Mathematical Methods and its Applications

Overview

Syllabus 12

1. Week 1
   • Introduction to linear differential equations
   • Linear dependence, independence and Wronskian of functions, Existence and Uniqueness theorem
   • Solution of second order homogeneous linear differential equations with constant coefficient-
   • Solution of second order homogeneous linear differential equations with constant coefficient-II
   • Methods of Undetermined coefficients
2. Week 2
   • Methods for finding particular integrals for second order linear differential equations with constant coefficients I
   • Methods for finding particular integrals for second order linear differential equations with constant coefficients II
   • Methods for finding particular integrals for second order linear differential equations with constant coefficients III
   • Cauchy-Euler equation
   • Methods of reduction for second order-linear differential equations
3. Week 3
   • Methods of variation of parameters
   • Solution of second order differential equations by changing dependent variables
   • Solution of second order differential equations by changing independent variable
   • Solution of higher-order homogeneous linear differential equations with constant coefficients
   • Methods for finding particular integral for higher-order linear differential equations with constant coefficient s
4. Week 4
   • Formulation of partial differential equations
   • Solution of Lagrange's equation I
   • Solution of Lagranage's equations II
   • Solution of first order non-linear equation I
   • Solution of first order non-linear equation II
5. Week 5
   • Solution of first order non linear equation III
   • Solution of first order non linear equation IV
   • Introduction to Laplace transforms
   • Laplace transforms of some standard functions
   • Existence theorem for Laplace transforms
6. 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
7. Week 7
   • Convolution Theorem for Laplace Transforms-II
   • Initial and Final Value Theorems for Laplace Transforms
   • Laplace Transforms of Periodic Functions
   • Laplace Transforms of Unit Step Function
   • Laplace Transforms of Dirac delta Functions
8. Week 8
   • Applications of Laplace Transforms-I
   • Applications of Laplace Transforms-II
   • Applications of Laplace Transforms-III
   • Z- transform and inverse Z- transform of elemetary functions
   • Properties of Z-transforms-I
9. Week 9
   • Properties of Z - transformatons - II
   • Initial and final theorems for Z- transforms
   • Convolution theorem for Z-transforms
   • Convergence of Z- transform
   • Application of Z-transforms-I
10. Week 10
    • Applications of Z- transform- II
    • Fourier Series and its Convergence-I
    • Fourier Series and its Convergence-II
    • Fourier Series of Even and Odd functions
    • Fourier half-range series
11. Week 11
    • Parseval’s Identity
    • Complex form of Fourier series
    • Fourier integrals
    • Fourier sine and cosine integrals
    • Fourier transforms
12. 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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Taught by

• Prof. P. N. AgarwalIIT Roorkee
• Prof. S. K. GuptaIIT Roorkee