Singularity-removing Chebyshev collocation methods for nonlinear fractional differential equations with blow-up

Two singularity-removing Chebyshev collocation methods are proposed for discretizing nonlinear fractional differential equation with singular solution at the initial and terminal points. By reformulating the problem into the Volterra integral equation of the second kind, we obtain the fractional series solution around the origin via successive approximation, which depicts the initial singularity of the unknown solution accurately. By removing the singularity at the origin, we design two Chebyshev collocation methods to solve the equivalent Volterra integral equation and Hadamard finite-part integral equation on a regular interval. The convergence of the scheme for the equivalent Volterra integral form is proved. For blow-up problem, we perform the Padé technique for the obtained series solution to detect the blow-up time. We also explore the blow-up behavior of the solution for three typical types of nonlinear term. By removing the dominant term of the blow-up behavior from the equation, the accuracy of the collocation methods is improved significantly. Finally, numerical examples demonstrate the high efficiency and exponential convergence of the singularity-removing Chebyshev collocation methods.