Fractional Parabolic Equations Research Articles

Existence results for fractional evolution equations with superlinear growth nonlinear terms

We investigate the initial value problem to a class of fractional evolution equations with superlinear growth nonlinear functions in Banach spaces. When the linear operator generates an extendable compact $ C_0 $-semigroup of contractions and the nonlinearity $ f:[0,T]\times X\to Y $ is Carathéodory continuous, with $ X $ and $ Y $ two real Banach spaces such that $ X\subseteq Y $, the existence of mild solutions to such problems are established without assuming that the nonlinear term satisfies the transversality condition by combining an approximation technique with Leray-Schauder continuation principle. The obtained results allow to treat, as an application, fractional parabolic equations with continuous superlinear growth nonlinear term.

Uniqueness for fractional parabolic and elliptic equations with drift

We investigate uniqueness, in suitable weighted Lebesgue spaces, of solutions to a class of linear, nonlocal parabolic problems with a drift. More precisely, the problem is nonlocal due to the presence of the fractional Laplacian as diffusion operator. The drift term is driven by a smooth enough, possibly unbounded vector field $ b $ which satisfies a suitable growth condition in the set $ \{x\in {\mathbb{R}}^N:\langle b(x), x\rangle> 0\} $. In general, our uniqueness class includes unbounded solutions; in particular, we get uniqueness of bounded solutions. Furthermore, we show sharpness of the hypothesis on the drift term $ b $; in fact we show that, if the drift term $ b $ violates, in an appropriate sense, the mentioned growth condition (see (2.5)), then infinitely many bounded solutions to the problem exist. Finally, we also investigate uniqueness of a linear, nonlocal elliptic equation with a drift term obtaining similar results.

Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations

<abstract><p>In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by $ H^1 $-Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and $ L1 $ scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples.</p></abstract>

Asymptotic symmetry and monotonicity of solutions for weighted fractional parabolic equations

Asymptotic symmetry and monotonicity of solutions for weighted fractional parabolic equations

A new fast predictor-corrector method for nonlinear time-fractional reaction-diffusion equation with nonhomogeneous terms

Time-fractional reaction-diffusion (TFRD) equation is an important fractional parabolic equation, and the research of its numerical solution has scientific significance and engineering application value. In this paper, a new fast predictor-corrector (FP-C) scheme is constructed based on fast L1 approximation of Caputo fractional derivative for solving nonlinear TFRD equation with nonhomogeneous terms. The linearized implicit difference scheme is used for the predictor step and Crank-Nicolson(C-N) scheme is used for the corrector step. Theoretical analysis proves that FP-C scheme is convergent and stable unconditionally for nonlinear TFRD equation. Numerical analysis and experiments show that the computational accuracy of FP-C scheme is $ O(\tau^{2-\alpha}+h^2) $ under the strong regularity condition and $ O(\tau^{\alpha}+h^2) $ under the weak regularity condition. Compared with the classical predictor-corrector (P-C) scheme based on standard L1 approximation, the FP-C scheme improves the computational efficiency without losing the computational accuracy. It is shown that the new FP-C scheme is an efficient method to solve nonlinear TFRD equation with nonhomogeneous terms.

Mittag-Leffler stability of numerical solutions to linear homogeneous time fractional parabolic equations

<abstract><p>In a bounded domain, the solution of linear homogeneous time fractional parabolic equation is known to exhibit polynomial type decay rate (the so-called Mittag-Leffler stability) over time, which is quite different from the exponential decay of classical parabolic equation. We firstly use the finite element method or finite difference method to discretize the parabolic equation in space to obtain fractional ordinary differential equation, and then use fractional linear multistep method (F-LMM) to discretize in time to obtain a fully discretized schemes. We prove that the strongly $ A $-stable F-LMM method combined with appropriate spatial discretization can accurately maintain the long-term optimal algebraic decay rate of the original continuous equation. Numerical examples are included to confirm the correctness of our theoretical analysis.</p></abstract>

Nonuniform difference schemes for multi-term and distributed-order fractional parabolic equations with fractional Laplacian

In this paper, the multi-term temporal fractional order and temporal distributed-order parabolic equations with fractional Laplacian are numerically investigated. Several unconditional stable difference schemes based on non-uniform meshes for solving these differential equations are provided. We find that the constructed nonuniform difference schemes are convergent and it has been shown that the temporal convergence rate is faster and more accurate compared to the uniform difference schemes in case of nonsmooth solutions with respect to time. Some numerical examples are given to verify the theoretical findings.

Solvability of superlinear fractional parabolic equations

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.

Sliding‐mode boundary control for perturbed time fractional parabolic systems with spatially varying coefficients using backstepping

AbstractIn this paper, we consider the boundary stabilization of an uncertain time fractional parabolic systems governed by time fractional parabolic partial differential equations (PDEs) with a boundary input disturbance and spatially varying coefficients (nonconstant coefficients) using a fractional‐order sliding‐mode controller. For this, the backstepping approach is used to transform an original system into a target system with a new manipulable input and perturbation. Then, the fractional‐order sliding‐mode algorithm is employed to design this new discontinuous boundary input to achieve the asymptotical stabilization of the target system (and, therefore, of the original system as well) by the fractional Lyapunov method. Apart from this, the well‐posedness of the fractional parabolic system is analyzed theoretically. Fractional‐order numerical simulations are provided to validate the developed technique.

Harnack inequality and self-similar solutions for fully nonlinear fractional parabolic equations

Harnack inequality and self-similar solutions for fully nonlinear fractional parabolic equations

The quasi-reversibility method for an inverse source problem for time-space fractional parabolic equations

In this paper, we apply the quasi-reversibility method to solve an inverse source problem for a time-space fractional parabolic equation. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. The theoretical error estimates are confirmed with numerical tests for one and two dimensional equations.

Numerical analysis of fourth-order compact difference scheme for inhomogeneous time-fractional Burgers-Huxley equation

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O(τ2−α+h4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O(τα+h4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation.

Nonlocal fractional elliptic equations and applications

AbstractThe Lp‐coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in Lp space and also is a generator of an analytic semigroup. Moreover, by using the Lp‐separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.

Space-like strong unique continuation for some fractional parabolic equations

In this paper we establish the space-like strong unique continuation for nonlocal equations of the type (∂t−Δ)su=Vu, for 0<s<1. The proof of our main result, Theorem 1.1, is achieved via a conditional elliptic type doubling property for solutions to the appropriate extension problem, followed by a blowup analysis.

Ancient solutions to nonlocal parabolic equations

In this paper, we develop a systematical approach in applying the method of moving planes to study qualitative properties of solutions for nonlocal parabolic equations. We first establish a series of key ingredients in the proofs, such as maximum principles for antisymmetric functions, narrow region principles, maximum principles near infinity, and the Hopf lemma, then we demonstrate how these new tools can be employed to derive the radial symmetry and monotonicity of ancient positive solutions to fractional parabolic equations in the whole space. Our results are in particular applicable to entire solutions, by which we mean solutions defined for all t∈R. We believe that the new ideas and methods introduced here can be adapted to study many other nonlocal parabolic problems with more general operators and nonlinear terms.

Monotone iterative technique for solving finite difference systems of time fractional parabolic equations with initial/periodic conditions

This article is devoted to a numerical analysis of two types of fractional nonlinear reaction-diffusion problems with periodic conditions or with initial conditions. We also consider two types of derivative in time, the first one is that of Caputo and the second is the conformable derivative; we discretize the problem via a finite difference method. We construct two iterative schemes by the upper and lower solutions method which converges monotonically towards a maximal solution or a minimal solution of the considered problem when the mesh decreases to zero, depending on whether the initial iteration is an upper solution or a lower solution. A comparison lemma for the different monotone sequences is also proved. The presented iterative scheme is used to show that the finite difference system converges to continuous solutions of the fractional reaction-diffusion problem. Finally to validate the theoretical results some examples with numerical simulations of reaction-diffusion problem are also presented and discussed in detail.

On a Fractional Parabolic Equation with Regularized Hyper-Bessel Operator and Exponential Nonlinearities

Recent decades have witnessed the emergence of interesting models of fractional partial differential equations. In the current work, a class of parabolic equations with regularized Hyper-Bessel derivative and the exponential source is investigated. More specifically, we examine the existence and uniqueness of mild solutions in Hilbert scale-spaces which are constructed by a uniformly elliptic symmetry operator on a smooth bounded domain. Our main argument is based on the Banach principle argument. In order to achieve the necessary and sufficient requirements of this argument, we have smoothly combined the application of the Fourier series supportively represented by Mittag-Leffler functions, with Hilbert spaces and Sobolev embeddings. Because of the presence of the fractional operator, we face many challenges in handling proper integrals which appear in the representation of mild solutions. Besides, the source term of an exponential type also causes trouble for us when deriving the desired results. Therefore, powerful embeddings are used to limit the growth of nonlinearity.

A Third-Order Two-Stage Numerical Scheme for Fractional Stokes Problems: A Comparative Computational Study

Abstract A third-order numerical scheme is proposed for solving fractional partial differential equations (PDEs). The first explicit stage can converge fast, and the second implicit stage is responsible for enlarging the stability region. The fourth-order compact scheme is employed to discretize spatial derivative terms. The stability of the scheme is given for the standard fractional parabolic equation, whereas convergence of the proposed scheme is given for the system of fractional parabolic equations. Mathematical models for heat and mass transfer of Stokes first and second problems using Dufour and Soret effects are given in a set of linear and nonlinear PDEs. Later on, these governing equations are converted into dimensionless PDEs. It is shown that the proposed scheme effectively solves the fractional forms of dimensionless models numerically, and a comparison is also conducted with existing schemes. If readers want it, a computational code for the discrete model system suggested in this paper may be made accessible to them for their convenience.

Lower semicontinuity and pointwise behavior of supersolutions for some doubly nonlinear nonlocal parabolic p-Laplace equations

We discuss pointwise behavior of weak supersolutions for a class of doubly nonlinear parabolic fractional p-Laplace equations which includes the fractional parabolic p-Laplace equation and the fractional porous medium equation. More precisely, we show that weak supersolutions have lower semicontinuous representative. We also prove that the semicontinuous representative at an instant of time is determined by the values at previous times. This gives a pointwise interpretation for a weak supersolution at every point. The corresponding results hold true also for weak subsolutions. Our results extend some recent results in the local parabolic case, and in the nonlocal elliptic case, to the nonlocal parabolic case. We prove the required energy estimates and measure theoretic De Giorgi type lemmas in the fractional setting.

Distributed optimal control problems driven by space-time fractional parabolic equations

Abstract We study distributed optimal control problems, governed by space-time fractional parabolic equations (STFPEs) involving time-fractional Caputo derivatives and spatial fractional derivatives of Sturm-Liouville type. We first prove existence and uniqueness of solutions of STFPEs on an open bounded interval and study their regularity. Then we show existence and uniqueness of solutions to a quadratic distributed optimal control problem. We derive an adjoint problem using the right-Caputo derivative in time and provide optimality conditions for the control problem. Moreover, we propose a finite difference scheme to find the approximate solution of the considered optimal control problem. In the proposed scheme, the well-known L1 method has been used to approximate the time-fractional Caputo derivative, while the spatial derivative is approximated using the Grünwald-Letnikov formula. Finally, we demonstrate the accuracy and the performance of the proposed difference scheme via examples.