Galerkin Methods for Two-Point Boundary Value Problems for First Order Systems

1983 ◽  
Vol 20 (1) ◽  
pp. 161-171 ◽  
Author(s):  
William J. Layton
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Galerkin Methods ◽  
Point Boundary ◽  
First Order

Node Selection for Two-Point Boundary-Value Problems

1985 ◽  
Vol 107 (3) ◽  
pp. 364-369 ◽  
Author(s):  
C. M. Ablow ◽  
S. Schechter ◽  
W. H. Zwisler
Keyword(s):  
Boundary Value Problems ◽  
Boundary Layers ◽  
Boundary Value ◽  
System Size ◽  
Transformation Method ◽  
Point Boundary ◽  
First Order ◽  
Standard Problem ◽  
Tem Method ◽  
Irregular Spacing

The solutions of two-point boundary-value problems often have boundary layers, narrow regions of sharp variation, that can occur in any part of the interval between the points. A finite difference method of numerical solution will generally require more closely spaced nodes in the boundary layers than elsewhere. An automatic method is needed for achieving the irregular spacing when the location of the boundary layer is not known in advance. Several automatic node-insertion or node-movement methods have been proposed. A new node-movement method is presented that is optimal under the criterion of producing the least sum of squares of the truncation errors at the nodes. For the Keller box scheme applied to a system of N coupled first-order differential equations this truncation-error minimizing (TEM) method increases the system size to N+6 equations. The campylotropic coordinate transformation method and other published methods based on heuristically derived monitor functions are node-movement methods that involve systems of only N+1 or N+2 first order equations. A comparison is made of the accuracies of several such methods and the TEM method in the solution of a standard problem.

scholarly journals A Peek into Theory of Automation with Applications to Various Branches of Science

2020 ◽  
Vol 9 (09) ◽  
pp. 25156-25160
Author(s):  
Divyalaxmi N.
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Least Square ◽  
Uniqueness Of Solution ◽  
Characteristic Matrix ◽  
Point Boundary ◽  
Difference System ◽  
Lattice Vector ◽  
First Order

In this paper, we shall be concerned with the existence and uniqueness of solution to three- point boundary value problems associated with a system of first order matrix difference system. Shortest and Closest Lattice vector methods are used as a tool to obtain the best least square solution of the  three-point boundary value problems when  the characteristic matrix D is rectangular. In this paper, we shall be concerned with the existence and uniqueness of solution to three- point boundary value problems associated with a system of first order matrix difference system. Shortest and Closest Lattice vector methods are used as a tool to obtain the best least square solution of the  three-point boundary value problems when  the characteristic matrix D is rectangular. 

Some Collocation-Galerkin Methods for Two-Point Boundary Value Problems

1976 ◽  
Vol 13 (5) ◽  
pp. 720-733 ◽  
Author(s):  
Roderick J. Dunn, Jr. ◽  
Mary Fanett Wheeler
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Galerkin Methods ◽  
Point Boundary

scholarly journals Kronecker Product of matrices and their applications to self-adjoint two-point boundary value problems associated with first order matrix differential systems

2021 ◽  
Vol 10 (10) ◽  
pp. 25399-25407
Author(s):  
Sriram Bhagavatula ◽  
Dileep Durani Musa ◽  
Murty Kanuri
Keyword(s):  
Boundary Value Problems ◽  
Kronecker Product ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Least Square ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Differential Systems ◽  
First Order ◽  
Product Of Matrices

In this paper, we shall be concerned with Kronecker product or Tensor product of matrices and develop their properties in a systematic way. The properties of the Kronecker product of matrices is used as a tool to establish existence and uniqueness of solutions to two-point boundary value problems associated with system of first order differential systems. A new approach is described to solve the Kronecker product linear systems and establish best least square solutions to the problem. Several interesting examples are given to highlight the importance of Kronecker product of matrices. We present adjoint boundary value problems and deduce a set of necessary and sufficient conditions for the Kronecker product boundary value problem to be self-adjoint.

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