Exponentially fitted non-polynomial cubic spline method for time-fractional singularly perturbed convection-diffusion...

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Applied Mathematics

Background:

• Singularly perturbed convection-diffusion problems are crucial in modeling various physical phenomena.
• These problems often exhibit boundary layers and require specialized numerical techniques for accurate solutions.
• Large temporal lags introduce additional complexity, necessitating advanced computational approaches.

Purpose of the Study:

• To develop and analyze an exponentially fitted non-polynomial cubic spline method.
• To address time-fractional singularly perturbed convection-diffusion problems with significant temporal lags.
• To establish the parameter-uniform convergence properties of the proposed numerical scheme.

Main Methods:

• The time-fractional derivative is discretized using the backward Euler method.
• A non-polynomial cubic spline scheme is constructed for spatial discretization on a uniform mesh.
• An exponential fitting factor is incorporated to handle the perturbation parameter effectively.

Main Results:

• The proposed method is rigorously proven to be parameter-uniform convergent.
• The scheme achieves an order of convergence of O(Δt + h^2).
• Numerical experiments validate the theoretical findings and demonstrate superior accuracy compared to existing methods.

Conclusions:

• The exponentially fitted non-polynomial cubic spline method provides an accurate and efficient approach for solving time-fractional singularly perturbed convection-diffusion problems with large temporal lags.
• The method exhibits excellent convergence properties, independent of the perturbation parameter.
• This work contributes a valuable tool for the numerical simulation of complex diffusion-convection phenomena.