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The time-fractional diffusion-wave equation with the Caputo derivative is considered. The typical features of the solution to the Cauchy problem for this equation are discussed depending on values of the order of fractional derivative.
Content available remote Anomalous diffusion equation and diffusive stresses
Essentials of the Riemann-Liouville fractional calculus are recalled. Nonlocal generalizations of the Fourier law of the classical theory of heat conduction relating the heat flux vector to the temperature gradient and of the Fick law of the classical theory of diffusion relating the matter flux vector to the concentration gradient lead to non-classical theories. The time-nonlocal dependence between the flux vectors and corresponding gradients with “long-tale” power kernel can be interpreted in terms of fractional integrals and derivatives and yields the time-fractional diffusion equation.
Content available remote Two-dimensional and One-dimensional Quations and Their Applications
An interfacial region and a three-phase line region are considered as two dimensional and one-dimensional continua. Equations of the linear momentum balance and moment-of-momentum balance generalize the Laplace equation for surfaces and the Young equation for lines. Balance equations for surface dislocations and disc1inations are also considered. The motor analysis is used for a description of continua with couple stresses.
Content available remote Modeling of Crystal Defects in Nonlocal Elasticity : A Review
A review of studies concerning models of crystal defects in solids is presented. The emphasis is on describing imperfections in nonlocal elastic continuum. Nonlocal theory reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit.
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