The complex number system. Extended complex plane. Analytic functions. Cauchy-Riemann conditions. Mappings by elementary functions. Conformal mappings and Mobius Transformation. Complex integration. Cauchy-Goursat theorem. Cauchy integral formula. The Homotopic version of Cauchy's theorem and simple connectivity. Morera’s and Liouvile’s theorems. Uniform convergence of sequences and series. Taylor's and Laurent's series. Singularities, zeros and Poles. Isolated singularities and residues. Cauchy residue theorem. Evaluation of real integrals. The Argument Principle and Rouche's theorem. Maximum Modulus Theorem.

Introduction to PDE. First order quasi-linear equations. Nonlinear equations. Cauchy-Kowalewski’s theorem. Higher order equations and characteristics. Classification of second order equations. Riemann’s method and applications. One dimensional wave equation and De’Alembert’s method. Solution of three dimensional wave equation. Method of decent and Duhamel’s principle. Solutions of equations in bounded domains and uniqueness of solutions. BVPs for Laplace’s and Poisson’s equations. Maximum principle and applications. Green’s functions and properties. Existence theorem by Perron’s method. Heat equation, Maximum principle. Uniqueness of solutions via energy method. Uniqueness of solutions of IVPs for heat conduction equation. Green’s function for heat equation.

Introduction to probability theory, Probability and counting, Some applications.

Limit theorems: Probability spaces, random variables, independence, Kolmogorov’s 0 − 1 law, Borel-Cantelli lemma, Integration, Expectation, Variance, The weak law of large numbers, The probability distribution function, Convergence of random variables, The strong law of large numbers, Weak convergence, The central limit theorem, Markov operators, Characteristic functions.

Discrete Stochastic Processes: Conditional Expectation, Martingales, Doob’s convergence theorem, Doob’s decomposition of a stochastic process, L^(p) inequality, Random walks, A discrete Feynman-Kac formula, Markov processes.

Continuous Stochastic Processes: Brownian motion, Stopping times, Continuous time martingales, Recurrence of Brownian motion, Feynman-Kac formula revisited, The Ito integral for Brownian motion, Processes of bounded quadratic variation, The Ito integral for martingales, Stochastic differential equations.

Definition and sources of errors, solutions of nonlinear equations; Bisection method, Newton's method and its variants, fixed point iterations, convergence analysis; Newton's method for non-linear systems; Finite differences, polynomial interpolation, Hermite interpolation, spline interpolation; Numerical integration - Trapezoidal and Simpson's rules, Gaussian quadrature, Richardson extrapolation; Initial value problems - Taylor series method, Euler and modified Euler methods, Runge-Kutta methods, multistep methods and stability; Boundary value problems - finite difference method, collocation method.

Topological spaces, Basis for a topology, Limit points and closure of a set, Continuous and open maps, Homeomorphisms, Subspace topology, Product and quotient topology.

Connected and locally connected spaces, Path connectedness, Components and path components, Compact and locally compact spaces, One point compactification.

Countability axioms, Separation axioms, Urysohn’s Lemma, Urysohn’s metrization theorem, Tietze extension theorem, Tychonoff’s theorem, Completely Regular Spaces, Stone-Cech Compactification.

Normed linear spaces, C0, C, lp, Lp, 1≤ p ≤ ∞, C[a,b], dimension, linear transformations -continuity and boundedness, linear functional-continuity, compactness of unit ball of finite dementional spaces, equivalence of norms and continuity of inear transformations of finite dimensional spaces, dual spaces duals of C0, lp, Lp, 1≤ p ≤ ∞, separability, non-separability of l∞. reflexive spaces. Horn-Banach theorem for real and complex normed linear spaces, Uniform Boundedness Principle and its applications. Closed Graph Theorem, Open Mapping Theorem and their applications. Inner product spaces, Hilbert spaces. Orthonormal basis, Projection theorem and Riesz Representation Theorem.

Concept and calculation of Green's function, Approximate Green's function, Green's function method for differential equations, Fourier Series, Generalized Fourier series, Fourier Cosine series, Fourier Sine series, Fourier integrals. Fourier transform, Laplace transform, Z-transform, Hankel transform, Mellin transform. Solution of differential equation by Laplace and Fourier transform methods.

Introduction to optimization, Formulation of linear Optimization problems, Convex set. Linear Programming model, Graphical method, Simplex method, Finding a feasible basis – Big M and two phase Simplex method, revised simplex method. Duality in Linear Program. Primal-dual relationship & economic interpretation of Duality, Dual Simplex Algorithm, Sensitivity analysis.

Network analysis: Transportation & Assignment problem, Integer programming problem: Formulation, Branch& Bound and Cutting Plane methods. Dynamic Programming (DP).

Non-linear Programming: Lagrange multipliers and Kuhn - Tucker conditions, convex optimization.

Numerical optimization techniques: line search methods, gradient methods, Newton's method, conjugate direction methods, quasi-Newton methods, projected gradient methods, penalty methods.