Boundary Conditions, Which Simulates the Cauchy Problem (Transparent Boundary Conditions) for Finite-Difference Approximations of Basic Mathematical Physics Equations

To formulate a correct mixed boundary problem, boundary conditions that describe physical processes happening on the boundary are needed. However, sometimes there are no such special processes occurring. Initial conditions outside of the computational area are set approximately. Boundary conditions that provide the same solution for mixed boundary problem and the Cauchy’s problem for any initial condition usually are nonlocal. Such boundary conditions can be obtained by including integral operators of convolution type. A core of the convolution operator can be found for linear finite-difference equations too. A number of arithmetic operation required to calculate the convolution linearly depends on integration time. Technics based on Hermite – Pade approximations allow decreasing the number of arithmetic operations required to calculate one border point value. Such boundary conditions are found for finite-difference schemes that approximate some basic mathematical physics equations: wave, diffusion, Schrodinger equation. Theoretical base for small transverse vibrations of a rod equation is introduced