Syllabus
Basic examples of PDE's and their numerical solution by the finite difference method. Cauchy problem for a quasilinear PDE of the first order, transport equation, characteristics.
Von Neumann stability analysis of numerical schemes for Cauchy problems. Numerical solution of transport equation: CFL condition, upwinding, maximum principle, truncation error and approximation error, dissipation and dispersion.
Real analytic functions, Cauchy-Kowalevska Theorem, characteristic surfaces, classification of semilinear PDE's of the second order, transformation to canonical form.
Heat equation (fundamental solution, Cauchy problem, problem in bounded domain), wave equation (fundamental solution, Cauchy problem, energy methods).
Numerical solution of the mixed problem for heat equation: implicit and explicit schemes, theta-scheme, Fourier error analysis, maximum principle and convergence.
Relation between consistence, convergence and stability: general scheme for equations of the first order in time, Lax equivalence theorem.
Elliptic equations of the second order: fundamental solution of Laplace equation, Green's representation formula, Dirichlet problem for Laplace equation, mean value theorems, maximum principles.
Numerical solution of elliptic equations of the second order: approximation of general diffusion equation, derivation of schemes in irregular nodes, maximum principle and convergence.
Annotation
An introductory course in partial differential equations for bachelor's program in General Mathematics.
Recommended for specializations Mathematical Analysis and Mathematical Modelling and Numerical Analysis.