Galerkin-finite difference method for fractional parabolic partial differential equations.

Area of Science:

• Numerical Analysis
• Fractional Calculus
• Partial Differential Equations

Background:

• Fractional diffusion equations model anomalous transport phenomena.
• Existing methods for fractional PDEs have limitations in efficiency and accuracy.
• Super-diffusion and sub-diffusion behaviors are captured by fractional orders.

Purpose of the Study:

• To develop and analyze a hybrid numerical method for solving time and space fractional parabolic partial differential equations.
• To apply the proposed method to the fractional Black-Scholes model and other numerical examples.
• To assess the accuracy and convergence of the new technique.

Main Methods:

• A combined approach using the Galerkin weighted residual method with modified Bernoulli polynomials for spatial discretization.
• Finite difference approximation for the temporal fractional derivative.
• Convergence analysis for the temporal component of the method.

Main Results:

• The proposed hybrid method effectively approximates solutions for fractional parabolic PDEs.
• Numerical experiments, including the fractional Black-Scholes model, demonstrate considerable accuracy.
• Results are presented using tabular data and 3D visualizations for clarity.

Conclusions:

• The integrated Galerkin and finite difference method provides a robust and accurate approach for fractional PDEs.
• The technique shows significant potential for applications in finance and physics.
• The study validates the method's efficacy through comparative analysis and visualization.