Spatially fractional-order viscoelasticity, non-locality and a new kind of anisotropy

Spatial non-locality of space-fractional viscoelastic equations of motion is studied. Relaxation effects are accounted for by replacing second-order time derivatives by lower-order fractional derivatives and their generalizations. It is shown that space-fractional equations of motion of an order strictly less than 2 allow for a new kind anisotropy, associated with angular dependence of non-local interactions between stress and strain at different material points. Constitutive equations of such viscoelastic media are determined. Explicit fundamental solutions of the Cauchy problem are constructed for some cases isotropic and anisotropic non-locality

Distributed Order Derivatives and Relaxation Patterns

We consider equations of the form $(D_{(\rho)}u)(t)=-\lambda u(t)$, $t>0$, where $\lambda >0$, $D_{(\rho)}$ is a distributed order derivative, that is the Caputo-Dzhrbashyan fractional derivative of order $\alpha$, integrated in $\alpha\in (0,1)$ with respect to a positive measure $\rho$. Such equations are used for modeling anomalous, non-exponential relaxation processes. In this work we study asymptotic behavior of solutions of the above equation, depending on properties of the measure $\rho$

A complex ray-tracing tool for high-frequency mean-field flow interaction effects in jets

This paper presents a complex ray-tracing tool for the calculation of high-frequency Green’s functions in 3D mean field jet flows. For a generic problem, the ray solution suffers from three main deficiencies: multiplicity of solutions, singularities at caustics, and the determining of complex solutions. The purpose of this paper is to generalize, combine and apply existing stationary media methods to moving media scenarios. Multiplicities are dealt with using an equivalent two-point boundary-value problem, whilst non-uniformities at caustics are corrected using diffraction catastrophes. Complex rays are found using a combination of imaginary perturbations, an assumption of caustic stability, and analytic continuation of the receiver curve. To demonstrate this method, the ray tool is compared against a high-frequency modal solution of Lilley’s equation for an off-axis point source. This solution is representative of high-frequency source positions in real jets and is rich in caustic structures. A full utilization of the ray tool is shown to provide excellent results<br/

A time-fractional Borel-Pompeiu formula and a related hypercomplex operator calculus

In this paper we develop a time-fractional operator calculus in fractional Clifford analysis. Initially we study the $L_p$-integrability of the fundamental solutions of the multi-dimensional time-fractional diffusion operator and the associated time-fractional parabolic Dirac operator. Then we introduce the time-fractional analogues of the Teodorescu and Cauchy-Bitsadze operators in a cylindrical domain, and we investigate their main mapping properties. As a main result, we prove a time-fractional version of the Borel-Pompeiu formula based on a time-fractional Stokes' formula. This tool in hand allows us to present a Hodge-type decomposition for the forward time-fractional parabolic Dirac operator with left Caputo fractional derivative in the time coordinate. The obtained results exhibit an interesting duality relation between forward and backward parabolic Dirac operators and Caputo and Riemann-Liouville time-fractional derivatives. We round off this paper by giving a direct application of the obtained results for solving time-fractional boundary value problems.The work of M. Ferreira, M.M. Rodrigues and N. Vieira was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT–Fundação para a Ciência e a Tecnologia, within project UID/MAT/04106/2019. The work of the authors was supported by the project New Function Theoretical Methods in Computational Electrodynamics / Neue funktionentheoretische Methoden für instationäre PDE, funded by Programme for Cooperation in Science between Portugal and Germany (“Programa de Ações Integradas Luso-Alemãs 2017” - DAAD-CRUP - Acção No. A-15/17 / DAAD-PPP Deutschland-Portugal, Ref: 57340281). N. Vieira was also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).publishe