## Application of the variational iteration method to inverse heat source problems.(English)Zbl 1189.65216

Summary: This paper investigates the inverse problem of determining a heat source in the parabolic heat equation using the usual conditions. The numerical solution is developed by using the variational iteration method. This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional. Using this method a rapid convergent sequence can be obtained which tends to the exact solution of the problem. Furthermore, the variational iteration method does not require the discretization of the problem. Thus the variational iteration method is suitable for finding the approximation of the solution without discretization of the problem. Two numerical examples are presented to illustrate the strength of the method.

### MSC:

 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
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### References:

 [1] Cannon, J. R.; Duchateau, P., Structural identification of an unknown source term in a heat equation, Inverse Problems, 14, 535-551 (1998) · Zbl 0917.35156 [2] Fatullayev, A. G., Numerical solution of the inverse problem of determining an unknown source term in a two-dimensional heat equation, Applied Mathematics and Computation, 152, 659-666 (2004) · Zbl 1077.65107 [3] Savateev, E. G., On problems of determining the source function in a parabolic equation, Journal of Inverse and Ill-Posed Problems, 3, 83-102 (1995) · Zbl 0828.35142 [4] Trong, D. D.; Long, N. T.; Alain, P. N., Nonhomogeneous heat equation: Identification and regularization for the inhomogeneous term, Journal of Mathematical Analysis and Applications, 312, 93-104 (2005) · Zbl 1087.35095 [5] Farcas, A.; Lesnic, D., The boundary-element method for the determination of a heat source dependent on one variable, Journal of Engineering Mathematics, 54, 375-388 (2006) · Zbl 1146.80007 [6] Ling, L.; Yamamoto, M.; Hon, Y. C.; Takeuchi, T., Identification of source locations in two-dimensional heat equation, Inverse Problems, 22, 1289-1305 (2006) · Zbl 1112.35147 [7] Yi, Z.; Murio, D. A., Source term identification in 1-d IHCP, Computers and Mathematics with Applications, 47, 1921-1933 (2004) · Zbl 1063.65102 [8] Yan, L.; Fu, C. L.; Yang, F. L., The method of fundamental solutions for the inverse heat source problem, Engineering Analysis with Boundary Elements, 32, 3, 216-222 (2008) · Zbl 1244.80026 [9] He, J. H., Variational iteration method-a kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics, 34, 699-708 (1999) · Zbl 1342.34005 [10] He, J. H., Variational iteration method for autonomous ordinary differential system, Applied Mathematics and Computation, 114, 115-123 (2000) · Zbl 1027.34009 [11] He, J. H., Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B, 20, 1141-1199 (2006) · Zbl 1102.34039 [12] He, J. H.; Wu, X. H., Construction of solitary solution and compaction-like solution by variational iteration method, Chaos, Solitons and Fractals, 29, 108-113 (2006) · Zbl 1147.35338 [13] He, J. H., A new approach to nonlinear partial differential equations, Communications in Nonlinear Science and Numerical Simulation, 2, 4, 230-235 (1997) · Zbl 0923.35046 [14] He, J. H., Variational approach for nonlinear oscillators, Chaos, Solitons and Fractals, 34, 5, 1430-1439 (2007) · Zbl 1152.34327 [15] He, J. H., Variational iteration method—Some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207, 1, 3-17 (2007) · Zbl 1119.65049 [16] Wang, S. Q.; He, J. H., Variational iteration method for solving integro-differential equations, Physics Letters A, 367, 3, 188-191 (2007) · Zbl 1209.65152 [17] Lu, J. F., Variational iteration method for solving two-point boundary value problems, Journal of Computational and Applied Mathematics, 207, 92-95 (2007) · Zbl 1119.65068 [18] Xu, L., The variational iteration method for fourth order boundary value problems, Chaos, Solitons and Fractals (2007) [19] He, J. H., Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos, Soliton and Fractals, 19, 847-851 (2004) · Zbl 1135.35303 [20] Yusufoglu, E., Variational iteration method for construction of some compact and noncompact structures of Klein-Gordon equations, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 153-158 (2007) [21] Tari, H.; Ganji, D. D.; Rostamian, M., Approximate solutions of K (2, 2) KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 203-210 (2007) [22] Abdou, M. A.; Soliman, A. A., Variational iteration method for solving Burger’s and coupled Burger’s equations, Journal of Computational and Applied Mathematics, 181, 245-251 (2005) · Zbl 1072.65127 [23] Ganji, D. D.; Sadighi, A., Application of He’s homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 4, 411-418 (2006) [24] Momani, S.; Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos, Soliton and Fractal, 27, 1119-1123 (2006) · Zbl 1086.65113 [25] Odibat, Z. M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 27-34 (2006) · Zbl 1401.65087 [26] Shou, D. H.; He, J. H., Application of parameter-expanding method to strongly nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 1, 121-124 (2007) [27] Bildik, N.; Konuralp, A., The use of variational iteration method differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1, 65-70 (2006) · Zbl 1401.35010 [28] Soliman, A. A.; Abdou, M. A., Numerical solution of nonlinear evolution equations using variational iteration method, Journal of Computational and Applied Mathematics, 207, 111-120 (2007) · Zbl 1120.65111 [29] Sweilam, N. H., Fourth order integro-differential equations using variational iteration method, Computers and Mathematics with Applications, 54, 7, 1086-1091 (2007) · Zbl 1141.65399 [30] Biazar, J.; Ghazvini, H., He’s variational iteration method for solving linear and non-linear systems of ordinary differential equations, Applied Mathematics and Computation, 191, 287-297 (2007) · Zbl 1193.65144
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