Explicit scheme for solving variable-order time-fractional initial boundary value problems.

Area of Science:

• Numerical Analysis
• Fractional Calculus
• Applied Mathematics

Background:

• Fractional calculus is crucial for modeling systems with memory and hereditary properties.
• Variable-order temporal fractional derivatives offer a more flexible approach to modeling complex phenomena.
• Initial boundary value problems (IBVPs) are fundamental in many scientific and engineering disciplines.

Purpose of the Study:

• To develop an explicit finite difference scheme for linear and semi-linear IBVPs with variable-order temporal fractional derivatives.
• To establish the stability of the proposed numerical scheme.
• To demonstrate the scheme's efficacy through numerical examples.

Main Methods:

• Development of an explicit finite difference scheme.
• Utilization of the Caputo fractional derivative.
• Fourier stability analysis of the numerical scheme.
• Numerical simulations using MATLAB.

Main Results:

• The explicit finite difference scheme effectively resolves the targeted fractional IBVPs.
• Fourier analysis confirms the conditional stability of the scheme.
• Numerical examples validate the accuracy and applicability of the method.

Conclusions:

• The proposed explicit finite difference scheme is a viable tool for solving variable-order fractional differential equations.
• The Caputo derivative is well-suited for capturing memory effects in these problems.
• The study provides a foundation for further research in numerical methods for fractional calculus.