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

Abstract

MSC: Keywords:Nonlinear boundary value problems Sextic B-spline method Collocation Convergence Green's function a b s t r a c t A numerical method based on B-spline is developed to solve the general nonlinear twopoint boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green's function approach. O(h 6 ) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.

Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method

“…Now, the unknown parameters are determined by solving the system of equation in (14) by direct method and substituting these values into (9) yields the approximate solution ( ) y x of the DE in (4) satisfying the given boundary conditions in (6). Consider a BVP of type I.…”

“…Among this, a spline solution of two point boundary value problems introduced in [6], a method for solutions of nonlinear second order multi-point boundary value problems produced in [7], in [8] linear and non-linear differential equations were solved numerically by Galerkin method using a Bernstein polynomials basis, in [9] a numerical method is established to solved second order ordinary differential equation with Neumann and Cauchy boundary conditions using Hermite polynomials, in [10] a parametric cubic spline solution of two point boundary value problems were obtained, a second-order Neumann boundary value problem with singular nonlinearity for exact three positive solutions were solved [11], a Numerical solution of a singular boundary-value problem in non-Newtonian fluid mechanics were established [12], a Fourth Order Boundary Value Problems by Galerkin Method with Cubic B-splines were solved by considering different cases on the boundary condition [13] and a special successive approximations method for solving boundary value problems including ordinary differential equations were proposed. [14] In this paper Galerkin method will be applied to the linear second order ordinary differential equation of the form …”

On exact and approximate solutions of (2+1)-dimensional Konopelchenko-Dubrovsky equation via modified simplest equation and cubic B-spline schemes

“…Recently, for studying the exact traveling wave solutions, the modified simplest equation method has been proposed to investigate many kinds of solutions such as rational, exponential, hyperbolic, trigonometric, kink, rogue, lump, bilinear, and solitary [16][17][18][19][20]. Moreover, the complexion solutions had been investigated such that it defines as an interaction of exponential and trigonometric waves [21][22][23][24][25] while the B-spline schemes have been being used to study the numerical solutions of many various forms of non-linear PDE [26][27][28][29][30].…”

An elegant operational matrix based on harmonic numbers: Effective solutions for linear and nonlinear fourth-order two point boundary value problems

“…The aim of spectral methods is to approximate functions (solutions of differential equations) by means of truncated series of orthogonal polynomials. There are three well-known methods of spectral methods, namely, tau, collocation and Galerkin methods (see, for example, [5,8,13,19,29]). The choice of the suitable used spectral method suggested for solving the given equation c Vilnius University, 2016 depends certainly on the type of the differential equation and also on the type of the boundary conditions governed by it.…”