Nonlinear Fractional Diffusion Equation Research Articles

Iterative learning algorithms for boundary tracing problems of nonlinear fractional diffusion equations

<abstract><p>In this paper, the iterative learning control technique is extended to distributed parameter systems governed by nonlinear fractional diffusion equations. Based on $ P $-type and $ PI^{\theta} $-type iterative learning control methods, sufficient conditions for the convergences of systems are given. Finally, numerical examples are presented to illustrate the efficiency of the proposed iterative schemes. The numerical results show that the closed-loop iterative learning control scheme converges faster than the open-loop iterative learning control scheme and the $ PI^{\theta} $-type iterative learning control scheme converges faster than the $ P $-type and the $ PI $-type iterative learning control scheme.</p></abstract>

Well-posedness results for nonlinear fractional diffusion equation with memory quantity

We study the well-posedness for solutions of an initial-value boundary problem on a two-dimensional space with source functions associated to nonlinear fractional diffusion equations with the Riemann-Liouville derivative and nonlinearities with memory on a two-dimensional domain. In order to derive the existence and uniqueness for solutions, we mainly proceed on reasonable choices of Hilbert spaces and the Banach fixed point principle. Main results related to the Mittag-Leffler functions such as its usual lower or upper bound and the relationship with the Mainardi function are also applied. In addition, to set up the global-in-time results, $ L^p-L^q $ estimates and the smallness assumption on the initial data function are also necessary to be applied in this research. Finally, the work also considers numerical examples to illustrate the graphs of analytic solutions.

High-order numerical algorithm for fractional-order nonlinear diffusion equations with a time delay effect

<abstract><p>In this paper, we examine and provide numerical solutions to the nonlinear fractional order time-space diffusion equations with the influence of temporal delay. An effective high-order numerical scheme that mixes the so-called Alikhanov $ L2-1_\sigma $ formula side by side to the power of the Galerkin method is presented. Specifically, the time-fractional component is estimated using the uniform $ L2-1_{\sigma} $ difference formula, while the spatial fractional operator is approximated using the Legendre-Galerkin spectral approximation. In addition, Taylor's approximations are used to discretize the term of the nonlinear source function. It has been shown theoretically that the suggested scheme's numerical solution is unconditionally stable, with a second-order time-convergence and a space-convergent order of exponential rate. Furthermore, a suitable discrete fractional Grönwall inequality is then utilized to quantify error estimates for the derived solution. Finally, we provide a numerical test that closely matches the theoretical investigation to assess the efficacy of the suggested method.</p></abstract>

Numerical solution of Atangana–Baleanu–Caputo time-space fractional diffusion equation

In this article, the time-space fractional diffusion equation is solved by using the fractional operator in Atangana–Baleanu–Caputo (ABC) sense based on the Mittag-Leffler function involving non-singular and non-local kernels. This study mainly develop the numerical method for time-space linear and non-linear ABC fractional diffusion equation by utilising centred difference approximation with second-order accuracy of Riesz fractional derivative in space and trapezoidal formula for fractional integral approximation for a system of ABC integral equations. Also, we examine the stability and show that the proposed scheme converges at the rate of having space mesh size h and time step size t. Numerical simulations are presented based on the presented method. We finished the article with a conclusion.

Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities \(\mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\,\gamma ,\mu \in \mathbb {R},\,m>0,q>1,\) involving time-fractional Caputo derivative \(\mathcal {D}_{0|t}^{\alpha }\) and space-fractional p-Laplacian operator \((-\varDelta )^s_p\). We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of \(\gamma ,\mu ,m\) and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.

Refined Asymptotic Expansions of Solutions to Fractional Diffusion Equations

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.

Solutions of a Nonlinear Diffusion Equation with a Regularized Hyper-Bessel Operator

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag–Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces.

An Efficient Local Meshfree Method for Signal Smoothing by a Time–Fractional Nonlinear Diffusion Equation

An Efficient Local Meshfree Method for Signal Smoothing by a Time–Fractional Nonlinear Diffusion Equation

A meshless method to solve the variable-order fractional diffusion problems with fourth-order derivative term

Using the meshless collocation method, a scheme for solving nonlinear variable-order fractional diffusion equations with a fourth-order derivative term is presented. Here the fractional derivative term is approximated by weighted and shifted Grünwald difference (WSGD) approximation formula. The difficulty caused by the nonlinear terms is carefully handled by quasilinearization technique. Using the radial basis functions, the solution of the problem is written in terms of the primary approximation, and the related correcting functions at each time step. Then the approximation is substituted back to the governing equations where the unknown parameters can be determined. Finally, the method is supported by several numerical experiments on irregular domains.

Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations

In this article, an efficient two-grid finite element method is proposed for solving the nonlinear time fractional variable coefficient diffusion equations. This algorithm firstly solves a nonlinear system to get the numerical solution uHn on the coarse grid with size H, then based on the initial iterative solution uHn on the coarse grid, the linearized finite element system is solved on the fine grid with size h to get the numerical solution Uhn, in which the temporal direction is approximated by the L2−1σ scheme. Besides, the stability and priori error estimates of standard finite element method and two-grid method are given. Finally, the validity and efficiency of the two-grid algorithm are verified by two numerical experiments.

Large deviations for stochastic equations in Hilbert spaces with non-Lipschitz drift

We prove a Freidlin–Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space–time white noise.

Global solutions of nonlinear fractional diffusion equations with time-singular sources and perturbed orders

In a Hilbert space, we consider a class of nonlinear fractional equations having the Caputo fractional derivative of the time variable t and the space fractional function of the self-adjoint positive unbounded operator. We consider various cases of global Lipschitz and local Lipschitz source with time-singular coefficient. These sources are generalized of the well–known fractional equations such as the fractional Cahn–Allen equation, the fractional Burger equation, the fractional Cahn–Hilliard equation, the fractional Kuramoto–Sivashinsky equation, etc. Under suitable assumptions, we investigate the existence, uniqueness of maximal solution, and stability of solution of the problems with respect to perturbed fractional orders. We also establish some global existence and prove that the global solution can be approximated by known asymptotic functions as \(t\rightarrow \infty \).

Partially explicit time discretization for nonlinear time fractional diffusion equations

Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses a coarse spatial grid for spatial resolution. For temporal discretization, implicit methods are often used. For implicit methods, though the time step can be relatively large, the equations are difficult to compute due to the nonlinearity and the fact that one deals with large-scale systems. On the other hand, the discrete systems in explicit methods are easier to compute but it requires small time steps. In this work, we propose a partially explicit scheme following earlier works on developing partially explicit methods for nonlinear diffusion equations. In these schemes, the diffusion term is treated partially explicitly and the reaction term is treated fully explicitly. With the appropriate construction of spaces and stability analysis, we find that the required time step in our proposed scheme scales as the coarse mesh size, which creates a computational saving. The main novelty of this work is the extension of our earlier works for diffusion equations to time fractional diffusion equations. For the case of fractional diffusion equations, the constraints on time steps are more severe and the proposed methods alleviate this problem since the time step in the partially explicit method scales as the coarse mesh size. We present stability analysis. Numerical results are presented where we compare our proposed partially explicit methods with a fully implicit approach. We show that our proposed approach provides similar results while treating many degrees of freedom in nonlinear terms explicitly.

Stability of fractional order of time nonlinear fractional diffusion equation with Riemann–Liouville derivative

In this paper, we investigate an equation of nonlinear fractional diffusion with the derivative of Riemann–Liouville. Firstly, we determine the global existence and uniqueness of the mild solution. Next, under some assumptions on the input data, we discuss continuity with regard to the fractional derivative order for the time. Our key idea is to combine the theories Mittag–Leffler functions and Banach fixed‐point theorem. Finally, we present some examples to test the proposed theory.

On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

<abstract><p>In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.</p></abstract>

High-Accuracy Time Discretization of Stochastic Fractional Diffusion Equation

A high-accuracy time discretization is discussed to numerically solve the nonlinear fractional diffusion equation forced by a space-time white noise. The main purpose of this paper is to improve the temporal convergence rate by modifying the semi-implicit Euler scheme. The solution of the equation is only Hölder continuous in time, which is disadvantageous to improve the temporal convergence rate. Firstly, the system is transformed into an equivalent form having better regularity than the original one in time. But the regularity of nonlinear term remains unchanged. Then, combining Lagrange mean value theorem and independent increments of Brownian motion leads to a higher accuracy discretization of nonlinear term which ensures the implementation of the proposed time discretization scheme without loss of convergence rate. Our scheme can improve the convergence rate from $${\min \{\frac{\gamma }{2\alpha },\frac{1}{2}\}}$$ to $${\min \{\frac{\gamma }{\alpha },1\}}$$ in the sense of mean-squared $$L^2$$ -norm. The theoretical error estimates are confirmed by extensive numerical experiments.

Long-time dynamics of fractional nonclassical diffusion equations with nonlinear colored noise and delay on unbounded domains

This paper deals with the long-time dynamics for a class of highly nonlinear fractional nonclassical diffusion equations with nonlinear colored noise and time delay defined on the whole space Rn. The existence and uniqueness of tempered pullback random attractors of the equations are established in C([−ρ,0],Hα(Rn)) (ρ>0 and α∈(0,1)) for polynomial growth drift and diffusion terms as well as Lipschitz time-delay terms. In a special case where the nonlinear diffusion term depends only on the space variable, the approximation of those random attractors is also investigated in C([−ρ,0],Hα(Rn)) when the correlation time of the colored approaches zero. The pullback asymptotical compactness of the solutions in C([−ρ,0],Hα(Rn)) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates developed by Wang (1999) [59] in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains as well as the weakly dissipative distinguishing structures of the equations.

Growing solutions of the fractional [formula omitted]-Laplacian equation in the Fast Diffusion range

We establish existence, uniqueness as well as quantitative estimates for solutions u(t,x) to the fractional nonlinear diffusion equation, ∂tu+Ls,p(u)=0, where Ls,p=(−Δ)ps is the standard fractional p-Laplacian operator. We work in the range of exponents 0<s<1 and 1<p<2, and in some sections we need sp<1. The equation is posed in the whole space x∈RN. We first obtain weighted global integral estimates that allow establishing the existence of solutions for a class of large data that is proved to be roughly optimal. We use the estimates to study the class of self-similar solutions of forward type, that we describe in detail when they exist. We also explain what happens when possible self-similar solutions do not exist. We establish the dichotomy positivity versus extinction for nonnegative solutions at any given time. We analyse the conditions for extinction in finite time.

On Fractional Diffusion Equation with Caputo-Fabrizio Derivative and Memory Term

In this paper, we examine a nonlinear fractional diffusion equation containing viscosity terms with derivative in the sense of Caputo-Fabrizio. First, we establish the local existence and uniqueness of lightweight solutions under some assumptions about the input data. Then, we get the global solution using some new techniques. Our main idea is to combine theories of Banach’s fixed point theorem, Hilbert scale theory of space, and some Sobolev embedding.

An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation

This paper presents the fractional formulation and numerical solution of a non-linear fractional diffusion equation with advection and reaction terms. The fractional derivative employed in this equation is of non-singular type with Mittag-Leffler kernel. To investigate this fractional equation numerically, we adopted a collocation method with the Legendre operational matrix. The numerical value of Atangana-Baleanu Caputo (ABC) fractional differentiation of polynomial function f(ϑ)=ϑk is determined by using the definition of Mittag-Leffler kernel fractional differentiation. The operational matrix of fractional differentiation of Mittag-Leffler kernel fractional derivative is determined numerically employing a polynomial function. Within the above context, we solve the fractional equation in-hand by using this matrix and using the collocation method based upon Legendre polynomial. The effectiveness of these techniques is illustrated through different cases that include fractional diffusion, advection-diffusion, and reaction-diffusion equation. The feasibility of the proposed derived ABC fractional differentiation operational matrix is clearly demonstrated by validating the numerical resolutions against exact solutions. The results demonstrated that the method is capable of resolving the non-linear fractional diffusion equation with different fractional order.