Related papers: Evolution, its Fractional Extension and Generaliza…
A fractional diffusion equation based on Riemann-Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of $H$-functions. It differs from the known…
In this paper, a fractional generalization of the wave equation that describes propagation of damped waves is considered. In contrast to the fractional diffusion-wave equation, the fractional wave equation contains fractional derivatives of…
Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$,…
The time evolution of the universe is usually mathematically described under a continuous time and thus time reversible. Here, the consequences of studying the evolution of a homogenous isotropic universe by time continuous reversible…
In special relativity theory the physical quantities are generally expressed as function of the velocity. In the particular case of an extended object, the factor 1/gamma of Lorentz contraction of its length in the direction of motion is…
A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…
In this paper, we introduce a new classical fractional particle model incorporating fractional first derivatives. This model represents a natural extension of the standard classical particle with kinetic energy being quadratic in fractional…
We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such…
We study the effect of time evolution on galaxy bias. We argue that at any order in perturbations, the galaxy density contrast can be expressed in terms of a finite set of locally measurable operators made of spatial and temporal…
We describe a method for removing the numerical errors in the modeling of linear evolution equations that are caused by approximating the time derivative by a finite difference operator. The method is based on integral transforms realized…
Each scheme of state reconstruction comes down to parametrize the state of a quantum system by expectation values or probabilities directly measurable in an experiment. It is argued that the time evolution of these quantities provides an…
We introduce an $L_q(L_p)$-theory for the quasi-linear fractional equations of the type $$ \partial^{\alpha}_t u(t,x)=a^{ij}(t,x)u_{x^i x^j}(t,x)+f(t,x,u), \quad t>0, \,x\in \mathbf{R}^d. $$ Here, $\alpha\in (0,2)$, $p,q>1$, and…
Moments are expectation values of products of powers of position and momentum, taken over quantum states (or averages over a set of classical particles). For free particles, the evolution in the quantum case is closely related to that of a…
For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of the operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the…
A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the linearised system has frequencies 1 and…
We discuss initial value problems for time evolution equations in one dimensional space which are expressed by the lattice operators and propose some new equations to which complexity of solutions is of polynomial class. Novel type of…
A time dependent variational approach is considered to derive the equations of movement for the $\lambda \phi^4$ model. The temporal evolution of the model is performed numerically in the frame of the Gaussian approximation in a lattice of…
Following the recent study on the emergent Friedmann equation from the expansion of cosmic space for a flat universe, we apply this method to a nonflat universe, and modify the evolution equation to lead to the Friedmann equation. In order…
Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…