A new spline technique for the time fractional diffusion-wave equation.

Area of Science:

• Numerical Analysis
• Applied Mathematics
• Partial Differential Equations

Background:

• The fractional diffusion wave equation (FDWE) models complex phenomena.
• Existing numerical methods often lack sufficient accuracy or efficiency.
• Accurate solutions are crucial for understanding wave propagation in fractional media.

Purpose of the Study:

• To propose a novel, high-order approximate solution for the FDWE.
• To analyze the theoretical error bounds of the proposed method.
• To demonstrate the superiority of the new method over existing techniques.

Main Methods:

• A new collocation method utilizing cubic B-splines is developed.
• The Caputo fractional derivative is employed for the time direction.
• Theoretical error analysis is performed using L∞ and H1 norms.

Main Results:

• The proposed method achieves a high order of convergence: O(Δt³ + h⁴).
• Error analysis confirms the method's reliability and accuracy.
• Numerical comparisons show superior accuracy compared to existing methods.

Conclusions:

• The cubic B-spline collocation method provides an accurate and efficient solution for the FDWE.
• The method's high order of convergence and compact stencil are significant advantages.
• This technique offers a valuable tool for researchers in fractional calculus and wave phenomena.