E155B Mathematical and Computational Methods for Engineers
Fourier series, transforms, and applications, partial differential equations arising in science and engineering, analytical solutions of partial differential equations, numerical linear algebra, least squares, coupled systems of ordinary differential equations, eigensystem analysis, normal modes, numerical methods for solution of partial differential equations, iterative techniques, stability and convergence, time advancement, implicit methods, von Neumann stability analysis. Prerequisites: E155A. 5 units
Course Outline
Fourier Series, Transforms, and Applications
- Fourier series, orthogonal functions, convergence
- Fourier series for even and odd functions, half-range periodic extensions
- Solution of ordinary differential equations using Fourier series, resonance
- Complex Fourier series, Fourier transforms
- Solution of ordinary differential equations using Fourier transforms
Partial Differential Equations
- Classification of PDEs, modeling: heat conduction, diffusion, signal transmission
- Laplace, Poisson, and diffusion equations
- Separation of variables, boundary conditions, steady-state heat conduction and diffusion in 1D and 2 D
- Time-dependent solutions of the heat conduction equation in 1D and 2D
- Solution of inhomogeneous PDEs, steady-state solutions
- Laplace equation in polar coordinates
- Wave equation in 1D and 2D
- Modeling: vibration of a string/membrane, longitudinal vibrations in an elastic medium, separation of variables, traveling wave solutions
- Solution of the wave equation in 2D, eigenfunctions, eigenvalues, normal modes
- Inhomogeneous wave equation, solution using eigenfunction expansion
Linear Algebra and Applications
- Review of linear algebra: matrix operations and basic properties
- Systems of algebraic equations, Gaussian elimination, rank, undertermined and overdetermined systems
- Linear independence, determinants, properties of determinants
- Cramer's rule, solution of systems of algebraic equations
- Gauss-Jordan elimination, procedure for finding an inverse
- Roundoff error, partial pivoting
- Norms, condition numbers, ill-conditioning
- Operation counts
- Least squares, normal equation, applications
Eigenvalues, Eigenvectors, and Applications
- Eigenvalue problem applied to coupled systems of ODEs
- Real and complex eigenvalues, modeling of coupled systems, normal modes
- Applications: Markov chains, stochastic matrices, linear transformations
- Properties of eigenvalues and eigenvectors, symmetric matrices, similar matrices, diagonalization
- Power of a matrix, normal coordinates
Numerical Methods
- Numerical differentiation, finite differences, discretization error
- Discretization of second-order ODEs, eigenvalue problem
- Iterative methods, error propagation, stability and rate of convergence
- Jacobi and Gauss-Seidel iterations
- Numerical solution of Laplace/Poisson equation in 1D and 2D using iterative methods
- Unsteady PDEs, time marching
- Explicit schemes, von Neumann stability analysis
- Numerical solution of heat conduction and wave equations, stability criteria
- Implicit methods, Crank-Nicholson scheme, examples
