1,721,010 research outputs found
Confinement of a hot temperature patch in the modified SQG model
No full textIn this paper we study the time evolution of a temperature patch in R 2 according to the modified Surface Quasi-Geostrophic (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation
On the generalized Hardy–Hardy–Maurer model with memory effects
No full textIn this paper, we reconsider the Hardy–Hardy–Maurer model for the heat propagation in nonlinear rigid conductors in the framework of fractional thermoelasticity, taking into account memory effects. We therefore obtain nonlinear time-fractional telegraph-type equations that are linearizable by change in variable. We discuss in detail two different models in the context of the more general theory of Gurtin and Pipkin of heat propagation with memory. We finally show that a similar derivation of linearizable fractional telegraph-type equations of higher order can be obtained also in the physics of dielectrics
SOME APPLICATIONS OF WRIGHT FUNCTIONS IN FRACTIONAL DIFFERENTIAL EQUATIONS
No full textIn this note we prove some new results about the application of Wright functions of the first kind to solve fractional differential equations with variable coefficients. Then, we consider some applications of these results in order to obtain some new particular solutions for nonlinear fractional partial differential equations
On fractional Cattaneo equation with partially reflecting boundaries
No full textIn this paper we study the time-fractional Cattaneo equation in a bounded domain with semi-reflecting conditions. In particular, we are able to find the Laplace transform of the probability density function of the absorption time and therefore the mean-time to absorption. We show the crucial role of the time-fractional formulation. Indeed, in this case we have that the mean-time to absorption diverges due to the fact that the generalized Cattaneo equation is based on the application of integral operators with a long-tail memory kernel. We also consider the time-fractional diffusion and wave limits behaviour, recovering some previous results obtained in the literature. Finally, a section is devoted to the generalized Cattaneo equation in unbounded domain. In this case we are able to discuss the characterization of the mean square displacement for short times and asymptotically by using the Fourier-Laplace transform of the solution
Run-and-tumble motion in one dimension with space-dependent speed
No full textWe consider a particle performing run-and-tumble dynamics with space-dependent speed. The model has biological relevance as it describes motile bacteria or cells in heterogeneous environments. We give exact expression for the probability density function in the case of free motion in unbounded space. We then analyze the case of a particle moving in a confined interval in the presence of partially absorbing boundaries, reporting the probability density in the Laplace (time) domain and the mean time to absorption. We also discuss the relaxation to the steady state in the case of confinement with reflecting boundaries and drift effects due to direction-dependent tumbling rates, modeling taxis phenomena of cells. The case of diffusive particles with spatially variable diffusivity is obtained as a limiting case
A note on differential equations of logistic type
Full text linkLogistic equations play a pivotal role in the study of any nonlinear evolution process exhibiting growth and saturation. The interest for the phenomenology they rule goes well beyond physical processes and covers many aspects of ecology, population growth, economy. According to such a broad range of applications, there are different forms of functions and distributions which are recognized as generalized logistics. Sometimes they are obtained by fitting procedures. Therefore, criteria might be needed to infer the associated nonlinear differential equations, useful to guess “hidden” evolution mechanisms. In this article we analyze different forms of logistic functions and use simple means to reconstruct the differential equation they satisfy. Our study includes also differential equations containing nonstandard forms of derivative operators, like those of the Laguerre type
Nonlinear heat conduction equations with memory: Physical meaning and analytical results
No full textWe study nonlinear heat conduction equations with memory effects within the framework of the fractional calculus approach to the generalized Maxwell-Cattaneo law. Our main aim is to derive the governing equations of heat propagation, considering both the empirical temperature-dependence of the thermal conductivity coefficient (which introduces nonlinearity) and memory effects, according to the general theory of Gurtin and Pipkin of finite velocity thermal propagation with memory. In this framework, we consider in detail two different approaches to the generalized Maxwell-Cattaneo law, based on the application of long-tail Mittag-Leffler memory function and power law relaxation functions, leading to nonlinear time-fractional telegraph and wave-type equations.We also discuss some explicit analytical results to the model equations based on the generalized separating variable method and discuss their meaning in relation to some well-known results of the ordinary case
Alternative probabilistic representations of barenblatt-type solutions
Full text linkA general class of probability density functions (1 (‖x‖) )γ β u(x, t) = Ct−αd − ctα, x ∈ Rd,t >0, + is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions u(x, t) are proposed. In the one-dimensional case, by means of this approach, u(x, t) can be connected with the wave propagation
Some probabilistic properties of fractional point processes
No full textIn this article, the first hitting times of generalized Poisson processes Nf(t), related to Bernštein functions f are studied. For the space-fractional Poisson processes, Nα(t), t > 0 (corresponding to f = xα), the hitting probabilities P{Tαk < ∞} are explicitly obtained and analyzed. The processes Nf(t) are time-changed Poisson processes N(Hf(t)) with subordinators Hf(t) and here we study N(∑nj=1 Hfj(t)) and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form N(gH,v(t)) where (gH,v(t)) are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space–time Poisson process is no longer a renewal process
A note on Hadamard fractional differential equations with varying coefficients and their applications in probability
No full textIn this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions
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