Lahore University of Management Sciences

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1 MATH 414/514 Symmetry methods for Differential Equations / Symmetry method, Conservation Laws and Exact Solutions for Differential Equations Fall Instructor Imran Naeem Room No. 124 Office Hours Without appointment (any time) imran.naeem@lums.edu.pk Telephone Secretary/TA TBA TA Office Hours TBA Course URL (if any) Math.lums.edu.pk/moodle COURSE OBJECTIVES Find symmetries of differential manually and by using symbolic packages as well Deduce new solutions from known solutions Solve ODEs and systems of ODEs using Lie symmetries Reduce partial differential by reduction of the number of independent variables Linearize differential by invertible transformation and exact solutions by using integration strategy Compute first integrals/conservation laws for differential and systems of differential Construct solutions of ODEs and PDEs by using first integrals/ conservation laws Course Basics Credit Hours 3 Lecture(s) Nbr of Lec(s) Per Week 2 Duration 75min Recitation/Lab (per week) Nbr of Lec(s) Per Week Duration Tutorial (per week) Nbr of Lec(s) Per Week Duration Course Distribution Core Elective Open for Student Category Close for Student Category To all fields of science and people who are interested to learn differential in details All students None COURSE DESCRIPTION Lie symmetry analysis of differential was initiated by the Norwegian mathematician Sophus Lie ( ). Today, this area of research is actively engaged. In this course, we trace the mathematical idea of symmetry and provide the salient features on Lie theory of transformation groups with applications to ordinary and partial differential. The Lie approach is a systematic way of unraveling exact solutions of ordinary and partial differential. It works for linear as well as for nonlinear differential

2 COURSE PREREQUISITE(S) MATH 210 or Math 310 Learning Outcomes Grading Breakup and Policy Students will learn to: Compute symmetries of ODEs and PDEs manually and by using computer packages Find invariants, canonical variables using the symmetries of differential Solve ODEs, PDEs and systems using classical methods Linearize the ODE and system of ODEs using point symmetries Compute the first integrals for ODEs and conservation laws for PDEs using suitable methods Find exact solutions of ODEs and PDEs via first integrals and conservation laws Solve ODEs and PDEs using computer packages Assignment(s) and Quiz(s): 25% Class Participation: Attendance: Midterm Examination: 35% Project: Final Examination: 40% Examination Detail Midterm Exam Yes/No: Yes Combine/Separate: Duration: 90 min Preferred Date: 14 Oct 2012 Exam Specifications: No notes/no books/no calculators Final Exam Yes/No: Yes Combine Separate: Duration: 180min Exam Specifications: No notes/no books/no calculators

3 COURSE OVERVIEW Week 1 Topics Ordinary differential a) Preliminaries b) Point transformations and their generators Recommended Readings Objectives/ Application Basic theory of symmetries and point transformations for differential c) One parameter group d) The group generators in new variables e) Canonical variables f) Invariants, Exercises g) Extended generators, Prolongation formulas Lie point symmetries of ordinary differential Group of transformations using symmetries Fundamental operators and invariants of differential Computation of symmetry generators for ODEs a) The definition of a symmetry: first and second formulation b) How to find Lie symmetries of an ordinary differential c) Exercises including, first order to sixth order ODEs d) systems of second and third order ODEs How to Use Lie point Symmetries to find exact solutions: Differential with one symmetry a) First order (Linear and nonlinear) b) Higher order differential (Linear and nonlinear) c) Exercises Computation of symmetry generators for ODEs Applications to ODEs and system of ODEs from different fields of life 8 Mid Semester How to Use Lie point Symmetries to find exact solutions: Differential admitting groups G2 and G3 a) First order, higher order ODEs (Linear and nonlinear)

4 9 Lahore University of Management Sciences b) The first integration strategy: normal forms of generators in the space of variables c) The second integration strategy: normal forms of generators in the space of first integrals d) Examples and exercises e) Exact solutions for systems of second order ODEs by using symmetries Applications Linearization of ODEs a) Linearization of second-order and systems b) Lie s linearization criterium c) Examples, Applications to nonlinear of laser systems Partial differential a) Lie point transformation and symmetries b) How to determine the point symmetries of partial differential (First order fourth order) and linearization Construction of symmetry generators for PDEs 13 c) How to use Lie point symmetries of PDEs: similarity variables and reduction of the number of variables Solutions of partial differential l and applications First order fourth order PDEs Laplace equation, heat equation, wave equation, Cauchy Riemann equation, special Euler poison- Darboux equation, Klein-Gordon equation, Helmholtz equation, generalized axisymmetric Laplace equation, Schrodinger equation, isotropic harmonic oscillator, two dimensional harmonic oscillator, Maxwell in vacuum, of glacier mechanics, of breezes and monsoons, of soil water motions, Navier stokes, two dimensional boundary layer system, Burgers equation, Camassa- Holm equation, Dullin- Gottwald equation, generalized Dullin- Gottwald equation 14 Exact solutions of ODEs and PDEs from first integrals/conservation laws a) Noether s approach, partial Noether approach, Multipliers approach b) Solutions of linear and nonlinear ODEs and PDEs from first integrals/ conservation laws Solutions of differential

5 Textbook(s)/Supplementary Readings There is no required text but the following texts will be used for reference. 1) CRC handbook of Lie group analysis of differential, Volume 1: Symmetries, exact solutions and conservation laws by Nail H. Ibragimov (CRC Press) 2) Applications of Lie groups to differential by Peter J Olver (Springer - Verlag) 3) Differential : Their solution using symmetries by Hans Stephani (Cambridge University Press) 4) Handouts Handouts on topics will also been uploaded on the LUMS website Symbolic Packages: YaLie, Math Lie, GEM, Crack software for PDEs, SADE, Maple

Lahore University of Management Sciences. MATH 210 Introduction to Differential Equations

Lahore University of Management Sciences. MATH 210 Introduction to Differential Equations MATH 210 Introduction to Differential Equations Fall 2016-2017 Instructor Room No. Office Hours Email Telephone Secretary/TA TA Office Hours Course URL (if any) Ali Ashher Zaidi ali.zaidi@lums.edu.pk Math.lums.edu.pk/moodle