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He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation. (English) Zbl 1131.65084
Summary: The well known variational iteration method is used for finding the solution of a semi-linear inverse parabolic equation. This method is based on the use of Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method a rapid convergent sequence is produced which tends to the exact solution of the problem. Thus the variational iteration method is suitable for finding the approximation of the solution without discretization of the problem. We change the main problem to a direct problem which is easy to handle the variational iteration method. To show the efficiency of the present method, several examples are presented. Also it is shown that this method coincides with Adomian decomposition method for the studied problem.

MSC:
65M32Inverse problems (IVP of PDE, numerical methods)
65M70Spectral, collocation and related methods (IVP of PDE)
35K15Second order parabolic equations, initial value problems
35R30Inverse problems for PDE
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References:
[1] Abdou, M. A.; Soliman, A. A.: Variational iteration method for solving burger’s and coupled burger’s equations. J comput appl math 181, 245-251 (2005) · Zbl 1072.65127
[2] Abdou, M. A.; Soliman, A. A.: New applications of variational iteration method. Physica D 211, 1-8 (2005) · Zbl 1084.35539
[3] Cannon, J. R.; Yin, H. M.: Numerical solution of some parabolic inverse problems. Numer methods partial differen equat 2, 177-191 (1990) · Zbl 0709.65105
[4] Cannon, J. R.; Lin, Y.; Wang, S.: Determination of source parameter in parabolic equations. Meccanica 27, 85-94 (1992) · Zbl 0767.35105
[5] Cannon, J. R.; Yin, H. M.: On a class of nonlinear parabolic equations with nonlinear trace type functionals inverse problems. Inverse probl 7, 149-161 (1991) · Zbl 0735.35078
[6] Cannon, J. R.; Yin, H. M.: On a class of non-classical parabolic problems. J differen equat 79, 266-288 (1989) · Zbl 0702.35120
[7] Cannon, J. R.; Lin, Y.: Determination of parameter $p(t)$ in holder classes for some semilinear parabolic equations. Inverse probl 4, 595-606 (1988) · Zbl 0688.35104
[8] Cannon, J. R.; Lin, Y.: Determination of parameter $p(t)$ in some quasi-linear parabolic differential equations. Inverse probl 4, 35-45 (1988) · Zbl 0697.35162
[9] Cannon, J. R.; Lin, Y.; Wang, S.: Determination of a control parameter in a parabolic partial differential equation. J aust math soc ser B 33, 149-163 (1991) · Zbl 0767.93047
[10] Cannon, J. R.; Lin, Y.; Xu, S.: Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations. Inverse probl 10, 227-243 (1994) · Zbl 0805.65133
[11] Dehghan, M.: Determination of a control parameter in the two-dimensional diffusion equation. Appl numer math 37, 489-502 (2001) · Zbl 0982.65103
[12] Dehghan, M.: Fourth-order techniques for identifying a control parameter in the parabolic equations. Int J engrg sci 40, 433-447 (2002) · Zbl 1211.65120
[13] Dehghan, M.: Numerical solution of a non-local boundary value problem with Neumann’s boundary conditions. Commun numer methods engrg 19, 1-12 (2003) · Zbl 1014.65072
[14] Dehghan, M.: Finding a control parameter in one-dimensional parabolic equations. Appl math comput 135, 491-503 (2003) · Zbl 1026.65079
[15] Dehghan, M.: Determination of a control function in three-dimensional parabolic equations. Math comput simul 61, 89-100 (2003) · Zbl 1014.65097
[16] Dehghan, M.: The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure. Int J comput math 81, 979-989 (2004) · Zbl 1056.65099
[17] Day, W. A.: Extension of a property of the heat equation to linear thermoelasticity and other theories. Quart appl math 40, 319-330 (1982) · Zbl 0502.73007
[18] He, J. H.: Variational iteration method for delay differential equations. Commun nonlinear sci numer simul. 2, 235-236 (1997)
[19] He, J. H.: Approximate solution of nonlinear differential equations with convolution product non-linearities. Comput methods appl mech engrg 167, 69-73 (1998)
[20] He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput methods appl mech eng 167, 57-68 (1998) · Zbl 0942.76077
[21] He, J. H.: Variational iteration method: a kind of non-linear analytical technique: some examples. Int J nonlin mech 34, 699-708 (1999) · Zbl 05137891
[22] He, J. H.: Variational iteration method for autonomous ordinary differential systems. Appl math comput 114, 115-123 (2000) · Zbl 1027.34009
[23] He, J. H.; Wan, Y. Q.; Guo, Q.: An iteration formulation for normalized diode characteristics. Int J circ theory appl 32, 629-632 (2004) · Zbl 1169.94352
[24] Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in non-linear mathematical physics. Variational methods in the mechanics of solids, 156-162 (1978)
[25] Lin, Y.: An inverse problem for a class of quasilinear parabolic equations. SIAM J math anal 22, No. 1, 146-156 (1991) · Zbl 0739.35106
[26] Macbain, J. A.; Bendar, J. B.: Existence and uniqueness properties for one-dimensional magnetotelluric inversion problem. J math phys 27, 645-649 (1986)
[27] Macbain, J. A.: Inversion theory for a parametrized diffusion problem. SIAM J appl math 18, 1386-1391 (1987) · Zbl 0664.35075
[28] Momani, S.; Abuasad, S.: Application of he’s variational iteration method to Helmholtz equation. Choas, solitons & fractals 27, 1119-1123 (2006) · Zbl 1086.65113
[29] Prilepko, A. I.; Orlovskii, D. G.: Determination of the evolution parameter of an equation and inverse problems of mathematical physics, part I. J differen equat 21, 119-129 (1985)
[30] Prilepko, A. I.; Soloev, V. V.: Solvability of the inverse boundary value problem of finding a coefficient of a lower order term in a parabolic equation. Differen equat 23, No. 1, 136-143 (1987)
[31] Rundell, W.: Determination of an unknown non-homogenous term in a linear partial differential equation from overspecified boundary data. Appl anal 10, 231-242 (1980) · Zbl 0454.35045
[32] Wang S. Numerical solutions of two inverse problems for identifying control parameters in 2-dimensional parabolic partial differential equations. Modern developments in numerical simulation of flow and heat transfer, HTP-194, 1992. p. 11-6.
[33] Dehghan, M.: Finite difference schemes for two-dimensional parabolic inverse problem with energy overspecification. Int J comput math 75, 339-349 (2000) · Zbl 0966.65068
[34] Dehghan, M.: Implicit solution of a two-dimensional parabolic inverse problem with temperature overspecification. J comput anal appl 3, No. 4, 383-398 (2001)
[35] Dehghan, M.: Determination of an unknown parameter in a semi-linear parabolic equation. Math probl eng 8, No. 2, 111-122 (2002) · Zbl 1050.65085