Two reliable wavelet methods to Fitzhugh-Nagumo (FN) and fractional FN equations. (English) Zbl 1286.65131
Summary: Fractional reaction-diffusion equations serve as more relevant models for studying complex patterns in several fields of nonlinear sciences. In this paper, we have developed the wavelet methods to find the approximate solutions for the Fitzhugh-Nagumo (FN) and fractional FN equations. The proposed method techniques provide the solutions in rapid convergence series with computable terms. To the best of our knowledge, until now there is no rigorous wavelet solutions have been reported for the FN and fractional FN equations arising in gene propagation and model. With the help of Laplace operator and Legendre wavelets operational matrices, the FN equation is converted into an algebraic system. Finally, we have given some numerical examples to demonstrate the validity and applicability of the wavelet methods. The power of the manageable method is confirmed. Moreover, the use of the wavelet methods is found to be accurate, efficient, simple, low computation costs and computationally attractive.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 35K57 | Reaction-diffusion equations |
| 26A33 | Fractional derivatives and integrals |
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 35R11 | Fractional partial differential equations |
Keywords:
Fitzhugh-Nagumo equations; fractional FN equation; Haar wavelets; Laplace transform method; Legendre wavelets; operational matrices; homotopy analysis methodReferences:
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