Solving Multi-Point Boundary Value Differential Equations with a Novel Semi-Analytical Algorithm

To address the general multi-point boundary value problems, we present a novel semi-analytical algorithm. This scheme is applicable to both linear and nonlinear fractional-order and integer-order differential equations under multi-point boundary value and/or initial value conditions. The proposed algorithm involves the introduction of auxiliary fractional equations derived from the main equation. Subsequently, a linear combination of the solutions to the auxiliary equations is formed to meet the boundary conditions of the main equation. The combination includes some unknown coefficients. In respect with these unknown coefficients, we optimize the relevant residual function to achieve its minimum value. This is done by using the method of weighted residuals. The convergence of the method is established and proved. The algorithm is employed to obtain approximate solutions for multi-point boundary and initial value problems. Additionally, it effectively addresses problems with solutions that exhibit weak singularities at the origin. Numerical results, including solving two bridge models, reveal that the proposed method demonstrates a substantial improvement in performance over existing methods.