Hybrid compact finite volume method for nonlinear two-point boundary value problem with logarithmic singularity

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h 6) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h 8) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.

In this paper, we discuss the numerical solution of second-order nonlinear two-point fuzzy boundary value problems (TPFBVP) by combining the finite difference method with Newton’s method. Numerical example using the well-known nonlinear TPFBVP is presented to show the capability of the new method in this regard and the results are satisfied the convex triangular fuzzy number. We also compare the numerical results with the exact solution, and it shows that the proposed method is good approximation for the analytic solution of the given TPFBVP.

In this paper, we report on a non-uniform mesh smart alternating group explicit (SMAGE) parallel algorithm for the solution of non-linear singular two-point boundary value (BV) problems. The proposed method requires three non-uniform grid points and is applicable to both singular and non-singular problems. We also discuss the Newton-SMAGE parallel algorithm for the non-linear difference equations. The error analysis of the method is discussed and numerical tests are performed to demonstrate the utility of the proposed SMAGE iterative methods.

We propose a third-order-accurate variable-mesh two-parameter alternating group explicit (TAGE) iteration method for the numerical solution of the two-point singular boundary value problem subject to boundary conditions u(0)=A, u(1)=B, where A and B are finite constants. We also discuss a Newton–TAGE iteration method for the third-order numerical solution of a two-point non-linear boundary value problem. The proposed method is applicable to singular and non-singular problems and is suitable for use on parallel computers. The convergence analysis is briefly discussed. Computational results are provided to illustrate the proposed TAGE iterative methods.

In this paper, from a Hamiltonian point of view, the nonlinear optimal control problems are transformed into nonlinear two-point boundary value problems, and a symplectic adaptive algorithm based on the dual variational principle is proposed for solving the nonlinear two-point boundary value problem. The state and the costate variables within a time interval are approximated by using the Lagrange polynomial and the costate variables at two ends of the time interval are taken as independent variables. Then, based on the dual variational principle, the nonlinear two-point boundary value problems are replaced by a system of nonlinear equations which can preserve the symplectic structure of the nonlinear optimal control problem. Furthermore, the computational efficiency of the proposed symplectic algorithm is improved by using the adaptive multi-level iteration idea. The performance of the proposed algorithm is tested by the problems of Astrodynamics, such as the optimal orbital rendezvous problem and the optimal orbit transfer between halo orbits.

Numerical solution of nonlinear second order two-point boundary value problems based on Sinc-collocation method, developed in this work. We first apply the method to the class of nonlinear two-point boundary value problems in general and specifically solved special problem that is arising in chemical reactor theory. Properties of the Sinc-collocation method are utilized to reduce the solution of nonlinear two-point boundary value problem to some nonlinear algebraic equations. By solving such system we can obtain the numericalsolution. We compared the obtained numerical result with the previous methods so far, such as Adomiad method, shooting method, Sinc Galerkin method and contraction mapping principle method.

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

A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM

Three new and applicable approaches based on quasi-linearization technique, wavelet-homotopy analysis method, spectral methods, and converting two-point boundary value problem to Fredholm–Urysohn integral equation are proposed for solving a special case of strongly nonlinear two-point boundary value problems, namely Troesch problem. A quasi-linearization technique is utilized to reduce the nonlinear boundary value problem to a sequence of linear equations in the first method. Second method is devoted to applying generalized Coiflet scaling functions based on the homotopy analysis method for approximating the numerical solution of Troesch equation. In the third method we use an interesting technique to convert the boundary value problem to Urysohn–Fredholm integral equation of the second kind; afterwards generalized Coiflet scaling functions and Simpson quadrature are employed for solving the obtained integral equation. Introduced methods are new and computationally attractive, and applications are demonstrated through illustrative examples. Comparing the results of the presented methods with the results of some other existing methods for solving this kind of equations implies the high accuracy and efficiency of the suggested schemes.

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

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

Existence and uniqueness of solutions for nth-order nonlinear two-point boundary value problems

Numerical solutions for nonlinear two-point boundary value problems by the integral equation method

Sinc-galerkin solution for nonlinear two-point boundary value problems with applications to chemical reactor theory

On Numerov's method for a class of strongly nonlinear two-point boundary value problems