Identification of a spacewise dependent heat source

This paper presents a meshless numerical scheme for recovering the time-dependent heat source in general three-dimensional (3D) heat conduction problems. The problem considered is ill-posed and the determination of the unknown heat source is achieved here by using the boundary condition, initial condition and the extra measured data from a fixed point placed inside the domain. The extra measured data are used to guarantee the uniqueness of the solution. The generalized finite difference method (GFDM), a recently-developed meshless method, is then adopted to solve the resulting time-dependent boundary-value problem. In our computations, the second-order Crank–Nicolson scheme is employed for the temporal discretization and the proposed GFDM for the spatial discretization. Several benchmark test problems with both smooth and piecewise smooth geometries have been studied to verify the accuracy and efficiency of the proposed method. No need to apply any well-known regularization strategy, the accurate and stable solution could be obtained with a comparatively large level of noise.

The generalized finite difference method for an inverse time-dependent source problem associated with three-dimensional heat equation

An inverse time-dependent source problem for the heat equation with a non-classical boundary condition

In this paper, a regularized optimization method is proposed for identifying the space-dependent source and the initial value simultaneously in an inverse parabolic equation problem from two over-specified measurements at different instants of time. The solvability of the direct problem is presented and then the inverse problem is formulated into a regularized optimization problem for the stable identification of both the source term and the initial value. Based on a sequence of well-posed direct problems solved by the finite element method, a numerical scheme formulated into a linear system is proposed to implement the regularized optimization problem. Numerical results of three examples show that the proposed method is efficient and robust with respect to data noise.

A regularized optimization method for identifying the space-dependent source and the initial value simultaneously in a parabolic equation

The objective of this work is to reconstruct, for the first time, numerically the time‐dependent potential coefficient in a fourth‐order pseudo‐hyperbolic equation with initial and boundary conditions from additional measurement as an overdetermination condition. This inverse identification problem is an ill‐posed problem but has a unique solution. For the numerical realization, we apply the Quintic B‐spline (QB‐spline) collocation method to discretize the direct problem and the Tikhonov regularization to find a stable and accurate solution. The resulting nonlinear minimization problem is approximated using the MATLAB optimization toolbox routine lsqnonlin. The obtained numerical results are the evidence of the stable and accurate solutions. The von Neumann stability analysis is also carried out.

Identification of time‐dependent potential in a fourth‐order pseudo‐hyperbolic equation from additional measurement

Inverse source problems is a classical ill-posed problem. There exist several literatures on inverse heat source problems. In this study we use some stable methods to deal with the inverse problems of determining space-wise and time-wise dependent heat source in the heat diffusion equation. The problems of inverse heat source sources are transferred to the problems of numerical differentiation. By simple and effective methods which are used to regularize the problems of numerical differentiation before, we can obtain stable solutions. The numerical results show that our proposed procedure yields stable and accurate approximation.

A direct numerical method for solving inverse heat source problems

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A direct numerical method for solving inverse heat source problems