Fractional Poisson Equation Research Articles

Time fractional Poisson equations: Representations and estimates

In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator L and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel q(t,x,y), which we call the fundamental solution for the time fractional Poisson equation, if the semigroup for L has an integral kernel. We further show that q(t,x,y) can be expressed as a time fractional derivative of the fundamental solution for the homogeneous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution q(t,x,y) through explicit estimates of transition density functions of subordinators.

A Finite-Difference Approximation for the One- and Two-Dimensional Tempered Fractional Laplacian

This paper provides a finite-difference discretization for the one- and two-dimensional tempered fractional Laplacian and solves the tempered fractional Poisson equation with homogeneous Dirichlet boundary conditions. The main ideas are to, respectively, use linear and quadratic interpolations to approximate the singularity and non-singularity of the one-dimensional tempered fractional Laplacian and bilinear and biquadratic interpolations to the two-dimensional tempered fractional Laplacian. Then, we give the truncation errors and prove the convergence. Numerical experiments verify the convergence rates of the order $$O(h^{2-2s})$$.

A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions

This paper investigates the use of radial basis function (RBF) interpolants to estimate a function's fractional Laplacian of a given order through a mesh-free pseudospectral method. The mesh-free approach yields an algorithm that can be implemented in high dimensional settings without adjustment. Moreover, the fractional Laplacian is defined in terms of the Fourier transform, and the symmetry of RBFs can be exploited to simplify the estimation problem. Convergence rates are established for RBFs when the function whose fractional Laplacian to be estimated is compactly supported. Further results demonstrate convergence when a function is in the native space for a Wendland RBF (i.e. a Sobolev space) and satisfies a certain L1 condition. Numerical experiments demonstrate the developed method by estimating the fractional Laplacian of several functions and by solving a fractional Poisson equation with extended Dirichlet condition in one and two dimensions.

Symmetry analysis of space-time fractional Poisson equation with a delay

In this paper, we extended Lie symmetry theory to the class of space-time fractional differential equation with a delay and carried out a complete group classification of space-time fractional Poisson equation with a constant delay. The admitted symmetries found are used to obtained some exact solutions.

Legendre wavelets optimization method for variable-order fractional Poisson equation

In this study, the Poisson equation is generalized with the concept of variable-order (V-O) fractional derivatives called variable-order fractional Poisson equation (V-OFPE). In order to find an accurate solution of this system, we establish an optimization method through the Legendre wavelets (LWs). To carry out the method, we firstly derive an operational matrix (OM) of V-O fractional derivative for the LWs to be employed in expanding the unknown solution. Then, the function of residual is applied to reform the V-OFPE to an optimization problem which leads to choose the unknown coefficients optimally. In the final step, we implement the constrained extremum method which adjoins the objective function implied from the two-norm of residual function and the constraints corresponded to the given boundary conditions by a set of Lagrange multipliers. Accordingly, the final optimal conditions are actually the algebraic equations including the expansion coefficients and Lagrange multipliers. Theoretical convergence and error analysis of the approximation procedure using the LWs are investigated. In addition, the applicability and computational efficiency are experimentally examined for some illustrative examples.

A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem

In this paper, we develop a novel finite difference method to discretize the fractional Laplacian (−Δ)α/2 in hypersingular integral form. By introducing a splitting parameter, we formulate the fractional Laplacian as the weighted integral of a weak singular function, which is then approximated by the weighted trapezoidal rule. Compared to other existing methods, our method is more accurate and simpler to implement, and moreover it closely resembles the central difference scheme for the classical Laplace operator. We prove that for u∈C3,α/2(R), our method has an accuracy of O(h2)uniformly for anyα∈(0,2), while for u∈C1,α/2(R), the accuracy is O(h1−α/2). The convergence behavior of our method is consistent with that of the central difference approximation of the classical Laplace operator. Additionally, we apply our method to solve the fractional Poisson equation and study the convergence of its numerical solutions. The extensive numerical examples that accompany our analysis verify our results, as well as give additional insights into the convergence behavior of our method.

Solving Poisson Equation within Local Fractional Derivative Operators

In this paper, we are concerned with finding approximate solutions to local fractional Poisson equation by using the reduced differential transform method (RDTM) and homotopy perturbation transform method (HPTM). The presented methods are considered in the local fractional operator sense. Illustrative examples for handling the local fractional Poisson equation are given. The obtained results are given to show the sample and efficient features of the presented techniques to implement partial differential equations with local fractional derivative operators.

Some new operational matrices and its application to fractional order Poisson equations with integral type boundary constrains

Enormous application of fractional order partial differential equations (FPDEs) subjected to some constrains in the form of nonlocal boundary conditions motivated the interest of many scientists around the world. The prime objective of this article is to find approximate solution of a general FPDEs subject to nonlocal integral type boundary conditions on both ends of the domain. The proposed method is based on spectral method. We construct some new operational matrices which have the ability to handle integral type non-local boundary constrains. These operational matrices can be effectively applied to convert the FPDEs together with its integral types boundary conditions to easily solvable matrix equation. The accuracy and efficiency of proposed method are demonstrated by solving some bench mark problems. The proposed method has the ability to solve non-local FPDEs with high accuracy and low computational cost. Different aspects of presented approach are compared with two other recently developed methods, Haar wavelets collocation method and a family of collocation methods which are based on Radial base functions. It is observed that the proposed method is highly accurate, robust, efficient and stable as compared to these methods.

Local fractional functional decomposition method for solving local fractional Poisson equation in steady heat-conduction problem

The steady heat-conduction problem via local fractional derivative is investigated in this paper. The analytical solution of the local fractional Poisson equation is obtained. The local fractional functional decomposition method is proposed to find the analytical solution of the partial differential equation in the steady heat-conduction problem.

Hydrodynamic Nambu brackets derived by geometric constraints

A geometric approach to derive the Nambu brackets for ideal two-dimensional (2D) hydrodynamics is suggested. The derivation is based on two-forms with vanishing integrals in a periodic domain, and with resulting dynamics constrained by an orthogonality condition. As a result, 2D hydrodynamics with vorticity as dynamic variable emerges as a generic model, with conservation laws which can be interpreted as enstrophy and energy functionals. Generalized forms like surface quasi-geostrophy and fractional Poisson equations for the stream-function are also included as results from the derivation. The formalism is extended to a hydrodynamic system coupled to a second degree of freedom, with the Rayleigh–Bénard convection as an example. This system is reformulated in terms of constitutive conservation laws with two additive brackets which represent individual processes: a first representing inviscid 2D hydrodynamics, and a second representing the coupling between hydrodynamics and thermodynamics. The results can be used for the formulation of conservative numerical algorithms that can be employed, for example, for the study of fronts and singularities.

Local Fractional Poisson and Laplace Equations with Applications to Electrostatics in Fractal Domain

From the local fractional calculus viewpoint, Poisson and Laplace equations were presented in this paper. Their applications to the electrostatics in fractal media are discussed and their local forms in the Cantor-type cylindrical coordinates are also obtained.

Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

The local fractional Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

Singular fractional differential equations

AbstractFractional differential equations have received increasing attention during recent years since the behavior of many physical systems can be properly described using the fractional order system theory.By fractional analog for Duhamel principle we give the existence‐uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions. In order to find the explicit solutions we use integral representation of the solution obtained via Laplace and Fourier transforms in succession and their inverses. We deal with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes, extending the results in viscoelasticity, continuum random walk, seismology, continuum mechanics and many other branches of life and science. The main task is finding existence‐uniqueness results like in the case of evolution equations with entire derivatives. By examining the fractional evolution equations it turns out that they lead to till now known results from the evolution equations with entire derivatives in limiting case. They give more, behavior of the solution when order of derivatives are inside the intervals of entire points. In this way we can follow the influence of the operators generated by entire derivative in many fractional time evolution PDEs especially with singular initial data, and non‐Lipschitz's nonlinear term.Apart from evolution equations we prove also an existence‐uniqueness result for an initial value problem with singularities for linear and nonlinear fractional elliptic equation of Helmholtz type and fractional order α, where 1 < Re(α) ≤ 2, with respect to the one variable from R+. As a framework, we employ also Colombeau algebra of generalized functions containing fractional derivatives and operations among them in order to deal with the fractional equations with singularities. We apply the same techniques to the fractional Laplace and Poisson equation linear and nonlinear ones. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions

Abstract In this paper, the two-dimensional Legendre wavelets are applied for numerical solution of the fractional Poisson equation with Dirichlet boundary conditions. In this way, a new operational matrix of fractional derivative for the Legendre wavelets is derived and then this operational matrix has been employed to obtain the numerical solution of the above-mentioned problem. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problem. The convergence of the two-dimensional Legendre wavelets expansion is investigated. Also the power of this manageable method is illustrated.

Gross–Pitaevskii–Poisson Equations for Dipolar Bose–Einstein Condensate with Anisotropic Confinement

Ground states and dynamical properties of dipolar Bose-Einstein condensate are analyzed based on the Gross-Pitaevskii-Poisson system (GPPS) and its dimension reduction models under anisotropic confining potential. We begin with the three-dimensional (3D) Gross-Pitaevskii-Poisson system and review its quasi-2D approximate equations when the trap is strongly confined in $z$-direction and quasi-1D approximate equations when the trap is strongly confined in $x$-, $y$-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-\Delta)^{1/2}$ is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both equations and finite time blowup in 2D are analyzed. Finally, we rigorously prove the convergence and linear convergence rate between the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in weak interaction regime.

A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A) = A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b ≈ β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick‐restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0 0 anda j >0 and Σ j=1 ∞ a j =1. The operator calculus justifies the notation(I−T) α :=I−T α (e.g., (I−T 1/2)2=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) α x=y, i.e.,y is a coboundary forT α . The fractional Poisson equation forT is the Poisson equation forT α . We show that if(I−T)X is not closed, then(I−T) α X strictly contains(I−T)X (but has the same closure).

On the Dependence of the Convergence Rate in the Strong Law of Large Numbers for Stationary Processes on the Rate of Decay of the Correlation Function

Previous article Next article On the Dependence of the Convergence Rate in the Strong Law of Large Numbers for Stationary Processes on the Rate of Decay of the Correlation FunctionV. F. GaposhkinV. F. Gaposhkinhttps://doi.org/10.1137/1126078PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] André Blanc-Lapierre and , Albert Tortrat, Sur la loi forte des grands nombres pour les fonctions aléatoires stationnaires du second ordre, C. R Acad. Sci. Paris Sér. A-B, 267 (1968), A740–A743 39:4924 0177.46103 Google Scholar[2] I. N. Verbitskaya, On conditions for the applicability of the strong law of large numbers to wide-sense stationary processes, Theory Prob. Appl., 11 (1966), 632–636 LinkGoogle Scholar[3] V. V. Petrov, The strong law of large numbers for a stationary sequence, Dokl. Akad. Nauk SSSR, 213 (1973), 42–44, (In Russian.) 49:4083 0303.60032 Google Scholar[4] R. J. Serfling, Moment inequalities for the maximum cumulative sum, Ann. Math. Statist., 41 (1970), 1227–1234 42:3835 0272.60013 CrossrefGoogle Scholar[5] V. F. Gaposhkin, Convergence of series that are connected with stationary sequences, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 1366–1392, 1438, (In Russian.) 53:6694 Google Scholar[6] V. F. Gaposhkin, Criteria for the strong law of large numbers for some classes of second-order stationary processes and homogeneous random fields, Theory Prob. Appl., 22 (1977), 286–310 0377.60033 LinkGoogle Scholar[7] V. F. Gaposhkin, Exact estimates of the rate of convergence in the strong law of large numbers for classes of stationary (in the wide sense) sequences and processes, Uspehi Mat. Nauk, 31 (1976), 233–234, (In Russian.) 55:9240 Google Scholar[8] V. F. Gaposhkin, Estimates of means for almost all realizations of stationary processes, Sibirsk. Mat. Zh., 20 (1979), 978–989, 1165, (In Russian.) 82d:60064 0447.60028 Google Scholar[9] G. Aleksich, Problems in the Convergence of Orthogonal Series, IL, Moscow, 1963, (In Russian.) Google Scholar[10] A. Zygmund, Trigonometrical Series, 2 vols., Cambridge University Press, Cambridge, 1959 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Exponent of Convergence of a Sequence of Ergodic Averages26 August 2022 | Mathematical Notes, Vol. 112, No. 1-2 Cross Ref Weighted laws of large numbers and convergence of weighted ergodic averages on vector valued $$L_p$$-spaces5 July 2021 | Advances in Operator Theory, Vol. 6, No. 3 Cross Ref Constructive approach to limit theorems for recurrent diffusive random walks on a stripAsymptotic Analysis, Vol. 122, No. 3-4 Cross Ref Almost everywhere convergence of ergodic series6 October 2015 | Ergodic Theory and Dynamical Systems, Vol. 37, No. 2 Cross Ref Pointwise equidistribution with an error rate and with respect to unbounded functions4 April 2016 | Mathematische Annalen, Vol. 367, No. 1-2 Cross Ref An effective Ratner equidistribution result for SL(2,R)⋉R2Duke Mathematical Journal, Vol. 164, No. 5 Cross Ref On Estimates of the Convergence Rate in the Strong Law of Large Numbers for Stationary Sequences (in Some Classes $S_w$ and $R_\Phi$)V. F. Gaposhkin10 November 2011 | Theory of Probability & Its Applications, Vol. 55, No. 4AbstractPDF (194 KB)POINTWISE ERGODIC THEOREMS WITH RATE WITH APPLICATIONS TO LIMIT THEOREMS FOR STATIONARY PROCESSES21 November 2011 | Stochastics and Dynamics, Vol. 11, No. 01 Cross Ref Pointwise ergodic theorems with rate and application to the CLT for Markov chainsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol. 45, No. 3 Cross Ref Extensions of the Menchoff-Rademacher theorem with applications to ergodic theoryIsrael Journal of Mathematics, Vol. 148, No. 1 Cross Ref Fractional Poisson equations and ergodic theorems for fractional coboundariesIsrael Journal of Mathematics, Vol. 123, No. 1 Cross Ref Volume 26, Issue 4| 1982Theory of Probability & Its Applications History Submitted:18 December 1979Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1126078Article page range:pp. 706-720ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics