A Novel Numerical Approach for Odd Higher Order Boundary Value Problems

In this paper, we investigate numerical solutions of odd higher order differential equations, particularly the fifth, seventh and ninth order linear and nonlinear boundary value problems (BVPs) with two point boundary conditions.  We exploit Galerkin weighted residual method with Legendre polynomials as basis functions. Special care has been taken to satisfy the corresponding homogeneous form of boundary conditions where the essential types of boundary conditions are given. The method is formulated as a rigorous matrix form. Several numerical examples, of both linear and nonlinear BVPs available in the literature, are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods. Keywords: Galerkin method, fifth, seventh and ninth order linear and nonlinear BVPs, Legendre Polynomials

B-Spline collocation method for numerical solution of nonlinear kawahara and modified kawahara equations

In this paper, a collocation method is applied for solving the Kawahara and modified Kawahara equations. For the spatial discretization, we use the sextic B-spline collocation (SBSC) method on uniform meshes, finite difference scheme is employed for the time discretization. The stability analysis of the collocation methods are examined by the Von Neumann approach. Numerical results demonstrate the efficiency and accuracy of the proposed methods.Publisher's Versio

Two Legendre-Dual-Petrov-Galerkin Algorithms for Solving the Integrated Forms of High Odd-Order Boundary Value Problems

Two numerical algorithms based on dual-Petrov-Galerkin method are developed for solving the integrated forms of high odd-order boundary value problems (BVPs) governed by homogeneous and nonhomogeneous boundary conditions. Two different choices of trial functions and test functions which satisfy the underlying boundary conditions of the differential equations and the dual boundary conditions are used for this purpose. These choices lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost. The various matrix systems resulting from these discretizations are carefully investigated, especially their complexities and their condition numbers. Numerical results are given to illustrate the efficiency of the proposed algorithms, and some comparisons with some other methods are made

A cubic spline collocation method for solving Bratu’s Problem

This paper, we develop a numerical method for solving a Bratu-type equations by using the cubic splinecollocation method (CSCM) and the generalized Newton method. This method converges quadratically if arelation-ship between the physical parameter ? and the discretization parameter h is satisfied. An error estimatebetween the exact solution and the discret solution is provided. To validate the theoretical results, Numericalresults are presented and compared with other collocation methods given in the literature.Keywords: Bratu-type equations, Boundary value problems, Cubic spline collocation method

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

This paper analyzes the solution of fourth-order linear and nonlinear two point boundary value problems. The suggested method is quite innovative and it is completely different from all previous methods used for solving such kind of boundary value problems. The method is based on employing an elegant operational matrix of derivatives expressed in terms of the well-known harmonic numbers. Two algorithms are presented and implemented for obtaining new approximate solutions of linear and nonlinear fourth-order boundary value problems. The two algorithms rely on employing the new introduced operational matrix for reducing the differential equations with their boundary conditions to systems of linear or nonlinear algebraic equations which can be efficiently solved by suitable solvers. For this purpose, the two spectral methods namely, Petrov-Galerkin and collocation methods are applied. Some illustrative examples are considered aiming to ascertain the wide applicability, validity, and efficiency of the two proposed algorithms. The obtained numerical results are satisfactory and the approximate solutions are very close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature

Development and implementation of a tenth-order hybrid block method for solving fifth-order boundary value problems

A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator

Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms

Evaluation of the stress singularities of plane V-notches in bonded dissimilar materials

According to the linear theory of elasticity, there exists a combination of different orders of stress singularity at a V-notch tip of bonded dissimilar materials. The singularity reflects a strong stress concentration near the sharp V-notches. In this paper, a new way is proposed in order to determine the orders of singularity for two-dimensional V-notch problems. Firstly, on the basis of an asymptotic stress field in terms of radial coordinates at the V-notch tip, the governing equations of the elastic theory are transformed into an eigenvalue problem of ordinary differential equations (ODEs) with respect to the circumferential coordinate h around the notch tip. Then the interpolating matrix method established by the first author is further developed to solve the general eigenvalue problem. Hence, the singularity orders of the V-notch problem are determined through solving the corresponding ODEs by means of the interpolating matrix method. Meanwhile, the associated eigenvectors of the displacement and stress fields near the V-notches are also obtained. These functions are essential in calculating the amplitude of the stress field described as generalized stress intensity factors of the V-notches. The present method is also available to deal with the plane V-notch problems in bonded orthotropic multi-material. Finally, numerical examples are presented to illustrate the accuracy and the effectiveness of the method