The hyperbolic equation with fractional derivative in modelling the polymer concrete behaviour under constant load

Abstract

The paper considers an inhomogeneous partial differential equation of hyperbolic type containing both second order partial derivatives and fractional derivatives of an order lower than the second order on the spatial variable. The fractional derivative is understood in the Riemann-Liouville sense. The paper first part presents analytical solution of the boundary value problem of the first kind with arbitrary initial conditions of the equation in question by the Fourier method (separation of variables). The solution is given in the form of two functions, one of which characterizes the stationary state of the process, and the second is treated as a deviation from the stationary state. The second function is found using eigenfunctions and eigenvalues of the corresponding two-point Dirichlet problem, while the first is written out as a Mittag-Leffler function. In the second part of the paper, the solution application examples of the boundary value problem of the equation under consideration in modelling the behaviour of polymer concrete based on polyester resin (dian and dichloroanhydride-1,1-dichloro-2,2-diethylene) under load are considered. The examples give specific values for the parameters of the equation and graphs of the solutions and the error resulting from replacing a series by its partial sum. All calculations were performed in the MATLAB application package.