An effective method for numerical solution and numerical derivatives for sixth order two-point boundary value problems

Abstract

In this paper, we study an effective quintic polynomial spline method for numerical solu tion, and first order to fifth order numerical derivatives of the analytic solution at the knots for a class of sixth order two point boundary value problems. Our new method is based on a quintic spline inter polation problem. It is easy to implement and is able to provide sixth order accurate numerical results at the knots. Numerical tests show that our method is very practical and effective.

Convergence of the class of methods for solutions of certain sixth-order boundary value problems

“…Proof : We can write the error equation (4.12) in the following form where M14 = max|u (14) (ξ (1) ) , B and t (1) from above relations in (4.15) and simplifying we obtain (4.17)…”

“…Jha et al [11] introduced an ecient algorithm based on non-polynomial spline approximations on a geometric mesh for the numerical solution of linear and non-linear two-point boundary value problems. Lang et al [14] used quintic spline and Arshad Khan et al [12,13] applied parametric quintic spline and septic splines for the solution of sixth-order boundary value problems.…”

Bernoulli polynomial and the numerical solution of high-order boundary value problems

“…Also, these type of equations which occurs in torsional vibration of uniform beams have been investigated by Bishop et al [10]. They are often used in the simulation of a lot of physical phenomena such as astrophysics or heat transfer problems which are known to be simulated by a sixth-order boundary value problem [11,33,48]. Also, in the modeling of viscoelastic or inelastic flows and deformation of beams, a fourth-order boundary value problem is used to model the effect of these flows or these deformations [15].…”

“…However, there have been few numerical techniques which treats this types of problems. These methods include collocation methods [5], finite difference method [12,14], sinc Galerkin [20], non-polynomial spline functions [46], octic spline functions [47], quintic spline [33], Adomian decomposition method [55,56], reproducing kernel space method [4], differential quadrature method [35], variational iteration method [1,45], differential transformation method [22], Homotopy perturbation method [25,40,43], and the Galerkin residual technique with Bernstein and Legendre polynomials [30]. Recently, El-Gamel and Abdrabou [16] proposed a new approach for solving eight-order boundary value problems via sinc-Galerkin.…”

Existence Results and Numerical Method for a Fourth Order Nonlinear Problem

“…was studied in many works, e.g., [2], [4], [16], [17], [25], [26]. The authors of these works were interested only in finding the solution of the problem without attention on the investigation of its qualitative aspects such as the existence, uniqueness and properties of solutions.…”