aim of this paper is to find the time-dependent term numerically in a two-dimensional heat equation using initial and Neumann boundary conditions and nonlocal integrals as over-determination conditions.This is a very interesting and challenging nonlinear inverse coefficient problem with important applications in various fields ranging from radioactive decay,melting or cooling processes,electronic chips,acoustics and geophysics to medicine.Unique solvability theo-rems of these inverse problems are supplied.However,since the problems are still ill-posed(a small modification in the input data can lead to bigger impact on the ultimate result in the output solution)the solution needs to be regularized.Therefore,in order to obtain a stable solution,a regularized objective function is minimized in order to retrieve the unknown coefficient.The two-dimensional inverse problem is discretized using the forward time central space(FTCS)finite-difference method(FDM),which is conditionally stable and recast as a non-linear least-squares minimization of the Tikhonov regularization function.Numerically,this is effectively solved using the MATLAB subroutine lsqnonlin.Both exact and noisy data are inverted.Numerical results for a few benchmark test examples are presented,discussed and assessed with respect to the FTCS-FDM mesh size discretisation,the level of noise with which the input data is contaminated,and the choice of the regularization parameter is discussed based on the trial and error technique.
文摘The inverse problem of reconstructing the time-dependent thermal conductivity and free boundary coefcients along with the temperature in a two-dimensional parabolic equation with initial and boundary conditions and additional measurements is,for the rst time,numerically investigated.This inverse problem appears extensively in the modelling of various phenomena in engineering and physics.For instance,steel annealing,vacuum-arc welding,fusion welding,continuous casting,metallurgy,aircraft,oil and gas production during drilling and operation of wells.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 alternating direction explicit method along with the Tikhonov regularization to nd a stable and accurate numerical solution of nite differences.The root mean square error(rmse)values for various noise levels p for both smooth and non-smooth continuous time-dependent coef-cients Examples are compared.The resulting nonlinear minimization problem is solved numerically using the MATLAB subroutine lsqnonlin.Both exact and numerically simulated noisy input data are inverted.Numerical results presented for two examples show the efciency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.
文摘In this paper,we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition.We obtain sufficient conditions for the unique solvability of the inverse problem.The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem.This inverse problem appears extensively in the modelling of various phenomena in engineering and physics.For example,seismology,medicine,fusion welding,continuous casting,metallurgy,aircraft,oil and gas production during drilling and operation of wells.In addition,the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences.The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin.Both analytical and numerically simulated noisy input data are inverted.The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared.Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.Furthermore,the choice of the regularization parameter is also discussed based on the trial and error technique.
文摘In this paper,we consider solving numerically for the first time inverse problems of determining the time-dependent thermal diffusivity coefficient for a weakly degenerate heat equation,which vanishes at the initial moment of time,and/or the convection coefficient along with the temperature for a one-dimensional parabolic equation,from some additional information about the process(the so-called over-determination conditions).Although uniquely solvable these inverse problems are still ill-posed since small changes in the input data can result in enormous changes in the output solution.The finite difference method with the Crank-Nicolson scheme combined with the nonlinear Tikhonov regularization are employed.The resulting minimization problem is computationally solved using the MATLAB toolbox routine lsqnonlin.For both exact and noisy input data,accurate and stable numerical results are obtained.
