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An efficient Legendre wavelet-based approximation method for a few Newell-Whitehead and Allen-Cahn equations.
1Department of Mathematics, School of Humanities & Sciences, Sastra University, Thanjavur, 613401, Tamilnadu, India, hariharang2011@gmail.com.
The Journal of Membrane Biology
|March 7, 2014
Summary
This study introduces a novel Legendre wavelet method for solving the Newell-Whitehead and Allen-Cahn equations, offering a rigorous approach previously unavailable for these complex differential equations.
Area of Science:
- Numerical Analysis
- Applied Mathematics
- Computational Physics
Background:
- The Newell-Whitehead (NW) and Allen-Cahn (AC) equations are fundamental in modeling phenomena like pattern formation and phase transitions.
- Existing numerical methods may face challenges with accuracy and convergence for these nonlinear partial differential equations.
- A rigorous wavelet-based solution for NW and AC equations using Legendre wavelets has been lacking.
Purpose of the Study:
- To introduce and validate a novel wavelet-based approximation method for solving the NW and AC equations.
- To establish a rigorous Legendre wavelets solution for these specific differential equations.
- To demonstrate the method's applicability and accuracy through numerical examples.
Main Methods:
- The highest derivative of the NW and AC equations is approximated using Legendre series expansion.
- Integration of the approximated derivative is performed, incorporating boundary conditions via integration constants.
- Legendre wavelets operational matrices are employed to transform the differential equations into an algebraic system.
- Block pulse functions are utilized to handle the nonlinear terms within the wavelet framework.
Main Results:
- The proposed Legendre wavelet method successfully converts the NW and AC equations into a solvable algebraic system.
- Convergence analysis confirms the stability and reliability of the developed numerical technique.
- Numerical examples validate the accuracy and effectiveness of the wavelet-based approach.
Conclusions:
- The introduced Legendre wavelet approximation method provides a rigorous and effective tool for solving the Newell-Whitehead and Allen-Cahn equations.
- This work fills a gap in the literature by presenting the first reported rigorous Legendre wavelets solution for these equations.
- The method demonstrates significant potential for application in various scientific and engineering fields involving similar differential equations.
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