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Unit – I (18 Hours)Symmetries of Planar Objects  Symmetries of the Simplest ODE  The Symmetry Condition for First  Order ODEs  Lie Symmetries Solve FirstOrder ODEs.Unit – II (18 Hours)The Action of Lie Symmetries on the Plane  Canonical Coordinates  How to Solve ODEs with Lie Symmetries  The Linearized Symmetry Condition  Symmetries and Standard Methods  The Infinitesimal Generator.Unit –III (18 Hours)The Symmetry Condition  The Determining Equations for Lie Point Symmetries  Linear ODEs  Justification of the Symmetry Condition.Unit – IV (18 Hours)Reduction of order by Using Canonical Coordinates  Variational Symmetries, Invariant Solutions.Unit – V (18 Hours)Differential Invariants and Reduction of Order  The Lie Algebra of Point Symmetry Generators  Stepwise Integration of ODEs.Text Book : Symmetry Methods for Differential Equations – A Beginners Guide  Peter E. Hydon, Cambridge University Press, 2000.
Reference Book: Symmetries and Differential Equations  George W. Bluman and Sukeyuki Kumei, SpringerVerlag, New York, 1989. THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2013 batch onwards) Course : M.Sc. Code No. : Semester : IV No. of hours allotted : 6 Paper : Elective III No. of credits : 5 Title of the Paper : Algorithmic Graph Theory Course Objective: To study the advance Algorithmic approach in Graph theoretical problems and their efficiency by means of computational complexity. Unit  I (18 Hours) Graphs and their complements : Introduction – degree sequence – analysis. Paths and Walks: Introduction complexity – walks – the shortest path problem – weighted graphs and Dijkstra’s algorithm – data structures – Floyd’s algorithm. Unit  II (18 Hours) Trees and Cycles: Spanning tree algorithms – Prim’s algorithm data structure – Kruskal’s algorithm data structure and complexity – The Cheriton – Tarjan algorithm. Unit  III (18 Hours) Connectivity: Introduction – blocks – finding blocks of a graph – the DFS (Depth First Search) – complexity. Unit  IV (18 Hours) Hamiltonian cycles: Introduction – the crossover algorithm – complexity – the Hamiltonian closure – the extended multi graph algorithm – data structures for the segments – decision problems – NP completeness the travelling salesman problem – the TSPChristofides algorithm. Unit  V (18 Hours) Network Flows: Introduction – The FordFulkerson algorithm – Matching and flows – Menger’s theorems – disjoint paths and separating sets. Text Book: Graphs, Algorithms and Optimization – William Kocay, Donald L. Kresher, Chapman & Hall/CRC, 2005
Reference Book: Graphs: Theory and Algorithms – K. Thulasiraman, M.N.S. Swamy, Jhon Wiley & Sons, 1992 THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2013 batch onwards) Course : M.Sc. Code No. : Semester : IV No. of hours allotted : 6 Paper : Elective III No. of credits : 5 Title of the Paper : Numerical Analysis Course objective : To study Numerical computational skills, errors in Numerical Computation, Interpolation, Fourier Transform, Numerical Differentiation and Integration and numerical solution of PDE. Unit  I (18 Hours) Errors in Numerical calculations: Introduction – Computer and Numerical Software, computer languages, software packages – mathematical preliminaries – errors and their computation – a general error formula – error in a series approximation. Solution of algebraic and transcendental equations: Introduction – Muller method – Graffe’s rootsquaring method – LinBairstow’s method. Unit  II (18 Hours) Interpolation: Introduction – errors in polynomial interpolation – central differences – central difference interpolation formulae – Gauss central difference interpolation formula – Striling’s formula – Bessel’s formula – Everett’s formula – relationship between Bessel’s formula and Everett’s formula – Interpolation with unevenly spaced points – Hermite’s interpolation formula. Unit  III (18 Hours) Fourier Transform: Fourier approximation the Fourier transform – the fast Fourier transform – CooleyTukey algorithm – SadeyTukey algorithm – computation of inverse DFT. Approximation of functions – Chebyshev polynomials – economization o power series. Unit  IV (18 Hours) Numerical differentiation and Integration: Introduction – numerical differentiation – errors in numerical differentiation – the cubic spline method – maximum minimum values of a tabulated function. Numerical differentiation Boole’s and Weddle’s rules – Romberg integration – NewtonCotes integration formula. Numerical calculation of Fourier integrals – Trapezoidal rule Filon’s formula – the cubic spline method. Unit – V (18 Hours) Numerical solution of PDEs: Introduction – finite difference approximation to derivatives – Laplace’s equations – parabolic equations – iterative method for the solution iof equations – hyperbolic equations. Text Book: Introductory Methods of Numerical Analysis – S.S. Sastry, Prentice Hall, Fourth Edition, 2009
Reference book: Elementary Numerical Analysis Samuel D. Conte / Carl de Boor Tata McGrawHill, Third edition, 2009. THIAGARAJAR COLLEGE (AUTONOMOUS), MADURAI – 625 009 (Re Accredited with 'A' Grade by NAAC) Department of Mathematics (From 2011 – 2013 batch onwards) Course : M.Sc. Code No. : Semester : IV No. of hours allotted : 6 Paper : Elective III No. of credits : 5 Title of the Paper : Computer Algorithms and Data Structures Course objective : To study the various design techniques and general methods for solving problems using computer programming. To apply them as algorithms to solve specific problems and analyze and compare these algorithms. 