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# Mathematical methods and its applications

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## Overview

This course is a basic course offered to UG student of Engineering/Science background. It contains ODE,PDE, Laplace transforms, Z-transforms, Fourier series and Fourier transforms. It plays an important role for solving various engineering sciences problems.Therefore, it has tremendous applications in diverse fields in engineering sciences.

## Syllabus

Week 1: Introduction to linear differential equations ,Linear dependence, independence and Wronskian of functions,Solution of second-order homogeneous linear differential equations with constant coefficients-I,Solution of second-order homogeneous linear differential equations with constant coefficients-II,Method of undetermined coefficients

Week 2:Methods for finding Particular Integral for second-order linear differential equations with constant coefficients-I,Methods for finding Particular Integral for second-order linear differential equations with constant coefficients-II, Methods for finding Particular Integral for second-order linear differential equations with constant coefficients-III,Euler-Cauchy equations,Method of reduction for second-order, linear differential equations

Week 3: Method of variation of parameters , Solution of second order differential equations by changing dependent variable, Solution of second order differential equations by changing independent variable,Solution of higher-order homogenous linear differential equations with constant coefficients, Methods for finding Particular Integral for higher-order linear differential equations

Week 4: Formulation of Partial differential equations, Solution of Lagrange’s equation-I, Solution of Lagrange’s equation-II,Solution of first order nonlinear equations-I,Solution of first order nonlinear equations--II

Week 5: Solution of first order nonlinear equations-III,Solution of first order nonlinear equations-IV,Introduction to Laplace transforms,Laplace transforms of some standard functions,Existence theorem for Laplace transforms

Week 6: Properties of Laplace transforms--I, Properties of Laplace transforms--II, Properties of Laplace transforms--III, Properties of Laplace transforms--IV, Convolution theorem for Laplace transforms--I

Week 7: Convolution theorem for Laplace transforms--II,Initial and final value theorems for Laplace transforms, Laplace transforms of periodic functions, Laplace transforms of Heaviside unit step function,Laplace transforms of Dirac delta function

Week 8: Applications of Laplace transforms-I, Applications of Laplace transforms-II, Applications of Laplace transforms-III,   Z – transform and inverse Z-transform of elementary functions, Properties of Z-transforms-I

Week 9: Properties of Z-transforms-II, Initial and final value theorem for Z-transforms, Convolution theorem for Z- transforms, Applications of Z- transforms--I,Applications of Z- transforms-II

Week 10: Applications of Z- transforms--III, Fourier series and its convergence--I, Fourier series and its convergence--II, Fourier series of even and odd functions,Fourier half-range series

Week 11: Parsevel’s Identity, Complex form of Fourier series, Fourier integrals, Fourier sine and cosine integrals, Fourier transforms

Week 12: Fourier sine and cosine transforms,Convolution theorem for Fourier transforms,Applications of Fourier transforms to BVP-I,Applications of Fourier transforms to BVP-II,Applications of Fourier transforms to BVP-III

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