On special finite difference approximations for solving second order differential equations

The described special methods are applicable for various mathematical physics problems with second-order differential equations involving periodic boundary conditions (PBCs) and first-order homogenous boundary conditions (FBCs). Solutions of some linear and nonlinear problems for parabolic type partial differential equations (PDEs) with FBCs are obtained, using the method of lines (MOL) to approach the PDEs in the time and the discretization in space applying the finite difference scheme with exact spectrum (FDSES). For PBCs we use the finite difference scheme (FDS) for locally approximating periodic function's derivatives in a 2n+1, n>=1 -point stencil, obtaining higher order accuracy approximation. This method in the uniform grid with N mesh points is used to approximate the differential operator of the second and the first-order derivatives in the space. In this paper, we show that the approximation using the FDSES method is equivalent to the spectral differentiation matrix method based on trigonometric (Fourier) interpolant. Considering, that the solutions obtained in solving nonlinear problems can be very significantly different from classical solutions, for example, mathematical modelling of processes where temperature or energy is concentrated in a very narrow interval or around a point, again causes increased interest in such areas of application as laser technology, military sphere, etc. In this regard, also in the given publication, the solution of the "blow-up" phenomenon of the boundary problem of the nonlinear heat conduction equation has been studied and obtained with the above-mentioned high-accuracy solving methods.