Solution of the initial boundary value problem in symbolic form

Algorithms for finding a symbolic-numerical solution of the initial- boundary value problem for a continuum transport equation are discussed. Analytical solution of such equations, as a rule, is impossible; therefore, approximate methods of solution that provide the condition of approximation, stability, and convergence are being actively developed. This article proposes a symbolic solution which is more convenient than a numerical one to be used, for example, in the synthesis of control systems. The algorithm is based on the approximation of partial derivatives with respect to one of the variables by a difference relation and the application of the Laplace transform to the resulting system of differential-difference equations. A block diagram of the algorithm is presented. The description of the structure of the software package based on the developed algorithm is carried out. The software package is developed in the Java programming language. To enter the initial data of the initial boundary value problem and output the solution, a web interface is used. The web interface of the software package is based on the Spring framework. An example of solving an initial boundary value problem with initial and boundary conditions using this software package is considered.

The results are of interest to researchers in applied areas related to heat transfer through a network coolant, transportation of viscous liquids through a network hydraulic carrier, and diffusion processes in biophysics. The developed algorithm can be used to solve some problems of automatic control.