Related papers: Fractional Electromagnetic Waves
In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for diffusion-wave equation with the Caputo fractional derivative in a domain bounded by smooth curves. The uniqueness…
The following material was created with the idea of being used for an introductory fractional calculus course. A recapitulation of the history of fractional calculus is presented, as well as the different attempts at fractional derivatives…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
In this work, a new relationship is established between the solutions of higher fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As…
In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1.…
The solution of a causal fractionary wave equation in an infinite potential well was obtained. First, the so-called "free particle" case was solved, giving as normalizable solutions a superposition of damped oscillations similar to a wave…
Time-fractional semilinear and quasilinear parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are considered, solutions of which exhibit a singular behaviour at an initial time of type $t^\sigma$ for any fixed…
In practice many problems related to space/time fractional equations depend on fractional parameters. But these fractional parameters are not known a priori in modelling problems. Hence continuity of the solutions with respect to these…
We develop a theory of the Cauchy problem for linear evolution systems of partial differential equations with the Caputo-Dzrbashyan fractional derivative in the time variable $t$. The class of systems considered in the paper is a fractional…
We describe and implement a numerical method for modelling the frequency-dependent power-law absorption of ultrasound in tissue, as governed by the first order linear wave equations with a loss taking the form of a fractional time…
Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a…
Distributed-order time-fractional wave equations appear in the modeling of wave propagation in viscoelastic media. The material characteristics of the medium are modeled through constitutive functions or distributions in the…
This paper is concerned with the study of the well-posedeness for the initial boundary value problem to the time-fractional wave equation with acoustic boundary conditions. The problem is considered in a bounded and connected domain $\Omega…
Properties of the phase space of the standard map with memory are investigated. This map was obtained from a kicked fractional differential equation. Depending on the value of the parameter of the map and the fractional order of the…
We consider time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$. For such equations, we give an elementary proof of the weak maximum principle under no assumptions on the sign of the reaction…
We consider a class of fractional time stochastic equation defined on a bounded domain and show that the presence of the time derivative induces a significant change in the qualitative behaviour of the solutions. This is in sharp contrast…
We consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable $t$, the fractional order of the self-adjoint positive definite unbounded operator in a Hilbert space and a singular…
A boundary integral equation formulation is presented for the electromagnetic transmission problem where an incident electromagnetic wave is scattered from a bounded dielectric object. The formulation provides unique solutions for all…
The transformation of the partial fractional derivatives under spatial rotation in $R^2$ are derived for the Riemann-Liouville and Caputo definitions. These transformation properties link the observation of physical quantities, expressed…
Fractional dynamics of relativistic particle is discussed. Derivatives of fractional orders with respect to proper time describe long-term memory effects that correspond to intrinsic dissipative processes. Relativistic particle subjected to…