In this paper, a variational iteration method (VIM), which is a well-known method for solving nonlinear equations, has been employed to solve an Inverse parabolic partial differential equation. Inverse problems in partial differential equations can be used to model many real problems in engineering and other physical sciences. The VIM is to construct correction functional using general Lagrange multipliers identified optimally via the variational theory. This method provides a sequence of function which converges to the exact solution of the problem. This technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Numerical examples are examined to show the efficiency of the technique.

In this work, we focus on the final value problem of an Inverse problem for the pseudo-parabolic equation. This study aims to provide a regularization method for this equation, once the measurement data are obtained at the final time in $L^{r}(0,\pi)$. We obtain an approximated solution using the Fourier method and the final input data $L^{r}(0,\pi)$ for $r \neq 2$. Using embedding between $L^{r}(0,\pi)$ and Hilbert scales $\mathcal{H}^{\rho}(0,\pi)$, this study is the error between the exact and regularized solutions to be estimated in $L^{r}(0,\pi)$.

In this paper, a nonlinear Inverse problem of parabolic type, is considered. By reducing this Inverse problem to a system of Volterra integral equations the existence, uniqueness, and stability of the solution will be shown.

In this article, a linear Inverse problem for approximating the right hand side of a fourth order parabolic equation is studied. In this problem, it is assumed that the homogeneous boundary conditions along with an integral condition on the time domain and a local condition at a point of the space domain are known. In the first step, we show that this problem has a unique classical solution. Then, we convert the initial problem into a new problem by using suitable transformations, in which the time-dependent unknown function is transferred to the boundary conditions, and then we provide a spectral approximation based on the Ritz method to detect the unknown functions. The discretization of the problem using the presented technique leads to a system of linear algebraic equations which is solved by employing the Tikhonov's regularization method. The numerical simulation results confirm the acceptable accuracy and stability of the approximate solution.

On the basis of a reproducing kernel space, an iterative algorithm for solving the Inverse problem for heat equation with a nonlocal boundary condition is presented. The analytical solution in the reproducing kernel space is shown in a series form and the approximate solution un is constructed by truncating the series to n terms. The convergence of un to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such Inverse problems.

In this work, the Inverse quasi-linear pseudo-parabolic problem was investigated. We demonstrated the solution by the Fourier approximation. The Inverse problem was first examined by linearizing and then used implicit finite difference schema for the numerical solution.

In this article, we compare two procedures for solving a parabolic Inverse heat problem subject to the over specified boundary condition .The first procedure is obtained by introducing transformation of unknown function and solving transformed problem by finite difference method. And the second method based on trace-type functional (TTF) formulation is examined on the solving of considered problem. Some numerical examples are presented.

The determination of an unknown boundary condition, in a nonlinaer Inverse diffusion problem is considered. For solving these ill-posed Inverse problems, Galerkin method based on Sinc basis functions for space and time will be used. To solve the system of linear equation, a noise is imposed and Tikhonove regularization is applied. By using a sensor located at a point in the domain of x, say x = a′, and determining u (a′, t) a stable solution will be achived. An illustrative example is provided to show the ability and the efficiency of this numerical approach.