scholarly journals A new algorithm based on compressed Legendre polynomials for solving boundary value problems

2021 ◽  
Vol 7 (3) ◽  
pp. 3277-3289
Author(s):  
Hui Zhu ◽  
◽  
Liangcai Mei ◽  
Yingzhen Lin
Keyword(s):  
Boundary Value Problems ◽  
Numerical Algorithm ◽  
Legendre Polynomials ◽  
Orthonormal Basis ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Runge Phenomenon ◽  
Order Polynomial ◽  
Complex Boundary ◽  
Linear And Nonlinear

<abstract><p>In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.</p></abstract>

scholarly journals Alternative methods for solving nonlinear two-point boundary value problems

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 371-374
Author(s):  
Fateme Ghomanjani ◽  
Stanford Shateyi
Keyword(s):  
Numerical Solution ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Bernstein Polynomials ◽  
Nonlinear Problems ◽  
Alternative Methods ◽  
Point Boundary ◽  
Numerical Examples ◽  
Curve Method ◽  
Linear And Nonlinear

Abstract In this sequel, the numerical solution of nonlinear two-point boundary value problems (NTBVPs) for ordinary differential equations (ODEs) is found by Bezier curve method (BCM) and orthonormal Bernstein polynomials (OBPs). OBPs will be constructed by Gram-Schmidt technique. Stated methods are more easier and applicable for linear and nonlinear problems. Some numerical examples are solved and they are stated the accurate findings.

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

2015 ◽  
Vol 62 (2) ◽  
pp. 103-108 ◽  
Author(s):  
Md Bellal Hossain ◽  
Md Shafiqul Islam
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Legendre Polynomials ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Equivalent Form ◽  
Linear And Nonlinear

In this paper, Galerkin weighted residual method is presented to find the numerical solutions of the general fourth order linear and nonlinear differential equations with essential boundary conditions. For this, the given differential equations and the boundary conditions over arbitrary finite domain [a, b] are converted into its equivalent form over the interval [0, 1]. Here the Legendre polynomials, over the interval [0, 1], are chosen as trial functions satisfying the corresponding homogeneous form of the Dirichlet boundary conditions. Details matrix formulations are derived for solving linear and nonlinear boundary value problems (BVPs). Numerical examples for both linear and nonlinear BVPs are considered to verify the proposed formulation and the results obtained are compared. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21973 Dhaka Univ. J. Sci. 62(2): 103-108, 2014 (July)

scholarly journals A new method for high-order boundary value problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yingchao Zhang ◽  
Liangcai Mei ◽  
Yingzhen Lin
Keyword(s):  
Approximate Solution ◽  
Boundary Value Problem ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Numerical Algorithm ◽  
Time Complexity ◽  
Numerical Experiments ◽  
Orthonormal Basis ◽  
Boundary Value ◽  
High Order

AbstractThis paper presents a numerical algorithm for solving high-order BVPs. We introduce the construction method of multiscale orthonormal basis in $W^{m}_{2}[0,1]$ W 2 m [ 0 , 1 ] . Based on the orthonormal basis, the numerical solution of the boundary value problem is obtained by finding the ε-approximate solution. In addition, the convergence order, stability, and time complexity of the method are discussed theoretically. At last, several numerical experiments show the feasibility of the proposed method.

scholarly journals Application of Optimal Homotopy Asymptotic Method to Some Well-Known Linear and Nonlinear Two-Point Boundary Value Problems

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Muhammad Asim Khan ◽  
Shafiq Ullah ◽  
Norhashidah Hj. Mohd Ali
Keyword(s):  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Homotopy Perturbation Method ◽  
Boundary Value ◽  
Point Boundary ◽  
Small Perturbations ◽  
Homotopy Perturbation ◽  
Optimal Homotopy Asymptotic Method ◽  
Linear And Nonlinear ◽  
Semianalytical Method

The objective of this paper is to obtain an approximate solution for some well-known linear and nonlinear two-point boundary value problems. For this purpose, a semianalytical method known as optimal homotopy asymptotic method (OHAM) is used. Moreover, optimal homotopy asymptotic method does not involve any discretization, linearization, or small perturbations and that is why it reduces the computations a lot. OHAM results show the effectiveness and reliability of OHAM for application to two-point boundary value problems. The obtained results are compared to the exact solutions and homotopy perturbation method (HPM).

scholarly journals On Triple Series Equations Involving Series of Jacobi Polynomials

1967 ◽  
Vol 15 (3) ◽  
pp. 221-231 ◽  
Author(s):  
K. N. Srivastava
Keyword(s):  
Boundary Value Problems ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mixed Boundary Value Problems ◽  
Dual Series Equations

Recently Collins (2) has studied triple series equations involving series of Legendre polynomials. These equations arise in the study of mixed boundary value problems and can be regarded as extensions of the dual series equations considered by Collins in (1).

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