Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 573932, 9 pages
http://dx.doi.org/10.1155/2015/573932
Research Article

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

Received 3 November 2014; Revised 18 December 2014; Accepted 19 December 2014

Academic Editor: Bo-Wei Chen

Copyright © 2015 Guanglu Zhou et al. 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

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

1. Introduction

Various inverse problems in a parabolic partial differential equation are widely encountered in modeling physical phenomena [13]. 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.

The aim of this paper is to find in the parabolic equationwhere is only a function with respect to or .

When , the boundary conditions and an extra measurement of (1) are as follows:where , , , , , , and are known functions. This equation is 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 are very expensive to be measured. The existence and uniqueness of the solution to this problem are discussed in [4, 5].

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 first- and 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.

When , the boundary conditions and an extra measurement of (1) are as follows:where , , , and are known functions. It is widely known that this model describes the heat conduction procedure in a given inhomogeneous medium with some input source and the coefficient represents a heat conduction property, namely, the heat capacity. There are various numerical methods to solve (1) and (3)–(6) or similar problems. Deng et al. [14] applied the gradient iteration algorithm for obtaining the approximate solutions. Kansa method is used by [15] to solve problem (1) and (2) and the stable experiments are given. Authors in [16] give an iterative fixed point projection method for this problem. In addition, there are other methods [1722].

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 [2528]. The reliability of the method and the reduction in the burden of computational work give this method wider application [2932]. 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.

2. 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].

2.1. Recovering Time-Dependent Coefficients

Using (1) and (2), we obtain

Assuming that , we have

Therefore the inverse problem (1) and (2) is equivalent to the following problem:

From (9),

Constructing a correction function for the above equation:

In the following, we determine the Lagrange multiplier via variation theory:

Applying , thenso

Thus ; this gives the iterative formula:

Now, take and as an initial value. By (18), we can obtain the -order approximate solution of (9). Puttingthenand its derivative about :and the derivative of the above about :

Inserting , we obtain

From (18), one can infer thatsowhich leads to the following:so as to deduce

Therefore, by (8), the approximate solution to can be expressed in the following form:

2.2. Recovering Space-Dependent Coefficients

Using (1) and (3)–(6), we obtainthenputting and integrating the above equation with from 0 to ,applying condition (4),thus

Assuming that , we have

Therefore, the inverse problem (1) and (3)–(6) is equivalent to the following problem:with the initial conditionand 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.

From (35),

Constructing a correction function for the above equation,

In the following, we determine the Lagrange multiplier via variation theory:

Applying , thenso

Thus ; this gives the iterative formula

Now, take and as an initial value. By (43), we can obtain the -order approximate solution of (35).

If , then we can approximate to by the following:

Now, we prove that .

By (43), , which leads to the following:so as to deduce

3. Numerical Examples

Example 1. Considering a special case of (1) and (2) with [9, 13],with , for which the exact solution is

Beginning withwhere , are the unknown parameters to be further determined, according to (18), one can obtain the first-order approximation , and we find

Incorporating the initial condition , , of Example 1 into , the unknown parameters , can be obtained. Therefore, the first-order approximationis obtained, which is the exact solution of Example 1. From (28), we havewhich is equal to the exact of Example 1.

Example 2. Finding in (1) and (2) with [9, 13],where . The true solution is while .

Beginning withwhere , are the unknown parameters to be further determined, according to (18), one can obtain the first-order approximation , and we find

Incorporating the initial condition , , of Example 2 into , the unknown parameters , can be obtained. Therefore, the first-order approximation is obtained and , which is the exact solution of Example 2. From (28), we have , which is equal to the exact of Example 2.

Example 3. We solve the problem (1) and (2) with [9, 13]:whose true solution is while , .

Beginning withwhere , are the unknown parameters to be further determined, according to (18), one can obtain the first-order approximation , and we find

Incorporating the initial condition , , of Example 3 into , the unknown parameters , can be obtained. Therefore, the first-order approximation is obtained and , which is the exact solution of Example 3. From (28), we have , which is equal to the exact of Example 3.

The above three examples are about time-dependent coefficient; in the following we take space-dependent coefficient examples.

Applying the above VIM, we begin with , where , are the unknown parameters to be further determined. Incorporating the initial and boundary condition (36) and (37) into , the unknown parameters , can be obtained. According to (43), one can obtain the first-order approximation . Here, .

Example 4. We take the boundary conditions, initial condition, and additional specification function (3)–(6) as [14, 15]with the exact solution asand the identifying coefficient as

Therefore, the first-order approximation is obtained. We can determine , , so which is the exact solution of Example 4. From (44), we have which is equal to the exact of Example 4.

Example 5. Finding in (1) and (3)–(6) in [15]The true solution is while .
From (43), the first-order approximate solutionIncorporating the initial conditions, we determine , . Therefore, the first-order approximation is obtained and which is the exact solution of Example 5. From (44), we have which is equal to the exact of Example 5.

Example 6. We solve the problem (1) and (3)–(6) in [15]:whose true solution is while .

We can determine , . Therefore, the first-order approximation , which is the exact solution of Example 6. From (44), we have which is equal to the exact of Example 6.

In order to imply the stability of this method, we perturb the additional specification data aswith ; the reconstruction results are also stable, see Figure 1.

Figure 1: Numerical solutions of Example 6 with .

4. 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 space-dependent 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.

Appendix

To imagine the basic idea behind He’s VIM, we consider the following general differential equation:where is the highest order derivative that is assumed to be easily invertible, is a linear differential operator of order less than , represents the nonlinear terms, and is a source term. The basic characteristic of He’s method is to construct a correction function for (A.1), which readswhere is a Lagrange multiplier which can be identified optimally via variation theory, is the nth approximate solution, and denotes a restricted variation; that is, .

To solve (A.1) by He’s VIM, we first determine the Lagrange multiplier that can be identified optimally via variation theory. Then, the successive approximations , , of the solution can be readily obtained upon using the obtained Lagrange multiplier and any selective function . Consequently, the exact solution may be obtained by using

In summary, we have the following variation iteration formula:where is an arbitrary function satisfying initial and boundary conditions.

It should be specially pointed out that the more accurate the identification of the multiplier is, the faster the approximations converge to their exact solutions.

Remark 7. We cite an integrate of in (A.2) as an example; one needs an integrate of by a similar method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the unknown referees for their careful reading and helpful comments.

References

  1. M. Dehghan, “An inverse problem of finding a source parameter in a semilinear parabolic equation,” Applied Mathematical Modelling, vol. 25, no. 9, pp. 743–754, 2001. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Dehghan, “Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement,” Numerical Methods for Partial Differential Equations, vol. 21, no. 3, pp. 611–622, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. M. Dehghan, “Parameter determination in a partial differential equation from the overspecified data,” Mathematical and Computer Modelling, vol. 41, no. 2-3, pp. 197–213, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. J. R. Cannon and W. Rundell, “Recovering a time dependent coefficient in a parabolic differential equation,” Journal of Mathematical Analysis and Applications, vol. 160, no. 2, pp. 572–582, 1991. View at Publisher · View at Google Scholar · View at Scopus
  5. A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, vol. 1, Marcel Dekker, New York, NY, USA, 2000.
  6. J. R. Cannon, “Determination of an unknown coefficient in a parabolic differential equation,” Duke Mathematical Journal, vol. 30, pp. 313–323, 1963. View at Publisher · View at Google Scholar · View at MathSciNet
  7. J. R. Cannon and H. M. Yin, “Numerical solutions of some parabolic inverse problems,” Numerical Methods for Partial Differential Equations, vol. 2, pp. 177–191, 1990. View at Google Scholar
  8. H. Azari, W. Allegretto, Y. Lin, and S. Zhang, “Numerical procedures for recovering a time dependent coefficient in a parabolic differential equation,” Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, vol. 11, no. 1-2, pp. 181–199, 2004. View at Google Scholar · View at Scopus
  9. M. Dehghan, “Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement,” Numerical Methods for Partial Differential Equations, vol. 21, no. 3, pp. 611–622, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. M. Dehghan and M. Tatari, “Solution of a parabolic equation with a time-dependent coefficient and an extra measurement using the decomposition procedure of adomian,” Physica Scripta, vol. 72, no. 6, pp. 425–431, 2005. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Shamsi and M. Dehghan, “Recovering a time-dependent coefficient in a parabolic equation from overspecified boundary data using the pseudospectral legendre method,” Numerical Methods for Partial Differential Equations, vol. 23, no. 1, pp. 196–210, 2007. View at Publisher · View at Google Scholar · View at Scopus
  12. W. Liao, M. Dehghan, and A. Mohebbi, “Direct numerical method for an inverse problem of a parabolic partial differential equation,” Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 351–360, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. M. Lakestani and M. Dehghan, “The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement,” Journal of Computational and Applied Mathematics, vol. 235, no. 3, pp. 669–678, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. Z.-C. Deng, L. Yang, J.-N. Yu, and G.-W. Luo, “Identifying the diffusion coefficient by optimization from the final observation,” Applied Mathematics and Computation, vol. 219, no. 9, pp. 4410–4422, 2013. View at Publisher · View at Google Scholar · View at Scopus
  15. K. Parand and J. A. Rad, “Kansa method for the solution of a parabolic equation with an unknown spacewise-dependent coefficient subject to an extra measurement,” Computer Physics Communications, vol. 184, no. 3, pp. 582–595, 2013. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Golayoglu Fatullayev and S. Cula, “An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1033–1037, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. F. Hettlich and W. Rundell, “Identification of a discontinuous source in the heat equation,” Inverse Problems, vol. 17, no. 5, pp. 1465–1482, 2001. View at Publisher · View at Google Scholar · View at Scopus
  18. Z. Yi and D. A. Murio, “Source term identification in 1-D IHCP,” Computers and Mathematics with Applications, vol. 47, no. 12, pp. 1921–1933, 2004. View at Publisher · View at Google Scholar · View at Scopus
  19. A. Hasanov, “Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: weak solution approach,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 766–779, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. L. Yang, Z. C. Deng, J. N. Yu, and G. W. Luo, “Two regularization strategies for an evolutional type inverse heat source problem,” Journal of Physics A: Mathematical and Theoretical, vol. 42, Article ID 365203, p. 16, 2009. View at Google Scholar
  21. L. Yang, Z.-C. Deng, J.-N. Yu, and G.-W. Luo, “Optimization method for the inverse problem of reconstructing the source term in a parabolic equation,” Mathematics and Computers in Simulation, vol. 80, no. 2, pp. 314–326, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. J. Peralta and L. E. Olivar, “Regularization algorithm within two parameters for the identification of the heat conduction coefficient in the parabolic equation,” Mathematical and Computer Modelling, vol. 57, no. 7-8, pp. 1990–1998, 2013. View at Publisher · View at Google Scholar · View at Scopus
  23. J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. J.-H. He, “Variational iteration method—some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. Y. Aksoy, M. Pakdemirli, S. Abbasbandy, and H. Boyaci, “New perturbation-iteration solutions for nonlinear heat transfer equations,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 22, no. 7, pp. 814–828, 2012. View at Publisher · View at Google Scholar · View at Scopus
  26. A. Yildirim and H. Koçak, “Rational approximation solution of the foam drainage equation with time- and space-fractional derivatives,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 22, no. 4, pp. 512–525, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. S. M. Mehdi Hosseini, S. T. Mohyud-Din, and H. Ghaneai, “Variational iteration method for Hirota-Satsuma coupled KDV equation using auxiliary Parameter,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 22, no. 3, pp. 277–286, 2012. View at Publisher · View at Google Scholar · View at Scopus
  28. H. Ghaneai, M. M. Hosseini, and S. T. Mohyud-Din, “Modified variational iteration method for solving a neutral functional-differential equation with proportional delays,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 22, no. 8, pp. 1086–1095, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. F. Geng and Y. Lin, “Application of the variational iteration method to inverse heat source problems,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2098–2102, 2009. View at Publisher · View at Google Scholar · View at Scopus
  30. D. D. Ganji and A. Sadighi, “Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 24–34, 2007. View at Publisher · View at Google Scholar · View at Scopus
  31. M. Dehghan, M. Tatari, and A. Azizi, “The solution of the Falkner-Skan equation arising in the modelling of boundary-layer problems via variational iteration method,” International Journal of Numerical Methods for Heat and Fluid Flow, vol. 21, no. 2, pp. 136–149, 2011. View at Publisher · View at Google Scholar · View at Scopus
  32. G. E. Drăgănescu and V. Căpălnăşăn, “Nonlinear relaxation phenomena in polycrystalline solids,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 4, no. 3, pp. 219–225, 2003. View at Publisher · View at Google Scholar · View at Scopus
  33. J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters. A, vol. 375, no. 38, pp. 3362–3364, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. J.-H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet