Fractional Parabolic Equations Research Articles

Stabilization of coupled delayed nonlinear time fractional reaction diffusion systems using sampled‐in‐space sensing and actuation

AbstractThis paper is considered with the asymptotic stabilization of coupled delayed nonlinear time fractional reaction diffusion systems (FRDSs) governed by fractional parabolic partial differential equations (PDEs) with space‐dependent coefficients under sampled‐data in space control. It is assumed that state measurements can be averaged measurements (AMs) or point measurements (PMs), and a finite number of sensing and actuation devices are located in a spaced manner along the spatial domain of the interest. With the proposed sampled‐data in space controller, the closed‐loop stability is obtained. Tuning rules of system parameters and control parameters are derived using the fractional Halanay's inequality and the fractional Lyapunov method. Subsequently, the dual problem of observer design is formulated. Fractional examples are used to valid the theoretical result. Discussions on the extension of sampled‐data boundary feedback stabilization are provided finally.

Identification an Unknown Sources in a Time-Space Fractional Parabolic Equation

Identification an Unknown Sources in a Time-Space Fractional Parabolic Equation

On the Solvability of a Singular Time Fractional Parabolic Equation with Non Classical Boundary Conditions

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

Sharp analysis of [formula omitted] method on graded mesh for time fractional parabolic differential equation

In this paper, we study the L1−2 method for solving numerically a class of time fractional parabolic differential equations on a graded mesh. Based on the nonsmooth regularity assumptions, the stability and convergence of the proposed numerical method are investigated. Using rigorous and sharp theoretical analysis, we obtain the optimal graded parameter r=(3−α)/(1+α) of the graded mesh. Numerical experiments are given to demonstrate that even if the α is small, the graded mesh based on the optimal graded parameter is still able to recover the optimal convergence order of the L1−2 method.

The direct RBF-based partition of unity method for solving nonlinear fractional parabolic equations

This paper aims to analyze a novel localized radial basis function method known as the ’direct RBF-based partition of unity method’ for solving nonlinear fractional parabolic equations. In the proposed method, the weight functions are not operated on by the differential operators, resulting in a decrease in computational cost and algorithmic complexity. Another advantage of the direct RBF-based partition of unity method compared to the standard RBF-based partition of unity method is that in the direct method, it is possible to use discontinuous weights, which is not applicable in the standard method. Furthermore, we conduct a convergence analysis of the method and derive an error bound for the local approximation. This error bound is determined by considering conditions on the eigenvalues of the Laplacian operator matrix. To ensure the accuracy and reliability of our proposed approach, we conducted a numerical simulation and provided two numerical examples for comparison. We evaluated the consistency between the theoretical and numerical results. Our comparison between the standard RBF-based partition of unity and our proposed framework shows that the direct method is more efficient computationally, but both versions have relatively the same accuracy.

Decay at infinity for solutions to some fractional parabolic equations

For $s\in [\tfrac {1}{2},\, 1)$ , let $u$ solve $(\partial _t - \Delta )^s u = Vu$ in $\mathbb {R}^{n} \times [-T,\, 0]$ for some $T>0$ where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} < \infty$ . We show that if for some $0<\mathfrak {K} < T$ and $\epsilon >0$ \[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \] then $u \equiv 0$ in $\mathbb {R}^{n} \times [-T,\, 0]$ .

Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

Abstract In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

Monotonicity and one-dimensional symmetry of solutions for the weighted fractional parabolic equations on the whole space

In this paper, we investigate the monotonicity and one-dimensional symmetry of solutions for parabolic equations related to the weighted fractional Laplacian on the whole space. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, and then carry out the sliding method to obtain monotonicity and one-dimensional symmetry of solutions.

On fractional parabolic equations with Hardy-type potentials

A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren–Poon monotonicity formula combined with a blow-up analysis.

A Liouville theorem and radial symmetry for dual fractional parabolic equations

In this paper, we first study the dual fractional parabolic equation [Formula: see text] subjected to the vanishing exterior condition. We show that for each [Formula: see text], the positive bounded solution [Formula: see text] must be radially symmetric and strictly decreasing about the origin in the unit ball in [Formula: see text]. To overcome the challenges caused by the dual nonlocality of the operator [Formula: see text], some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space [Formula: see text] We first prove a maximum principle in unbounded domains for antisymmetric functions to deduce that [Formula: see text] must be constant with respect to [Formula: see text] Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation [Formula: see text] To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator [Formula: see text], we bring in some new ideas and simpler approaches. Instead of disturbing the antisymmetric function, we employ a perturbation technique directly on the solution [Formula: see text] itself. This method provides a more concise and intuitive way to establish the Liouville theorem for one-sided operators [Formula: see text], including even more general Marchaud time derivatives.

BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions

In this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integro-differential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a second-order backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a second-order fractional quadrature rule proposed by Lubich for the Riemann-Liouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.

Symmetry of Ancient Solution for Fractional Parabolic Equation Involving Logarithmic Laplacian

In this research, we focus on the symmetry of an ancient solution for a fractional parabolic equation involving logarithmic Laplacian in an entire space. In the process of studying the property of a fractional parabolic equation, we obtained some maximum principles, such as the maximum principle of anti-symmetric function, narrow region principle, and so on. We will demonstrate how to apply these tools to obtain radial symmetry of an ancient solution.

A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations

A computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.

Regularity theory for a new class of fractional parabolic stochastic evolution equations

AbstractA new class of fractional-order parabolic stochastic evolution equations of the form $$(\partial _t + A)^\gamma X(t) = {\dot{W}}^Q(t)$$ ( ∂ t + A ) γ X ( t ) = W ˙ Q ( t ) , $$t\in [0,T]$$ t ∈ [ 0 , T ] , $$\gamma \in (0,\infty )$$ γ ∈ ( 0 , ∞ ) , is introduced, where $$-A$$ - A generates a $$C_0$$ C 0 -semigroup on a separable Hilbert space H and the spatiotemporal driving noise $${\dot{W}}^Q$$ W ˙ Q is the formal time derivative of an H-valued cylindrical Q-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process X are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of A. In addition, the covariance of X and its long-time behavior are analyzed. These abstract results are applied to the cases when $$A:= L^\beta $$ A : = L β and $$Q:={\widetilde{L}}^{-\alpha }$$ Q : = L ~ - α are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle–)Matérn fields to space–time.

Time fractional parabolic equations with partially SMO coefficients

We present the unique solvability in Sobolev spaces of time fractional parabolic equations in divergence and non-divergence forms. The leading coefficients are merely measurable in (t,x1) for aij, 1≤i,j≤d, (i,j)≠(1,1). The coefficient a11 is merely measurable locally either in t or x1. As functions of the remaining variables, the coefficients have small mean oscillations. We consider mixed norm Sobolev spaces with Muckenhoupt weights. Our results generalize previous work on parabolic equations with time fractional derivatives to a much larger class of coefficients and solution spaces.

Uniform large deviations of fractional stochastic equations with polynomial drift on unbounded domains

In this paper, we first prove a uniform contraction principle for verifying the uniform large deviation principles of locally Hölder continuous maps in Banach spaces. We then show the local Hölder continuity of the solutions of a class of fractional parabolic equations with polynomial drift of any order defined on [Formula: see text]. We finally establish the large deviation principle of the fractional stochastic equations uniformly with respect to bounded initial data, despite the solution operators are not compact due to the non-compactness of Sobolev embeddings on unbounded domains.

Global existence and convergence results for a class of nonlinear time fractional diffusion equation

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−.

Numerical Analysis of Direct and Inverse Problems for a Fractional Parabolic Integro-Differential Equation

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed.

Nonexistence of solutions to fractional parabolic problem with general nonlinearities

In this content, we investigate a class of fractional parabolic equation with general nonlinearities ∂z(x,t)∂t-(Δ+λ)β2z(x,t)=a(x1)f(z),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\frac{\\partial z(x,t)}{\\partial t}-(\\Delta +\\lambda )^{\\frac{\\beta }{2}}z(x,t)=a(x_{1})f(z), \\end{aligned}$$\\end{document}where a and f are nondecreasing functions. We first prove that the monotone increasing property of the positive solutions in x1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_{1}$$\\end{document} direction. Based on this, nonexistence of the solutions are obtained by using a contradiction argument. We believe these new ideas we introduced will be applied to solve more fractional parabolic problems.

Identifying an unknown coefficient in the fractional parabolic differential equation

In this study, we considered a fractional parabolic equation for identifying the unknown diffusion coefficient from the noisy measurement of the ultimate time solution. It is an inverse problem involving a nonlocal operator that is nonlinear and poorly formulated. By demonstrating the existence of this inverse problem singular solution with regard to the final observed data, we demonstrate the identifiability of this problem. The inverse problem is expressed as a regularized optimization problem that minimizes a cost function of the least-squares kind. We have covered various theoretical and practical difficulties pertaining to the problem under consideration. It has been demonstrated that a unique stable solution to the optimization problem exists. The Morozov discrepancy principle and the conjugate gradient approach are used to develop an iterative reconstruction procedure. The accuracy and efficiency of the suggested method are illustrated using a few numerical examples.