The OSC solver for the fourth-order sub-diffusion equation with weakly singular solutions

Abstract

A high-order method based on orthogonal spline collocation (OSC) method is formulated for the solution of the fourth-order subdiffusion problem on the rectangle domain in 2D with sides parallel to the coordinate axes, whose solutions display a typical weak singularity at the initial time. By introducing an auxiliary variable v=Δu, the fourth-order problem is reduced into a couple of second-order system. The L1 scheme on graded mesh is considered for the Caputo fractional derivatives of order α∈(0,1) by inserting more grid points near the initial time. By virtue of some properties, such as complementary discrete convolution kernel and discrete fractional Grönwall inequality, we establish unconditional stability and convergence for the original unknown u and auxiliary variable v. Some numerical experiments are provided to further verify our theoretical analysis.