A numerical algorithm for nonlinear multi-point boundary value problems

AbstractIn this paper, an algorithm is presented for solving second-order nonlinear multi-point boundary value problems (BVPs). The method is based on an iterative technique and the reproducing kernel method (RKM). Two numerical examples are provided to show the reliability and efficiency of the present method

Shape deformation analysis from the optimal control viewpoint

A crucial problem in shape deformation analysis is to determine a deformation of a given shape into another one, which is optimal for a certain cost. It has a number of applications in particular in medical imaging. In this article we provide a new general approach to shape deformation analysis, within the framework of optimal control theory, in which a deformation is represented as the flow of diffeomorphisms generated by time-dependent vector fields. Using reproducing kernel Hilbert spaces of vector fields, the general shape deformation analysis problem is specified as an infinite-dimensional optimal control problem with state and control constraints. In this problem, the states are diffeomorphisms and the controls are vector fields, both of them being subject to some constraints. The functional to be minimized is the sum of a first term defined as geometric norm of the control (kinetic energy of the deformation) and of a data attachment term providing a geometric distance to the target shape. This point of view has several advantages. First, it allows one to model general constrained shape analysis problems, which opens new issues in this field. Second, using an extension of the Pontryagin maximum principle, one can characterize the optimal solutions of the shape deformation problem in a very general way as the solutions of constrained geodesic equations. Finally, recasting general algorithms of optimal control into shape analysis yields new efficient numerical methods in shape deformation analysis. Overall, the optimal control point of view unifies and generalizes different theoretical and numerical approaches to shape deformation problems, and also allows us to design new approaches. The optimal control problems that result from this construction are infinite dimensional and involve some constraints, and thus are nonstandard. In this article we also provide a rigorous and complete analysis of the infinite-dimensional shape space problem with constraints and of its finite-dimensional approximations

Picard-Reproducing Kernel Hilbert Space Method for Solving Generalized Singular Nonlinear Lane-Emden Type Equations

An iterative method is discussed with respect to its effectiveness and capability of solving singular nonlinear Lane-Emden type equations using reproducing kernel Hilbert space method combined with the Picard iteration. Some new error estimates for application of the method are established. We prove the convergence of the combined method. The numerical examples demonstrates a good agreement between numerical results and analytical predictions

Numerical investigation of Differential Biological-Models via GA-Kansa Method Inclusive Genetic Strategy

In this paper, we use Kansa method for solving the system of differential equations in the area of biology. One of the challenges in Kansa method is picking out an optimum value for Shape parameter in Radial Basis Function to achieve the best result of the method because there are not any available analytical approaches for obtaining optimum Shape parameter. For this reason, we design a genetic algorithm to detect a close optimum Shape parameter. The experimental results show that this strategy is efficient in the systems of differential models in biology such as HIV and Influenza. Furthermore, we prove that using Pseudo-Combination formula for crossover in genetic strategy leads to convergence in the nearly best selection of Shape parameter.Comment: 42 figures, 23 page

Performance of modified non-linear shooting method for simulation of 2nd order two-point BVPS

In this research article, numerical solution of nonlinear 2nd order two-point boundary value problems (TPBVPs) is discussed by the help of nonlinear shooting method (NLSM), and through the modified nonlinear shooting method (MNLSM). In MNLSM, fourth order Runge-Kutta method for systems is replaced by Adams Bashforth Moulton method which is a predictor-corrector scheme. Results acquired numerically through NLSM and MNLSM of TPBVPs are discussed and analyzed. Results of the tested problems obtained numerically indicate that the performance of MNLSM is rapid and provided desirable results of TPBVPs, meanwhile MNLSM required less time to implement as comparable to the NLSM for the solution of TPBVPs

New numerical scheme for solving Troesch’s Problem

In this paper, we will manipulate the cubic spline to develop a collocation method (CSCM) and the generalized Newton method for solving the nonlinear Troesch problem. This method converges quadratically if a relation-ship between the physical parameter and the discretization parameter is satisfied. An error estimate between the exact solution and the discret solution is provided. To validate the theoretical results, Numerical results are presented and compared with other collocation methods given in the literature. Keywords: Troesch problem, Boundary value problems, Cubic spline collocation method