Efficient High-Accuracy Numerical Scheme for the Solution of Time Fractional Parabolic Partial Differential Equations With Application in Financial Modeling

Parabolic partial equations, particularly the Black–Scholes equation, are fundamental in mathematical finance for option pricing and risk management. Despite their widespread use, efficiently solving these equations remains a challenge, especially in complex financial scenarios. This paper presents a new numerical algorithm that combines shifted Chebyshev cardinal polynomials with a finite difference scheme to solve time-fractional parabolic partial equations. We first introduce a finite difference method to discretize the temporal variable, providing a detailed analysis of the stability and convergence properties. The shifted Chebyshev cardinal polynomials are employed at collocation points to accurately discretize spatial variables, enabling the precise representation of the unknown function. Extensive numerical experiments are conducted to evaluate the accuracy and robustness of the proposed method, demonstrating its effectiveness in solving the Black–Scholes model. The preeminent efficacy of the proposed methodology in comparison to other techniques is underscored via an exhaustive analysis of the numerical findings.