The Mixed Boundary Value Problems and Chebyshev Collocation Method for Caputo-Type Fractional Ordinary Differential Equations

The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval 0 <= x <= 1 was considered. First, the BVP was converted into an equivalent differential-integral equation merging the boundary conditions. Then, the shifted Chebyshev polynomials and the collocation method were used to solve the differential-integral equation. Varying coefficients were also decomposed into the truncated shifted Chebyshev series such that calculations of integrals were only for polynomials and can be carried out exactly. Finally, numerical examples were examined and effectiveness of the proposed method was verified.

Automated summary

This study focuses on the boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to mixed boundary conditions on the interval 0 <= x <= 1. The BVP is transformed into an equivalent differential-integral equation by merging the boundary conditions. The shifted Chebyshev polynomials and the collocation method are employed to solve the differential-integral equation. The varying coefficients are decomposed into truncated shifted Chebyshev series, allowing accurate calculations of integrals. Numerical examples are provided to verify the effectiveness of the proposed method.