MT-505 : Ordinary Differential Equations
Review : General remarks on solutions of differential equations, Families of curves,
Orthogonal trajectories.
1. Second order linear equations : The general solution of the homogeneous equations,
Use of a known solution to find another solution, Homogeneous equations with constant
coefficients.The method of undetermined coefficients.The method of variation ofparameters.
2. Qualitative Properties of solutions of ordinarydifferential equations of order two :
Sturm separation theorem. Normal form, Standard form, Sturm's comparison theorem.
3. Power Series solutions : Review of power series,Series solutions of first order
equations, Second order linear equations, Ordinarypoints, Regular singular points,
Indicial equations, Gauss's Hypergeometric equation, The point at infinity.
4. Systems of first order equations : General remarks on systems, Linear systems,
Homogenous linear systems with constant coefficient. Non-linear systems, Volterra's
Prey-Predator equations.
5. Non-linear equations : Autonomous systems, Critical points, Stability, Liapunov's
direct method, Nonlinear mechanics, Conservative systems.
6. The existence and uniqueness of solutions. The method of successive
approximations, Picard's theorem, Systems, The second order linear equations.
Text Book : G.F. Simmons : Differential equations with applicat
ions and Historical Notes, SecondEdition (Mc-Graw Hill). Sections : 15 to 19, 24 to 31, 54 to 63, 68 to 70.
Reference Book :
1. G. Birkhoff and G.C. Rota : Ordinary differential equations. (John Wiley and Sons)
2. E. A. Coddington : Ordinary differential equations. Prentice Hall of India.
3. S. G. Deo, V. Lakshmikantham, V. Raghvendra. Text book of Ordinary Differential Equations. Second edition.Tata Mc-Graw Hill.


                                 MT-504: Numerical Analysis
0. Preliminaries :
Convergence, Floating Point Number Systems, Floating Point Arithmetic.
1. Root finding methods :
Fixed Point Interaction Schemes, Newton’s Method, Secant Method, Accelerating
Convergence.
2. System of Equations :
Formation of Systems of Equations, Gaussian Elimination, Pivoting Strategies, Errors
Estimates and Condition Number, LU decomposition, Direct Factorization, Iterative
Techniques for Linear Systems, Nonlinear Systems ofEquations.
3. Eigenvalues and Eigenvectors :
The Power Method, The Inverse Power Method, Reduction to Symmetric Tridiagonal
form, Eigenvalues of Symmetric Tridiagonal Matrices.
4. Differentiation and Integration:
Numerical differentiation, using Lagrange’s Interpolating polynomial, Numerical Integration, Newton-Cotes Quadrature, Composite Newton-Cotes Quadrature.
5. Initial Value Problems of Ordinary Differential Equations :
Euler’s Method, Runge-Kutta Methods, Multistep Methods, Convergence and Stability Analysis
Text Book: 1. Brian Bradie, A Friendly Introduction to Numerical Analysis, Pearson Prentice Hall 2007.
Articles from the Text Book : 1.2 – 1.4, 2.3 – 2.6, 3.1, 3.2, 3.4 -3.6, 3.8, 3.10, 4.1,
4.2, 4.4, 4.5, 6.1, 6.2, 6.4-6.6, 7.2-7.6.
2. John H. Mathews; Kurtis D. Fink, NUMERICAL METHO
DS Using Matlab, 4thEd., Pearson Education (Singapore) Pte. Ltd., Indian Branch, Delhi 2005
(SciLab commands similar to MatLab commands can beused for problems)
Reference Books: 1. K .E. Atkinson: An Introduction to Numerical Analysis.
2. J. I. Buchaman and P. R. Turner, Numerical Methods and Analysis.
3.M.K. Jain, S.R.K. Iyengar, R.K. Jain, Numerical Methods for scientific & Engineering Computation, 5th Edition New Age International Publication.

                                      MT-503 : Group Theory 

1. Revision of definition and examples of groups, subgroups. 

2. Cyclic Groups, Classification of subgroups of cyclic groups. 

3. Permutation Groups - Revision 

4. Isomorphism, Cayley’s theorem, properties of isomorphisms, automorphism, 

5. Revision of Cosets and Lagrange’s theorem. Orbit-stabilizer theorem, the rotation group of a cube and a soccer ball 

6. External Direct Products 

7. Normal subgroups and factor groups, Internal direct products 

8. Group Homomorphism 

9. Fundamental theorem of finite abelian groups 

10. Sylow theorems 

11. Finite simple groups 

Text Books: Joseph Gallian – Contemporary Abstract Algebra (Narosa Publishing House). Chapter 2 to 11, 24, 25. 

Reference Books: 1. I.S. Luthar and I.B.S. Passi : Algebra (Volume 1) Groups (Narosa Publishing House ) 2. I.N. Herstein : Topics in Algebra (Wiley -Eastern Ltd) 3. M. Artin : Algebra (Prentice Hall) 4. N.S. Gopala Krishnan : University Algebra (Wiley-Eastern Ltd) 5. Fraleigh : A First Course in Abstract Algebra 6. Dummit and Foote : Abstract Algebra ( Wiley-Eastern Ltd)

                                         MT - 502: Advanced Calculus 

1. Derivative of a scalar field with respect to a vector, Directional derivative, Gradient of a scalar field, Derivative of a vector field, Matrix form of the chain rule, Inverse function theorem and Implicit function theorem. 

2. Path and line integrals, The concept of work as a line integral, Independence of path, The first and the second fundamental theorems of calculus for line integral, Necessary condition for a vector field to be a gradient. 

3. Double integrals, Applications to area and volume, Green's Theorem in the plane, Change of variables in a double integral, Transformation formula, Change of variables in an n-fold integral. 

4. The fundamental vector product, Area of a parametric surface, Surface integrals, The theorem of Stokes, The curl and divergence of a vector field, Gauss divergence theorem, Applications of the divergence theorem. 

Text Book: T. M. Apostol: Calculus, Vol. II (2nd edition) (John Wiley and Sons, Inc.) Chapter 1: Sections 81 to 8.22 Chapter 2: Sections 10.1 to 10.11 and 10.14 to 10.16 Chapter 3: Sections 11.1 to 11.5 and 11.19 to 11.22 and 11.26 to 11.34 Chapter 4: Sections 12.1 to 12.15, 12.18 to 12.21 (For Inverse function theorem and Implicit function theorem refer the book “Mathematical Analysis” by T. M. Apostol.)

Reference Books : 1. T. M. Apostol: Mathematical Analysis (Narosa publishing house) 2. W. Rudin: Principles of Mathematical Analysis (Mc-Graw Hill) 3. A. Devinatz: Advanced Calculus, (Holt, Rinehart and Winston), 1968

                                                              MT - 501: Real Analysis 

1. Measure Theory: Preliminaries, Exterior Measure, Measurable Sets and Lebesgue Measure, Measurable Functions.

 2. Integration Theory: The Lebesgue Integral, basic properties and convergence theorems. The space L^1 of integrable functions, Fubini’s theorem. 

3. Differentiation and Integration: Differentiation of the integral, Good kernels and approximation to the identity, differentiation of functions. 

Text Book: Real Analysis, E. Stein and R. Shakharchi, New Age International Publishers, Princeton Lecture Notes III. Chapter 1 - Sections 1 to 4, Chapter 2 - Sections 1 to 3, Chapter 3 - Sections 1 to 3. 

Reference Books: 1. Karen Saxe : Beginning Functional Analysis (Springer International Edition) 2. N. L. Carothers: Real Analysis (Cambridge University Press) 3. W. Rudin : Principles of Mathematical Analysis (Mc-Graw Hill) 4. H. Royden, Real Analysis, McMillan Publishing Company