We show that nonlinear Liouville theorems does not hold in general for indefinite problems on half spaces. Thus, in order to use blow-up method to obtain a priori estimates of indefinite elliptic equations, one has to impose assumptions on the nodal set of nonlinearity. The counter example is constructed by shooting method in one-dimensional case and then extended to higher dimensions.

Assumption 2 (Smoothness of the control problem) We assume that the function f : Rnx×nq 7→ Rx is sufficiently smooth. Note that Lipschitz continuity guarantees the unique existence of a solution x(·) of the differential equations for fixed q by virtue of the Picard-Lindelöf theorem. As we assume unique sensitivities in the following, also the partial derivative functions ∂f ∂x , ∂f ∂q , ∂f ∂x2 ...

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...

In this paper we give a mathematical proof of the existence of the time independent and spherically symmetric solution to the ’t HooftPolyakov model of magnetic monopole by using 2D-shooting method. ∗Current address: Department of Mathematics, University of Californis, Davis, CA95616. e-mail: [email protected].

The difficulties with the simple shooting method for the solution of ordinary differential boundary value problems are well known, and there have been many suggested modifications to overcome them. Notable among them are the discrete orthonormalization process of Godunov and Conte (1966), the multiple shooting method (Keller(1968) and Osborne(1969)) and the Riccati transformation method (Scott(...

αu(0)−β u(0) = 0, γu(1)+ δu(1) = 0, (2) where α,β ,γ,δ ≥ 0,λ ∈ R and ρ := γβ + αγ + αδ > 0; p ∈ C((0,1), [0,∞)) and may be singular at t = 0,t = 1. When λ < 0, see [3,4,7,8] for the result concerning the above problem. When λ > 0, [6] shows the existence and uniqueness to (1) and (2) in the case of β = δ = 0 by means of the shooting method. For the following problem u + p(t)u−λ(t)+ q(t)u(t) = 0...

In this work, we solve multipoint boundary value problems where the boundary value conditions are equations using the Newton-Broyden Shooting method (NBSM).The proposed method is tested upon several problems from the literature and the results are compared with the available exact solution. The experiments are given to illustrate the efficiency and implementation of the method. Keywords—Boundar...

We propose a sequential quadratic programming (SQP) method for the optimal control of large-scale dynamical systems. The method uses modified multiple shooting to discretize the dynamical constraints. When these systems have relatively few parameters, the computational complexity of the modified method is much less than that of standard multiple shooting. Moreover, the proposed method is demons...

In this thesis we prove the existence of m-armed spiral wave solutions for the complex Ginzburg-Landau equation in the circular and spherical geometries. Instead of applying the shooting method in the literature, we establish a functional approach and generalize the known results of existence for rigidly-rotating spiral waves. Moreover, we prove the existence of two new patterns: frozen spirals...