A new collocation algorithm for solving even-order boundary value problems via a novel matrix method
type: Item
instance of: scholarly article
title: A new collocation algorithm for solving even-order boundary value problems via a novel matrix method
zbMATH Open document ID: 1380.65131
author: Anna Napoli
author: Waleed M. Abd-Elhameed
published in: Mediterranean Journal of Mathematics
publication date: 2017-11-08T00:00:00Z
review text: The authors analyse the application of the collocation method for solving the boundary value problem of even order \[ u^{2n}(x)= f(x, \mathbf{U}(x)),\qquad x\in [a,b], \] \[ u^{r}(a)=a_{r}, u^{r}(b)=b_{r}, \qquad r=0, 1, \dots,n-1, \] where \(\mathbf{U}(x)=(u(x),u^{(1)}(x),\dots, u^{(2n-1)}(x))\). The construction of an approximate scheme of this problem uses the system of Legendre polynomials \( P_{n}(x)\) defined by recursive formulas. Using the properties of the derivatives and repeated integration of \(P_{n}(x)\) this scheme is presented as a system of linear equations in matrix form. This fact allows to apply the software Mathematica for some examples to illustrate the behavior of the error, compared with those obtained by other known methods (spline method and reproducing kernel method).
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L60
Mathematics Subject Classification ID: 65L70
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 6803991
zbMATH Keywords: even order differential equation
zbMATH Keywords: collocation
zbMATH Keywords: Legendre polynomials
zbMATH Keywords: error bounds
zbMATH Keywords: numerical example
zbMATH Keywords: comparison of methods
zbMATH Keywords: spline method
zbMATH Keywords: reproducing kernel method
reviewed by: Ivan Secrieru
describes a project that uses: Chebfun
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: s00009-017-0973-z
OpenAlex ID: W2736661884
cites work: On Solving Linear and Nonlinear Sixth-Order Two PointBoundary Value Problems Via an Elegant HarmonicNumbers Operational Matrix of Derivatives
cites work: Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials
cites work: Q3755784
cites work: Numerical solution of eighth-order boundary value problems in reproducing Kernel space
cites work: On coupled bending and torsional vibration of uniform beams
cites work: Q4039908
cites work: Q5518219
cites work: Finite difference methods for two-point boundary value problems involving high order differential equations
cites work: A new spectral method for a class of linear boundary value problems
cites work: Q5756872
cites work: Collocation for high order differential equations with two-points Hermite boundary conditions
cites work: Spectral Galerkin methods for the primary two-point boundary value problem in modelling viscoelastic flows
cites work: Finite Difference Collocation Methods for Nonlinear Two Point Boundary Value Problems
cites work: On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds
cites work: New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods
cites work: Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations
cites work: New wavelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth kinds
cites work: Integrals of Bernstein polynomials: an application for the solution of high even-order differential equations
cites work: Q4199478
cites work: Homotopy perturbation method: a new nonlinear analytical technique
cites work: An efficient approach to approximate solutions of eighth-order boundary-value problems
cites work: Q4585424
cites work: Q3995847
cites work: An innovative harmonic numbers operational matrix method for solving initial value problems
cites work: Variational iteration technique for solving higher order boundary value problems
cites work: Solution of eighth-order boundary value problems using the non-polynomial spline technique
cites work: Solutions of 12th order boundary value problems using non-polynomial spline technique
cites work: Spline solutions of linear eighth-order boundary-value problems
cites work: Spline solutions of linear twelfth-order boundary-value problems
cites work: Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability
DOI: 10.1007/S00009-017-0973-Z
description: scientific article; zbMATH DE number 6803991