Mixed Collocation for Fractional Differential Equations

We present the mixed collocation method for numerical integration of fractional differential equations of the type Dβu=Φ(u,t). Given a regular mesh with constant discretization step, the unknown u(t) is considered as continuous and affine in each cell, and the dynamics Φ(u,t) as a constant. After a fractional integration, the equation is written strongly at the mesh vertices and the dynamics weakly in each cell. The “Semidif” software has been developed for the particular case of numerical integration of order 1/2. The validation for analytical results and published solutions is established and experimental convergence as the mesh size tends to zero is obtained. Good results are obtained for a nonlinear model with a strong singularity.