Numerical Solutions of General Fourth Order Two Point Boundary Value Problems by Galerkin Method with Legendre Polynomials

Hossain, Bellal; Islam, Shafiqul

doi:10.3329/dujs.v62i2.21973

Cited by 8 publications

(3 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So far we are not aware of any specific methods for the numerical solution of problem ( 1)-( 2) even in the case r = 2 (see [7,22,23,25,[30][31][32]35,37,41,[45][46][47] and references therein). The reason for this could be the non-symmetry of the boundary conditions.…”

Section: Numerical Examplesmentioning

confidence: 99%

Lidstone–Euler interpolation and related high even order boundary value problem

Costabile

Gualtieri

Napoli

2021

Calcolo

View full text Add to dashboard Cite

We consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent polynomial sequence. Particularly, we consider the fourth-order case, arising in various physical models. Finally, we present some numerical examples and we compare the proposed method with a modified decomposition method for a tenth-order problem. The numerical results confirm the theoretical and computational ones.

show abstract

Section: Numerical Examplesmentioning

confidence: 99%

Lidstone–Euler interpolation and related high even order boundary value problem

Costabile

Gualtieri

Napoli

2021

Calcolo

View full text Add to dashboard Cite

show abstract

“…Many techniques have provided alternatives to standard Green function techniques to convert a boundary value problem to a fixed-point problem. For instance, see Haq and Ali [11] or Hossain and Islam [12] for numerical solutions to boundary value problems using Haar wavelets and the Galerkin method, respectively. Another approach is an S-iteration process for quasi-contractive mappings to find a solution to a nonlinear boundary value problem; see Kumar, Latif, Rafiq, and Hussain [13] or Thenmozhi and Marudai [14].…”

Section: Introductionmentioning

confidence: 99%

Iteration with Bisection to Approximate the Solution of a Boundary Value Problem

Avery,

Anderson,

Lyons

2024

Axioms

View full text Add to dashboard Cite

Due to the restrictive growth and/or monotonicity requirements inherent in their employment, classical iterative fixed-point theorems are rarely used to approximate solutions to an integral operator with Green’s function kernel whose fixed points are solutions of a boundary value problem. In this paper, we show how one can decompose a fixed-point problem into multiple fixed-point problems that one can easily iterate to approximate a solution of a differential equation satisfying one boundary condition, then apply a bisection method in an intermediate value theorem argument to meet a second boundary condition. Error estimates on the iterates are also established. The technique will be illustrated on a second-order right focal boundary value problem, with an example provided showing how to apply the results.

show abstract

“…Many Scholars have developed numerous numerical methods for highly accurate approximate solutions of Fourth order boundary value problems. For example Hossain and Islam (2014) applied Galerkin weighted residual method for the numerical solutions of the general fourth order linear and nonlinear differential equations. Adeyeye and Omar, (2017).…”

Section: Introductionmentioning

confidence: 99%

Eleventh Degree Polynomial Series Solution Approach of Special Non-linear Fourth Order Boundary Value Problems

Mutawakilu,

Bello

2024

USci

View full text Add to dashboard Cite

In this paper a new eleventh degree polynomial series solution approach is developed for the solution of special non-linear fourth order boundary value problems. The method consist first of obtaining a linear differentials system of twelve equations from the boundary conditions, governing equation and its three successive derivatives which were evaluated at boundary points. Secondly we assume the approximate solution in the form of a polynomial of degree eleventh with twelve unknown coefficients. To determine the unknown coefficients we incorporate the assumed solution into linear differentials systems of twelve equations which results into a linear systems of twelve equations with twelve unknown and which can be solve uniquely. It is clear from the tables and figures that the method is in good agreement with the exact and with some existing results in the literatures. Also it can be seen from example 3.3 that the exact solution is reproduced which is an added advantage of the method.

show abstract

Numerical Solutions of General Fourth Order Two Point Boundary Value Problems by Galerkin Method with Legendre Polynomials

Cited by 8 publications

References 19 publications

Lidstone–Euler interpolation and related high even order boundary value problem

Lidstone–Euler interpolation and related high even order boundary value problem

Iteration with Bisection to Approximate the Solution of a Boundary Value Problem

Eleventh Degree Polynomial Series Solution Approach of Special Non-linear Fourth Order Boundary Value Problems

Contact Info

Product

Resources

About