Unknown Time-dependent Heat Source Research Articles

Identification of the unknown heat source terms in a 2D parabolic equation

The objective of this paper is to reconstruct the unknown time-dependent heat source terms numerically, for the first time, in a two-dimensional parabolic equation in the rectangular domain with initial and Neumann boundary conditions supplemented by the temperature data as over-determination conditions. Although, the problem is ill-posed (in the sense of Hadamard) but has a unique solution. We apply the forward time central space finite difference scheme along with the Tikhonov regularization to find a stable and accurate numerical solution. The MATLAB subroutine lsqnonlin is used to solve the resulting nonlinear minimization problem. The obtained results show that accurate and stable solutions are achieved. Computational efficiency of the method is investigated by small values of CPU-time.

Determination of an Unknown Time-dependent Heat Source from A Nonlocal Measurement by Finite Difference Method

In this paper, we consider an inverse time-dependent source problem of heat conduction equation. Firstly, the ill-posedness and conditional stability of this inverse source problem is analyzed. Then, a finite difference inversion method is proposed for reconstructing the time-dependent source from a nonlocal measurement. The existence and uniqueness of the finite difference inverse solutions are rigorously analyzed, and the convergence is proved. Combined with the mollification method, the proposed finite difference inversion method can obtain more stable reconstructions from the nonlocal data with noise. Finally, numerical examples are given to illustrate the efficiency and convergence of the proposed finite difference inversion method.

A multi-resolution collocation procedure for time-dependent inverse heat problems

In this paper, a Haar wavelet collocation method (HWCM) is developed for PDEs related to the framework of so-called inverse problem. These include PDEs with unknown time dependent heat source and unknown solution in interior of the domain. To this end, a transformation is used to eliminate the unknown heat source to obtain a PDE without a heat source. After elimination of unknown non-homogeneous term, an implicit finite-difference approximations is used to approximate the time derivative and Haar wavelets are used for approximation of the space derivatives. Several numerical experiments related to one- and two-dimensional heat sources are included to validate small condition number of coefficient matrix, accuracy and simple applicability of the proposed approach.

Identification of separable sources for advection-diffusion equations with variable diffusion coefficient from boundary measured data

This paper considers the inverse problems of identifying either the unknown space-dependent heat source or the unknown time-dependent heat source of the variable coefficient advection–diffusion equation with separable sources of the form from supplementary time-dependent temperature measurement at the right boundary of the domain. We proved that solutions of both inverse source problems can be identified uniquely under some regularity assumptions. Two non-iterative inversion algorithms were developed and numerically implemented for the identification of the unknown space-wise and time-wise dependent sources. The robustness and limitations of the two algorithms are investigated through numerical examples related to the reconstruction of continuous and discontinuous sources.

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.

The multiple-scale polynomial Trefftz method for solving inverse heat conduction problems

The polynomial Trefftz method consists of the polynomial type solutions as bases, providing a cheap boundary-type meshless method to solve the heat conduction equation, since the bases automatically satisfy the governing equation. In order to stably solve the backward heat conduction problem (BHCP), and the inverse heat source problem (IHSP) together with the boundary condition recovery problem by a polynomial Trefftz method, which are both known to be highly ill-posed, we introduce a multiple-scale post-conditioner in the resultant linear system to reduce the condition number. Then the conjugate gradient method (CGM) is used to solve the post-conditioned linear system to determine the unknown expansion coefficients. In the multiple-scale polynomial Trefftz method (MSPTM) the scales are determined a priori by the collocation points on space–time boundary, which can retrieve the missing initial data, the unknown time-dependent heat source as well as the boundary condition rather well. Several numerical examples of the inverse heat conduction problems demonstrate that the MSPTM is effective and accurate, even for those of severely ill-posed inverse problems under very large noises.

Resolution of Unknown Heat Source Inverse Heat Conduction Problems Using Particle Swarm Optimization

The main objective of this article is to solve inverse heat conduction problems with the particle swarm optimization method. An enhanced particle swarm optimization (EPSO) algorithm is proposed to overcome the shortcoming of earlier convergence of standard PSO algorithms. The EPSO is used to estimate the unknown time-dependent heat source in complex regions. Numerical experiments indicate the validity and stability of the EPSO method.

On-line detecting heat source of a nonlinear heat conduction equation by a differential algebraic equation method

We consider an inverse problem for detecting an unknown time-dependent heat source in real-time for a nonlinear heat conduction equation, with the aid of an extra measurement of temperature at an internal point. After a finite difference discretization of governing equation into ordinary differential equations, we recast them and the measured data as a set of differential algebraic equations (DAEs), which is a novel view of the inverse heat source problem. Then we solve the resultant DAEs by a GL(n,R) Lie-group method, which can be used as an on-line estimator to detect unknown heat source of nonlinear heat conduction equation, by using only a real-time measurement of internal temperature under a randomly noisy disturbance. The estimated results obtained by the novel Lie-group differential algebraic equations (LGDAE) method are quite promising and robust enough against large random noises.

Identification of an unknown time-dependent heat source term from overspecified Dirichlet boundary data by conjugate gradient method

The inverse problem of identifying the unknown time-dependent heat source H(t) of the variable coefficient heat equation ut=(k(x)ux)x+F(x)H(t), with separable sources of the form F(x)H(t), from supplementary temperature measurement h(t)≔u(0,t) at the left end of the rod, is investigated. The Fourier method is employed to illustrate the comparison of spacewise (F(x)) and time-dependent (H(t) heat source identification problems. An explicit formula for the Fréchet gradient of the cost functional J(H)=‖u(0,⋅;H)−h‖L2(0,Tf)2 is derived via the unique solution of the appropriate adjoint problem. The Conjugate Gradient Algorithm, based on the gradient formula for the cost functional, is then proposed for numerical solution of the inverse source problem. The algorithm is examined through numerical examples related to reconstruction of continuous and discontinuous heat sources H(t), when heat is transferred through non-homogeneous as well as composite structures. Numerical analysis of the algorithm applied to the inverse source problem in typical classes of source functions is presented. Computational results, obtained for random noisy output data, show how the iteration number of the Conjugate Gradient Algorithm can be estimated. Based on these results it is shown that this iteration number plays a role of a regularization parameter. Numerical results illustrate bounds of applicability of the proposed algorithm, and also its efficiency and accuracy.

A Lie-group shooting method for reconstructing a past time-dependent heat source

We consider an inverse problem for the reconstruction of a past unknown time-dependent heat source H( t), t < t f , in a heat conduction equation T t ( x, t) = T xx ( x, t) + H( t) with the aid of an extra measurement of temperature gradient on the left-boundary, where a final time condition is measured at the present terminal time t f . This inverse problem is quite difficult to be solved numerically owing to a twofold ill-posedness, as a combination of the backward heat conduction problem and the inverse heat source identification problem, which is abbreviated as inverse heat source/backward heat conduction problem (IHSBHCP). The new method proposed here, namely the Lie-group shooting method (LGSM), is examined through the tests by several numerical examples. Although the recovery of an unknown heat source is carried out under a presently measured temperature at a final time and under a large measurement noise both imposed on the final time data and all boundary data, the LGSM still works effectively and accurately. The accuracy in the reconstruction of H( t) is almost uneffected by different levels of noise.

Two regularization methods to identify time-dependent heat source through an internal measurement of temperature

This paper deals with an inverse problem for identifying an unknown time-dependent heat source in a one-dimensional heat equation, with the aid of an extra measurement of temperature at an internal point. Since this problem is ill-posed, two regularization solutions are obtained by employing a Fourier truncation regularization and a Quasi-reversibility regularization. Furthermore, the Hölder type stability estimate between the regularization solutions and the exact solution, are obtained, respectively. Numerical examples show that these regularization methods are effective and stable.

A two-stage LGSM to identify time-dependent heat source through an internal measurement of temperature

We consider an inverse problem for estimating an unknown time-dependent heat source H ( t ) in a heat conduction equation T t ( x , t ) = T xx ( x , t ) + H ( t ) , with the aid of an extra measurement of temperature at an internal point. The Lie-group shooting method (LGSM) was used in the solution of this inverse problem; however, when the data are acquired at an internal point we require to develop a two-stage Lie-group shooting method (TSLGSM) to solve it. This novel approach is examined through numerical examples to convince that it is a rather accurate and efficient method, whose estimation error is small even for the identification of discontinuous and oscillatory heat source under large noise.

Determination of an Unknown Heat Source from Overspecified Boundary Data

Determination of an Unknown Heat Source from Overspecified Boundary Data