Multiple Positive Solutions of Nonlinear Two-Point Boundary Value Problems

Solution of nonlinear boundary value problems — I method of third order convergence for solution of nonlinear two-point boundary value problems single second order equation

B-spline collocation for solution of two-point boundary value problems

The boundary value problems of ordinary differential equations play an important role in many fields. Here, we implement the He-Laplace method for the solution of linear and nonlinear two-point boundary value problems. The aim of this paper is to compare the performance of the He-Laplace method with shooting method. As a result, for the same number of terms, our method provides relatively more accurate results with rapid convergence than other methods.

The global warming and the environmental destruction are caused from the mass consumption of energy in a factory in which robotic manipulators are used. It is necessary to take the trajectory planning into considerations in order to save the consumption energy of the manipulator controlled with Point-to-Point (PTP) motion. An optimal trajectory that minimizes the consumption energy can be obtained by solving a two-point boundary value problem. The optimal trajectory can be analyzed easily if the system is linear. However, the dynamical equation of the multi-link manipulator with Coulomb friction has a strong non-linearity. Then the Genetic Algorithm (GA) is applied to obtain the solution of two-point boundary value problem with the strong non-linearity. The GA is the method for searching discrete solution space by genes. However, the trajectory function is continuous with respect to time. In this paper, the coefficients of the approximation functions are determined as the gene’s phenotype. This method has disadvantages that are long computation time and increase the possibility of lapsing into the local optimal solution. So, a new method is proposed in order to shorten the computation time and to lessen the possibility of lapsing into the local optimal solution. It is seen from application to the minimization of the consumption energy of a manipulator with Coulomb friction that a proposed method is effective.

Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu’s equation, Troesch’s problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.

Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations

In this paper, the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method together with Newton scheme namely Newton-HSSOR is investigated in solving the nonlinear systems generated from the fourth-order half-sweep finite difference approximation equation for nonlinear two-point boundary value problems. The Newton scheme is proposed to linearize the nonlinear system into the form of linear system. On top of that, we also present the basic formulation and implementation of Newton-HSSOR iterative method. For comparison purpose, combinations between the Full-Sweep Gauss-Seidel (FSGS) and Full-Sweep Successive Over-Relaxation (FSSOR) iterative methods with Newton scheme, which are indicated as Newton-FSGS and Newton-FSSOR methods respectively have been implemented numerically. Numerical experiments of two problems are given to illustrate that the Newton-HSSOR method is more superior compared with the tested methods.

High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form

A collocation method for solving a class of singular nonlinear two-point boundary value problems

Positive solutions of nonlinear singular third-order two-point boundary value problem

In the present work, we propose a two-phase fourth-order method for the approximate numerical solution for second-order non-linear two-point boundary value problems with Dirichlet boundary conditions. Our numerical approach is based on a finite difference and the solution of the problem at discrete points. Our method generates a system of equations, and the solution of the system of equations is considered an approximate solution to the problem. An essential analysis of the method is considered to ensure the performance of the method. A numerical experiment is carried out with model problems to test the performance in terms of efficiency and accuracy of the proposed method.

Numerical solution of two-point boundary value problems

Positive solutions to a multi-point boundary value problem with delay

In this paper we investigate the existence of positive solution of the following nonlinear discrete third-order two-point boundary value problem. whereis continuous and there existssuch that . Our approach relies on the Krasnosel'skii fixed point theorem. An example is given to demonstrate the application of the theorem obtained.

General nonlinear problems in the abstract form F ( χ ) = 0 F(\chi ) = 0 and corresponding families of approximating problems in the form F h ( χ h ) = 0 {F_h}({\chi _h}) = 0 are considered (in an appropriate Banach space setting). The relation between "isolation" and "stability" of solutions is briefly studied. The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximating problem has a stable solution which is close to the exact solution. Error estimates are obtained and Newton’s method is shown to converge quadratically. These results are then used to justify a broad class of difference schemes (resembling linear multistep methods) for general nonlinear two-point boundary value problems.