Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

1Department of Computer Science, Harbin Institute of Technology at Weihai, Weihai, Shandong 264209, China 2Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China 3Beijing Key Laboratory of Mobile Computing & Pervasive Device, Institute of Computing Technology, Beijing, China 4Department of Multimedia, Sungkyul University, Anyō, Gyeonggi 100190, Republic of Korea


Introduction
Various inverse problems in a parabolic partial differential equation are widely encountered in modeling physical phenomena [1][2][3].There are three kinds of inverse parameter problems of parabolic partial differential equations, including determining an unknown time-dependent coefficient, an unknown space-dependent coefficient, and an unknown source term.
There are various numerical methods to solve (1) and (2) or similar problems.Now we give a quick review of the previous work placed to our problem.Cannon [6] reduced the problem to a nonlinear integral equation for the coefficient ().This approach works well for a parabolic equation in one space variable but does not easily extend to higher-dimensional problems because it depends on the explicit form of the fundamental solution of the heat operator.In Cannon and Yin [7], this problem was studied from a different point of view.The authors first transformed a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functional on the solution and its derivatives subject to some initial and boundary conditions.For the resulted nonclassical problem, they introduced a variation form by defining a new function; then both continuous and discrete Galerkin procedures are employed to the nonclassical problem.Authors of [8] presented the backward Euler finite difference scheme.It is shown that this scheme is stable in the maximum norm and error estimation was obtained.In [9], several firstand second-order finite difference numerical schemes have been developed to solve the nonclassical problem which is obtained by applying the transformation technique in [7] to problem (1) and (2).Also, a method is proposed in [10] to solve this problem which is based on a semianalytical approach.Authors of [11] used the pseudospectral Legendre method to solve this problem.An unconditionally stable efficient fourth-order numerical algorithm based on the functional transformation, the Pade approximation, and the Richardson extrapolation is proposed in [12] to compute the main function and the unknown time-dependent coefficient in (1).The Chebyshev cardinal functions are employed in [13] to recover the unknown coefficient.These schemes are efficient and easy to implement but the convergence order is low.
Although there are many methods for recovering the above inverse problems, those methods only give approximate solution.So it is worth noting that the variational iteration method can give the exact solution.
Professor He proposed variational iteration method (VIM) firstly in 1998 [23] and developed quickly VIM in 2006 and 2007.Based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional, VIM gives rapidly convergent successive approximations of the exact solution if such a solution exists.There are three standard variational iteration algorithms [24], called VIM-I, VIM-II, and VIM-III, for solving differential difference equations, integrodifferential equations, fractional differential equations, and fractal differential equations.These three forms of VIM have been proved by many authors to be a powerful mathematical tool for addressing various kinds of linear and nonlinear problems [25][26][27][28].The reliability of the method and the reduction in the burden of computational work give this method wider application [29][30][31][32].In addition, some reviews can be found in He [24,33,34].Since the applications of VIM in inverse problems are very few, we use VIM-I to recover the unknown coefficients here.Furthermore, VIM gives the exact solution of this problem.Thus the variational iteration method is suitable for finding the approximation solution of the problem.
The rest of the paper is organized in four sections including Introduction.Section 2 gives the detailed progress and proof for recovering the unknown coefficients by applying VIM.In Section 3, numerical examples and a stable experiment are presented to imply the accuracy of VIM.Finally, a brief conclusion ends this paper.

Application of He's Variational Iteration Method
In this section, we will apply He's variational iteration method (VIM) to recover time-or space-dependent coefficient problems.The detailed introduction of VIM can be found in [24,33,34].

Recovering Time-Dependent Coefficients. Using (1) and
(2), we obtain Assuming that   ( * , ) ̸ = 0, we have Therefore the inverse problem (1) and ( 2) is equivalent to the following problem: From ( 9 In the following, we determine the Lagrange multiplier  via variation theory:  Applying ũ  = 0, then so Thus () =  − ; this gives the iterative formula: Now, take  0 (, ) and    0 ( * , ) →   ( * , ) as an initial value.By (18), we can obtain the -order approximate solution   (, ) of ( 9).Putting then and its derivative about : Inserting  =  * , we obtain From (18), one can infer that which leads to the following: so as to deduce Therefore, by ( 8), the approximate solution   () to () can be expressed in the following form: Assuming that   (, ) =   1 () ̸ = 0, we have Therefore, the inverse problem ( 1) and ( 3)-( 6) is equivalent to the following problem: with the initial condition and boundary conditions Next, we are concerned with the approximate solutions of ( 35)-(37) by the variational iteration method.Applying the variation theory, we can construct an iteration formula.
The above three examples are about time-dependent coefficient; in the following we take space-dependent coefficient examples.
In order to imply the stability of this method, we perturb the additional specification data  1 () as with  = 1%; the reconstruction results are also stable, see Figure 1.

Conclusion
The VIM has been applied in solving a variety of equations, but it was rarely applied in inverse problems.Here, we develop the new application area of VIM; our contribution is that we apply VIM to solve the inverse problem of time-or spacedependent coefficients in a parabolic partial differential equation and obtain the exact solution.The numerical results fully demonstrate the superiority of VIM for these inverse problems.