A modified scheme to the multiple shooting method for BVPs

A significant modification of the multiple shooting method for solving nth order boundary value problems (BVPs) is presented. Initially, the mathematical foundation of this modification is elucidated. The method discretizes the domain of the original problem into N subintervals to construct semi-analytical approximate solutions that are continuously differentiable up to the (n−1)-th order. The accuracy of the solution is improved by minimizing the L2-norm of the residual functions within each subinterval. The resulting piecewise continuous solution is represented by polynomials or basis functions, ensuring smoothness and derivative continuity at the junctions. The effectiveness of the proposed method is demonstrated through various test problems, including a ninth-order linear problem, a tenth-order nonlinear problem, and several stiff problems, with results showing superior performance compared to the standard shooting method and the modified decomposition method. Additionally, the empirical convergence orders are computed, and the computation times required for each test problem are reported, further highlighting the efficiency and accuracy of the approach.