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In this article, we discuss a numerical method for solving some nonlinear inverse parabolic partial differential equations with Dirichlet's boundary conditions. The approach used, is based on collocation of cubic B-splines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. We apply cubic B-splines for spatial variable and derivatives, which produce an ill-posed system. We solve this system using the Tikhonov regularization method. The accuracy of the proposed method is demonstrated by applying it on two test problems. The figures and comparisons have been presented for clarity. Also the stability of this method has been discussed. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

In this paper, we will first study the existence and uniqueness of the solution of an inverse initial-boundary-value problem, via an auxiliary problem. Furthermore, we propose a stable numerical approach based on the use of the solution to the auxiliary problem as a basis function for solving this problem in the presence of noisy data. Also note that the inverse problem has a unique solution, but this solution is unstable and hence the problem is ill-posed. This instability is overcome using the Tikhonov regularization method. The effectiveness of the algorithm is illustrated by numerical example.

A numerical approach combining the use of the finite difference method with the solution of ordinary differential equation (ODE) is proposed for the determination of boundary conditions in an inverse parabolic problem. The least-squares method is adopted to modify the solution. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.

This study is intended to provide a new numerical algorithm for solving a one-dimensional inverse nonlinear parabolic problem. At the beginning of the study, Taylor’s series expansion is employed to linearize nonlinear terms and then finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate unknown functions. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown coefficient in the present study, thus, it is classified as the function estimation. Results show that an excellent estimation on the unknown functions of the inverse problem which can be obtained within a couple of minutes CPU time at Pentium IV-2.4 GHz PC.

In this paper, we consider an inverse heat conduction problem (IHCP). The given heat conduction equation, the boundary condition, and the initial condition are presented in a dimensionless form. A set of temperature measurements at a single sensor location inside the heat conduction body is required. Using a linear transformation, the ill-posed IHCP becomes a Cauchy problem. This problem will be solved by applying Legendre polynomials. Results show that an excellent estimation can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.

In this paper, a numerical solution of an inverse non-dimensional heat conduction problem will be considered. By using a sensor located at a point inside the body and measuring the temperature at a point x = x 1 , 0 x 1

A numerical procedure based on a fully implicit finite difference method for an inverse problem of identification of an unknown source in a heat equation is presented. The approach of the proposed method is to approximate unknown function from the solution of the minimisation problem based on the overspecified data. This problem is ill-posed, in the sense that the solution (if it exist) does not depend continuously on the data. To regularise this ill-conditioned, we apply the Tikhonov regularisation 0th, 1st and 2nd method to obtain the stable numerical approximation to the solution. A stability analysis shows that this numerical scheme approximation is unconditionally stable. Numerical results for two inverse source identification problems show that the proposed numerical algorithm is simple, accurate, stable and computationally efficient.

In this paper, we will first study the existence and uniqueness of the solution for a one dimensional inverse heat conduction problem IHCP via an auxiliary problem. Then the present work is motivated by desire to obtain numerical approach for solving this IHCP. Our method begins with the utilisation of some transformations. These transformations allow us to eliminate an unknown term from parabolic equation to obtain an inverse parabolic problem with two unknown boundary conditions. To solve this inverse problem, we use the fundamental solution method. The effectiveness of the algorithm is illustrated by numerical example.

The problem of identifying the solution kx, t,Ux, t in an inverse semilinear wave problem is considered. It is shown that under certain conditions of data φ, ψ, there exists a unique solution kx, t,Ux, t of this problem. Furthermore a numerical algorithm for solving the inverse semilinear wave problem is proposed. The approach for this inverse problem is given by using the semi-discretisation method. A polynomial function is proposed to approximate Ux, t then the finite difference method is applied to approximate unknown kx, t. Numerical results show efficiency of our method.

In this paper, we will first study the existence and uniqueness of the solution of a one-dimensional inverse problem for an inhomogeneous linear wave equation with initial and boundary conditions via an auxiliary problem. Then a stable numerical method consisting of zeroth-, first-, and second-order Tikhonov regularization to the matrix form of Duhamel's principle for solving this inverse problem is presented. The stability and accuracy of the scheme presented is evaluated by comparison with the Singular Value Decomposition method. Some numerical experiments confirm the utility of this algorithm as the results are in good agreement with the exact data.

In this paper, we will first study the existence and uniqueness of the solution for a one-dimensional nonlinear inverse diffusion problem via an auxiliary problem and the Schauder fixed-point theorem. Furthermore, a numerical algorithm based on the finite-difference method and the least-squares scheme for solving a nonlinear inverse problem is proposed. At the beginning of the numerical algorithm, Taylor's series expansion is employed to linearize nonlinear terms and then the finite-difference method is used to discretize the problem domain. The present approach is to rearrange the matrix forms of the differential governing equations and estimate the unknown diffusion coefficient. The least-squares method is adopted to find the solution. It is assumed that no prior information is available on the functional form of the unknown diffusion coefficient in the present study, and thus, it is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the diffusion coefficient can be obtained within a couple of minutes of CPU time on a Pentium IV-2.4GHz PC.