Advanced Mathematics for Engineers (N4)

Introduction to partial differential equations. 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. Brief review of basic mathematical tools in partial differential equations. Elliptic type equations (Laplace and Helmholtz) and boundary value problems. Solution with the method of separation of variables and eigenfunctions in orthogonal curvilinear coordinate systems with applications to physical problems. Spatial Fourier transform, fundamental solutions of differential operators, field integral representations. Parabolic type equations (diffusion), problems of boundary and initial values. Solution of homogeneous problems with the method of separation of variables. Non-homogeneous problems of diffusion in the presence of source, solution with the methods of asymptotic kernels and eigenfunctions. Solution of diffusion problems with the use of the integral transforms Fourier (in space) and Laplace (in time). Hyperbolic type equations (wave), basic concepts of wave propagation, problems of boundary and initial values. Solution of homogeneous problems with the method of separation of variables, finite and infinite string. Solution of wave problems with the use of the integral transforms Fourier (in space) and Laplace (in time). Characteristic cone and wave field energy.