Fractional Parabolic Equations Research Articles

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.

A Numerical Scheme for Fractional Mixed Convection Flow Over Flat and Oscillatory Plates

Abstract A fractional scheme is proposed to solve time-fractional partial differential equations. According to the considered fractional Taylor series, the scheme is compact in space and provides fourth-order accuracy in space and second-order accuracy in fractional time. The scheme is conditionally stable when applied to the scalar fractional parabolic equation. The convergence of the scheme is demonstrated for the system of fractional parabolic equations. Moreover, a fractional model for heat and mass transfer of mixed convection flow over the flat and oscillatory plate is given. The radiation effects and chemical reactions are also considered. The scheme is tested on this model and the nonlinear fractional Burgers equation. It is found that it is more accurate than considering existing schemes in most of the regions of the solution domain. The compact scheme with exact findings of spatial derivatives is better than considering linearized equations. The error obtained by the proposed scheme with the determination of exact spatial derivatives is better than that obtained by two explicit existing schemes. The main advantage of the proposed scheme is that it is capable of providing the solution for convection-diffusion equations with compact fourth-order accuracy. Still, the corresponding implicit compact scheme is unable to find the solution to convection-diffusion problems.

Solving a category of two‐dimensional fractional optimal control problems using discrete Legendre polynomials

AbstractIn this paper, we formulate a numerical method to approximate the solution of two‐dimensional optimal control problem with a fractional parabolic partial differential equation (PDE) constraint in the Caputo type. First, the optimal conditions of the optimal control problems are derived. Then, we discretize the spatial derivatives and time derivatives terms in the optimal conditions by using shifted discrete Legendre polynomials and collocations method. The main idea is simplifying the optimal conditions to a system of algebraic equations. In fact, the main privilege of this new type of discretization is that the numerical solution is directly and globally obtained by solving one efficient algebraic system rather than step‐by‐step process which avoids accumulation and propagation of error. Several examples are tested and numerical results show a good agreement between exact and approximate solutions.

Inverse Problem for a Time Fractional Parabolic Equation with Nonlocal Boundary Conditions

This article considers an inverse problem of time fractional parabolic partial differential equations with the nonlocal boundary condition. Dirichlet-measured output data are used to distinguish the unknown coefficient. A finite difference scheme is constructed and a numerical approximation is made. Examples and numerical experiments, such as man-made noise, are provided to show the stability and efficiency of this numerical method.

A numerical method for solving retrospective inverse problem of fractional parabolic equation

An effective numerical method for solving the inverse problem of time fractional parabolic equation is constructed in this paper. We use implicit finite difference method to discretize the problem and for the inverse problem we propose a conjugate gradient type regularization method to solve the discretized ill-posed linear systems. By comparing the different errors and the results with different perturbed data in several numerical experiments, our method is shown to solve the inverse problem, even with some noisy measurements, efficiently and stably.

Regularity properties of nonlocal fractional differential equations and applications

Abstract The regularity properties of nonlocal fractional elliptic and parabolic equations in vector-valued Besov spaces are studied. The uniform B p , q s B_{p,q}^{s} -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. In particular, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, the maximal regularity properties of the nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional parabolic equations are studied.

Well-posed results for nonlocal fractional parabolic equation involving Caputo-Fabrizio operator

In this paper, we study the parabolic problem associated with non-local conditions, with the Caputo-Fabrizio derivative. Equations on the sphere have many important applications in physics, phenomena, and oceanography. The main motivation for us to study non-local boundary value problems comes from two main reasons: the first reason is that current major interest in several application areas. The second reason is to study approximation for the terminal value problem. With some given data, we prove that the problem has only the solution for two cases. In case \(\epsilon = 0,\) we prove the problem has a local solution. In case \(\epsilon > 0,\) then the problem has a global solution. The main tools and techniques in our demonstration are of using Banach's fixed point theorem in conjunction with some Fourier series analysis involved some evaluation of spherical harmonic function. Several upper and lower upper limit techniques for the Mittag-Lefler functions are also applied.

A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model

&lt;abstract&gt; &lt;p&gt;This contribution proposes a numerical scheme for solving fractional parabolic partial differential equations (PDEs). One of the advantages of using the proposed scheme is its applicability for fractional and integer order derivatives. The scheme can be useful to get conditions for obtaining a positive solution to epidemic disease models. A COVID-19 mathematical model is constructed, and linear local stability conditions for the model are obtained; afterward, a fractional diffusive epidemic model is constructed. The numerical scheme is constructed by employing the fractional Taylor series approach. The proposed fractional scheme is second-order accurate in space and time and unconditionally stable for parabolic PDEs. In addition to this, convergence conditions are obtained by employing a proposed numerical scheme for the fractional differential equation of susceptible individuals. The scheme is also compared with existing numerical schemes, including the non-standard finite difference method. From theoretical analysis and graphical illustration, it is found that the proposed scheme is more accurate than the so-called existing non-standard finite difference method, which is a method with notably good boundedness and positivity properties.&lt;/p&gt; &lt;/abstract&gt;

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations