In this paper, a new analytical method, which is based on the fixed point concept in functional analysis (namely, the fixed point analytical method (FPM)), is proposed to acquire the explicit analytical solutions of nonlinear differential equations. The key idea of this approach is to construct a contractive map which replaces the nonlinear differential equation into a series of linear differential equations. Usually, the series of linear equations can be solved relatively easily and have explicit analytical solutions. The FPM is different from all existing analytical methods, such as the well-known perturbation technique applied in weakly nonlinear problems, because it is independent of any small physical or artificial parameters at all; thus, it can handle more nonlinear problems, including strongly nonlinear ones. Two typical cases are investigated by FPM in detail and the comparison with the numerical results shows that the present method is one of high accuracy and efficiency.
Fixed Point Analytical Method for Nonlinear Differential Equations
Coal Technology & Engineering Group Corp.,
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 26, 2011; final manuscript received March 3, 2012; published online June 14, 2012. Assoc. Editor: J. A. Tenreiro Machado.
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Xu, D., and Guo, X. (June 14, 2012). "Fixed Point Analytical Method for Nonlinear Differential Equations." ASME. J. Comput. Nonlinear Dynam. January 2013; 8(1): 011005. https://doi.org/10.1115/1.4006337
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