syllabus for Mathematics III

syllabus for Mathematics III in the third semester of an engineering program:

Unit I: Partial Differential Equations

• Formation: By elimination of arbitrary constants and functions
• Solutions: Standard types of first-order PDEs, Lagrange’s linear equation
• Higher-Order PDEs: Linear PDEs with constant coefficients

Unit II: Fourier Series

• Introduction: Dirichlet’s conditions, general Fourier series
• Types: Odd and even functions, half-range sine and cosine series
• Applications: Parseval’s identity, harmonic analysis

Unit III: Boundary Value Problems

• Classification: Second-order quasi-linear PDEs
• Solutions: One-dimensional wave equation, heat equation
• Steady-State Solutions: Two-dimensional heat equation using Fourier series

Unit IV: Fourier Transforms

• Fourier Integral Theorem: Without proof
• Transforms: Fourier transform pair, sine and cosine transforms
• Properties: Convolution theorem, Parseval’s identity
• Applications: Transforms of simple functions

Unit V: Z-Transforms

• Introduction: Definition and properties
• Inverse Z-Transform: Methods of finding inverse
• Applications: Solving difference equations using Z-transforms

Unit VI: Complex Variables

• Analytic Functions: Cauchy-Riemann equations, harmonic functions
• Complex Integration: Cauchy’s integral theorem and formula
• Series: Taylor and Laurent series, residue theorem, applications

Unit VII: Numerical Methods

• Solutions of Equations: Bisection method, Newton-Raphson method
• Interpolation: Lagrange’s and Newton’s interpolations
• Numerical Integration: Trapezoidal and Simpson’s rules
• Numerical Solutions of ODEs: Euler’s method, Runge-Kutta methods