Abstract
In this paper, a wavelet-based approximation method is introduced for solving the Newell–Whitehead (NW) and Allen–Cahn (AC) equations. To the best of our knowledge, until now there is no rigorous Legendre wavelets solution has been reported for the NW and AC equations. The highest derivative in the differential equation is expanded into Legendre series, this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the aforesaid equations are converted into an algebraic system. Block pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed methods is proved. Finally, we have given some numerical examples to demonstrate the validity and applicability of the method.
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Acknowledgments
This work was supported by the Naval Research Board (DNRD/05/4003/NRB/322), Government of India. The author is very grateful to the referees for their valuable suggestions. Our hearty thanks are due to Prof. R. Sethuraman, Vice-Chancellor, SASTRA University, Dr. S. Vaidhyasubramaniam, Dean/Planning and development and Dr. S. Swaminathan, Dean/Sponsored research for their kind encouragement and for providing good research environment.
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Hariharan, G. An Efficient Legendre Wavelet-Based Approximation Method for a Few Newell–Whitehead and Allen–Cahn Equations. J Membrane Biol 247, 371–380 (2014). https://doi.org/10.1007/s00232-014-9638-z
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DOI: https://doi.org/10.1007/s00232-014-9638-z