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.

Week

Topics

1

Introduction to linear differential equations, Linear dependence, independence and Wronskian of functions, Solution of second-order homogenous linear differential equations with constant coefficients, Method of undetermined coefficients

2

Methods for finding Particular Integral for second-order linear differential equations with constant coefficients, Euler-Cauchy equations, Method of reduction for second-order linear differential equations

3

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

4

Formulation of Partial differential equations, Solution of Lagrange equation, Solution of first order nonlinear equations

5

Solution of first order nonlinear equations, Introduction to Laplace transforms, Laplace transforms of some standard functions, Existence theorem for Laplace transforms

6

Properties of Laplace transforms, Convolution theorem for Laplace transforms

7

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

8

Applications of Laplace transforms, transform and inverse Z-transform of elementary functions, Properties of Z-transforms

9

Properties of Z-transforms, Initial and final value theorem for Z-transforms, Convolution theorem for Z- transforms, Applications of Z- transforms

10

Applications of Z- transforms, Fourier series and its convergence, Fourier series of even and odd functions, Fourier half-range series

11

Parsevel Identity, Complex form of Fourier series, Fourier integrals, Fourier sine and cosine integrals, Fourier transforms

12

Fourier sine and cosine transforms, Convolution theorem for Fourier transforms, Applications of Fourier transforms to BVP

1. E. Kreyszig, Advanced Engineering Mathematics, 10th edition, John Wiley and Sons, Inc., U.K. (2011) 2. M. D. Weir, J. Hass, F.R. Giordano, Thomas Calculus, 11th Edition, Pearson Education (2008) 3. R.K. Jain and S.R.K Iyengar, Advanced Engineering Mathematics, Narosa Publishing House (2009) 4. G. F. Simmons and S. G. Krantz, Differential Equations:Theory, Technique and Practice , Tata McGraw-Hill Edition (2007) 5. K. S. Rao Introduction to Partial Differential Equations, PHI Learning Pvt. Ltd. (II Edition) (2010)

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