Identifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method

In this research, the He’s variational iteration technique is used for computing an unknown time-dependent parameter in an inverse quasilinear parabolic partial differential equation. Parabolic partial differential equations with overspecified data play a crucial role in applied mathematics and physics, as they appear in various engineering models. The He’s variational iteration method is an analytical procedure for finding solutions of differential equations, is based on the use of Lagrange multipliers for identification of an optimal value of a parameter in a functional. To show the efficiency of the new approach, several test problems are presented for one-, two- and three-dimensional cases.

[1]  Ji-Huan He.  SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS , 2006 .

[2]  Ji-Huan He.  Variational approach to the Lane-Emden equation , 2003, Appl. Math. Comput..

[3]  Yanping Lin,et al.  Determination of source parameter in parabolic equations , 1992 .

[4]  Mehdi Dehghan,et al.  Fourth-order techniques for identifying a control parameter in the parabolic equations , 2002 .

[5]  Nasser Hassan Sweilam,et al.  Variational iteration method for one dimensional nonlinear thermoelasticity , 2007 .

[6]  John A. Macbain,et al.  Inversion theory for parameterized diffusion problem , 1987 .

[7]  Mehdi Dehghan.  Determination of a control function in three-dimensional parabolic equations , 2003, Math. Comput. Simul..

[8]  M. A. Abdou,et al.  The solution of nonlinear coagulation problem with mass loss , 2006 .

[9]  D. Kaya.  An application for the higher order modified KdV equation by decomposition method , 2005 .

[10]  A. A. Soliman,et al.  A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations , 2006 .

[11]  Ji-Huan He,et al.  Construction of solitary solution and compacton-like solution by variational iteration method , 2006 .

[12]  Shaher Momani,et al.  Application of He’s variational iteration method to Helmholtz equation , 2006 .

[13]  Suheil A. Khuri,et al.  A new approach to Bratu's problem , 2004, Appl. Math. Comput..

[14]  A. A. Soliman,et al.  Variational iteration method for solving Burger's and coupled Burger's equations , 2005 .

[15]  Ji-Huan He.  Approximate solution of nonlinear differential equations with convolution product nonlinearities , 1998 .

[16]  Mehdi Dehghan,et al.  Identification of a time‐dependent coefficient in a partial differential equation subject to an extra measurement , 2005 .

[17]  Mehdi Dehghan,et al.  An inverse problem of finding a source parameter in a semilinear parabolic equation , 2001 .

[18]  Ji-Huan He.  Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parametrized perturbation technique , 1999 .

[19]  H. Sekine,et al.  General Use of the Lagrange Multiplier in Nonlinear Mathematical Physics1 , 1980 .

[20]  Yanping Lin,et al.  Determination of parameter p(t) in Holder classes for some semilinear parabolic equations , 1988 .

[21]  Ji-Huan He,et al.  Variational iteration method for delay differential equations , 1997 .

[22]  Mehdi Dehghan.  The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure , 2004, Int. J. Comput. Math..

[23]  Ji-Huan He,et al.  A simple perturbation approach to Blasius equation , 2003, Appl. Math. Comput..

[24]  A. A. Soliman,et al.  New applications of variational iteration method , 2005 .

[25]  Mehdi Dehghan,et al.  Numerical solution of one-dimensional parabolic inverse problem , 2003, Appl. Math. Comput..

[26]  Ji-Huan He.  Variational approach to the Thomas-Fermi equation , 2003, Appl. Math. Comput..

[27]  M. A. Abdou,et al.  Nonlinear fluid flows in pipe-like domain problem using variational-iteration method , 2007 .

[28]  Y. Fung,et al.  Variational Methods in the Mechanics of Solids , 1982 .

[29]  Ji-Huan He,et al.  Variational theory for one-dimensional longitudinal beam dynamics , 2006 .

[30]  Mehdi Dehghan,et al.  Parameter determination in a partial differential equation from the overspecified data , 2005, Math. Comput. Model..

[31]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[32]  William Alan Day,et al.  Extensions of a property of the heat equation to linear thermoelasticity and other theories , 1982 .