A comparison of some inverse methods for estimating the initial condition of the heat equation

In this work we analyze two explicit methods for the solution of an inverse heat conduction problem and we confront them with the least-squares method, using for the solution of the associated direct problem a classical finite difference method and a method based on an integral formulation. Finally, the Tikhonov regularization connected to the least-squares criterion is examined. We show that the explicit approaches to this inverse heat conduction problem will present disastrous results unless some kind of regularization is used.     

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non‐linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed‐point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

A nonlinear model and viscosity solution for the temperature field of an oil-immersed self-cooled three-phrase transformer

In this paper, on the basis of the characteristics of an oil-immersed self-cooled three-phrase transformer, we establish a mathematical model of the three-dimensional temperature field. But because the specific heat, density, heat sources and coefficient of heat transfer are discontinuous and non-differentiable, the problem has no analytical solution. We decompose the problem into seven subproblems, and prove the existence and uniqueness of a viscosity solution for every subproblem, by combining Perron’s method with the technique of coupled solutions.     

FOURIER REGULARIZATION FOR DETERMINING SURFACE HEAT FLUX FROM INTERIOR OBSERVATION BASED ON A SIDEWAYS PARABOLIC EQUATION

In this paper we consider a non-standard inverse heat conduction problem for determining surface heat flux from an interior observation which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. A Fourier method is applied to formulate a regularized approximation solution, and some sharp error estimates are also given.     

Leak identification in a saturated unsteady flow via a Cauchy problem

In this initial study, we propose a numerical method for identifying multiple leak zones in a saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modeled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes because the leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we modify it and employ a previously proposed iterative regularizing method. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint to obtain a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a finite element method and the numerical results indicate that the leak zones can be identified with the proposed method.     

A modified quasi-boundary value method for solving the radially symmetric inverse heat conduction problem

In this paper, we consider a radially symmetric inverse heat conduction problem of determining the surface heat flux distribution from a fixed location inside a cylinder. This problem is ill-posed in the Hadamard sense and a conditional stability estimate is given for it. A modified quasi-boundary value regularization method is applied to formulate a regularized solution, and a sharp error estimate between the approximate solution and the exact solution is established by choosing a suitable regularization parameter. A numerical example is presented to verify the efficiency of the regularization method.     

反向热传导问题的一种新的正则化方法

讨论一个高维反向热传导问题,这是一个经典的严重不适定问题.关于这一问题我们给出一种新的正则化方法-改进的Tikhonov正则化方法,以恢复解对数据的连续依赖性.通过构造一个重要的不等式和提高先验光滑条件,获得正则解在0相似文献   

A new regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation in a one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a new regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under the a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Numerical solution of the problem of optimal control of the heating of a thermoelastic plate by internal heat sources

We give the statement and numerical solution of the problem of optimal control (in the sense of rapidity of response) of the heating of an unbounded plate by internal heat sources in the presence of restrictions on the maximal (in absolute value) thermal stresses. In constructing the solution of this nonlinear optimal control problem we use the method of the inverse heat conduction problem.     

Splitting-based conjugate gradient method for a multi-dimensional linear inverse heat conduction problem

In this paper we consider a multi-dimensional inverse heat conduction problem with time-dependent coefficients in a box, which is well-known to be severely ill-posed, by a variational method. The gradient of the functional to be minimized is obtained by the aid of an adjoint problem, and the conjugate gradient method with a stopping rule is then applied to this ill-posed optimization problem. To enhance the stability and the accuracy of the numerical solution to the problem, we apply this scheme to the discretized inverse problem rather than to the continuous one. The difficulties with large dimensions of discretized problems are overcome by a splitting method which only requires the solution of easy-to-solve one-dimensional problems. The numerical results provided by our method are very good and the techniques seem to be very promising.     

Spectral regularization method for solving a time-fractional inverse diffusion problem

In this paper, we consider an inverse problem for a time-fractional diffusion equation with one-dimensional semi-infinite domain. The temperature and heat flux are sought from a measured temperature history at a fixed location inside the body. We show that such problem is severely ill-posed and further apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical method is effective.     

Solution of the stationary problem of heat conduction of laminated anisotropic inhomogeneous plates by the method of initial functions

The problem of stationary heat conduction of laminated plates of constant and variable thickness is formulated in the three-dimensional statement. We reduce the three-dimensional problem to a twodimensional one by the method of initial functions. For plates with layers of variable thickness, a system of resolving equations with variable coefficients is obtained. The obtained two-dimensional boundary-value problems are analyzed. For plates with homogeneous layers of constant thickness, we construct a solution in an analytic form. It is shown that this solution coincides with a solution obtained by the method of separation of variables.     

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.     

热传导(对流-扩散)方程源项识别的粒子群优化算法

提出了利用粒子群优化(PSO)算法反演热传导方程与对流-扩散方程源项的一种新方法,在已有文献方法的基础上,求解出这两类方程正问题的解析解,再把源项识别问题转化为最优化问题,结合粒子群优化算法寻优求解.通过数值模拟与统计检验,结果表明,此方法可快速有效地实现热传导方程与对流-扩散方程源项的识别,并可推广应用到其它数学物理方程的源项或参数的反演识别.     

Solution of a three-dimensional boundary-value problem for a fractional differential heat conduction equation

We consider a three-dimensional axisymmetric problem for a differential heat conduction equation with fractional time derivatives. Using the method of homogeneous solutions and integral transformations, we obtain an asymptotic and a numerical solution of the problem. The results of calculations are presented.     

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.     

Approximate analytic solution of heat conduction problems with a mismatch between initial and boundary conditions

We consider a heat conduction problem for an infinite plate with a mismatch between initial and boundary conditions. Using the method of integral relations, we obtain an approximate analytic solution to this problem by determining the temperature perturbation front. The solution has a simple form of an algebraic polynomial without special functions. It allows us to determine the temperature state of the plate in the full range of the Fourier numbers (0≤F<∞) and is especially effective for very small time intervals.     

Analytic solutions to heat transfer problems on a basis of determination of a front of heat disturbance

With the use of additional boundary conditions in integral method of heat balance, we obtain analytic solution to nonstationary problem of heat conductivity for infinite plate. Relying on determination of a front of heat disturbance, we perform a division of heat conductivity process into two stages in time. The first stage comes to the end after the front of disturbance arrives the center of the plate. At the second stage the heat exchange occurs at the whole thickness of the plate, and we introduce an additional sought-for function which characterizes the temperature change in its center. Practically the assigned exactness of solutions at both stages is provided by introduction on boundaries of a domain and on the front of heat perturbation the additional boundary conditions. Their fulfillment is equivalent to the sought-for solution in differential equation therein. We show that with the increasing of number of approximations the accuracy of fulfillment of the equation increases. Note that the usage of an integral of heat balance allows the application of the given method for solving differential equations that do not admit a separation of variables (nonlinear, with variable physical properties etc.).     

Static thermoelasticity problems for layered thermosensitive plates with cubic dependence of the coefficients of heat conductivity on temperature

We describe an analytic-numerical method of solution of one-dimensional static thermoelasticity problems for layered plates, heated in different ways. We take into account the cubic dependence of the coefficients of heat conductivity and arbitrary nature of the dependence of other physicomechanical parameters on temperature. Here, using the constructed exact solution of an auxiliary problem, we have reduced the heat conduction problems, irrespective of the number of layers, to the solution of one or a system of two nonlinear algebraic equations. We have also studied the temperature fields and stresses in four-layer plates under conditions of complex heat exchange.     

特征值问题的应用简述

从矩阵的特征问题入手,引出常系数线性齐次微分方程求解的特征方程方法;利用分离变量法求解热传导方程,引入拉普拉斯方程的特征问题,给出求解过程,并给出热方程的解的渐近稳定性.