Setting of specific problems of continuum mechanics often leads to initial boundary value problems for differential equations with fractional derivatives. Fractional calculus is being used increasingly frequently to take into account hereditary properties and fractal structure of real materials. Such problems can be solved analytically; however, numerical methods are widely in use. In this paper, we consider several numerical methods based on different definitions of fractional derivatives and their application to solutions of specific problems of heat conduction (diffusion). Conducted analysis allows us to select the most promising definitions and methods for an adequate description of actual diffusion processes in fractal media. A number of constitutive equations with derivatives of fractional order are analyzed. The problem of quasi-static and dynamic stretching of a thin rod is solved with Maxwell-type viscoelastic body model with fractional derivatives.