Robust numerical techniques for solving nonlinear ordinary and fractional partial differential problems

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Abstract

This manuscript presents novel spectral techniques based on shifted orthonormal normalized Jacobi polynomials (SONJPs) for solving two types of fractional differential equations. The first method uses the tau technique to solve the nonlinear boundary value problem (NBVP), and the second method uses the collocation approach to solve the time-fractional generalized Fisher problem (TFGFP) that has importance in mathematical biology, population dynamics, and economic modeling. We derive new relations that form the foundation for the proposed numerical techniques, thereby reducing the problem and its governing conditions to a nonlinear algebraic system that can be solved efficiently using Newton’s method or another suitable iterative technique. We present numerical results with comparisons to illustrate the suggested algorithm’s high accuracy and applicability.

Keywords

Jacobi polynomials / spectral methods / nonlinear boundary value problems / time-fractional generalized Fisher equation

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