Course objectives
- Develop the ability to solve differential equations using series methods and special functions applicable to engineering problems. 2. Introduce Laplace and Fourier transform techniques for analyzing differential equations and signal-processing applications. 3. Build a strong foundation in complex variable theory and its applications in engineering and applied sciences. 4. Understand the concepts of sequences and series, convergence tests, and their role in mathematical modeling. 5. Enable students to apply advanced mathematical techniques to solve real-world problems in computer science, electronics, communication, and engineering.
Unit I
Series Solution & Special Functions: Ordinary and singular point, Series solution about an ordinary point, Solution of equation by Frobenius method to solve second order linear differential equations, formation of Indicial equation, 1. Determination of indicial roots, cases of indicial roots: Distinct roots, equal roots, roots differ by an integer and not differ by an integer. Applications: Fluid mechanics, quantum mechanical wave equation, Heat transfer, Solving electromagnetic field equation.
Unit II
Laplace Transform: Definition of Laplace Transform, Existence Conditions, Linearity Property, Laplace Transform of some standard functions, I and II Shifting theorems (without proof), Change of Scale Property, Laplace transform of derivatives and integrals, Periodic Functions, Unit Step Function, Inverse Laplace Transform, Partial Fraction Method, Convolution theorem (without proof), Application of Laplace transform to solve ordinary differential equation. Applications : Transfer Function Determination, Circuit Equations, Step Response.