Approximate Collocation Solution for the Time-Fractional Newell-Whitehead-Segel Equation

Document Type : Research Paper

Authors
Ahmed Gamal Atta 1
Waleed Mohamed Abd-Elhameed 2
Youssri Hassan Youssri 2
1 Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt
2 Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
10.22055/jacm.2024.47269.4686
Abstract
This study offers a novel method for solving the partial differential equation known as the time-fractional Newell-Whitehead-Segel equation (TFNWSE), which has important applications in several different domains. We use a linear combination of certain basis functions that fulfill the homogeneous boundary and initial conditions to expand the exact solution in both temporal and spatial variables. An analytical framework is made possible by the combination of fourth-kind Chebyshev polynomials (CP4K) that make up the selected basis functions. By rigorous mathematical derivations, we obtain explicit forms of all required derivatives (both fractional and integer derivatives involving hypergeometric functions). We then discretize the differential equation using the spectral collocation approach, which reduces it to an algebraic system of equations that can be solved numerically. A thorough error analysis is carried out to evaluate our suggested method’s accuracy and dependability. We exhibit the applicability and efficiency of our method in solving the TFNWSE through four thorough numerical examples, supported by tables and figures, highlighting its potential for useful application in many real-world circumstances.
Keywords
Chebyshev polynomials of the fourth kind
Collocation method
Error analysis
Fractional Newell-Whitehead-Segel equation
Subjects

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Journal of Applied and Computational Mechanics
Volume 11, Issue 2
Spring 2025
Pages 529-540