scholarly journals The stability of LU-decompositions of block tridiagonal matrices

1984 ◽  
Vol 29 (2) ◽  
pp. 177-205 ◽  
Author(s):  
R. M. M. Mattheij
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Solution Space ◽  
Lu Decomposition ◽  
Tridiagonal Matrices ◽  
Separated Boundary Conditions ◽  
The Stability

An investigation is made of the stability of block LU-decomposition of matrices A arising from boundary value problems of differential equations, in particular of ordinary differential equations with separated boundary conditions. It is shown that for such matrices the pivotal growth can be bounded by constants of the order of ‖A‖ and, if solution space is dichotomic, often by constants of order one. Furthermore a method to estimate the growth of the pivotal blocks is given. A number of examples support the analysis.

scholarly journals Analysis and Application of Quadratic B-spline Interpolation for Boundary Value Problems

2020 ◽  
pp. 11-21
Author(s):  
Md. Asaduzzaman ◽  
Liton Chandra Roy ◽  
Md. Musa Miah
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Spline Interpolation ◽  
Boundary Value ◽  
Data Interpolation ◽  
B Spline ◽  
B Splines ◽  
Numerical Example

B-splines interpolations are very popular tools for interpolating the differential equations under boundary conditions which were pioneered by Maria et.al.[16] allowing us to approximate the ordinary differential equations (ODE). The purpose of this manuscript is to analyze and test the applicability of quadratic B-spline in ODE with data interpolation, and the solving of boundary value problems. A numerical example has been given and the error in comparison with the exact value has been shown in tabulated form, and also graphical representations are shown. Maple soft and MATLAB 7.0 are used here to calculate the numerical results and to represent the comparative graphs.

scholarly journals On the Stability of Solutions of Linear Boundary Value Problems for a System of Ordinary Differential Equations

1994 ◽  
Vol 1 (2) ◽  
pp. 115-126
Author(s):  
M. Ashordia
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Stability Of Solutions ◽  
Linear Boundary ◽  
Boundary Values ◽  
Small Perturbations ◽  
The Stability ◽  
Stability Of The Solution

Abstract Linear boundary value problems for a system of ordinary differential equations are considered. The stability of the solution with respect to small perturbations of coefficients and boundary values is investigated.

scholarly journals Numerical solution for initial and boundary value problems of high-order ordinary differential equations using Hermite neural network algorithm with improved extreme learning machine

2021 ◽  
Author(s):  
Yanfei Lu ◽  
Futian Weng ◽  
Hongli Sun
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Extreme Learning Machine ◽  
Boundary Value ◽  
Initial Boundary ◽  
High Order ◽  
Learning Machine

Abstract In this paper we put forth Hermite neural network (HNN) algorithm with improved extreme learning machine (IELM) to solve initial/boundary value problems of high-order ordinary differential equation(ODEs) and high-order system of ordinary differential equations (SODEs). The model function was expressed as a sum of two terms, where the first term contains no adjustable parameters but satisfies the initial/boundary conditions, the second term involved a single-layered neural network structure with IELM and Hermite basis functions to be trained. The approximate solution is presented in closed form by means of HNN algorithm, whose parameters are obtained by solving a system of linear equations utilizing IELM, which reduces the complexity of the problem. Numerical results demonstrate that the method is effective and reliable for solving high-order ODEs and high-order SODEs with initial and boundary conditions.Mathematics Subject Classification (2020) 34A30 ; 65D15

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