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In this paper, we present a reliable algorithm to calculate positive solutions of homogeneous nonlinear boundary value problems (BVPs). The algorithm converts the nonlinear BVP to an equivalent nonlinear Fredholm– Volterra integral equation.We employ the multistage Adomian decomposition method for BVPs on two or more subintervals of the domain of validity, and then solve the matching equation for the flux at the interior point, or interior points, to determine the solution. Several numerical examples are used to highlight the effectiveness of the proposed scheme to interpolate the interior values of the solution between boundary points. Furthermore we demonstrate two novel techniques to accelerate the rate of convergence of our decomposition series solutions by increasing the number of subintervals and adjusting the lengths of subintervals in the multistage Adomian decomposition method for BVPs.

A reliable algorithm for positive solutions of nonlinear boundary value problems by the multistage Adomian decomposition method