Efficient higher-order approximations for a nonlinear time-fractional biharmonic equation with initial singularity

This paper introduces a high-order efficient algorithm for approximating a nonlinear time-fractional biharmonic equation with an initial singularity. The Caputo fractional derivative is employed, and a second-order scheme is developed to discretize the time derivative on nonuniform time steps, effectively addressing the initial singularity. For the spatial derivative, a high-order non-polynomial parametric quintic spline method is considered. The proposed approach efficiently handles the initial singularity and reduces computational cost through a fast nonuniform time discretization scheme. The resulting method is computationally efficient, with a complexity of approximately $$\mathcal{O}(MN \log^2 N)$$ and storage requirements of $$\mathcal{O}(M \log^2 N)$$ , where $$N$$ and $$M$$ denote the total number of grid points in the time and spatial directions, respectively. Furthermore, the method is proven to be unconditionally stable and convergent, with an error of order $$\mathcal{O}\left(h^{4.5} + N^{-\min\{r\mu, 2\}}+\epsilon\right)$$ , where $$\mu$$ is the fractional derivative order, $$r$$ is the mesh grading parameter, $$h$$ is the spatial mesh size and $$\epsilon$$ represents the approximation tolerance introduced by the fast sum-of-exponentials technique. Numerical experiments are presented to validate the theoretical analysis and demonstrate the method’s effectiveness in achieving high accuracy and computational efficiency.