MSc mathematics fourth semester syllabus: Model science college Jabalpur

 Govt. Model Science College (Auto), Jabalpur

Department of Mathematics & Computer Science

M.Sc. IV Semester (Mathematics)

Unit Paper I : Fuzzy Sets and their Applications - II MM : 35

I. Fuzzy sets: Basic Definitions, α-level sets, Convex fuzzy set, Basic operations on fuzzy sets, types of fuzzy sets, Extensions: Types of fuzzy sets, Further operations on fuzzy sets, Cartesian product, Algebraic products, Bounded sum and Difference, t-norm & t-conorm.

II. Extension principle and applications, Zadeh extension principle, image and inverse image of fuzzy sets, fuzzy numbers, algebraic operations with fuzzy numbers, extended operation and its properties, Special extended operation, addition, subtraction, product and division of fuzzy numbers.

III. Fuzzy relations on fuzzy sets, The union & intersection of fuzzy relations, Composition of fuzzy relations, max-* and max-product compositions, minmax composition and its properties, reflexivity, symmetry, transitivity, and their examples, special fuzzy relations, similarity relation.

IV. Fuzzy graphs: Definition and Examples, Fuzzy sub-graph, Spanning subgraph, path in a fuzzy graph, strength and length of a path,μ -length and μ- distances, connected nodes, fuzzy forest, fuzzy tree, Examples, Fuzzy Analysis: Fuzzy functions on fuzzy sets, classical function, fuzzy function, Examples.

V. Fuzzy Logic; An overview of classical logic, Its connectives, Tautologies, Contradiction, Fuzzy logic, logical connectives for fuzzy logic, Examples, Approximate reasoning, its rules, examples, other of implication operations, Linguistic hedges, Fuzzy quantifiers, Examples.

Unit Paper II - ADVANCED SPECIAL FUNCTION MM : 35

I. Legendre polynomials : Definition of Pn(x), Generating functions, recurrence relations, Beltrami`s result Christoffels summation formula , Murphy formula, Rodrigues formula Bateman's generating relations and other generating relations.

II. Legendre differential equation and its solutions . Laplace first and second integral for Pn(x) . Orthogonal properties of Legendre polynomials. Expansion involving Legendre polynomials . Fourier - Legendre Expansion

III. Bessel functions : Definition of Jn (z), Generating functions Bessel's differential equation, recurrence relations , Bessel's integral with index half and an odd integer, Orthogonality of Bessel functions

IV. Hermite polynomial : Definition of Hermite polynomials Hn(x), Pure recurrence relations, Differential recurrence relations, Rodrigue's formula, Other generating functions, Orthogonality, Expansion of polynomials, more generating functions

V. Laguerre Polynomials : The Laguerre Polynomials Ln(X), Generating functions, Pure recurrence relations, Differential recurrence relation, Rodrigues formula, Orthogonality, Expansion of polynomials, Special properties, Other generating functions.

Unit Paper - III : Advanced Numerical Analysis II MM : 35

I. Extrapolation Methods for Numerical Differentiation. Multistep methods for Numerical Solution of initial value problems. Explicit and implicit Multi step methods.

II. General Multistep methods : r (x) ans s (x) for linear multiple step methods . Convergence of Multi step methods. Predictor correctior methods.

III. Stability analysis of Multistep methods: First order differential Equations, Stability of Predictor- Corrector Methods. Stability of PMp CMc methods , second Ordinary Differential.

IV. Ordinary differential Equations: Three kind of Boundary conditions . Finite Difference methods, Linear second order differential Equations, Non linear second order differential Equations

V. Finite element method : Finite element Ritz Finite element method methods, Linear Boundary Value Problems, mixed boundary conditions

Unit Paper - IV : Operations Research MM : 35

I. Replacement Problems : Replacement of Items that deteriorate , Replacement policy for items whose maintenance cost increase with time and money value is constant. Money value . present worth factor (PWF) and discount rate . Replacement policy for items whose maintenance cost increase with time and money value with constant rate. Individual Replacement policy, Mortality theorem, Group replacement policy.

II. Assignment problems : Mathematical formulation, Fundamental theorems, Hungarian method for assignment problem. Unbalanced assignment problem , The Travelling Salesman (Routing) problem, Job sequencing Processing n Jobs through 2 machines , Processing n Jobs through 3 machines, a graphical method.

III. Transportation problems : North - West Corner Method Least - Cost Method. Vogel's Approximation Method, MODI Method, Exceptional cases and problem of degeneracy.

IV. Network analysis, constraints in Network, Construction of network, Critical Path Method (CPM) PERT, PERT Calculation, Resource Leveling by Network Techniques and advances of network (PERT/CPM)

V. Game theory - Two persons, Zero - Sum Games, Maximix - Minimax principle, games without saddle points -Mixed strategies, Graphical solution of 2xm and mx2 games, Solution by Linear Programming, Non-Linear programming Techniques - Kuhn - Tucker Conditions, Non - negative Constraints

Unit Optional Paper - V : Integral Transforms-II MM : 35

I Application of Laplace Transform to boundary value problems. 

II Electric Circuits problems, related to application of Electric Circuits. Application to dynamics, application to heat conduction equation, application to wave equations, Application to Beams.

III The complex Fourier Transform, Inversion Formula, Fourier cosine and sine transform.

IV Properties of Fourier Transforms, Convolution & Parseval's identity. 

V Fourier Transform of the derivatives, Finite Fourier Sine & Cosine Transform, Inversion Operational and combined properties Fourier transform