Author: G.W. Bluman
Publisher: Springer Science & Business Media
ISBN: 1461263948
Category : Mathematics
Languages : en
Pages : 343
Book Description
The aim of this book is to provide a systematic and practical account of methods of integration of ordinary and partial differential equations based on invariance under continuous (Lie) groups of trans formations. The goal of these methods is the expression of a solution in terms of quadrature in the case of ordinary differential equations of first order and a reduction in order for higher order equations. For partial differential equations at least a reduction in the number of independent variables is sought and in favorable cases a reduction to ordinary differential equations with special solutions or quadrature. In the last century, approximately one hundred years ago, Sophus Lie tried to construct a general integration theory, in the above sense, for ordinary differential equations. Following Abel's approach for algebraic equations he studied the invariance of ordinary differential equations under transformations. In particular, Lie introduced the study of continuous groups of transformations of ordinary differential equations, based on the infinitesimal properties of the group. In a sense the theory was completely successful. It was shown how for a first-order differential equation the knowledge of a group leads immediately to quadrature, and for a higher order equation (or system) to a reduction in order. In another sense this theory is somewhat disappointing in that for a first-order differ ential equation essentially no systematic way can be given for finding the groups or showing that they do not exist for a first-order differential equation.
Similarity Methods for Differential Equations
Author: G.W. Bluman
Publisher: Springer Science & Business Media
ISBN: 1461263948
Category : Mathematics
Languages : en
Pages : 343
Book Description
The aim of this book is to provide a systematic and practical account of methods of integration of ordinary and partial differential equations based on invariance under continuous (Lie) groups of trans formations. The goal of these methods is the expression of a solution in terms of quadrature in the case of ordinary differential equations of first order and a reduction in order for higher order equations. For partial differential equations at least a reduction in the number of independent variables is sought and in favorable cases a reduction to ordinary differential equations with special solutions or quadrature. In the last century, approximately one hundred years ago, Sophus Lie tried to construct a general integration theory, in the above sense, for ordinary differential equations. Following Abel's approach for algebraic equations he studied the invariance of ordinary differential equations under transformations. In particular, Lie introduced the study of continuous groups of transformations of ordinary differential equations, based on the infinitesimal properties of the group. In a sense the theory was completely successful. It was shown how for a first-order differential equation the knowledge of a group leads immediately to quadrature, and for a higher order equation (or system) to a reduction in order. In another sense this theory is somewhat disappointing in that for a first-order differ ential equation essentially no systematic way can be given for finding the groups or showing that they do not exist for a first-order differential equation.
Publisher: Springer Science & Business Media
ISBN: 1461263948
Category : Mathematics
Languages : en
Pages : 343
Book Description
The aim of this book is to provide a systematic and practical account of methods of integration of ordinary and partial differential equations based on invariance under continuous (Lie) groups of trans formations. The goal of these methods is the expression of a solution in terms of quadrature in the case of ordinary differential equations of first order and a reduction in order for higher order equations. For partial differential equations at least a reduction in the number of independent variables is sought and in favorable cases a reduction to ordinary differential equations with special solutions or quadrature. In the last century, approximately one hundred years ago, Sophus Lie tried to construct a general integration theory, in the above sense, for ordinary differential equations. Following Abel's approach for algebraic equations he studied the invariance of ordinary differential equations under transformations. In particular, Lie introduced the study of continuous groups of transformations of ordinary differential equations, based on the infinitesimal properties of the group. In a sense the theory was completely successful. It was shown how for a first-order differential equation the knowledge of a group leads immediately to quadrature, and for a higher order equation (or system) to a reduction in order. In another sense this theory is somewhat disappointing in that for a first-order differ ential equation essentially no systematic way can be given for finding the groups or showing that they do not exist for a first-order differential equation.
Similarity Solutions of Nonlinear Partial Differential Equations
Author: Lawrence Dresner
Publisher: Pitman Advanced Publishing Program
ISBN:
Category : Mathematics
Languages : en
Pages : 148
Book Description
Publisher: Pitman Advanced Publishing Program
ISBN:
Category : Mathematics
Languages : en
Pages : 148
Book Description
Similarity Methods for Differential Equations
Author: George W. Bluman
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 332
Book Description
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 332
Book Description
Simple Methods for Classification and Construction of Similarity Solutions of Partial Differential Equations
Author: Douglas Eugene Abbott
Publisher:
ISBN:
Category : Boundary value problems
Languages : en
Pages : 58
Book Description
Publisher:
ISBN:
Category : Boundary value problems
Languages : en
Pages : 58
Book Description
Similarity Method for Differential Equations
Author: Boon-chang Tan
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 134
Book Description
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 134
Book Description
Similarity Solutions for Partial Differential Equations Generated by Finite and Infinitesimal Groups
Author: Henry Stott Woodard
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 106
Book Description
The problem of developing systematic methods for obtaining similarity variables is considered for partial differential equations. Similarity variables are a set of transformations which reduce a partial differential equation to an ordinary differential equation. This paper considers two methods of generating similarity variables. The first method uses a group of finite transformations and the second uses a group of infinitesimal transformations. The mathematical theory for both techniques is described and illustrated. The two methods of obtaining similarity variables are applied to the Burgers' equation u(sub y) + u(u sub x) = u sub xx and to the laminar boundary layer equations with a pressure gradient. In all cases considered, new types of similarity variables are found. In addition, the auxiliary conditions are discussed in the light of the new similarity variables obtained for the boundary layer equations. (Author).
Publisher:
ISBN:
Category : Differential equations
Languages : en
Pages : 106
Book Description
The problem of developing systematic methods for obtaining similarity variables is considered for partial differential equations. Similarity variables are a set of transformations which reduce a partial differential equation to an ordinary differential equation. This paper considers two methods of generating similarity variables. The first method uses a group of finite transformations and the second uses a group of infinitesimal transformations. The mathematical theory for both techniques is described and illustrated. The two methods of obtaining similarity variables are applied to the Burgers' equation u(sub y) + u(u sub x) = u sub xx and to the laminar boundary layer equations with a pressure gradient. In all cases considered, new types of similarity variables are found. In addition, the auxiliary conditions are discussed in the light of the new similarity variables obtained for the boundary layer equations. (Author).
Similarity Solutions of Systems of Partial Differential Equations Using MACSYMA.
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
A code has been written to use the algebraic computer system MACSYMA to generate systematically the infinitesimal similarity groups corresponding to systems of quasi-linear partial differential equations. The infinitesimal similarity groups can be used to find exact solutions of the partial differential equations. In an example from fluid mechanics the similarity method using the computer code reproduces immediately a solution obtained from dimensional analysis.
Publisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
A code has been written to use the algebraic computer system MACSYMA to generate systematically the infinitesimal similarity groups corresponding to systems of quasi-linear partial differential equations. The infinitesimal similarity groups can be used to find exact solutions of the partial differential equations. In an example from fluid mechanics the similarity method using the computer code reproduces immediately a solution obtained from dimensional analysis.
Simple Methods for Classification and Construction of Similarity Solutions of Partial Differential Equations
Author: Stanford University. Thermosciences Division. Thermosciences Division
Publisher:
ISBN:
Category : Boundary value problems
Languages : en
Pages : 58
Book Description
Publisher:
ISBN:
Category : Boundary value problems
Languages : en
Pages : 58
Book Description
Group and Potential Similarity Transformation Methods
Author: Magda Kassem
Publisher: LAP Lambert Academic Publishing
ISBN: 9783659185632
Category :
Languages : en
Pages : 116
Book Description
The object of this work is to obtain the similarity solution of a given problem by applying the Group and Potential similarity transformation methods where the governing partial differential equations are written in a conserved form to obtain new simpler system of partial differential equations. We then applied the group method which reduces the new system with the auxiliary condition to a system of ordinary differential equation with the appropriate corresponding conditions that can be solved analytically or numerically. Because of the absence of this combination between the potential and the group methods so far, this search is considered as an innovation in the field of mathematics
Publisher: LAP Lambert Academic Publishing
ISBN: 9783659185632
Category :
Languages : en
Pages : 116
Book Description
The object of this work is to obtain the similarity solution of a given problem by applying the Group and Potential similarity transformation methods where the governing partial differential equations are written in a conserved form to obtain new simpler system of partial differential equations. We then applied the group method which reduces the new system with the auxiliary condition to a system of ordinary differential equation with the appropriate corresponding conditions that can be solved analytically or numerically. Because of the absence of this combination between the potential and the group methods so far, this search is considered as an innovation in the field of mathematics
FINAL REPORT SIMILARITY ANALYSIS OF PARTIAL DIFFERENTIAL EQUATIONS
Author: TSUNG-YEN NA, DOUGLAS E. ABBOT, AND ARTHUR G. HANSEN
Publisher:
ISBN:
Category :
Languages : en
Pages : 136
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 136
Book Description