High order approximation on non-uniform meshes for generalized time-fractional telegraph equation

Area of Science:

• Numerical analysis
• Fractional calculus
• Partial differential equations

Background:

• The generalized fractional derivative (GFD) introduces scale and weight functions, impacting solution behavior.
• Understanding the GFTE is crucial for modeling various physical phenomena.

Purpose of the Study:

• To develop and analyze a high-order approximation scheme for the GFTE.
• To investigate the influence of scale and weight functions in GFD on GFTE solutions.
• To establish the stability and convergence properties of the proposed numerical scheme.

Main Methods:

• A high-order approximation scheme combining quadratic temporal discretization and compact finite difference spatial discretization.
• Utilization of non-uniform meshes to enhance numerical accuracy.
• Derivation of error estimates for GFD approximation on non-uniform meshes.
• Analysis of scheme stability and convergence.

Main Results:

• The developed numerical scheme achieves a convergence order of O(τ^2 + h^2).
• Error estimates for GFD approximation on non-uniform meshes were established.
• The stability and convergence of the scheme were rigorously examined.
• Numerical results demonstrated superior accuracy compared to existing methods.

Conclusions:

• The proposed high-order scheme provides an accurate and stable method for solving the GFTE.
• The use of non-uniform meshes and the specific discretization schemes contribute to improved accuracy.
• The findings offer a valuable tool for researchers working with fractional differential equations.