High-order scheme for determination of a control parameter in an inverse problem from the over-specified data
Introduction
The parameter determination in a parabolic partial differential equation from the over-specified data plays a crucial role in applied mathematics and physics. This technique has been widely used to determine the unknown properties of a region by measuring only data on its boundary or a specified location in the domain. These unknown properties such as the conductivity medium are important to the physical process but usually cannot be measured directly, or very expensive to be measured [1], [2]. The literature on the numerical approximation of solutions of parabolic partial differential equations with standard boundary conditions is large and still growing rapidly. While many finite difference, finite element, spectral, finite volume and boundary element methods have been proposed to approximate such solutions, there has been less research into the numerical approximation of parabolic partial differential equations with over-specified boundary data [3], [4], [5], [6]. Over the last few years, it has become increasingly apparent that many physical phenomena can be described in terms of parabolic partial differential equations with source control parameters [1]. The main contribution of this paper is to propose a high order method to approximate the solution of the following inverse problem, i.e. to find the unknown control parameter in the semi-linear time-dependent diffusion equation with initial condition and boundary conditions subject to either the over-specification at a point in the spatial domain (temperature over-specification), or the integral over-specification of the function over the spatial domain (energy over-specification), where and are known function while u and p are unknown functions. Also it is assumed that, for some constant , the kernel satisfies, The existence and uniqueness and continuous dependence of the solutions to some kind of these inverse problems are discussed in [7], [8], [9], [10], [11], [12], [13], [14], [15]. The applications of these inverse problems and some other similar parameter identification problems are discussed in [1]. The inverse problem (1.1) can be used to describe a heat transfer process with a source parameter , where for example (1.3) represents the temperature at a given point in a spatial domain at a time t, and u is the temperature distribution. Cannon et al. [16] formulated a backward Euler finite difference scheme via a transformation and proved the convergence of u with the convergence order of . Dehghan in [1], [17] presented four finite difference schemes for the problems (1.1), (1.2), (1.3), (1.4). These techniques are based on the three-point explicit, five-point explicit, three-point implicit and implicit Crandall's approximations. Also a numerical scheme based of method of radial basis functions for problem (1.1), (1.2), (1.3) is proposed in [18]. Dehghan and Saadatmandi in [19] proposed a tau method for the solution of problem (1.1), (1.2), (1.3). The method of lines based on applying the standard finite difference schemes and the Runge–Kutta formula presented in [20] for (1.1), (1.2), (1.3). Also a technique based on the moving least-square approximation (MLS) has been used in [21] for finding the solution of problem (1.1), (1.2), (1.3). Authors of [22] presented a method based on Legendre multiscaling functions for solving the one-dimensional parabolic inverse problem with a source control parameter.
In this paper we propose an efficient algorithm for solving problems (1.1), (1.2), (1.3), (1.4) which has fourth order accuracy in both space and time components. In this method we first discretize the spatial derivative with a fourth order compact finite difference (CFD) scheme and then apply a fourth order boundary value method for the solution of the resulting system of ordinary differential equations. We compare the numerical results obtained with other existing methods in the literature and show the efficiently and applicability of our proposed method.
The outline of this paper is as follows: In Section 2, we introduce a fourth-order compact finite difference scheme for discretizing spatial derivative of Eq. (1.1). In Section 3, we briefly introduce the boundary value methods proposed in [23], [24], [25] and present a fourth-order boundary value method for solving the initial value problems (1.1), (1.2), (1.3), (1.4). A numerical method based on a compact finite difference and a boundary value method for the solution of problem (1.1), (1.2), (1.3) is proposed in Section 4 and for problem (1.1), (1.2) and (1.4) is proposed in Section 5. The numerical results obtained from applying the new method on some test problems and comparison with analytical and other methods are shown in Section 6. We conclude this article with a brief conclusive discussion in Section 6.
Section snippets
A fourth-order CFD scheme
In this section we state the fourth-order finite difference scheme for the second order spatial derivative of (1.1) which we will use it in derivation of the new method.
In general consider the second order differential equation with boundary conditions We can design a fourth-order accurate compact difference scheme for Eq. (2.1), following the approach of [26]. If we denote the central difference operator as where h is spatial mesh
The boundary value methods
Boundary value methods (BVMs) are the recent classes of ordinary differential equation solvers which can be interpreted as a generalization of the linear multi-step methods (LMMs) [23], [25]. One of the aim of boundary value methods is to circumvent the known Dahlquist-barriers on convergence and stability and do not have barriers whatsoever [23]. In comparison with the other initial value solvers, BVMs achieve the advantage of both unconditional stability and high-order accuracy. In
The proposed method
In this section we propose a high order scheme for the solution of inverse problem (1.1), (1.2), (1.3), (1.4). For a positive integer n let denote the step size of spatial variable (x) and Δt for the step size of the time variable (t). So we define We first utilize a pair of the following transformations [1], [2], So we have, With this transformation, is disappeared
Numerical experiments
In this section we present the numerical results of the new method on several test problems. We performed our computations in Matlab 7 software on a Pentium IV, 2800 MHz CPU machine with 1 GB of memory. We tested the accuracy and stability of the method presented in this paper by performing the mentioned method for different values of h and Δt.
Conclusion
In this paper we proposed a high order scheme for determining the unknown control parameter and the unknown solution of parabolic inverse problem with both integral overspecialization and overspecialization at a point in the spatial domain. In these equations, we first approximated the spatial derivative with a fourth order compact scheme and reduced the problem to a system of ordinary differential equations (ODEs). Then we applied a fourth order boundary value method for the solution of the
Acknowledgements
The authors are very grateful to both referees for carefully reading this paper and for their comments and suggestions which have improved the paper.
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