Numerical Analysis of the Quadratic Spline Collocation Method and Averaged <i>L</i>1 Scheme on Graded Mesh for Nonlinear Time‐Fractional Fourth‐Order Reaction‐Diffusion Problem

ABSTRACT In this paper, we consider a time‐fractional fourth‐order reaction‐diffusion equation with a nonlinear reaction term in the two‐dimensional spatial domain, and combine the quadratic spline collocation (QSC) method and the formula to propose a QSC‐ scheme, where the graded mesh in time is employed to reduce the effect of initial time singularity, and the original problem with spatial fourth‐order derivative is changed into a second‐order coupled system to avoid the use of high‐order interpolation. The main novelties in numerical analysis lie in proposing an energy method to construct a recursive sequence of an innovatively defined function on the graded mesh, as well as its combination with the extrapolation method. With some proper assumptions on the regularity parameter of the solution, we prove that the QSC‐ scheme is uniquely solvable and convergent with order , where , and are spatial step size and number of time steps, respectively, and is temporal mesh grading parameter. Numerical experiments are attached to support the theoretical analysis.