Inverse source problems for degenerate time-fractional PDE

In this paper, we investigate two inverse source problems for degenerate time-fractional partial differential equation in rectangular domains. The first problem involves a space-degenerate partial differential equation and the second one involves a time-degenerate partial differential equation. Solutions to both problem are expressed in series expansions. For the first problem, we obtained solutions in the form of Fourier-Legendre series. Convergence and uniqueness of solutions have been discussed. Solutions to the second problem are expressed in the form of Fourier-Sine series and they involve a generalized Mittag- Leffler type function. Moreover, we have established a new estimate for this generalized Mittag-Leffler type function. The obtained results are illustrated by providing example solutions using certain given data at the initial and final time.Comment: 12 pages, 8 figure

Dynamics with Infinitely Many Time Derivatives and Rolling Tachyons

Both in string field theory and in p-adic string theory the equations of motion involve infinite number of time derivatives. We argue that the initial value problem is qualitatively different from that obtained in the limit of many time derivatives in that the space of initial conditions becomes strongly constrained. We calculate the energy-momentum tensor and study in detail time dependent solutions representing tachyons rolling on the p-adic string theory potentials. For even potentials we find surprising small oscillations at the tachyon vacuum. These are not conventional physical states but rather anharmonic oscillations with a nontrivial frequency--amplitude relation. When the potentials are not even, small oscillatory solutions around the bottom must grow in amplitude without a bound. Open string field theory resembles this latter case, the tachyon rolls to the bottom and ever growing oscillations ensue. We discuss the significance of these results for the issues of emerging closed strings and tachyon matter.Comment: 46 pages, 14 figures, LaTeX. Replaced version: Minor typos corrected, some figures edited for clarit

Fractional Calculus in Wave Propagation Problems

Fractional calculus, in allowing integrals and derivatives of any positive order (the term "fractional" kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. In this lecture we devote our attention to wave propagation problems in linear viscoelastic media. Our purpose is to outline the role of fractional calculus in providing simplest evolution processes which are intermediate between diffusion and wave propagation. The present treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades.Comment: 33 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1008.134

On the thermal conduction in tangled magnetic fields in clusters of galaxies

Thermal conduction in tangled magnetic fields is reduced because heat conducting electrons must travel along the field lines longer distances between hot and cold regions of space than if there were no fields. We consider the case when the tangled magnetic field has a weak homogeneous component. We examine two simple models for temperature in clusters of galaxies: a time-independent model and a time-dependent one. We find that the actual value of the effective thermal conductivity in tangled magnetic fields depends on how it is defined for a particular astrophysical problem. Our final conclusion is that the heat conduction never totally suppressed but is usually important in the central regions of galaxy clusters, and therefore, it should not be neglected.Comment: 16 pages, 4 figure

Guidance, flight mechanics and trajectory optimization. Volume 4 - The calculus of variations and modern applications

Guidance, flight mechanics, and trajectory optimization - calculus of variations and modern application