Efficient simulation of Time-Fractional Korteweg-de Vries equation via conformable-Caputo non-Polynomial spline...

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Fractional Calculus

Background:

• The time-fractional Korteweg-de Vries (KdV) equation models various phenomena, including shallow water waves.
• Existing numerical methods often face challenges with accuracy and stability for fractional-order derivatives.
• There is a need for robust and efficient numerical techniques to solve these complex equations.

Purpose of the Study:

• To introduce a novel numerical method for solving the time-fractional KdV equation.
• To enhance precision and modeling capabilities for fractional differential equations.
• To establish the stability and accuracy of the proposed method.

Main Methods:

• Development of a conformable-Caputo fractional non-polynomial spline method.
• Application of the Von Neumann stability analysis.
• Comparative analysis with existing numerical techniques using graphical and error norm assessments.

Main Results:

• The proposed method exhibits unconditional stability under specific parameters.
• Quantitative evaluation using L2 and L∞ error norms confirms the method's superiority.
• Graphical representations (contour, 2D/3D) validate the accuracy and efficiency compared to other approaches.

Conclusions:

• The conformable-Caputo fractional non-polynomial spline method provides a robust and accurate solution for the time-fractional KdV equation.
• The study validates the method's effectiveness through rigorous numerical analysis and comparative studies.
• This novel approach advances the numerical treatment of fractional differential equations.