Partial Differential Equations

Concept of partial differential equation and its solution, well pose of problem, analytical and numerical approach, hybrid methods of confrontation. Linear partial differential equations of first order and use of characteristic curves to obtain general solution, Cauchy’s conditions or data and models of applied problems. Differential equations with partial derivatives of second order, main applications to modern technology and mathematical physics. Dirac’s functional and Heaviside’s function. Bessel’s and Legendre’s special functions, spherical harmonics, orthogonality and recurrence formulae. General introduction to basic integral transforms. Elliptic type equations and boundary value problems. Laplace’s and Helmholtz’s equations, solution with the method of separating the variables and eigenfunctions in Cartesian, polar, cylindrical and spherical coordinates with applications to several physical problems and spatial Fourier’s transform. Parabolic type equations (diffusion equation) and method of separation of variables, non homogeneous problems and dealing with the methods of asymptotic solutions and expansion to eigenfunctions. Brief introduction to hyperbolic type equations (wave equation) and method of separation of variables, principal concepts of wave propagation, finite and infinite string. Solution of problems of parabolic and hyperbolic type with initial and boundary conditions using the Fourier’s in space and the Laplace’s in time integral transforms.