Expanded Lie Group Transformations and Similarity Reductions of Differential Equations (original) (raw)
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Reduction for Ordinary and Partial Differential Equations by Using Lie Group
JOURNAL OF UNIVERSITY OF BABYLON for pure and applied sciences
In this publication, We have done Lie group theory is applied to reduce the order of ordinary differential equations (ODEs) with 1-parameter and reduce a PDEs to ODEs . Also, set up algorithm to solve ODEs and PDEs to obtain the exact solution.
An algorithm for solutions of linear partial differential equations via Lie group of transformations
The Lie group of transformations associated with linear partial differential equations is derived using the invariance properties. Infinitesimals of Lie group of transformations with respect to independent and dependent variables along with invariance surface conditions are used to obtain generalized auxiliary equations. The main objective of this paper is to develop an easily applicable algorithm for linear partial differential equations, which can be used to solve the PDE or reduce the same to another PDE with fewer independent variables. The results are illustrated by considering suitable examples.
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SIAM Review, 1998
The importance of similarity transformations and their applications to partial differential equations is discussed. The theory has been presented in a simple manner so that it would be beneficial at the undergraduate level. Special group transformations useful for producing similarity solutions are investigated. Scaling, translation, and the spiral group of transformations are applied to well-known problems in mathematical physics, such as the boundary layer equations, the wave equation, and the heat conduction equation. Finally, a new transformation including the mentioned transformations as its special cases is also proposed.
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We construct a new type of symmetries using the regular Lie symmetries as the basis, which we call Modified symmetries. The contrast is that while Lie symmetries arise from point transformations, the Modified symmetries result from the transformations of the neighborhood of that point. The similarity is that as the neighborhood contracts to the central point, the two sets of symmetries become indistinguishable from one another, meaning the Modified symmetries will cease to exist if there were no Lie symmetries in the first place. The advantage is that the group invariant solutions are not affected by all these, because they result from ratios of the symmetries, and will therefore exist in the absence of Lie symmetries, i.e,. zero symmetries. Zero symmetries lead to 0/0, and no further. With the Modified symmetries we get f (x, ω)/g(x, ω) = 0/0 as ω goes to zero, and there are numerous mathematical techniques through which this can be resolved. We develop this concept using tensors a...
Reductions of PDEs to first order ODEs, symmetries and symbolic computation
Communications in Nonlinear Science and Numerical Simulation, 2015
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Two methods to reduce ODEs to generalized Abel or elliptic equations are provided. • New exact solutions for well-known PDEs are obtained. • Computer algebra codes for obtaining the reduced equations are included.
A Precise Definition of Reduction of Partial Differential Equations
Journal of Mathematical Analysis and Applications, 1999
We give a comprehensive analysis of interrelations between the basic concepts of Ž . the modern theory of symmetry classical and non-classical reductions of partial differential equations. Using the introduced definition of reduction of differential Ž . equations we establish equivalence of the non-classical conditional symmetry and Ž . direct Ansatz approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1 q 3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear wave equations.
This study will explicitly demonstrate by example that an unrestricted infinite and forward recursive hierarchy of differential equations must be identified as an unclosed system of equations, despite the fact that to each unknown function in the hierarchy there exists a corresponding determined equation to which it can be bijectively mapped to. As a direct consequence, its admitted set of symmetry transformations must be identified as a weaker set of indeterminate equivalence transformations. The reason is that no unique general solution can be constructed, not even in principle. Instead, infinitely many disjoint and thus independent general solution manifolds exist. This is in clear contrast to a closed system of differential equations that only allows for a single and thus unique general solution manifold, which, by definition, covers all possible particular solutions this system can admit. Herein, different first order Riccati-ODEs serve as an example, but this analysis is not restricted to them. All conclusions drawn in this study will translate to any first order or higher order ODEs as well as to any PDEs.
Reduction operators of linear second-order parabolic equations
Journal of Physics A: Mathematical and Theoretical, 2008
The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1 + 1)dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The "no-go" result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.
Two perspectives on reduction of ordinary differential equations
Mathematische Nachrichten, 2005
Key words Nonlinear differential equations, kinetic equations, multiple time scales, dimension reduction, slow manifold, normal form, computational singular perturbation, zero derivative principle MSC (2000) 34C20, 34E13, 34E15, 80A30, 80A25, 92C45