Superdiffusion Equation Research Articles

Inverse a time-dependent potential problem of a generalized time-fractional super-diffusion equation with a nonlinear source from a nonlocal integral observation

This paper focus on the problem of reconstructing the time-dependent potential in a class of generalized (including three special cases: the classical/multi-term/distributed order) time-fractional super-diffusion equations with nonlinear sources from a nonlocal integral observation. For such nonlinear equation, we investigate it for both the direct and inverse time-dependent potential problems. For the direct problem, given the time-dependent potential function p(t), we obtain the well-posedness of the corresponding initial–boundary value problem. For the inverse potential problem, by utilizing additional nonlocal measurement data and using the Arzelà–Ascoli theorem and Grönwall’s inequality, we prove the existence and uniqueness of the solution for such nonlinear problem. Meanwhile, we also demonstrate the ill-posedness of the inverse problem. Moreover, to validate the theoretical results, we numerically reconstruct the potential term from Bayesian perspective. Several numerical examples are presented to show the efficiency of the proposed method.

Well-Posedness of Mild Solutions for Superdiffusion Equations with Spatial Nonlocal Operators

Well-Posedness of Mild Solutions for Superdiffusion Equations with Spatial Nonlocal Operators

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Critical behavior of a semilinear time fractional diffusion equation with forcing term depending on time and space

In this paper we study the time fractional semilinear diffusion equation 0CDtαu(t,x)−Δu(t,x)=|u|p+tσw(x) with the initial conditions u(0,x)=u0(x) and ∂tu(0,x)=u1(x) for x∈RN, where α∈(0,1)∪(1,2), p>1, −1<σ<0 and w⁄≡0, u1=0 when 0<α<1. The novelty of this paper lies in considering semilinear time-fractional diffusion equations with a forcing term tσw(x) which depends on time and space. We show that there are critical exponents in the following cases. (i) For α+σ>0, the solution of the above subdiffusion equation blows up in finite time when 1<p<1+2ααN−2σ−2α and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0 and w belonging to certain Lebesgue spaces when p≥1+2ααN−2σ−2α. (ii) The solution of the above superdiffusion equation blows up in finite time when 1<p≤1+2ααN−2σ−2α and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0 and w belonging to certain Lebesgue spaces when p>1+2ααN−2σ−2α and u1=0. (iii) For α+σ≥1, the solution of the above superdiffusion equation blows up in finite time when 1<p≤1+2ααN−2σ−2α and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0, u1 and w belonging to certain Lebesgue spaces when p>1+2ααN−2σ−2α. (iv) For α+σ<1, the solution of the above superdiffusion equation blows up in finite time when 1<p<1+2ααN−2 and ∫RNw(x)dx>0, while the global solution exists for suitably small initial data u0, u1 and w belonging to certain Lebesgue spaces when p>1+2ααN−2. The critical exponent in (iv) is different from that in (iii) and (ii). This peculiarity is related to the fact the time order of the equation and the inhomogeneous are both fractional, and so the role played by the second data u1 becomes “unnatural” as α∈(1,2). Namely, the change of the critical exponent in (iv) is due to that α+σ<1 and u1⁄≡0.

A Stochastic Model of Anomalously Fast Transport of Heat Energy in Crystalline Bodies

In this work, a new method for constructing the infinite-dimensional Ornstein–Uhlenbeck stochastic process is introduced. The constructed process is used to perturb the harmonic system in order to model anomalously fast heat transport in one-dimensional nanomaterials. The introduced method made it possible to obtain a transition probability function that allows for a different approach to the analysis of equations with such a disturbance. This creates the opportunity to relax assumptions about temporal correlations for such a process, which may lead to a qualitatively different model of energy transport through vibrations of the crystal lattice and, as a result, to obtain the superdiffusion equation on a macroscopic scale with an order of the fractional Laplacian different from the value of 3/4 obtained so far in stochastic models. Simulations confirming these predictions are presented and discussed.

Chimera states in a chain of superdiffusively coupled neurons.

Two- and three-component systems of superdiffusion equations describing the dynamics of action potential propagation in a chain of non-locally interacting neurons with Hindmarsh-Rose nonlinear functions have been considered. Non-local couplings based on the fractional Laplace operator describing superdiffusion kinetics are found to support chimeras. In turn, the system with local couplings, based on the classical Laplace operator, shows synchronous behavior. For several parameters responsible for the activation properties of neurons, it is shown that the structure and evolution of chimera states depend significantly on the fractional Laplacian exponent, reflecting non-local properties of the couplings. For two-component systems, an anisotropic transition to full incoherence in the parameter space responsible for non-locality of the first and second variables is established. Introducing a third slow variable induces a gradual transition to incoherence via additional chimera states formation. We also discuss the possible causes of chimera states formation in such a system of non-locally interacting neurons and relate them with the properties of the fractional Laplace operator in a system with global coupling.

Numerical Method for Solving the Nonlinear Superdiffusion Equation with Functional Delay

For a space-fractional diffusion equation with a nonlinear superdiffusion coefficient and with the presence of a delay effect, the grid numerical method is constructed. Interpolation and extrapolation procedures are used to account for the functional delay. At each time step, the algorithm reduces to solving a linear system with a main matrix that has diagonal dominance. The convergence of the method in the maximum norm is proved. The results of numerical experiments with constant and variable delays are presented.

Super-diffusion and crossover from diffusive to anomalous transport in a one-dimensional system

We study transport in a one-dimensional lattice system with two conserved quantities - “volume” and energy. Considering a slowly evolving local equilibrium state that is slightly deviated from an underlying global equilibrium, we estimate the correction to the local equilibrium distribution. This correction arises mainly through the space-time correlations of some local currents. In the continuum limit, we show that the local equilibrium distribution along with the correction yields drift-diffusion equation for the “volume” and super-diffusion equation for the energy in the linear response regime as macroscopic hydrodynamics as one would obtain from non-linear fluctuating hydrodynamic theory. We find explicit expression of the super-diffusion equation. Further, we find diffusive correction to the super-diffusive evolution. Such a correction allows us to study a crossover from diffusive to anomalous transport. We demonstrate this crossover numerically through the spreading of an initially localized heat pulse in equilibrium as well as through the system size scaling of the stationary current in non-equilibrium steady state.

Percolation properties of the neutron population in nuclear reactors.

Reactor physics aims at studying the neutron population in a reactor core under the influence of feedback mechanisms, such as the Doppler temperature effect. Numerical schemes to calculate macroscopic properties emerging from such coupled stochastic systems, however, require us to define intermediate quantities (e.g., the temperature field), which are bridging the gap between the stochastic neutron field and the deterministic feedback. By interpreting the branching random walk of neutrons in fissile media under the influence of a feedback mechanism as a directed percolation process and by leveraging on the statistical field theory of birth death processes, we will build a stochastic model of neutron transport theory and of reactor physics. The critical exponents of this model, combined with the analysis of the resulting field equationinvolving a fractional Laplacian, will show that the critical diffusion equationcannot adequately describe the spatial distribution of the neutron population and shifts instead to a critical superdiffusion equation. The analysis of this equationwill reveal that nonnegligible departure from mean-field behavior might develop in reactor cores, questioning the attainable accuracy of the numerical schemes currently used by the nuclear industry.

α-Robust H1-norm analysis of a finite element method for the superdiffusion equation with weak singularity solutions

In this work, the superdiffusion equation with a Caputo derivative of order α∈(1,2) is considered. Some priori bounds on certain derivatives of the solution show that the solution exhibits a weak singularity at the initial time t=0. To resolve this initial singularity, we rewrite the superdiffusion equation as a coupled system by introducing a intermediate variable p:=Dtα/2(u−tu1), and adopt the L1 scheme and Alikhanov scheme on graded meshes in temporal direction. In spatial direction, the conforming finite element method is used. Furthermore, we derive the H1-norm stability result. It is worth noting that some priori bounds on certain derivatives of p are obtained, on basis of which, we derive an α-robust prior error estimate with optimal H1-norm convergence order. Finally, we provide the numerical experiment to further verify our theoretical analysis.

Numerical method for system of space-fractional equations of superdiffusion type with delay and Neumann boundary conditions

We consider a system of two space-fractional superdiffusion equations with functional general delay and Neumann boundary conditions. For this problem, an analogue of the Crank-Nicolson method is constructed, based on the shifted Grünwald-Letnikov formulas for approximating fractional Riesz derivatives with respect to a spatial variable and using piecewise linear interpolation of discrete prehistory with extrapolation by continuation to take into account the delay effect. With the help of the Gershgorin theorem, the solvability of the difference scheme and its stability are proved. The order of convergence of the method is obtained. The results of numerical experiments are presented.

Numerical methods for systems of diffusion and superdiffusion equations with Neumann boundary conditions and with delay

Numerical methods for systems of diffusion and superdiffusion equations with Neumann boundary conditions and with delay

Self-Similar Solutions in the Theory of Nonstationary Radiative Transfer in Spectral Lines in Plasmas and Gases

Radiative transfer (RT) in spectral lines in plasmas and gases under complete redistribution of the photon frequency in the emission-absorption act is known as a superdiffusion transport characterized by the irreducibility of the integral (in the space coordinates) equation for the atomic excitation density to a diffusion-type differential equation. The dominant role of distant rare flights (Lévy flights, introduced by Mandelbrot for trajectories generated by the Lévy stable distribution) is well known and is used to construct approximate analytic solutions in the theory of stationary RT (the escape probability method is the best example). In the theory of nonstationary RT, progress based on similar principles has been made recently. This includes approximate self-similar solutions for the Green’s function (i) at an infinite velocity of carriers (no retardation effects) to cover the Biberman–Holstein equation for various spectral line shapes; (ii) for a finite fixed velocity of carriers to cover a wide class of superdiffusion equations dominated by Lévy walks with rests; (iii) verification of the accuracy of above solutions by comparison with direct numerical solutions obtained using distributed computing. The article provides an overview of the above results with an emphasis on the role of distant rare flights in the discovery of nonstationary self-similar solutions.

Semilinear fractional-order evolution equations of Sobolev type in the sectorial case

ABSTRACT The local unique solvability of the Cauchy-type problem to a semilinear equation in a Banach space, which is solved with respect to the highest order Riemann–Liouville derivative, is proved. A linear unbounded operator at the unknown function in the equation generates an analytic in a sector resolving the family of operators of the linear homogeneous fractional-order equation. This result is applied to the study of initial-boundary value problems for a class of nonlinear partial differential equations, in particular, containing a nonlinear superdiffusion equation. Besides, it is used for the investigation of the local unique solvability of the Showalter–Sidorov type problem to a semilinear Sobolev-type equation in a Banach space with a sectorial pair of operators and with Riemann–Liouville derivatives. For this aim, we essentially use two types of condition on the nonlinear operator: the condition on the image of this operator or the condition of its independence of the elements of the degeneration subspace. These abstract results are demonstrated on examples of initial-boundary value problems for Sobolev-type nonlinear partial differential equations.

ON THE NATURE OF A CLASSICAL PSEUDODIFFERENTIAL EQUATION

The work is devoted to the study of the general nature of one classical parabolic pseudodi- fferential equation with the operator M.Rice of fractional differentiation. At the corresponding values of the order of fractional differentiation, this equation is also known as the isotropic superdiffusion equation. It is a natural generalization of the classical diffusion equation. It is also known that the fundamental solution of the Cauchy problem for this equation is the density distribution of probabilities of stable symmetric random processes by P.Levy. The paper shows that the fundamental solution of this equation is the distribution of probabilities of the force of local influence of moving objects in a nonstationary gravitational field, in which the interaction between masses is subject to the corresponding potential of M.Rice. In this case, the classical case of Newton’s gravity corresponds to the known nonstationary J.Holtsmark distribution.

A fourth order non-polynomial quintic spline collocation technique for solving time fractional superdiffusion equations

The purpose of this article is to present a technique for the numerical solution of Caputo time fractional superdiffusion equation. The central difference approximation is used to discretize the time derivative, while non-polynomial quintic spline is employed as an interpolating function in the spatial direction. The proposed method is shown to be unconditionally stable and O(h^{4}+Delta t^{2}) accurate. In order to check the feasibility of the proposed technique, some test examples have been considered and the simulation results are compared with those available in the existing literature.

Boundary Value Problems Governed by Superdiffusion in the Right Angle: Existence and Regularity

For α∈(1,2), we analyze a stationary superdiffusion equation in the right angle in the unknown u=u(x1,x2): Dx1αu+Dx2αu=f(x1,x2), where Dxα is the Caputo fractional derivative. The classical solvability in the weighted fractional Hölder classes of the associated boundary problems is addressed.

Global well-posedness and blow-up solutions of the Cauchy problem for a time-fractional superdiffusion equation

We study the following time-fractional nonlinear superdiffusion equation $$\begin{aligned} \left\{ \begin{array}{l} {}_0^CD_t^\alpha u-\triangle u=|u|^{p}, \ \ x\in \mathbb {R}^N,\ \ t>0, \\ u(0,x)=u_0(x),\ \ u_t(0,x)=u_1(x),\ \ x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$$ where $$1<\alpha <2$$ , $$p>1$$ , $$u_0,u_1\in L^q(\mathbb {R}^N)$$ ( $$q>1$$ ) and $${_0^CD_t^\alpha u}$$ denotes the Caputo fractional derivative of order $$\alpha .$$ The critical exponents of this problem are determined when $$u_1\equiv 0$$ and $$u_1\not \equiv 0, $$ respectively.

Differential quadrature method for space-fractional diffusion equations on 2D irregular domains

In mathematical physics, the space-fractional diffusion equations are of particular interest in the studies of physical phenomena modelled by L\'{e}vy processes, which are sometimes called super-diffusion equations. In this article, we develop the differential quadrature (DQ) methods for solving the 2D space-fractional diffusion equations on irregular domains. The methods in presence reduce the original equation into a set of ordinary differential equations (ODEs) by introducing valid DQ formulations to fractional directional derivatives based on the functional values at scattered nodal points on problem domain. The required weighted coefficients are calculated by using radial basis functions (RBFs) as trial functions, and the resultant ODEs are discretized by the Crank-Nicolson scheme. The main advantages of our methods lie in their flexibility and applicability to arbitrary domains. A series of illustrated examples are finally provided to support these points.

Existence and uniqueness of weak solutions for a class of fractional superdiffusion equations

In this paper, we consider the existence and uniqueness of weak solutions for a class of fractional superdiffusion equations with initial-boundary conditions. For a multidimensional fractional drift superdiffusion equation, we just consider the simplest case with divergence-free drift velocity u in L^{2}(Omega) only depending on the spatial variable x. Finally, exploiting the Schauder fixed point theorem combined with the Arzelà-Ascoli compactness theorem, we obtain the existence and uniqueness of weak solutions in the standard Banach space C([0,T]; H_{0}^{1}(Omega)) for a class of fractional superdiffusion equations.