MSc mathematics third semester syllabus: Model science college Jabalpur

 Govt. Model Science College (Auto), Jabalpur

Department of Mathematics & Computer Science

M.Sc. III Semester (Mathematics)

Unit Paper I : Fuzzy Sets and Their Application-I MM : 35

I. Idea of fuzzy set and membership function, Definition of a fuzzy set, membership function, representation of membership function, General definitions and properties of fuzzy sets, Support, height, equality of two fuzzy sets, containment, examples.

II. Union and Intersection of two fuzzy sets, Complement of a fuzzy set, normal fuzzy set, α-cut set of a fuzzy set, strong α-cut, convex fuzzy set, a necessary and sufficient condition for convexity of a fuzzy set (Theorem 1), Decomposition of fuzzy sets, Degree of sub sethood, Level set of a fuzzy set, Cardinality, fuzzy cardinality, examples.

III. Other important operations on fuzzy sets, Product of two fuzzy sets, Product of a fuzzy set with a crisp number, Power of a fuzzy set, Difference of two fuzzy sets, Disjunctive sum of two fuzzy sets, example.

IV. General properties of operations on fuzzy sets, Commutativity, associativity, distributivity, Idempotent law, identities for operations, Transitivity, involution, Demorgans laws, proofs and examples, Some important theorems on fuzzy sets, set inclusion of fuzzy sets and corresponding α-cuts and strong α-cuts (Theorem 1).

V. Comparison of α-cut and strong α-cut, Order relation of scalars α is inversely preserved by set inclusion of corresponding α-cuts and strong α-cuts, α-cut of union and intersection of two fuzzy sets, α-cut of complement of a fuzzy set (Theorem 2), Examples, α-cuts and strong α-cuts of union and intersection of arbitrary collection of fuzzy sets.

Unit Paper II : Advanced Special Function -I MM : 35

I. Gamma and Beta Functions : The Euler or Mascheroni Constant γ, Gamma Function, A series for Г' (z) / Г (z) , Difference equation Г(z+1) = zГ(z).

II. Beta function, value of Г(z) Г(1-z), Factorial Function, Legendre's duplication formula, Gauss multiplication theorem.

III. Hypergoemetric and Generalized Hypergeometric functions: Function 2F1 (a,b;c;z) A simple integral form evaluation of 2F1 (a,b;c;z) , Values of F (a,b;c;1) and F (-n, b;c;1) etc.

IV. Contiguous function relations, Hyper geometrical differential equation and its solutions, F (a,b;c;z) as function of its parameters.

V. Elementary series manipulations, Simple transformation, Relations between functions of z and 1-z

Unit Paper III : Advance Numerical Analysis -I MM : 35

I. Piece wise and spline interpolation: Piecewise Linear Interpolation, Piecewise Quadratic Interpolation, Piecewise cubic Interpolation, Piecewise cubic Interpolation using Hermitetype data, Quadratic and cubic spline Interpolation, Bivariate interpolation

II. Approximation : Least squares Approximation, Gram-schmitt orthogonalization process, chebsyshev polynomials ,legendre polynomials .

III. Uniform approximation : Uniform norm , uniform polynomial approximation, best Approximation, best Uniform approximation condition for uniform best approximation .

IV. Rational approximation, choice of method, Runge`s example. 

V. Numerical differentiation: Methods based on interpolation Method, Methods based on finite difference operators , methods based on undetermined coefficients, optimum choice of step length.

Unit Paper IV : Operation Research –I MM : 35

I. Operations Research and its scope, Origin and Development of Operations Research, Characteristics of Operations Research.

II. Model in Operations Research, Phase of Operations Research, Uses and Limitations of Operation Research, Linear Programming Problems.

III. Graphical procedure, Graphical solution of property behaved L.P problems. Graphical solution in some exceptional cases.

IV. General Linear Programming Problem : Simplex Method exceptional cases, artificial variable techniques ; Big M method, two phase Method and problem of degeneracy.

V. Concept of Duality : Definition of Primal-dual problems ,Symmetric Primal-dual problems, Unsymmetric Primal-dual problems, General rules for converting any primal into its dual. Fundamental theorem of duality.

Unit Paper V : INTEGRAL TRANSFORM-1 MM : 35

I. Problem related to Laplace transform Initial and bounding value problems, simultaneous ordinary differential equations. Problem related to solution of partial differential equations. Application of Laplace Transformed in Differential Equations

II. Two dimensional Laplace`s Equation (Cartesian and Polar form). Three dimensional Laplace`s Equation to related problems.

III. Notion of wave Equation. General solution of wave Equations. Solution by separation of variables. Solution of two dimensional wave equation , three dimensional wave equation.

IV. Definition: Integral Equations, problems related to Integral Equations of convolution type. Integral differential equation . Abel`s differential equation

V. Notion of Heat Equations. One and two dimensional heat conduction equation. Solution by separation of variables and problems based on it.