EGR 6110 - PDEs and Numerical Methods

• Employ the Taylor series for approximation and error analysis
• Formulate and apply numerical techniques for root finding, differentiation, and integration
• Write computer programs to solve engineering problems
• Recognize parabolic, elliptic, and hyperbolic types of partial differential equations
• Solve PDEs analytically via separation of variables and employing Fourier series where applicable
• Write programs and solve PDEs numerically under various initial and boundary conditions 
• Conduct an investigation (course project) and present results orally  

• Computer programming

• Taylor series, error propagation, numerical differentiation, forward-backward-central difference formulations of First and Second derivatives, Richardson’s extrapolation
• Numerical integration: Newton-Gregory forward formula for interpolation, trapezoidal rule, Simpson’s rules, Boole’s rule, Romberg integration
• Root finding methods: bisection, false position, fixed-point iteration, Newton-Raphson, Secant, modified Secant
• Partial differential equations: types- parabolic (heat conduction), elliptic (Laplace, Poisson) and hyperbolic (wave equation)
• Review of Fourier series; half-range expansions: Fourier sine and cosine series to solve PDEs analytically 
• Heat equation (one-d): explicit scheme, stability issues, implicit scheme, Crank-Nicholson’s scheme, non-homogeneous problem, non-linear equation; two-d heat equation treatment
• Elliptic PDE: Laplace equation, Jacobi iteration, Gauss-Seidel iteration, successive over-relaxation scheme, Poisson equation
• Wave equation (one-d): explicit method, Courant stability criterion, implicit method
• Course project based on student interest; use of flexpde or COMSOL is recommended for ease of solution