Course Catalogue

Ordinary Differential Equations: Degree and order of ordinary differential equations, formation of differential equations. Solution of first order differential equations by various methods. Solution of general linear equations of second and higher orders with constant coefficients. Solution of homogeneous linear equations. Solution of differential equations of the higher order when the dependent or independent variables are absent. Solution of differential equation by the method based on the factorization of the operators. Frobenius method. Partial Differential Equations: Introduction. Linear and non-linear first order equations. Standard forms. Linear equations of higher order. Equations of the second order with variable coefficients. Wave equations. Particular solution with boundary and initial conditions.

Laplace Transformations:
Laplace Transformation (LT), Inverse Laplace Transforms (ILT), Applications of LT to de, Complex Variable Theory: complex number system, polar form of complex numbers, operations in polar form, De Moivre’s theorem, roots of complex numbers, functions, limit, continuity, derivatives, Cauchy-Riemann equations, line integral formulas, Green’s theorem in the plane, Integrals, Cauchy’s theorem, Cauchy’s integral formulas, Taylor’s series, Singular points, Poles, Laurent’s series, residues, residues theorem, evaluation of definite integrals, Convergency and Divergency of infinite series: general properties of series, test of convergence and divergence of the series of constants, power series, uniform convergence, Cauchy criterion for uniform convergence. Power series solution of de: analytic and singular functions, ordinary and singular points of a differential equation, Solution in series of the de.

Fourier Analysis:
Beta and Gamma functions, Fourier series and Applications: Periodic functions and Trigonometric series, Definition of Fourier Series, Dirichlet conditions, Fourier sine and cosine series, Half-Range Fourier Sine and Cosine series, Perseval’s Identity, Integration and differentiation of Fourier series, Complex notation for fourier series, Some physical applications of fourier series, Fourier Integrals and Applications: Fourier integral, Fourier transforms, Fourier sine and cosine transforms, Persival’s Identities for Fourier Integrals, Convolution theorem for Fourier transforms, Applications of Fourier Integrals and transforms, Special functions: Legendere’s de and Legendere’s polynomials, Bessel’s de and Bessel functions, Hermite’s de and Hermit’s polynomials, Laguerre’s DE and Laguerre’s polynomials.

Laplace Transformations:
Laplace Transformation (LT), Inverse Laplace Transforms (ILT), Applications of LT to de, Complex Variable Theory: complex number system, polar form of complex numbers, operations in polar form, De Moivre’s theorem, roots of complex numbers, functions, limit, continuity, derivatives, Cauchy-Riemann equations, line integral formulas, Green’s theorem in the plane, Integrals, Cauchy’s theorem, Cauchy’s integral formulas, Taylor’s series, Singular points, Poles, Laurent’s series, residues, residues theorem, evaluation of definite integrals, Convergency and Divergency of infinite series: general properties of series, test of convergence and divergence of the series of constants, power series, uniform convergence, Cauchy criterion for uniform convergence. Power series solution of de: analytic and singular functions, ordinary and singular points of a differential equation, Solution in series of the de.

Fourier Analysis:
Beta and Gamma functions, Fourier series and Applications: Periodic functions and Trigonometric series, Definition of Fourier Series, Dirichlet conditions, Fourier sine and cosine series, Half-Range Fourier Sine and Cosine series, Perseval’s Identity, Integration and differentiation of Fourier series, Complex notation for fourier series, Some physical applications of fourier series, Fourier Integrals and Applications: Fourier integral, Fourier transforms, Fourier sine and cosine transforms, Persival’s Identities for Fourier Integrals, Convolution theorem for Fourier transforms, Applications of Fourier Integrals and transforms, Special functions: Legendere’s de and Legendere’s polynomials, Bessel’s de and Bessel functions, Hermite’s de and Hermit’s polynomials, Laguerre’s DE and Laguerre’s polynomials.

Basic Definitions and Terminology: differential equation (de), classifications of de, formation and solution of a de and further terminology, Ordinary de (ode), des of the first order and first degree: variable separable, homogeneous equations, exact equations, linear equations, Linear Equations with constant coefficients: linear and nonlinear de, solution of linear de, 2nd order des, 2nd and higher order homogeneous des, Method of Variation of Parameter; Method of undetermined coefficients; System of Linear de: operator method matrices and system of linear first order equations, homogeneous linear systems, undetermined coefficients, variation of parameters, Solution by Series. Errors in Numerical Calculations: numbers and their accuracy, errors and their computation, a general error formula, error in a series approximation, Solution of Algebraic and Transcendental Equations: bisection, iteration, false position, Newton-Raphson methods, Interpolation: finite difference, forward, backward and central differences, Newton's formula for interpolation, Stirlings formula, Lagrange's interpolation formula, divided differences and their properties, Numerical Differentiation and Integration, Matrices and Linear Systems of Equations, Numerical Solution of Ordinary Differential Equations.