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Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. (English) Zbl 1120.65108
Summary: Perturbation methods depend on a small parameter which is difficult to be found for real-life nonlinear problems. To overcome this shortcoming, two new but powerful analytical methods are introduced to solve nonlinear heat transfer problems in this article; one is J.-H. He’s variational iteration method [(VIM); Int. J. Non-Linear Mech. 34, No. 4, 699–708 (1999; Zbl 05137891)] and the other is J.-H. He’s homotopy-perturbation method [(HPM); Comput. Methods Appl. Mech. Eng. 178, No. 3–4, 257–262 (1999; Zbl 0956.70017)]. The VIM is to construct correction functionals using general Lagrange multipliers identified optimally via the variational theory, and the initial approximations can be freely chosen with unknown constants. The HPM deforms a difficult problem into a simple problem which can be easily solved. Nonlinear convective-radiative cooling equation, nonlinear heat equation (porous media equation) and nonlinear heat equation with cubic nonlinearity are used as examples to illustrate the simple solution procedures. Comparison of the applied methods with exact solutions reveals that both methods are tremendously effective.
MSC:
65M70Spectral, collocation and related methods (IVP of PDE)
35K55Nonlinear parabolic equations