GROUP THEORETIC AND VARIATIONAL TECHNIQUES FOR THE SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS OF PHYSICAL AND ENGINEERING SYSTEMS
G. CHANDRASEKARARI
A thesis submitted to the Indian Institute of Technology, Delhi
for the award of the degree of
DOCTOR OF PHILOSOPHY
Department of Mathematics
INDIAN INSTITUTE OF TECHNOLOGY, DELHI
MAY, 1988
DEDICATED TO. ,
MY FAMILY
AND
SUPERVISOR
CERTIFICATE
This to certify that the thesis entitled:
'GROUP THEORETIC AND VARIATIONAL TECHNIQUES FOR THE SOLUTION OF NONLINEAR DIFFERENTIAL EQUATIONS OF PHYSICAL AND ENGINEERING SYSTEMS',
which is being submitted by Mr. G.Chandrasekaran, Research Scholar, Mathematics Department to the Indian Institute of Technology, Delhi, for the award of the DEGREE OF DOCTOR OF PHILOSOPHY in MATHEMATICS, is a record of bonafide research work carried out by him under my guidance and supervision and has fulfilled all the requirements for
the submission of this thesis.
The results contained in this thesis have not been submitted in part or full, to any other University or Institute for the award of any degree or diploma.
0/0
O.P. BHUTANI Professor,
Department of Mathematics I.I.T., Delhi.
ACKNOWLEDGEMENT
I am profoundly grateful to Professor O.P. Bhutani, Department of Mathematics, Indian Institute of Technology, Delhi, for his valuable guidance throughout the preparation of this thesis.
But for his generous encouragement, keen interest and kind cooperation, this work would not have been possible.
I take this opportunity to thank the authorities of Indian Institute of Technology, Delhi and Professor M.M. Chawla, Head of the Department of Mathematics, Indian Institute of Technology, Delhi, for providing all necessary facilities to undertake this work.
I also express my sincere thanks to Dr.(Mrs.) Poornima Mital for the fruitful discussions carried out during my research work.
It is my duty to thank Professor G.Shanmugam, Principal, Mepco Schlenk Engineering College, Sivakasi,Dr.B.Rajendran and Mr.P.Tandian,Department of Mathematics, M.S.E.C. for their kind cooperation and encouragement.
Thanks are also due to my colleagues Mr.K.Sankarakumar, Mr. K.Vijay Kumar, Mr. P. Sampath Narayanan, Mr. Vikas Bist, Mr. Sarbeswar Raut and others.
It is with pleasure, I mention my friends - Dr.S.Balasundaran, Dr. S. Gopalsamy, Dr. R. Sekar, Mr. S.Selvakumar,Mr.R.Subramanian, Mr. C.Thangaraj, Mr. A.Thayumanavan, Mr. R.Venkatraj and
others who made my Shivalik Hostel life homely and enjoyable, during my stay in the IIT Campus.
Finally, Miss NeeZam Dhody deserves praise for her intelligent and expert typing of this thesis and I sincerely thank her for the same.
4A-U O. CHANDRASEKARAN
SYNOPSIS
The thesis entitled "Group Theoretic and Variational Techniques for the Solution of Nonlinear Differential Equations of Physical and Engineering Systems" can be envisaged as con- sisting of two parts. First part, comprising of four chapters is concerned with the problems of Existence and Formulation of Variational Principles, first integrals and conservation laws via invariant variational principles and the Quasi-variational principles for nonlinear differential equations of physical and engineering systems. Second part, which also contains four chapters, deals with the invariant (similar) solutions of
single/system of nonlinear partial differential equations and nonlinear initial value problems. We briefly outline the details of each of the chapters as follows:
CHAPTER-I: INTRODUCTION - THE SURVEY AND THE NECESSARY MATHEMATICAL TOOLS
Herein, the aim of this chapter is two fold in the sense that, besides giving a brief survey of the available
literature related to our work in Chapters II-IX, we have given a brief resume of the consistency conditions for establishing the existence of variational principles for any single nonlinear ordinary differential equation of second order and for system of two second order nonlinear partial differential equations, indicating the procedure for writing down the functional
whenever it exists. Herein, we have also stated the classical Noether's theorem with special reference to Rund's invariance identities for finding the one-parameter infinitesiiilal
transformation. Finally, we have given the short resume of Steinberg's symmetry method for finding similarity solutions of single/system of partial differential equations.
CHAPTER-II.: EXISTENCE AND FORMULATION OF VARIATIONAL
PRINCIPLES FOR SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS
In this chapter, we have utilized the consistenoy conditions given in Chapter-I, to study the existence and formulation of potential/ alternate potential principles for the following systems of nonlinear differential equations of physical interest:
(i) Flows governed by the generalised Korteweg-de Vries (K-dV) equation,
(ii) Nonlinear lateral vibrations of a beam, (iii) Nonlinear dispersive waves,
(iv) Time-dependent SU(2) Higg's model, (v) The equation of a relativistic string,
(vi) Coupled system of nonlinear Klein-Gordon equations, and
(vii) Finite axisymmetric deformations of thin shells of revolutions.
CHAPTER-III: ON THE EXACT SOLUTIONS OP NONLINEAR DIFFERENTIAL EQUATTONS OF NUCLEAR ENGINEERING SYSTEM VIA
INVARIANT VARIATIONAL PRINCIPLES
This chapter deals with the existence and formulation of the alternate potential principles for the nonlinear differential equations corresponding to the following physical situations of nuclear engineering system:
(i) The steady state heat conduction with nonlinear thermal conductivity and nonlinear source term, and
(ii) The reactor core optimization.
Further, the utilization of the invariance identities of. Rund involving the Laqrangian and the generators of the infinitesimal Lie group and Noether's theorem has been made to obtain the first integral. Also, through the repeated application of the invariance under the transformation obtained, the exact solutions of the
above mentioned differential equations have been generated. for various choices of the parameters involved.
CHAPTER-IV: CONSERVATION LAWS FOR PHYSICAL SYSTEMS VIA INVARIANT VARIATIONAL PRINCIPLES
After proving the invariance under a one-parameter
infinitesimal transformation grout:), we have used Noether's theorem for writing down the conservation laws for the following systems
of physical interest:
(i) Transonic gas flow equation,
(ii) (2+1)-dimensional nonlinear wave equation,
(iii) Flows governed by the generalised K-dV equation, (iv) Generalized Duffing problem and
(v) Nonlinear wave equation of Klein-Gordon's type.
CHAPTER-V: QUASI-VARIATIONAL PRINCIPLE AND.NOMLINEAR WAVES
The purpose of this chapter is to develop quasi- variational principle to construct approximate solution to nonlinear wave problems governing the following physical situations:
(i) The variable coefficients K-dV equation and (ii) The Benjamin-Bona and Mahoney (BBM) equation.
Here, the trial solution is chosen with the help of prior informations available. The solution found is slowly varying in the sense that higher order derivative terms are neglected.
CHAPTER-VI: INVARIANT SOLUTIONS OF SINGLE PARTIAL DIFFERENTIAL EQUATIONS BY SYmMETRY METHOD Herein, we have utilized the Steinberg's symmetry method for generating the invariant (similar) solutions of
the following nonlinear partial differential equations of physical interest:
(i) The generalised Boussinesq equation, (ii) The generalised Duffing problem, and
(iii) The nonlinear wave equation of Klein-Gordon's type.
Possible invariant solutions of these equations for physically realizable forms of the parameters involved have been obtained.
CHAPTER-VI': INVARIANT SOLUTIONS OF GENERALISED KORTEWEG- DE VRIES-BURGERS TYPE EQUATIONS
As in Chapter-VI, we have utilized the symmetry method to the generalised Korteweg-de-Vries-Burgers type equations
to generate the invariant solutions. Possible invariant solutions for physically realizable forms of the parameters involved are obtained.
CHAPTER-VIII: INVARIANT SOLUTIONS OF SYSTEMS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Unlike Chapters VI-VII, wherein we had generated the invariant solutions of single nonlinear partial differential equations, we have in this chapter, utilized the symmetry method for obtaining the invariant solutions of the following systems of nonlinear partial differential equations of physical
interest:
(i) Longitudinal waves in a nonlinear geometrically dispersive solid and
(ii) Generalised coupled K-dV equation.
CHAPTER-IX: CONTINUOUS TRANSFORMATION GROUPS AND SERIES SOLUTION OF NONLINEAR INITIAL VALUE PROBLEMS
In this chapter, we have utilized the theory of
continuous transformation groups, to construct the series form of the solution of the following initial value problems of physical interest, with the associated infinitesimal generator of the dynamical systems:
(i) Oscillator with nonlinear viscous resistance, (ii) The Vander Pol equation, and
(iii) The Duffing equation.
Tedious algebra associated with the substitution of power series with unknown coefficients and solution of nonlinear algebraic equations in solving an IVY' is thereby eliminated.
The following papers which form the part of the thesis have been published submitted:
1) On The Exact Solutions of Nonlinear Differential
Equations of Nuclear Engineering System Via Invariant Variational Principles. Int. J. Engng. Sci. 26(3)
243-248 (1988).
2) On Invariant Solutions of the Generalised Boussinesq Equation. Int. J.Engng. Sci. 26(3) 307-310 (1988).
3) Existence and Formulation of Variational Principles for Systems of Differential Equations. Presented at the 31st Congress of ISTAM, Gwalior (1986).
4 Conservation Laws for Physical Systems via Invariant Variational Principles. Presented at the 31st Congress of ISTAM Gwalior (1986).
5) On Invariant Solutions of the Generalised Korteweg- de-Vries-Burgers Type Equations-II. Communicated to J.Math. Phys.
CONTENTS
SYNOPSIS
CHAPTER-I: INTRODUCTION - THE SURVEY AND THE NECESSARY MATHEMATICAL TOOLS
Page No.
1 22 1.1 Introduction
1.2 Existence and Formulation of Variational Principle
1.3 Conservation Laws Via Noether's Theorem 25
1.4 Symmetry Method 27
CHAPTER-II: EXISTENCE AND FORMULATION OF VARIATIONAL PRINCIPLES FOR SYSTEMS OF NONLINEAR
DIFFERENTIAL EQUATIONS
2.1 Introduction 29
2.2 The Generalised Korteweg-de Vries Equation 30 2.3 Nonlinear Lateral Vibrations of a Beam 37 2.4 Nonlinear Dispersive Waves 38 2.5 Time-Dependent SU(2) Higg's Model 40 2.6 The Equation of a Relativistic String 41 2.7 Coupled System of Nonlinear Klein-Gordon
Equations 43
2.8 Finite Axisymmetric Deformation of thin
Shells of Revolution 45
CHAPTER-III: ON THE EXACT SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS OF NUCLEAR ENGINEERING SYSTEM VIA INVARIANT VARIATIONAL PRINCIPLES
3.1 Introduction 47
3.2 The Steady State Heat Conduction Equation 48 3.3 Reactor Core Optimization 56
CHAPTER-IV: CONSERVATION LAWS FOR PHySICAL SYSTEMS VIA INVARIANT VARIATIONAL PRINCIPLES
4.1 Introduction 60
4.2 Transonic Gas Flow 61
4.3 (2+1)-Dimensional Nonlinear Wave Equation 66 4.4 The Generalised Korteweg-de Vries Equation 69 4.5 The Generalised Duffing Problem 73 4.6 The Nonlinear Wave Equation of Klein-
Gordon's Type 75
CHAPTER-V: QUASI-VARIATIONAL PRINCIPLE AND NONLINEAR WAVES
5.1 Introduction 80
5.2 The Variable Coefficients K-dV Equation 81 5.3 The Benjamin-Bona and Mahoney Equation 87
CHAPTER-VI: INVARIANT SOLUTIONS OF SINGLE PARTIAL DIFFERENTIAL EQUATIONS BY SYMMETRY METHOD
6.1 Introduction 92
6.2 The Generalised Boussinesq Equation 94 6.3 Determination of the Solution 95 6.4 Generalised Duffing Problem 109 6.5 The Generalised Nonlinear Wave Equation
of Klein-Gordon's Type 115
Appendix A-4 122
CHAPTER-VII : INVARIANT SOLUTIONS OF GENERALISED
KORTEWEG-DE VRIES-BURGERS TYPE EQUATIONS
7.1 Introduction 125
7.2 Kortewg-de Vries-Burgers Type Equations 125 7.3 Determination of the Solution 127
Appendix A-II 144
CHAPTER-VIII: INVARIANT SOLUTIONS OF SYSTEMS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
8.1 Introduction 145
8.2 Longitudinal Waves in a Nonlinear
Geometrically Dispersive Solid 145 8.3 The Generalised Coupled Korteweg-de Vries
Equations 155
Appendix A-III 168
Appendix A-IV 169
CHAPTER-IX: CONTINUOUS TRANSFORMATION GROUPS AND SERIES SOLUTION OF NONLINEAR INITIAL VALUE PROBLEMS
9.1 Introduction 171
9.2 Series Solution of Initial Value Problems 172 9.3 Nonlinear Oscillator with Viscous
Resistance 175
9.4 The Vander Pol's Equation 177 9.5 The Duffing Equation 178 BIBLIOGRAPHY 180
