Fractional Stochastic Partial Differential Equations Research Articles

Space-time fractional stochastic partial differential equations with Lévy noise

AbstractWe consider non-linear time-fractional stochastic heat type equation$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg] \end{array} $$and$$\begin{array}{} \displaystyle \frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u = I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h)\stackrel{\cdot}{N }(t,x,h)\bigg] \end{array} $$in (d + 1) dimensions, where α ∈ (0, 2] and d < min{2, β−1}α, ν > 0, $\begin{array}{} \partial^\beta_t \end{array} $ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, $\begin{array}{} I^{1-\beta}_t \end{array} $ is the fractional integral operator, N(t, x) are Poisson random measure with Ñ(t, x) being the compensated Poisson random measure. σ : ℝ → ℝ is a Lipschitz continuous function. We prove existence and uniqueness of mild solutions to this equation. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in [16, 33]. Under the linear growth of σ, we show that the solution of the time fractional stochastic partial differential equation follows an exponential growth with respect to the time. We also show the nonexistence of the random field solution of both stochastic partial differential equations when σ grows faster than linear.

Neutral fractional stochastic partial differential equations with Clarke subdifferential

ABSTRACT By using fractional calculus, stochastic analysis theory and fixed point theorems, sufficient conditions for approximate controllability of nonlocal Sobolev-type neutral fractional stochastic differential equations with fractional Brownian motion and Clarke subdifferential are established. Finally, an example is given to illustrate the obtained results.

Operator-Based Uncertainty Quantification of Stochastic Fractional Partial Differential Equations

Abstract Fractional calculus provides a rigorous mathematical framework to describe anomalous stochastic processes by generalizing the notion of classical differential equations to their fractional-order counterparts. By introducing the fractional orders as uncertain variables, we develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations (SFPDEs), subject to additive random noise. We characterize different sources of uncertainty and then, propagate their associated randomness to the system response by employing a probabilistic collocation method (PCM). We develop a fast, stable, and convergent Petrov–Galerkin spectral method in the physical domain in order to formulate the forward solver in simulating each realization of random variables in the sampling procedure.

Traveling wave solutions of some important Wick-type fractional stochastic nonlinear partial differential equations

In this article, exact traveling wave solutions of a Wick-type stochastic nonlinear Schr\"{o}dinger equation and of a Wick-type stochastic fractional Regularized Long Wave-Burgers (RLW-Burgers) equation have been obtained by using an improved computational method. Specifically, the Hermite transform is employed for transforming Wick-type stochastic nonlinear partial differential equations into deterministic nonlinear partial differential equations with integral and fraction order. Furthermore, the required set of stochastic solutions in the white noise space is obtained by using the inverse Hermite transform. Based on the derived solutions, the dynamics of the considered equations are performed with some particular values of the physical parameters. The results reveal that the proposed improved computational technique can be applied to solve various kinds of Wick-type stochastic fractional partial differential equations.

Optimal control of fractional neutral stochastic differential equations with deviated argument governed by Poisson jumps and infinite delay

SummaryIn this work, the optimal control for a class of fractional neutral stochastic differential equations with deviated arguments driven by infinite delay and Poisson jumps is studied in Hilbert space involving the Caputo fractional derivative. The sufficient conditions for the existence of mild solution results are formulated and proved by the virtue of fractional calculus, characteristic solution operator, fixed‐point theorem, and stochastic analysis techniques. Furthermore, the existence of optimal control of the proposed problem is presented by using Balder's theorem. Finally, the obtained theoretical results are applied to the fractional stochastic partial differential equations and a stochastic river pollution model.

Blow-Up Results for Space-Time Fractional Stochastic Partial Differential Equations

Consider non-linear time-fractional stochastic reaction-diffusion equations of the following type, $$ \partial^{\beta}_{t}u_{t}(x)=-\nu(-{\Delta})^{\alpha/2} u_{t}(x)+I^{1-\beta}[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)] $$ in (d + 1) dimensions, where ν > 0,β ∈ (0, 1), α ∈ (0, 2]. The operator $\partial ^{\beta }_{t}$ is the Caputo fractional derivative while − (−Δ)α/2 is the generator of an isotropic α-stable Levy process and I1−β is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel {\cdot }{F}(t,x)$ is a Gaussian noise. These equations might be used as a model for materials with random thermal memory. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on b, σ and the initial condition. Our results complement those of P. Chow in (Commun. Stoch. Anal. 3(2):211–222, 2009), Chow (J. Differential Equations 250(5):2567–2580, 2011), and Foondun et al. in (2016), Foondun and Parshad (Proc. Amer. Math. Soc. 143(9):4085–4094, 2015) among others.

Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion

<p style='text-indent:20px;'>In this paper, we consider the numerical approximation for a class of fractional stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with hurst index <inline-formula><tex-math id="M1">\begin{document}$ H∈ (\frac{1}{2}, 1) $\end{document}</tex-math></inline-formula>. By using spectral Galerkin method, we analyze the spatial discretization, and we give the temporal discretization by using the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. Under some suitable assumptions, we prove the sharp regularity properties and the optimal strong convergence error estimates for both semi-discrete and fully discrete schemes.

Error estimates of finite element methods for nonlinear fractional stochastic differential equations

In this paper, we consider the Galerkin finite element approximations of the initial value problem for the nonlinear fractional stochastic partial differential equations with multiplicative noise. We study a spatial semidiscrete scheme with the standard Galerkin finite element method and a fully discrete scheme based on the Goreno–Mainardi–Moretti–Paradisi (GMMP) scheme. We establish strong convergence error estimates for both semidiscrete and fully discrete schemes.

On approximation for fractional stochastic partial differential equations on the sphere

This paper gives the exact solution in terms of the Karhunen-Lo\`{e}ve expansion to a fractional stochastic partial differential equation on the unit sphere $\mathbb{S}^{2}\subset \mathbb{R}^{3}$ with fractional Brownian motion as driving noise and with random initial condition given by a fractional stochastic Cauchy problem. A numerical approximation to the solution is given by truncating the Karhunen-Lo\`{e}ve expansion. We show the convergence rates of the truncation errors in degree and the mean square approximation errors in time. Numerical examples using an isotropic Gaussian random field as initial condition and simulations of evolution of cosmic microwave background (CMB) are given to illustrate the theoretical results.

An Efficient Solution for Stochastic Fractional Partial Differential Equations with Additive Noise by a Meshless Method

With respect to wide range of applications of stochstic partial differential equation (SPDE) and high ability of meshless methods to solve complicated problems, in this paper, an efficient numerical method for the time fractional SPDE, formulated with Caputo’s fractional derivative, based on meshless methods is presented. This article presents a meshless method based on the radial basis functions to solve one-dimensional stochastic heat and advection–diffusion equations. In here, first, we approximate the time fractional derivative of the mentioned equations by a scheme of order $$ \mathsf {O}(\tau ^{2-\alpha }) $$ , $$ 0<\alpha <1 $$ then the spatial derivatives are approximated by Kansa approach. Numerical examples are presented to show the efficiency and effectiveness of the proposed method in solving fractional SPDEs.

Efficient numerical schemes for the solution of generalized time fractional Burgers type equations

The main aim of this paper is to propose two semi-implicit Fourier pseudospectral schemes for the solution of generalized time fractional Burgers type equations, with an analysis of consistency, stability, and convergence. Under some assumptions, the unconditional stability of the schemes is shown. In implementation of these schemes, the fast Fourier transform (FFT) can be used efficiently to improve the computational cost. Various test problems are included to illustrate the results that we have obtained regarding the proposed schemes. The results of numerical experiments are compared with analytical solutions and other existing methods in the literature to show the efficiency of proposed schemes in both accuracy and CPU time. As numerical solution of fractional stochastic nonlinear partial differential equations driven by Brownian motions are among current related research interests, we report the performance of these schemes on stochastic time fractional Burgers equation as well.

Some properties of non-linear fractional stochastic heat equations on bounded domains

We consider the following fractional stochastic partial differential equation on a bounded, open subset B of Rd for d ≥ 1∂tut(x)=Lut(x)+ξσ(ut(x))F˙(t,x),where ξ is a positive parameter and σ is a globally Lipschitz continuous function. The stochastic forcing term F˙(t,x) is white in time but possibly colored in space. The operator L is fractional Laplacian which is the infinitesimal generator of a symmetric α-stable Lévy process in Rd. We study the behaviour of the solution with respect to the parameter ξ.We show that under zero exterior boundary conditions, in the long run, the pth-moment of the solution grows exponentially fast for large values of ξ. However when ξ is very small we observe eventually an exponential decay of the pth-moment of this same solution.

Hypothesis Testing for Fractional Stochastic Partial Differential Equations with Applications to Neurophysiology and Finance

Hypothesis Testing for Fractional Stochastic Partial Differential Equations with Applications to Neurophysiology and Finance

Optimal Controls for Fractional Stochastic Functional Differential Equations of Order $$\alpha \in (1, 2]$$ α ∈ ( 1 , 2

In this paper, we study optimal control problems for a class of fractional stochastic functional differential equations of order \(\alpha \in (1, 2]\) in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence of mild solutions are investigated by using the fractional calculus, stochastic analysis theory, and fixed point theorems with the strongly continuous \(\alpha \)-order cosine family. Then, the existence conditions of optimal pairs of systems governed by a fractional stochastic partial differential equations of order \(\alpha \in (1, 2]\) with infinite delay are presented. The results are obtained under the mixed Lipschitz and Caratheodory conditions. Finally, an example is presented to illustrate the main results.

Intermittency fronts for space-time fractional stochastic partial differential equations in [formula omitted] dimensions

We consider time fractional stochastic heat type equation ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[σ(u)W⋅(t,x)] in (d+1) dimensions, where ν>0, β∈(0,1), α∈(0,2], d<min{2,β−1}α, ∂tβ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, W⋅(t,x) is space-time white noise, and σ:R→R is Lipschitz continuous. Mijena and Nane proved in (Mijena and Nane, 2016) that: (i) absolute moments of the solutions of this equation grow exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. The last result was proved under the assumptions α=2 and d=1. In this paper we extend this result to the case α=2 and d∈{1,2,3}.

Intermittence and Space-Time Fractional Stochastic Partial Differential Equations

We consider time fractional stochastic heat type equation $$\partial^{\beta}_{t}u_{t}(x)=-\nu(-{\Delta})^{\alpha/2} u_{t}(x)+I^{1-\beta}_{t}[\sigma(u)\overset{\cdot}{W}(t,x)] $$ in (d + 1) dimensions, where ν > 0, β ∈ (0, 1), α ∈ (0, 2], \(d<\min \{2,\beta ^{-1}\}\alpha \), \(\partial ^{\beta }_{t}\) is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, \(\overset {\cdot }{W}(t,x)\) is space-time white noise, and \(\sigma :\mathbb {R}\to \mathbb {R}\) is Lipschitz continuous. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove: (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. These results extend the results of Foondun and Khoshnevisan (Electron. J. Probab. 14(21), 548–568, 2009) and Conus and Khoshnevisan (Probab. Theory Relat. Fields 152(3–4), 681–701, 2012) on the parabolic stochastic heat equations.

Space–time fractional stochastic partial differential equations

We consider non-linear time-fractional stochastic heat type equation ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[σ(u)W⋅(t,x)] in (d+1) dimensions, where ν>0,β∈(0,1), α∈(0,2] and d<min{2,β−1}α, ∂tβ is the Caputo fractional derivative, −(−Δ)α/2 is the generator of an isotropic stable process, It1−β is the fractional integral operator, W⋅(t,x) is space–time white noise, and σ:R→R is Lipschitz continuous. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove existence and uniqueness of mild solutions to this equation and establish conditions under which the solution is continuous. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in Foondun and Khoshnevisan (2009), Walsh (1986). In sharp contrast to the stochastic partial differential equations studied earlier in Foondun and Khoshnevisan (2009), Khoshnevisan (2014) and Walsh (1986), in some cases our results give existence of random field solutions in spatial dimensions d=1,2,3. Under faster than linear growth of σ, we show that time fractional stochastic partial differential equation has no finite energy solution. This extends the result of Foondun and Parshad (in press) in the case of parabolic stochastic partial differential equations. We also establish a connection of the time fractional stochastic partial differential equations to higher order parabolic stochastic differential equations.

Solving a Nonlinear Fractional Stochastic Partial Differential Equation with Fractional Noise

In this article, we will prove the existence, uniqueness and Holder regularity of the solution to the fractional stochastic partial differential equation of the form $$\begin{aligned} \frac{\partial }{\partial t}u(t,x)=\mathfrak {D}(x,D)u(t,x)+\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^2 W^H}{\partial t\partial x}(t,x), \end{aligned}$$ where \(\mathfrak {D}(x,D)\) denotes the Markovian generator of stable-like Feller process, \(f:[0,T]\times \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) is a measurable function, and \(\frac{\partial ^2 W^H}{\partial t\partial x}(t,x)\) is a double-parameter fractional noise. In addition, we establish lower and upper Gaussian bounds for the probability density of the mild solution via Malliavin calculus and the new tool developed by Nourdin and Viens (Electron J Probab 14:2287–2309, 2009).

Exact solution of the Wick-type stochastic fractional coupled KdV equations

Fractional differential equations are widely used to model many physical phenomena in science and engineering. This paper investigates the exact solutions of Wick-type stochastic fractional variable coefficients coupled KdV equations. By implementing fractional sub-equation method based on the Kudryashov technique, new families of exact travelling wave solutions are obtained. Moreover, the obtained white noise functional solutions can be expressed as exponential type. In particular, the stochastic fractional model is reduced into a deterministic fractional one by using the Hermite transform. The results reveal that the proposed technique is very effective and simple for obtaining exact solutions of stochastic fractional partial differential equations.

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.