M.Sc. Mathematics
Semester III

COURSE CODECOURSE NAMECREDITS

JMMH301
Functional Analysis:
UNITI
Normed linear spaces, Banach spaces, Examples and counter examples, Quotient space of normed linear spaces and its completeness; Equivalent norms.
UNITII
Reisz Lemma, Basic properties of finite dimensional normed linear spaces; Bounded linear transformations and normed linear spaces of bounded linear transformations; Uniform boundedness theorem and some of its applications.
UNITIII
Dual spaces, weak convergence, open mapping and closed graph theorems; Hahn Banch theoremfor real and complex linear spaces.
UNITIV
Inner product spaces, Hilbert spaces–Orthonormal sets; Bessel’s inequality, complete orthonormal sets and Perseval ’s identity.
UNITV
Structure of Hilbert spaces, Projection theorem, Riesz representation theorem, Adjoint of and operator on Hilbert space, Self adjoint operators, Normal and Unitary operators. Projections
Books:
 1. E. Kreyszig, Functional Analysis and its application, John Wiley and sons.
 2. J.N. Sharma & A. R. Vashistha, Functional Analysis, Krishana Publication.
 3. G. Bachman & L.Narici, Functional Analysis Academic Press.
 4. H.C. Goffman and G.Fedrick, First course in Functional Analysis, PHI.
 5. B.V. Limaye, Functional Analysis, New Age International Limited.
04 
JMMH302
Fluid Dynamics:
UNITI
Classification of fluid and its physical properties, Continuum hypothesis; Kinematics of fluids Methods of describing fluid motion, Translation, Rotation and deformation of fluid elements, Stream Lines, Path lines and Streak lines, concepts of Vortices.
UNITII
General theory of stress and rate of strain in a real fluid–Symmetry of stress tensor, Principal axes and Principle values of stress tensor, Constitutive equation for Newtonian fluid; Conservation laws Conservation of mass, momentum and energy.
UNITIII
One and two dimensional inviscid incompressible flowEquation of continuity and motion using stream tube, Circulation, Velocity potential, Irrotational flow; Some theorems about rotational and irrotational flows Stokes theorem, Kelvin’s minimum energy theorem, Gauss theorem, Kelvin’s circulation theorem.
UNITIV
Vortex motion and its elementary properties; Integration of Euler’s equation under different conditions;Bernoulli’s equation; Stream function in two dimensional motion; Complex variable technique; Flow past a circular cylinder; Blasius theorem; Milne’s circle theorem; Sources, Sinks and Doublets; Dynamical similarity; Buckingham’s π theorem; Nondimensional numbers and their physical significance.
UNITV
Incompressible viscous fluid flowsSteady flow between two parallel plates (nonporous and porous)Plane couette flow; Plane poiseuille flow, Generalized plane couette flow, Steady flow of two immiscible fluids between two rigid parallel plates; Steady flow through tube of uniform circular cross section, Steady flow through annulus under constant pressure gradient.
 1. S. W. Yuan, Foundations of fluid mechanics, Prentice Hall of India Prt. Limited.
 2. R. K. Rathy, An introduction of fluid dynamics, Oxford and IBH Pub Co.
 3. G. K. Betchelor, An introduction of fluid dynamics, Oxford University Books.
 4. F. Charlton, Text book of fluid dynamics, C.B.S. Publishers.
04 
JMMH303
Numerical Method:
UNITI
Errors in numerical calculations: Absolute, Relative and percentage errors, A general error formula, Error in a series approximation; Solutions of algebraic & transcendental equations: The Bisection method, The iteration method, RegulaFalsi method, Secant method, Newton Raphson method
UNITII
Interpolation: Errors in Polynomial interpolation; Finite differences: Forward, Backward and Central differences, Symbolic relations, Difference of polynomial, Newton’s formulae of interpolation, Central difference interpolation formulae: Gauss’s , Bessel’s & Stirling’s formulae, Interpolation with unevenly spaced points: Lagrange’s interpolation formula, Interpolation with cubic splines, Divided differences and their properties, Newton’s general interpolation formula, Inverse interpolation, Method of successive approximations.
UNITIII
Numerical differentiation and integration: Forward, Backward and Central difference formulae for first and second order derivatives; Errors in numerical differentiation; Numerical integration, Trapezoidal rule; Simpson’s 1/3 rule, Simpson’s 3/8 rule; Boole’s and Weddle’s rules; Newton’sCotes integration formulae.
UNITIV
Numerical solution of ordinary differential equations: Taylor’s series, Picard’s successive approximations, Euler’s, Modified Euler’s, RungeKutta & Milne’s PredictorCorrector methods; Simultaneous and higher order equations: Taylor’s series method and RungeKutta method, Boundary value problems: Finite differences method.
UNITV
Numerical solution of partial differential equations: Finite difference approximations to derivatives; Laplace’s equation: Jacobi’s method, Gauss Seidel method, The ADI method; Parabolic equations: Explicit scheme, CN scheme; Hyperbolic equations: Explicit scheme, Implicit scheme.
Books:
 1. S.S. Sastry, Introductory Methods of Numerical Analysis, Prentice Hall of India.
 2. Grewal B. S, Numerical Methods in Engineering and Science, Khanna Publishers.
 3. M.K. Jain, S.R.K Iyengar & R.K.Jain, Numerical methods of Scientific and Engineering Computation, New Age Pub.
Syllabus M.Sc.Mathematics Applicable w.e.f. Academic Session 201213 Page 16 M.Sc. – Mathematics
04 
JMMH304
Programming In C And Data Structures:
UNITI
Computer system introduction; Characteristics and classification of computers, CPU, ALU, Control unit, data & instruction flow, primary, secondary and cache memories; RAM, ROM, PROM, EPROM; Programming language classifications.
UNITII
CProgramming: Representation of integers, real, characters, constants, variables; Operators: Precedence & associative, Arithmetic, Relation and Logical operators, Bitwise operators, increment and decrement operators, comma operator, Arithmetic & Logical expression.
UNITIII
Assignment statement, Looping, Nested loops, Break and continue statements, Switch statement, goto statement; Arrays, String processing, functions, Recursion, Structures & unions.
UNITIV
Simple Data Structures: Stacks, queues, single and double linked lists, circular lists, trees, binary search tree. Cimplementation of stacks, queues and linked lists.
UNITV
Algorithms for searching, sorting and merging e.g., sequential search, binary search, insertion sort, bubble sort, selection sort, merge sort, quick sort, heap sort.
Books:
 1. Balaguruswami, Programming in C, Tata McGraw Hill.
 2. Y.P. Kanetkar, Let us C, BPB, India.
 3. Brian Kernighan and Dennis Ritchie, The C programming Language, PHI.
04 
JMMH352
Numerical Method Lab:
Write programs in C:
 1. To implement floating point arithmetic operations i.e., addition, subtraction, multiplication and division.
 2. Algebraic and transcendental equations using Bisection, Newton Raphson, Iterative method of false position, rate of conversions of roots in tabular form for each of these methods.
 3. Gauss Interpolation, flowchart C program and output.
 4. Implement numerical differentiation.
 5. Implement numerical integration using Simpson's 1/3 and 3/8 rules.
 6. Implement numerical integration using trapezoidal rule.
 7. Solution of differential equations using 4th order RungeKutta method.
 8. Numerical solution of ordinary first order differential equation Euler’s method with algorithm, flowchart C Program and output.
 9. Newton’s and Lagrange’s interpolation with algorithm, flowchart C Program and output.
02 
JMMH351
Programming In ‘C’ And Data Structures Lab:
Write programs in C:
 1. To search an element in array using Linear search.
 2. To search an element in the 2diamensional array using Linear search.
 3. To merge two sorted array into one sorted array.
 4. To perform the following operation in Matrix a:
 a) Addition
 b) Subtraction
 c) Multiplication
 d) Transpose.
 5. To perform the swapping of two numbers using call by value and cell by reference.
 6. To perform the following operation on strings using strings functions b:
 a) Addition
 b) Copying
 c) Reverse
 d) Length of string.
 7. To search an element in the array using Iterative Binary search.
 8. To search an element in the array using Recursive Binary search.
 9. To implement Bubble sort.
 10. To implement selection sort.
 11. To implement Insertion sort.
 12. To implement Quick sort.
 13. To implement Merge sort.
 14. To implement Stack using array.
 15. To implement Queue using array.
 16. To implement Linked List.
02 
Total Credits20