Advances in Mathematical Physics

Volume 2015 (2015), Article ID 256726, 8 pages

http://dx.doi.org/10.1155/2015/256726

## A Meshless Method Based on the Fundamental Solution and Radial Basis Function for Solving an Inverse Heat Conduction Problem

Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

Received 25 November 2014; Accepted 11 March 2015

Academic Editor: Alkesh Punjabi

Copyright © 2015 Muhammad Arghand and Majid Amirfakhrian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a new meshless method to solve a backward inverse heat conduction problem. The numerical scheme, based on the fundamental solution of the heat equation and radial basis functions (RBFs), is used to obtain a numerical solution. Since the coefficients matrix is ill-conditioned, the Tikhonov regularization (TR) method is employed to solve the resulted system of linear equations. Also, the generalized cross-validation (GCV) criterion is applied to choose a regularization parameter. A test problem demonstrates the stability, accuracy, and efficiency of the proposed method.

#### 1. Introduction

Transient heat conduction phenomena are generally described by the parabolic heat conduction equation, and if the initial temperature distribution and the boundary conditions are specified, then this, in general, leads to a well-posed problem which may easily be solved numerically by using various numerical methods. However, in many practical situations when dealing with a heat conducting body it is not always possible to specify the boundary conditions or the initial temperature. Hence, we are faced with an inverse heat conduction problem. Inverse heat conduction problems (IHCPs) occur in many branches of engineering and science. Mechanical and chemical engineers, mathematicians, and specialists in other sciences branches are interested in inverse problems. From another point of view, since the existence, uniqueness, and stability of the solutions of these problems are not usually confirmed, they are generally identified as ill-posed [1–4]. According to the fact that unknown solutions of inverse problems are determined through indirect observable data which contain measurement errors, such problems are naturally unstable. In other words, the main difficulty in the treatment of inverse problems is the unstability of their solution in the presence of noise in the measured data. Hence, several numerical methods have been proposed for solving the various kinds of inverse problems. In addition to, ill-posedness of these kinds of problems, ill-conditioning of the resulting discretized matrix from the traditional methods like the finite differences method (FDM) [5], the finite element method (FEM), and so forth [6, 7], is the main problem making all numerical algorithms for determining the solution of these kinds of problems. Accordingly, within recent years, meshless methods, as the method of fundamental solution (MFS), radial basis functions (RBFs) method, and some other methods, have been applied by many scientists in the field of applied sciences and engineering [8–15]. Kupradze and Aleksidze [16] first introduced MFS which defines the solution of the problem as a linear combination of fundamental solutions. Hon et al. [17–20] applied the MFS to solve some inverse heat conduction problems. In 1990s, Kansa applied RBFs method to solve the different types of partial differential equations [21, 22]. After that, Kansa and many scientists regarded RBFs method to solve different types of mathematical problems from partial or ordinary differential equations to integral equations [23–26]. Following their works, during recent years, many researchers have made some changes in RBFs and MFS methods and have developed advance methods to solve some of these kinds problems [27, 28]. Consequently, in this work, we will present a meshless numerical scheme, based on combining the radial basis function and the fundamental solution of the heat equation, in order to approximate the solution of a backward inverse heat conduction problem (BIHCP), the problem in which an unknown initial condition or/and temperature distribution in previous time will be determined. This kind of problem may emerge in many practical application areas such as archeology and mantle plumes [29]. On the other hand, since the system of the linear equations obtained from discretizing the problem in the presented method is ill-conditioned, Tikhonov regularization (TR) method is applied in order to solve it. The generalized cross-validation (GCV) criterion has been assigned to adopt an optimum amount of the regularization parameter. The structure of the rest of this work is organized as follows: In Section 2, we represent the mathematical formulation of the problem. The method of fundamental solutions, radial basis functions method, and method of fundamental solution-radial basis functions (MFSRBF) are described in Section 3. Section 4 embraces Tikhonov regularization method with a rule for choosing an appropriate regularization parameter. In Section 5, we present the obtained numerical results of solving a test problem. Section 6 ends in a brief conclusion and some suggestions.

#### 2. Mathematical Formulation of the Problem

In this section, we consider the following one-dimensional inverse heat conduction problem:with the following initial and boundary conditions:where and are considered as known functions and is a given positive constant, while and are regarded as unknown functions. So, in order to estimate and , we consider additional temperature measurements and heat flux given at a point , , as overspecified conditions:To solve the above problem, at first, we divide the problem (1)–(3) into two separate problems. The problem is as follows:and the problem is considered as follows: Obviously, the problems A and B are considered as IHCP, where , and , are unknown functions in the problems A and B, respectively.

#### 3. Method of Fundamental Solutions and Method of Radial Basis Functions

In this section, we introduce the numerical scheme for solving the problem (1)–(3) using the fundamental solutions and radial basis functions.

##### 3.1. Method of Fundamental Solutions

The fundamental solution of (1) is presented as below:where is Heaviside unit function. Assuming that is a constant, it can be demonstrated that the time shift functionis also a nonsingular solution of (1) in the domain .

In order to solve an IHCP by MFS, as the problem , since the basis function satisfies the heat equation (1) automatically, we assume that is a given set of scattered points on the boundary . An approximate solution is defined as a linear combination of as follows:where is given by (7) and ’s are unknown coefficients which can be determined by solving the following matrix equation:where and is a known vector. Also, as the fundamental functions are the solution of the heat equation, only the initial and boundary conditions are practiced to make the system of linear equations; that is, is a square matrix which is defined using the initial and the boundary conditions as follows:For more details, see [13, 18].

##### 3.2. Method of Radial Basis Functions

In this section, we consider RBF method for interpolation of scattered data. Suppose that and are a fixed point and an arbitrary point in , respectively. A radial function is defined via , where . That is, the radial function depends only on the distance between and . This property implies that the RBFs are radially symmetric about . Some well-known infinitely smooth RBFs are given in Table 1. As it is observed, these functions depend on a free parameter , known as the shape parameter, which has an important role in approximation theory using RBFs.