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Two and Three Dimensional Gelfand Equation

In: Homotopy Analysis Method in Nonlinear Differential Equations

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  • Shijun Liao

    (Shanghai Jiao Tong University)

Abstract

Using the two-dimensional (2D) and 3D Gelfand equation as an example, we illustrate that the homotopy analysis method (HAM) can be used to solve a 2nd-order nonlinear partial differential equation (PDE) in a rather easy way by transforming it into an infinite number of the 4th or 6th-order linear PDEs. This is mainly because the HAM provides us extremely large freedom to choose auxiliary linear operator and besides a convenient way to guarantee the convergence of solution series. To the best of our knowledge, such kind of transformation has never been used by other analytic/numerical methods. This illustrates the originality and great flexibility of the HAM for strongly nonlinear problems. It also suggests that we must keep an open mind, since we might have much larger freedom to solve nonlinear problems than we thought traditionally.

Suggested Citation

  • Shijun Liao, 2012. "Two and Three Dimensional Gelfand Equation," Springer Books, in: Homotopy Analysis Method in Nonlinear Differential Equations, chapter 0, pages 461-491, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-25132-0_14
    DOI: 10.1007/978-3-642-25132-0_14
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