M.Sc. Mathematics

Semester III

  • COURSE CODE
    COURSE NAME
    CREDITS
  • JMMH-301

    Functional Analysis:

    UNIT-I

    Normed linear spaces, Banach spaces, Examples and counter examples, Quotient space of normed linear spaces and its completeness; Equivalent norms.

    UNIT-II

    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.

    UNIT-III

    Dual spaces, weak convergence, open mapping and closed graph theorems; Hahn Banch theoremfor real and complex linear spaces.

    UNIT-IV

    Inner product spaces, Hilbert spaces–Orthonormal sets; Bessel’s inequality, complete orthonormal sets and Perseval ’s identity.

    UNIT-V

    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
  • JMMH-302

    Fluid Dynamics:

    UNIT-I

    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.

    UNIT-II

    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.

    UNIT-III

    One and two dimensional in-viscid incompressible flow-Equation 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.

    UNIT-IV

    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; Non-dimensional numbers and their physical significance.

    UNIT-V

    Incompressible viscous fluid flows-Steady flow between two parallel plates (non-porous 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
  • JMMH-303

    Numerical Method:

    UNIT-I

    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, Regula-Falsi method, Secant method, Newton- Raphson method

    UNIT-II

    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.

    UNIT-III

    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’s-Cotes integration formulae.

    UNIT-IV

    Numerical solution of ordinary differential equations: Taylor’s series, Picard’s successive approximations, Euler’s, Modified Euler’s, Runge-Kutta & Milne’s Predictor-Corrector methods; Simultaneous and higher order equations: Taylor’s series method and Runge-Kutta method, Boundary value problems: Finite differences method.

    UNIT-V

    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, C-N 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 2012-13 Page 16 M.Sc. – Mathematics
    04
  • JMMH-304

    Programming In C And Data Structures:

    UNIT-I

    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.

    UNIT-II

    C-Programming: 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.

    UNIT-III

    Assignment statement, Looping, Nested loops, Break and continue statements, Switch statement, goto statement; Arrays, String processing, functions, Recursion, Structures & unions.

    UNIT-IV

    Simple Data Structures: Stacks, queues, single and double linked lists, circular lists, trees, binary search tree. C-implementation of stacks, queues and linked lists.

    UNIT-V

    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
  • JMMH-352

    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 Runge-Kutta 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
  • JMMH-351

    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 2-diamensional 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 Credits
     
    20