SF1682 Analytical and Numerical Methods for Differential Equations

Completed basic course SF1626 Calculus in Several Variable.

• Equations: First and higher order ordinary differential equations and systems of these, partial differential equations (e.g. for heat and waves).
• Transforms: Fourier transform, Laplace transform and Fourier series.
• Analytical concepts: Initial value problems, boundary value problems, existence and uniqueness of solutions, autonomous equations, direction fields, phase portraits, solutions curves, oscillation phenomena, general solution, particular solution, stationary/critical points, stability, linearization of systems, the delta function, generalized derivatives.
• Numerical concepts: Apprximation, discretization, convergence, conditional number, accuracy, local linearization, stability, stiff systems, implicit and explicit methods, adaptivity.
• Analytic methods: Integrationg factor, separation of variables, variation of parameters, eigenvalue methods, transforms, spectral methods.
• Numerical methods: Newton's method for non-linear systems, Euler forward, Euler backwards, Runge_kutta methods, finite difference methods, spectral methods, fast Fourier transform (FFT), computational complexity.

After the course the student should be able to

• use concepts. theorems and methods to handle questions in analysis and numerical aspects of differential equations and transforms described by the course content,
• use analytical and numerical methods to solve the the differential equations described by the course content, and gain insights into the possibilities and limitiations of methods.
• read and comprehend mathematical text.