Fractional Parabolic Equations Research Articles

Sliding mode control to stabilization of coupled time fractional parabolic PDEs subject to disturbances

AbstractIn this article, the stabilization of coupled time fractional parabolic partial differential equations subject to external disturbances is investigated. By using sliding mode control method and backstepping approach, a boundary state feedback controller is designed to reject the matched disturbance and achieve the Mittag‐Leffler input‐to‐state stability of closed‐loop system. The existence of the generalized solution to the closed‐loop system is proven by Galerkin approximation scheme. Simulations are presented to illustrate the validity of our theoretical results.

A Hybrid Technique for Space-Time Fractional Parabolic Differential Equations

A Hybrid Technique for Space-Time Fractional Parabolic Differential Equations

Dual fractional parabolic equations with indefinite nonlinearities

In this paper, we consider the following indefinite dual fractional parabolic equation involving the Marchaud fractional time derivative∂tαu(x,t)+(−Δ)su(x,t)=a(x)f(u(x,t))inRn×R, where α,s∈(0,1), and the functions a and f are nondecreasing. We prove that there is no positive bounded solutions. To this end, we first show that all positive bounded solutions u(⋅,t) must be strictly monotone increasing along the direction determined by a(x). Then by mollifying the first eigenfunction for fractional Laplacian (−Δ)s and constructing an appropriate subsolution for the Marchaud fractional operator ∂tα−1, we derive a contradiction and thus obtain the non-existence of solutions.To overcome the challenges caused by the dual non-locality of the operator ∂tα+(−Δ)s, we introduce several new ideas and novel techniques. These novel approaches are not only applicable to the specific problem at hand but can also be extended to address various other fractional problems, be they elliptic or parabolic, including those featuring dual nonlocalities associated with the Marchaud time derivatives.

Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces

The maximal Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ B_{p,q}^{s}$$\\end{document} and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.

Nonuniqueness for fractional parabolic equations with sublinear power-type nonlinearity

We show that the parabolic equation ut+(−Δ)su=q(x)|u|α−1u posed in a time-space cylinder (0,T)×RN and coupled with zero initial condition and zero nonlocal Dirichlet condition in (0,T)×(RN∖Ω), where Ω is a bounded domain, has at least one nontrivial nonnegative finite energy solution provided α∈(0,1) and the nonnegative bounded weight function q is separated from zero on an open subset of Ω. This fact contrasts with the (super)linear case α≥1 in which the only bounded finite energy solution is identically zero.

Robin coefficient determination problem in a fractional parabolic differential equation

This article discusses the identification of an unknown Robin coefficient in a fractional parabolic equation using a noisy measurement of the ultimate solution over time. The problem is quite complex, involving a nonlocal, nonlinear, and ill‐posed operator. To solve this problem, the article proposes a regularized optimization approach that minimizes a least‐squares cost function. The article also examines the various conceptual and real‐world challenges associated with this problem. The article demonstrates the presence of a singular and stable solution for the optimization problem. It utilizes the Morozov discrepancy principle and the conjugate gradient method to streamline the iterative reconstruction process. The proposed method is accurate and efficient, as several numerical examples demonstrate.

Identification of unknown sources in time-space fractional parabolic equation

This study considers the issue for recognizing unknown source within a time-space fractional parabolic equation. This particular issue is characterized by severe ill-posedness, where the solution does not depend continuously on the data. To tackle this issue, the wavelet dual least squares method is extended to handle ill-posedness of the issue under priori rule. Additionally, a posterior wavelet regularization parameter selection rule is introduced to address the Cauchy problem. Furthermore, error estimates between the exact solution and its approximation are established using both a priori and a posteriori methodologies. Numerical examples are given to validate the effectiveness and stability of the proposed method.

Galerkin-finite difference method for fractional parabolic partial differential equations

The fractional form of the classical diffusion equation embodies the super-diffusive and sub-diffusive characteristics of any flow, depending on the fractional order. This study aims to approximate the solution of parabolic partial differential equations of fractional order in time and space. For this, firstly, we briefly discussed on some existing methods to solve Partial Differential Equations (PDEs) and fractional Differential Equations (DEs), then introduce a combined technique such as the Galerkin weighted residual method for the space fractional term with modified Bernoulli polynomials as basis functions, and the finite difference approximation for the time fractional term, respectively. The mathematical formulation of the proposed method is explained elaborately. Then we describe the order of convergence for the time fractional term only, as the convergence of the Galerkin method is obvious. We impose this technique on the fractional Black-Scholes model subsequently. Finally, we experimented with our proposed technique on some numerical problems. All the results are depicted in both tabular and 3D visualizations as well. We compare our results with the available methods in the literature, and our accuracy is considerable. To summarize: •The paper introduces an approach that integrates the Galerkin weighted residual method with modified Bernoulli polynomials to handle space fractional terms, alongside employing a finite difference approximation for time fractional terms.•The convergence analysis is focused.•The technique is implemented on the fractional Black-Scholes model and other numerical problems, with outcomes depicted through tables and 3D visualizations.

Doubly nonlinear isotropic fractional parabolic equation

Doubly nonlinear isotropic fractional parabolic equation

Numerical method for second order singularly perturbed delay differential equations with fractional order in time via fitted computational method

The numerical solution of time fractional parabolic differential equations with singular perturbations and delay is the subject of this article. An arbitrarily small perturbation parameter ɛ(0<ɛ<<1) is multiplied the highest order derivative term. The problem’s solution shows an exponential boundary layer on the right side of the spatial domain for small values of ɛ. The solution and its derivatives are discussed together with their properties and bounds. The Crank–Nicolson approach for time discretization and the fitted operator non-standard finite difference method for space discretization are used to solve the time fractional singularly perturbed time delay problem under consideration. The comparison principle and solution bound are used to examine and analyze the scheme’s stability. The scheme’s uniform convergence is examined and demonstrated. The proposed scheme has a uniform convergence order of Mx−1+(Δt)2−γ. Two numerical illustrations for various values of ɛ are taken into consideration to validate the theoretical analysis of the scheme. The constructed scheme provides a better order of convergence and more accurate results than methods found in the literature.

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.

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 $T&gt;0$ where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} &lt; \infty$ . We show that if for some $0&lt;\mathfrak {K} &lt; T$ and $\epsilon &gt;0$ \[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \] 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.