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.

Multiscale Method for Solving High-order BVPs

2020 ◽  
Author(s):  
Yingchao Zhang ◽  
Liangcai Mei ◽  
Yingzhen Lin
Keyword(s):  
Approximate Solution ◽  
Numerical Algorithm ◽  
Numerical Experiments ◽  
Orthonormal Basis ◽  
Construction Method ◽  
Multiscale Method ◽  
High Order ◽  
Approximate Theory ◽  
Convergence And Stability

This paper presents a numerical algorithm for solving high-order BVPs. We introduce the construction method of multiscale orthonormal basis in Wm[0; 1] by multiscale orthonormal basis in W1[0; 1]. We define approximate solution, and obtain the approximate solution of high-order BVPs by using the approximate theory. Moreover, the convergence and stability of the algorithm are improved. At last, several numerical experiments show the feasibility of the proposed method.

scholarly journals A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Minqiang Xu ◽  
Jing Niu ◽  
Li Guo
Keyword(s):  
Nonlinear System ◽  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Boundary Value ◽  
Linear Equations ◽  
Second Order ◽  
High Order ◽  
Reproducing Kernel Method

This paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.

scholarly journals ABOUT THE APPROXIMATE SOLUTION OF THE USUAL AND GENERALIZED HILBERT BOUNDARY VALUE PROBLEMS FOR ANALYTICAL FUNCTIONS

2000 ◽  
Vol 5 (1) ◽  
pp. 119-126
Author(s):  
V. R. Kristalinskii
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Cauchy Type ◽  
Analytical Functions ◽  
Method Of Solution ◽  
Cauchy Type Integrals ◽  
Hilbert Boundary Value Problem

In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the formproposed by the author. All methods are implemented with the Mathcad and Maple.

scholarly journals A New Numerical Algorithm for Two-Point Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Lihua Guo ◽  
Boying Wu ◽  
Dazhi Zhang
Keyword(s):  
Exact Solution ◽  
Error Analysis ◽  
Numerical Solution ◽  
Boundary Value Problems ◽  
Numerical Algorithm ◽  
Numerical Stability ◽  
Boundary Value ◽  
Numerical Results ◽  
Point Boundary ◽  
Stability And Error Analysis

We present a new numerical algorithm for two-point boundary value problems. We first present the exact solution in the form of series and then prove that then-term numerical solution converges uniformly to the exact solution. Furthermore, we establish the numerical stability and error analysis. The numerical results show the effectiveness of the proposed algorithm.

A Discreate Calculus with Applications of High-Order Discretizations to Boundary-Value Problems

2004 ◽  
Vol 4 (2) ◽  
pp. 228-261 ◽  
Author(s):  
Stanly Steinberg
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Energy Inequality ◽  
Numerical Test ◽  
Mixed Boundary Conditions ◽  
Discrete Analog ◽  
High Order ◽  
Sturm Liouville

AbstractWe develop a discrete analog of the differential calculus and use this to develop arbitrarily high-order approximations to Sturm–Liouville boundary-value problems with general mixed boundary conditions. An important feature of the method is that we obtain a discrete exact analog of the energy inequality for the continuum boundary-value problem. As a consequence, the discrete and continuum problems have exactly the same solvability conditions. We call such discretizations mimetic. Numerical test confirm the accuracy of the discretization. We prove the solvability and convergence for the discrete boundary-value problem modulo the invertibility of a matrix that appears in the discretization being positive definite. Numerical experiments indicate that the spectrum of this matrix is real, greater than one, and bounded above by a number smaller than three.

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 A Numerical Algorithm for a Kirchhoff-Type Nonlinear Static Beam

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jemal Peradze
Keyword(s):  
Approximate Solution ◽  
Boundary Value Problem ◽  
Galerkin Method ◽  
Numerical Algorithm ◽  
Boundary Value ◽  
Iteration Process ◽  
Beam Equation ◽  
Nonlinear Static ◽  
The Galerkin Method ◽  
Nonlinear Iteration

A boundary value problem is posed for an integro-differential beam equation. An approximate solution is found using the Galerkin method and the Jacobi nonlinear iteration process. A theorem on the algorithm error is proved.

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