Explicit Chebyshev Collocation Method for Multi-Order Fractional‎ ‎Nonlinear‎ ‎Boundary‎ ‎Value‎ ‎Problems‎ ‎in Mathematical Chemistry

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Faculty of Science‎, ‎Cairo University‎, ‎Giza 12613‎, ‎Egypt

2 Department of Mathematics‎, ‎Faculty of Education‎, ‎Ain Shams University‎, ‎Roxy‎, ‎Cairo 11341‎, ‎Egypt

10.22052/ijmc.2025.256602.1997

Abstract

‎This paper presents a numerical method for solving a class of nonlinear multi-order fractional differential equations using the first-kind Chebyshev polynomials‎. ‎The proposed approach is based on a collocation framework that incorporates operational matrices of derivatives specifically tailored to the spectral properties of the Chebyshev polynomials on the interval $[0,1]$‎. ‎Two cases of interest are considered‎: ‎the classical case with $\nu = 2$ and $\lambda = 1$‎, ‎and the fractional-order case with $1 < \nu \leq 2$ and $0 < \lambda \leq 1$‎. ‎To ensure high accuracy‎, ‎an appropriate set of the shifted Chebyshev basis functions that satisfy the boundary conditions is utilized‎. ‎The Caputo definition of fractional derivatives is adopted to handle the fractional operators‎. ‎The resulting nonlinear algebraic system is solved efficiently using Newton’s method‎. ‎Numerical experiments confirm the proposed method’s efficiency‎, ‎stability‎, ‎and accuracy in comparison with existing techniques‎.

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Main Subjects


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