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Anomalous diffusion equation and diffusive stresses

Povstenko J.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2007

Vol. 12

97-100

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.

Existence and uniqueness of the solution in the frequency domain for the reflection-transmission problem in a viscoelastic layer

Caviglia G., Morro A.

Archives of Mechanics

2004

Vol. 56, nr 1

59-82

Existence and uniqueness for the reflection-transmission process originated in a viscoelastic solid layer are investigated. Wave propagation is framed within the Fourier-transform domain and the oblique incidence is modelled by a factor involving a transverse wave vector. The backward-forward propagation in the axial direction is ascertained through the sign of an energy flux. Next, a connection is established between the energy flux and an Hermitian matrix whose eigenvalues are half positive and half negative. The proof is given that if the matrix has two diagonal blocks, one of which is positive definite and the other is negative definite, the solution to the reflection-transmission problem exists and is unique. The condition on the blocks is found to hold, e.g., for obliquely propagating homogeneous waves in anisotropic elasticity or normally propagating waves in isotropic viscoelasticity.