Thiagarajar college (autonomous), madurai – 9




НазваниеThiagarajar college (autonomous), madurai – 9
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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 First-Order 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.


Unit

Chapter/Section

I

Chapter 1 : 1.1 to 1.4

II

Chapter 2 : 2.1 to 2.6

III

Chapter 3 : 3.1 to 3.4

IV

Chapter 4 : 4.1 to 4.3

V

Chapter 5 : 5.1 to 5.3




Reference Book: Symmetries and Differential Equations

- George W. Bluman and Sukeyuki Kumei, Springer-Verlag, 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 closurethe extended multi graph algorithm – data structures for the segments – decision problems – NP completeness- the travelling salesman problem – the TSP-Christofides algorithm.

Unit - V (18 Hours)

Network Flows: Introduction – The Ford-Fulkerson 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


Unit

Chapter/Section

I

1: 1.1 to 1.3, 2: 2.1 to 2.7

II

4 : 4.4

III

6 : 6.1 to 6.4

IV

9 : 9.1 to 9.8

V

8: 8. 1 to 8.5



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 root-squaring method – Lin-Bairstow’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 – Cooley-Tukey algorithm – Sadey-Tukey 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 – Newton-Cotes 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


Unit

Chapter/Section

I

5.1 – 5.4

II

5.5 – 5.8

III

7(Full)

IV

6.1 – 6.5

V

6.6 – 6.10



Reference book: Elementary Numerical Analysis- Samuel D. Conte / Carl de Boor

Tata McGraw-Hill, 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.


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