Elsevier

Applied Mathematics and Computation

Volume 226, 1 January 2014, Pages 238-249
Applied Mathematics and Computation

Reconstructing an unknown time-dependent function in the boundary conditions of a parabolic PDE

https://doi.org/10.1016/j.amc.2013.10.074Get rights and content

Abstract

This work investigates the parabolic inverse problem of reconstructing an unknown time-dependent function in the boundary conditions from additional terminal, integral or point observations. A numerical scheme is developed for obtaining an approximate solution for this problem. The method consists of reducing the solution of the inverse problem to a system of integral equations. The properties of Sinc function are then utilized for replacing the resulting integral equations by a system of linear algebraic equations. Several numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.

Introduction

In many engineering contexts, there are many inverse problems for parabolic PDEs. These kinds of problems have a great deal of attention by mathematicians, physicists and engineers and they have been investigated by many researchers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. These inverse problems can be roughly divided into three principal classes.

  • 1.

    Backward or reversed-time problem: the initial condition is to be found [1], [2], [3].

  • 2.

    Coefficient inverse problem: this is a classical parameter problem where a multiplier in the governing equation is to be found [4], [5], [6], [7], [8], [9].

  • 3.

    Boundary inverse problem: some missing information at the boundary of the domain is to be found [10], [11], [12], [13].

In this paper, we consider the inverse problem of finding u(x,t) and the boundary function ψ(t) in the following parabolic equationut(x,t)=uxx(x,t)+q(x,t),(x,t)Q,subject to the initial conditionu(x,0)=p(x),0<x<1,boundary conditions-ux(0,t)+σ(t)u(0,t)=f0(t)ψ(t)+g0(t),0<ttf,ux(1,t)+σ(t)u(1,t)=f1(t)ψ(t)+g1(t),0<ttf,and for the additional information, we consider three cases as follows

Case I: an overspecification at a point in the spatial domainu(x,t)=E(t),x[0,1],t[0,tf],

Case II: the integral boundary observationλ0u(0,t)+λ1u(1,t)=E(t),t[0,tf],

Case III: the energy overspecified condition01u(x,t)dx=E(t),t[0,tf],where tf>0 is an arbitrary fixed time of interest, λ0 and λ1 are given constants, Q={(x,t)|x(0,1),t(0,tf]}, and q,p,σ,f0,f1,g0,g1 and E are known functions.

In [13], the boundary element method is used for the problem (1), (2), (3), (4) with the boundary measurements u(0,t), or u(1,t), or the integral boundary observation (6).

Sinc methods have been recognized as powerful tools for solving a wide class of problems arising from scientific and engineering applications including, inverse problems [6], [7], population growth [14], heat transfer [15], fluid mechanics [16] and medical imaging [17]. Unlike most numerical techniques, these methods are characterized by exponentially decaying errors [18], [19], [20], and also, they excel for problems whose solutions may have singularities, or infinite domains, or boundary layers [21], [22]. Moreover, due to their rapid convergence, Sinc numerical methods do not suffer from the common instability problems associated with other numerical methods [23]. Using Sinc functions for obtaining approximate solutions of ODEs, PDEs and integral equations is widely discussed in [24], [25].

In this paper, we make use of the Sinc method to determine the functions u and ψ which satisfy the problem (1), (2), (3), (4) with all cases additional information (5), (6), (7). The method consists of reducing the solution of the inverse problem to a system of integral equations. The properties of Sinc function are then utilized for replacing the resulting integral equations by a system of linear algebraic equations. In the following statement, we call

  • the problem (1), (2), (3), (4), (5) to Problem I,

  • the problem (1), (2), (3), (4) and (6) to Problem II, and

  • the problem (1), (2), (3), (4) and (7) to Problem III.

This paper is organized as follows. In Section 2, we briefly review the concept and some properties of the Sinc function and describe the collocation procedure by means of Sinc method for approximating convolution integrals. After we reduce the inverse problems into the systems of integral equations in Section 3 we propose the systems of algebraic equations for numerical solution of these integral equations in Section 4. Finally, in Section 5, we give some numerical examples.

Section snippets

Sinc function properties

In this section, we review some concepts of Sinc approximation necessary for collocating convolution integrals.

The kth Sinc function with step size h is given byS(k,h)(x)=sincx-khh,where kZ andsinc(x)=sin(πx)πx,x0,1,x=0.

To construct Sinc approximation on a finite interval (a,b), the “eye-shaped” region D={zC:|arg[(z-a)/(b-z)]|<d}, is transformed conformally onto the strip Dd={zC:|Im(z)|<d} via w=ϕ=log[(z-a)/(b-z)]. Let Sinc points be defined on (a,b) by zk=ϕ-1(kh)=(a+bekh)/(1+ekh), and Sinc

Reduction the inverse problems to integral equations

Consider the problem (1), (2), (3), (4) and assume that the source function q(x,t) is bounded over the domain Q, and that q(x,t) is uniformly Hölder continuous on each compact subset of Q. We shall also assume that the initial function p is piecewise-continuous. So, the bounded unique solution u of the problem (1), (2), (3), (4) is the form [27]u(x,t)=v(x,t)-20tθ(x,t-τ){σ(τ)φ1(τ)-f0(τ)ψ(τ)}dτ-20tθ(x-1,t-τ){σ(τ)φ2(τ)-f1(τ)ψ(τ)}dτ,where φ1,φ2 and ψ satisfy the following relationsφ1(t)=v(0,t)-20

Approximation based on Sinc collocation

The “Laplace transformation” of θ(x,t), can be determined asFx(s)=0θ(x,t)e-tsdt=s2+k=1scos(kπx)1+k2π2s,sΩ+,x[0,1].For x=0,1, we may writeF0(s)=s2coth1s,sΩ+,F1(s)=s2+k=1(-1)ks1+k2π2s,sΩ+.

Let Wα(D), with 0<α1, consists the family of functions gH1(D), such that|Θr(r,x,τ)|C1rξ-1(1+r)2ξ,r(0,tf),τD,x[0,1],where C1 and ξ are constants which are independent of r,τ and x, withΘ(r,x,τ)=0τθ(x,r+τ-η)g(η)dη,and Θ(r,·)Mα(D), uniformly, for r(0,tf).

Assume that φiσWα(D), using the Sinc

Numerical results

We present some results of numerical comparison of the approximations uN and ψN given in Section 4 and the corresponding exact solutions u and ψ of the problem. In all examples we take α=1 and d=π2, which yields h=π2N.

In practical applications, data contain random noise. We will illustrate the effect of the solution in virtue of the noisy datapδ(x)=p(x)(1+δsin50x),σδ(t)=σ(t)(1+δsin50t),Eδ(t)=E(t)(1+δsin50t),where δ is the noise parameter.

Example 1

Consider (1), (2), (3), (4) with tf=1, andp(x)=1+cosx,σ(t)

Conclusion

The exponential convergence rate of Sinc approximation make this approach very attractive and contributed to the good agreement between approximate and exact values. In this paper, after the boundary inverse problem is reduced to a system of integral equations, the Sinc approximation for convolution integrals was successfully employed to obtain an accurate approximate solution of the inverse problem. This scheme provides the numerical solution with closed form. The numerical tests showed that

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