Partial differentiation- Taylors series, Maxima and minima. Jacobians. Multiple Integrals, Gradient, divergence and curl, Normal and tangent to a surface, Line and surface integrals, Gauss and Stokes theorems. Differential Equations with constant coefficients. Cauchy-Euler’s equation Laplace Transform. Matrices- Row echelon form, Gauss elimination method, Rank, Eigenvalues and vectors, Quadratic forms.
Sets, Basic operations on sets. Mapping. Relation and their various representation, Equivalence and Partial order relation. Complex number and their properties. Roots of complex numbers. Limit and continuity. Differentiation. Tangent to a curve. Taylor’s series. Maxima and minima. Integrals of elementary functions. Substitution and partial fractions. Definite integral as a limit of sum. Application of integrals. Matrices and Determinants. Solution of equations.
Convergence of sequences and series, Second order linear differential equations, Solution in series, Bessel and Legendre functions, Partial differential equations, Equations of vibrating string, One dimensional wave and heat conduction equations, Functions of a complex variable, Analytic functions, Cauchy-Riemann equations, Conformal mapping, Poles and singularities, Complex Integration, Taylor’s and Laurent’s series, Cauchy residue theorem and applications.
Convergence and divergence of series. Fourier series. Two dimensional coordinate Geometry: Line, Circle, Ellipse, Parabola. Equation of a tangent to a curve. Vectors and their algebra. Direction cosines and direction ratios. Dot and cross products. Projection of a vector. Equations of a line, plane and sphere. Partial differentiation. Taylor’s series. Gradient of a scalar. Solution of first order differential equations. Initial and boundary value problems. Linear differential equations with constant coefficients. Bisection method, Newton-Raphson method. Interpolation. Trapezoidal and Simpson’s rule.
Representation of data, measures of central tendency, dispersion, skewness and kurtosis. Permutations and combinations. Axioms of probability, Conditional probability, Multiplication and addition theorems, Baye’s theorem. Random Variable, Discrete and continuous distributions (Binomial, Uniform, Normal and Poisson distributions). Elementary sampling theory. Test of hypothesis and significance. Curve fitting by the method of least squares. Correlation and regression. Time Series Analysis.
Conditional probability, Bayes theorem, random variables, properties: probability and cumulative density functions, MGF and CF, joint, marginal and conditional distributions, Probability distributions: Bernoulli, binomial, Poisson, negative binomial, geometric distributions. Uniform, exponential, normal, gamma, Earlang, Weibull distributions. Reliability: MTTF, system Reliability. Random processes: Averages, Stationary processes, Random walk, wiener process, Semi-random telegraph signal process, ergodic processes. PSDF, Poisson processes, Markov chains.
Relations, Lattices, Recursive functions, Generating functions, Recurrence relations, Z transform. Propositional calculus and Quantifiers. Graph Theory: Graphs, Subgraphs, Homeomorphic graphs, Isomorphism, Eulerian and Hamiltonian graph, Weighted graphs, Graph Coloring, Directed graphs, Trees, Rooted trees, Binary trees, Tree traversal. Shortest path algorithm. Algebraic Structures: Groups, Rings, Integral domains, Fields. Languages and Grammars, Finite State Machine, Finite State Automata.
Method of weighted residuals. Elements of calculus of variations. Application to ordinary differential. Derivation of element equations and their assembly. Imposition of boundary conditions and solution of assembled equations. Simple elements in two and three dimensions. Triangular, rectangular, quadrilateral elements. Serendipity, Sub-parametric and Iso-parametric elements. Applications for beam bending problems. Finite element modeling of a single variable problem. Time-dependent problems. Eigen-value Problems. Imposition of Boundary Conditions, Numerical integration, Natural coordinates.
Elementary row operations, row echelon forms. Linear system of equations, Gauss elimination, LU decomposition. Inversion of a matrix, Block matrices. Iterative methods. Eigen values and eigenvectors, modal matrix, linear independence, diagonalization. Power and inverse power methods, eigen systems of a Hermitian matrix. Jacobi method. Q-R algorithm. Generalized eigen value problem. Quardratic form, canonical form. Powers and functions of matrices. Application to solving discrete dynamical systems and system of differential. Norm of a vector and a matrix. Spectral radius. Gershgorin’s theorem. Condition number. Ill-conditioned system.
Laplace transform and applications to partial differential equations. Complex form of Fourier Series, Fourier integral theorem (without proof), Fourier Sine and Cosine transforms, applications to partial differential equations. The Hankel transform, application to partial differential equations. Z-transform applications to difference equations. Basic operational properties of Mellin Transform, applications of Mellin transform to summation of series, generalized Mellin Transform.
Measure of central tendency and dispersion, moments, skewness and kurtosis, Box and Whisker plot. Karl Pearson’s and Spearman’s rank correlation coefficient, regression lines. Sampling theory, simple random sampling, distribution of sample mean, proportion and variance, property of a good estimation, point and interval estimation, confidence intervals, null and alternative hypothesis, type-I and type II errors, testing of hypothesis for small and large samples, Factorization theorem, completeness, Rao-Blackwell theorem, Cramer-Rao inequality, Maximum likelihood method of estimation and method of moments.
The fundamental existence and uniqueness theorem, Sturm-Liouville problems, orthogonality of characteristic functions, the expansion of a function in a series of orthogonal functions. Matrix method for homogeneous linear systems with constant coefficients. Stress, strain, differential equation of equilibrium in a general three-dimensional stress system, principal stresses and strains, generalized Hook’s law, mechanical properties of different materials, applications. Complex variable method, finite difference method.
Fractions, simplification, HCF and LCM, ratio and proportion, percentage, partnership, age, average, profit and loss, simple and compound interest, time and work, time and distance. Identities, Venn diagram, addition principle, pigeon hole principle, functions, hashing function, characteristic function, Ackermann’s function. Binary, octal, hexadecimal, floating point representation of numbers. Permutation and combination, probability, binomial theorem. linear equations, quadratic equations, complex numbers, logarithms. Surds and indices, Inequalities, Mensuration, Geometry, Data Interpretation, Errors.
Divisibility, The greatest common divisor, coprime integers, The least common multiple, Linear Diophantine Equations, The Fundamental Theorem of Arithmetic, Prime Number Theorem, Goldbach and Twin Primes conjectures, Residue classes, Euclid's algorithm, Chinese Remainder, Wilson's and Fermat's Theorem, pseudoprimes. Greatest integer function, The Euler phi function, RSA Cyptosystem, arithmetic function, The Mobius function, Carmichael conjecture, The number-of-divisors and sum-of-divisors functions, Perfect numbers, characterization of even perfect numbers. Quadratic residues and non-residues, The Legendre symbol, Euler's Criterion, The law of quadratic reciprocity. Primitive roots.
Vector Space, Bases and dimension. Linear Transformation, Isomorphism, Rank-nullity theorem. Change of basis, Inverse of a linear transformation, Linear functional. Inner product space, Metric and normed spaces. Orthonormal basis, Gram-Schmidt orthogonalization. Eigen values and vectors, Modal matrix and diagonalization, Similarity Transformation, Eigen systems of real symmetric, orthogonal, Hermitian and unitary matrices, Quadratic forms, Norm of a matrix, Application to ordinary differential equations.
System of linear equations: Gauss-elimination and LU-Decomposition. Iterative methods: Gauss Seidel and successive-over relaxation. Power method, Jacobi method, Lagrange formula, Divided differences, Hermite interpolation, Least square approximation. Newton-Cote formulae, Gauss-Legendre quadrature formulae, Double integration Iterative methods.IVP: Runge-Kutta, predictor corrector method. BVP: FDM, Shooting methods, Numerical solutions of parabolic and elliptic PDE.
Metric space, Hausdorff distance, open and closed sets, continuous functions. Convergence and completeness, Cantor’s intersection theorem, completion of metric spaces. Countability, Baire’s category theorem. Compactness, bounded and totally bounded spaces, uniform continuity. Banach and Kannan contraction theorems, applications of Banach's contraction principle.
Simplex method and variants. Game theory, Queuing models, Minimal spanning tree algorithm, Shortest route problem, Dijkstra algorithm, Floyd’s algorithm, Maximal flow model, CPM and PERT. Sequencing problems. Discrete and continuous dynamic programming, Nonlinear programming: Golden section, Fibonacci search, Bisection method, One dimensional Newton's method, Steepest descent method, Multidimensional Newton's method.
Functional, variation and its properties, Euler's equation, the fundamental lemma of the calculus of variations, functionals in the form of integrals, functionals depending on the higher derivatives of the dependent variables, Conversion of Volterra Equation to ODE, successive approximation, successive substitution methods for Fredholm and Volterra integral equations. Finite difference method, explicit method, implicit method, Crank–Nicolson method and applications, comparison of FEM and FDM.
Vector Spaces and Subspaces ;Linear Transformations and their matrix representation. Rank -Nullity Theorem. Linear functionals, Inner Product Spaces: Orthogonality. Gram-Schmidt process. Normed spaces. Eigenvalues and Eigenvectors. Diagonalization. Quadratic forms. Solution of a system of linear differential equations. Convergence of Gauss-Seidel and other iterative methods. Cyclic Jacobi method, Householder method, LR and QR algorithms, singular value decomposition and its applications. Least squares solution.
Banach space, subspace and quotient space, linear operators and linear functionals, dual space. Hahn-Banach theorem, principle of uniform boundedness, strong and weak convergence, open mapping theorem, closed graph theorem. Hilbert Spaces, orthonormal basis, orthogonal projections, operators on Hilbert spaces, spectral decomposition. Sobolev spaces, Approximation by smooth functions, extension, imbedding and compactness theorems, dual, fractional order and trace spaces. Schauder fixed point theorem, Banach fixed point theorem and simple applications.
C Programming: arrays, Preparation of user defined functions for swapping of two numbers, interpolation and integration by Simpson’s rule. Functions involving structures. Programs involving built-in and user defined functions for strings. Dynamic memory allocation. Solution of a system of linear equations by Gauss-elimination. Creation of classes and objects and object oriented programming. I/O handling in C and C++. MATLAB Programming: Commands dealing with simple data types vectors and matrices. Built-in functions for vectors / matrices and mathematical functions. Element wise operations. Script files. Function files. Control structures and looping. 2D plotting. 3D plotting. Applications to numerical analysis.
Groups; Subgroups and cosets; Langrange’s theorem. Permutation groups, cyclic groups. Cayley Theorem; Normal subgroups, quotient groups. Isomorphisms. Series of groups, Sylow theorems. Applications of group theory in fast adding, image understanding and symmetry groups, Rings, Fields and Integral Domains. Isomorphisms. Rings of polynomials. Euclidean domains, unique factorization domains. Ideals, principal ideal ring. Eienstein’s criterion for irreducibility. Finite fields, extension field, algebraic and transcendental extensions. Separable and inseparable extensions. Automorphism of extensions. Solution of polynomial equations by radicals, insolvability of the general equation of degree 5 by radicals.
Divisibility and primes, fundamental theorem of arithmetic, Euclid’s theorem. Mobius, Euler’s totient, Mangoldt and Liouville functions, Dirichlet product of arithmetical functions, generalized convolutions, formal power series, Bell series. Derivatives of arithmetical functions. Asymptotic equality of functions, Euler’s summation formula, average order of functions. Chebychev’s functions, prime number theorem, Shapiro’s Tauberian theorem. Partial sums, Selberg’s asymptotic formula. Congruences, complete residue systems, linear congruences, reduced residue systems and Euler-Fermat, Lagrange and Chinese remainder theorems, polynomial congruences with prime power moduli.
Roots of Algebraic and Transcendental Equations: Bisection method, direct Iteration method, Newton Raphson method. Solutions of Simultaneous Equations: Gauss Elimination method, Gauss Seidel method. Eigenvalues and Eigenvectors: Power method, Jacobi Method. Interpolation: Newton Forward interpolation method, Newton Backward interpolation method, Stirling's formula, divided difference, Lagrange's Interpolation method. Integration: Trapezoidal formula, Simpson's 1/3 formula. Solution of Ordinary Differential Equation: Euler’s method, Runge-Kutta method for IVP’s, Finite Difference Method for BVP’s, finite difference method for Laplace Equation.
Introduction: Relations and functions. Finite Automata and Regular Languages: Finite automata, transition systems, regular expressions and finite automata, push-down automata, context free grammars, pumping lemma. Computability: Turing machine models, universal turing machines, limits on language acceptance, reducibility and unsolvability, functions computed by turing machines. Complexity Theory: Languages and problems, serial computational models, classification of decision problems (space and time hierarchies, time-bounded complexity classes, space-bounded complexity classes), hard and complete problems, NP-complete problems.
Linear programming problems and Duality - Introduction, Simplex method and its Variants, Revised simplex method, dual simplex method. Parametric and Sensitivity Analysis. Inventory Controls - Inventory models, Deterministic inventory problems with and without shortages. Network Analysis - Shortest path problems, PERT/CPM, Simulation techniques. Games and Strategies - Pure and mixed strategies, solution by graphical and linear programming methods. Multi-objective and Goal programming. Nonlinear programming - Convex functions and their properties, Kuhn-Tucker theory, Convex quadratic programming, Wolfe and Beale’s algorithm, Separable convex programming.
Green’s function, applications. Matrix method for homogeneous linear systems with constant coefficients equation. Sturm- Liouville problem. Existence and uniqueness of the solutions of the IVP, Existence and uniqueness of system of equations and higher order equations. First order nonlinear equations and their linearization, Abel’s equation, exact linearization of nonlinear second order equation via factoring, transformation of nonlinear equation to nonlinear integrable forms, reducing nonlinear second order equation to first order equation. Lagrange method for PDE, special methods for Laplace equation.
Basic data types. Arithmetic operators. Arithmetic expression. Logical and relational operators, scanf ( ) and printf ( ) functions. I/O using cin and cout. Simple programs. Control of flow using if, if … else, goto, do … while, while and for. Arrays and Strings. User defined functions, call by value, returning values. Pointers and their application. Call by reference. Structures. File I/O in C. Introduction to C++ programming. Basics of Simulation. Random variates and their generation. Monte-Carlo simulation.
Random Vectors and Matrices. Multivariate Normal Distribution: Univariate normal density function. Multivariate normal density function , Properties of the multivariate normal distribution. Distribution of Quadratic Forms in y: Distributions of quadratic forms, Noncentral Chi-square, F- and t- distributions. Independence of linear forms and quadratic Forms. Simple Linear Regression Model. Multiple Regression: Estimation, Tests of Hypotheses and Confidence Intervals. Multivariate Normal Regression Model. Analysis of Variance Models and Testing of hypothesis.
Laplace Transforms: Existence condition, differentiation and integration of Laplace transform. Inverse Laplace transform, convolution theorem. Applications to differential equations. Fourier Transform: Definition, basic properties of Fourier transforms. Applications to differential equations. Hankel Transforms: The Hankel transform, operational properties. Application of Hankel transform to partial differential equations. Mellin Transforms: Definition, properties. Applications to summation of series. Hilbert Transforms: Definition and properties, Hilbert transforms in complex plane. Applications of Hilbert transforms, asymptotic expansion of one sided Hilbert transforms. Z Transform: Dynamic linear systems and impulse response, Definition and properties of Z transforms. The inverse Z transform. Applications to finite difference equations.
Solution of Nonlinear Equations, Newton-Raphson method, Muller’s method, Graeffe’s method, Lin-Bairstow’s method. Interpolation using divided differences, Hermite interpolation, cubic splines. Method of least squares, orthogonal polynomials, Chebyshev polynimals, numerical integration, Romberg integration, Newton-Cotes integration formulae, Gaussian quadrature, Iterative methods for System of Equations, SOR method, Jacobi method, Householder’s method. Numerical Solution of Differential Equations: predictor-corrector methods. Finite difference methods for elliptic, parabolic and hyperbolic equations, Method of weighted residuals, Rayleigh-Ritz method, finite element method.
Preliminaries of Linear programming problems, Kuhn-Tucker conditions, Lagrange’s multipliers method, Convex programming problem. One-Dimensional Optimization: Unimodal functions, Numerical methods-Elimination methods, Interpolation method. Unconstrained Optimization Techniques-Direct search methods, Indirect search (Descent) methods. Constrained Optimization Techniques-: Direct methods, Indirect methods. Dynamic Programming - Deterministic and Probabilistic dynamic programming. Fuzzy Linear Programming. Multi objective linear programming, Multi objective fuzzy linear programming, Fuzzy dynamic programming.
Functional and Its Variation: Stationary values of a functional. Euler- Lagrange equations. Brachisto chrone problem. Constraints and Lagrange multipliers. Variable end points. Sturm-Liouville problems. Vibration problems. Hamilton principle. Lagrange equations. Method of Weighted Residuals : Collocation, Galerkin and Ritz methods. Fredholm and Volterra Type of Integral Equations: Green’s function, Fredholm equation with separable kernel, Iterative methods, Singular integral equations, Abel’s equation. Numerical Methods: The finite element method. One dimensional problems. Stiffness matrix. Assembly of equations . Handling of the boundary conditions. Two Dimensional Problems: Triangular and rectangular elements. Stiffness matrices and assembly. Comparison of FEM and FDM.
Review of metric spaces, completeness and compactness, Hutchinson operator, Banach contraction theorem, Hausdorff metric and the spaces of fractals, iterated function systems (IFS) and attractor of IFS. Fractal, some mathematical fractals, Box counting dimension, Hausdorff-Besicovitch dimension. Fractal Interpolation, fractal interpolation functions, fractal dimension of fractal interpolation functions. Dynamical systems, chaos, bifurcations, logistic map, Julia and Mandelbrot sets.
Basics of Fluid Dynamics: equation of continuity and motion, Navier-Stokes equations, various forms of N-S equations, dimensionless form, Energy Equation. Applied Numerical Methods: Numerical solution of ordinary differential equations, Runge Kutta method, finite differences, discretization, consistency, stability and fundamentals of fluid flow modelling. Applications in Heat Conduction and Convection: Discrete approximations, upwind corrected scheme, explicit schemes for the advection-diffusion equations, ADI method, quasi linearization Method for nonlinear boundary value problems. Incompressible Viscous Flow via FEM, Finite Volume Methods via FDM: 2-D problem, 3-D geometry structure, 3-D FVM equations, illustrative problems. Programming, Testing and Information Processing of Numerical Methods.