Vortrag: "Numerical Simulations for Time-Fractional Diffusion Equations using Exponential B-Splines"

This work focuses on the numerical solution of the Caputo time-fractional diffusion equation with weak initial singularity. To handle the weak singularity of the temporal derivative near t = 0, we utilize the L1 scheme on the graded mesh. The spatial derivatives are discretized using a collocation method based on exponential B-spline functions. The exponential B-spline is a very powerful numerical tool, composed of twice continuously differentiable functions. Although the spline methods are well known, the exponential splines are more general splines that define interpolants ranging from cubic splines to linear splines. The unconditional stability and convergence analysis of the proposed scheme are developed. The theoretical analysis is validated by performing numerical experiments for smooth and non-smooth solutions. The advantage of using graded mesh over a uniform mesh is shown by comparing the results for non-smooth solutions.