Abstract
Different methods for carrying out similarity analysis of partial differential equations were discussed in Chapter 3 with particular reference to the linear heat equation. The methods were classified into (1) direct methods and (2) group-theoretic methods. In the direct methods, the concept of group invariance is not explicitly invoked. They are straightforward and simple to apply. Since the direct methods are based on assumed transformations, the resulting solutions are restrictive. The group-theoretic methods on the other hand are based upon the invocation of invariance under groups of transformations of the partial differential equations and the auxiliary conditions. Group-theoretic methods such as Birkhoff-Morgan method and the Heliums-Churchill procedure start out by assuming a specific form of the group. Therefore, the resulting similarity solutions are restrictive. The simplicity of these methods is on account of the fact that only algebraic equations (resulting from invocation of invariance) need to be solved. On the other hand, deductive group procedures while being systematic and more rigorous and tedious.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abbott D.E. and Kline S.J.,Simple Methods for Construction of Similarity Solutions of Partial Differential Equations, AFOSR TN 60-1163, Report MD-6, Dept. of Mech. Eng., Stanford Univ. (1960).
Pai S.I., Viscous Flow Theory, Vol.1, p. 62, D. Van Nostrand Company, Inc.,Princeton,N.J. (1960).
Taulbee D.R.,Cozzarelli, F.A. and Dym, C.L.,“Similarity Solutioins to Some Nonlinear Impact Problems”, Int’l J. of Nonlinear Mechanics, Vol. 6 (1971).
Chand,R., Davy,D.T. and Ames,W.F.,“On the Similarity Solutions for a General Class of Nonlinear Dissipative Materials”,Int’l J. of Nonlinear Mechanics, Vol. 11 (1976).
Van Dyke M.,“Free Convection From a Vertical Needle”, Problems of Hydrodynamics and Continuum Mechanics, S.I.A.M. (1969).
Yih C.S.,“Free Convection Due to a Point Source of Heat”,Proc. 1st U.S. National Congress of Applied Mechanics (1951).
Morgan A.J.A.,“The Reduction of One of the Number of Independent Variables in Some Systems of Partial Differential Equations”,Quar. J. of Math., Oxford (2) (1952).
Kalthia N.L. and Jain,R.K., “Similarity Solutions of the Heated Jet With Variable Viscosity and Thermal Conductivity”, Symmetry, Similarity and Group Theoretic Methods in Mechanics, Calgary, Canada (1974).
Na T.Y., “Group-Theoretic Analysis of Unsteady One-Dimensional Gas Dyn. Eqs.”,Symmetry, Similarity and Group Theoretic Methods in Mechanics, Calgary, Canada (1974).
Mueller, von Ernst-August and Matschat, Klaus, Uber das Auffinden von Ahnlichkeitslosungen Partieller Differential Gleichungs Systeme unter Benutzung von Transformations Gruppen, mit Anwendungen auf Probleme der Stromungsphysik, Akademie-Verlag, Berlin (1962).
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer-Verlag New York Inc.
About this chapter
Cite this chapter
Seshadri, R., Na, T.Y. (1985). Application of Similarity Analysis to Problems in Science and Engineering. In: Group Invariance in Engineering Boundary Value Problems. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-5102-6_4
Download citation
DOI: https://doi.org/10.1007/978-1-4612-5102-6_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-9564-8
Online ISBN: 978-1-4612-5102-6
eBook Packages: Springer Book Archive