A problem of cauchy type for a fractal string

Abstract

We discuss the characteristics of the statement and solution of a problem of Cauchy type using the example of the plastic subsystem of a fractal string. To solve the basic dynamic equation in fractional derivatives we propose two approaches: reduction to a system of equations and the use of composition formulas for fractional derivative operators. The results obtained are generalized to the solution of the Cauchy problem in matrix form. Bibliography: 5 titles.We discuss the characteristics of the statement and solution of a problem of Cauchy type for a fractal string resuiting from the fractal properties and manifesting themselves most prominently in the course of the temporal and spatial evolution.For the plastic subsystem ofa fractal string the equation of dynamics has the form [1]where v and ct are the orders of the Riemann-Liouville fractional partial derivative operators D v and D ~ with respect to the variable t and the x-coordinate; F is the gamma-function; u(x, t) is an unknown function connected nordocally with the displacement u'(x, t) of the points of the string; 0, is the usual partial derivative with respect to the time variable t ; and V is a constant parameter interpreted as the velocity at a = v = 1. For the fractional parameters v, a ~ [0; 1] the interpretation as fractional dimensions along the respective axes is retained.Using the method of separation of variables u(x,/)= u~(t)u2(x), we reduce Eq.(1) to a set of two equations in the unknown functions u,, u 2 DVDVut(t)= ~.,u,(t); D=D=u2(x)= k2u2(x); k, = 3.2V z.(2)Here 3. I and 3. 2 are constants determined in the course of solving the spectral problem. If the constants k I and k 2 are given, the problem of finding solutions of Eqs.(2) reduces to a problem of Cauchy type for each of the equations, and two approaches are possible: the first is to reduce Eqs. (2) to a system of equations; the second is to use the composition formulas for fractional derivative operators [1 ].In order to compare the solutions and exhibit their characteristics we shall examine both approches using the example of the first equation of the system (2).By introducing the function f~(t) = D~u,(t) we obtain from (2) a system of differential equations of fractional order v D~u,(t) = fj(t); D~ fl(t)= k,u,(t).(3)Applying the fractional integration operator I v [2] on the left, we obtain from (3) the following system of coupled integral equations of fractional order: u, = I"fl +b,q~,; fl = ktUu, +b2q~,; q~, =lt-t'] v-' iV(v); t v-I I-v I I u,(t); d~/r(~); b:=Zt-~fl(t'), bt=