Browsing by Author "Gospodinov I.D."

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    Replacing the finite difference methods for nonlinear two-point boundary value problems by successive application of the linear shooting method
    (2019-10-01) Filipov S.M.; Gospodinov I.D.; Faragó I.
    This paper studies numerical solution of nonlinear two-point boundary value problems for second order ordinary differential equations. First, it establishes a connection between the finite difference method and the quasi-linearization method. We prove that using finite differences to discretize the sequence of linear differential equations arising from quasi-linearization (Newton method on operator level) leads to the usual iteration formula of the Newton finite difference method. From the provided derivation, it can easily be inferred that such a relation holds also for the Picard and the constant-slope methods. Based on this result, we propose a way of replacing the Newton, Picard, and constant-slope finite difference methods by respective successive application of the linear shooting method. This approach has a number of advantages. It removes the necessity of solving systems of algebraic equations, hence working with matrices, altogether. Compared to the usual finite difference method with general solver, it reduces the number of computational operations from O(N 3 ), where N is the number of mesh-points, to only O(N).
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    Shooting-projection method for two-point boundary value problems
    (2017-10-01) Filipov S.M.; Gospodinov I.D.; Faragó I.
    This paper presents a novel shooting method for solving two-point boundary value problems for second order ordinary differential equations. The method works as follows: first, a guess for the initial condition is made and an integration of the differential equation is performed to obtain an initial value problem solution; then, the end value of the solution is used in a simple iteration formula to correct the initial condition; the process is repeated until the second boundary condition is satisfied. The iteration formula is derived utilizing an auxiliary function that satisfies both boundary conditions and minimizes the H1 semi-norm of the difference between itself and the initial value problem solution.