Fractional evolution Dirac-like equations: Some properties and a discrete Von Neumann-type analysis

A system of fractional evolution equations results from employing the tool of the Fractional Calculus and following the method used by Dirac to obtain his well-known equation from Klein–Gordon’s one. It represents a possible interpolation between Dirac and diffusion and wave equations in one space dimension.     

A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

In this paper, we develop a fractional integro‐differential operator calculus for Clifford algebra‐valued functions. To do that, we introduce fractional analogues of the Teodorescu and Cauchy‐Bitsadze operators, and we investigate some of their mapping properties. As a main result, we prove a fractional Borel‐Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge‐type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann‐Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.     

Fundamental solution of the time‐fractional telegraph Dirac operator

In this work, we obtain the fundamental solution (FS) of the multidimensional time‐fractional telegraph Dirac operator where the 2 time‐fractional derivatives of orders α∈]0,1] and β∈]1,2] are in the Caputo sense. Explicit integral and series representation of the FS are obtained for any dimension. We present and discuss some plots of the FS for some particular values of the dimension and of the fractional parameters α and β. Finally, using the FS, we study some Poisson and Cauchy problems.     

Clifford analysis and the Navier-Stokes equations over unbounded domains

We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in-stationary Navier-Stokes equations over time-varying domains. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parabolic Dirac operators and the Navier–Stokes equations over time‐varying domains

We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in‐stationary Navier–Stokes equations over time‐varying domains. Copyright © 2005 John Wiley & Sons, Ltd.     

Functions of finite fractional variation and their applications to fractional impulsive equations

We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak σ-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a σ-additive term—we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures.     

Weak boundedness properties for maximal operators defined on weighted spaces

The purpose of this paper is to obtain characterizations of weak type (1,q) inequalities,q ≥ 1, for maximal operators defined on weighted spaces by means of the corresponding operator acting over Dirac deltas. We present a technical theorem which allows us to obtain characterizations for a pair of weights belonging to the classA ₁ of weights by means of the fractional maximal operator. Analogous results are obtained for the one-sided fractional maximal operator.     

Dirac γ-equation, classical gauge fields and Clifford algebra

An equation, we call Dirac γ-equation, is introduced with the help of the mathematical tools connected with the Clifford algebra. This equation can be considered as a generalization of the Dirac equation for the electron. Some features of Dirac γ-equation are investigated (plane waves, currents, canonical forms). Furthermore, on the basis of local gauge in variance regarding unitary group, a system of equations is introduced consisting of Dirac γ-equation and the Yang-Mills or Maxwell equations. This system of equations describes a Dirac’s field interacting with the Yang-Mills or Maxwell gauge field. Characteristics of this system of equations are studied for various gauge groups and the liaison between the new and the standard constructions of classical gauge fields is discussed. This paper is supported by the Russian Foundation for Basic Research, grant 95-10-00433a.     

Multi-component bi-Hamiltonian Dirac integrable equations

A specific matrix iso-spectral problem of arbitrary order is introduced and an associated hierarchy of multi-component Dirac integrable equations is constructed within the framework of zero curvature equations. The bi-Hamiltonian structure of the obtained Dirac hierarchy is presented be means of the variational trace identity. Two examples in the cases of lower order are computed.     

Dirac结构与Dirac流形

引入了Dirac结构的对偶特征对的概念,并给出了相应的可积性条件．利用这些结果,得到在Dirac流形的子流形上自然诱导出Dirac结构的条件,结果改进了Courant T．J．给出的相应条件;还得到Poisson流形在子流形上诱导出Poisson结构的条件,并改进了Weinstein A．和Courant T．J．所给出的相应条件;最后证明了预辛形式的可约Dirac结构与相应商流形上的辛结构之间存在一一对应的关系．     

Clifford–Fourier transform on hyperbolic space

In this paper, we introduce a new generalization of the Helgason–Fourier transform using the angular Dirac operator on both the hyperboloid and unit ball models. The explicit integral kernels of even dimension are derived. Furthermore, we establish the formal generating function of the even dimensional kernels. In the computations, fractional integration plays a key unifying role. Copyright © 2016 John Wiley & Sons, Ltd.     

Equivalence of Sobolev Spaces

We repair the proof of equivalence of certain L²-Sobolev spaces on manifolds with bounded curvature of all orders from [4]. The results are extended to generalized compatible Dirac operators, fractional order Sobolev spaces and weighted Sobolev spaces. A certain way of doing coordinate free computations is presented.     

Clifford algebraic spinor and the Dirac wave equations

The hyperbolic complex (HC) space is congruent with Minkowski space time.HC is a special kind of non-Euclidean space with continuous odd-points. The Clifford algebraic spinor and the Dirac wave equation can be introduced in the hyperbolic complex space. The Clifford algebraic spinor contains eight independent elements and the Dirac wave equations 64 coefficients. For Dirac particles 4×8 and for antiparticles 4×8 variables which are Hermitian conjugate to each other (on four dimensional space-time).     

An extension of the Dirac theory of constraints

Constructions introduced by Dirac for singular Lagrangians are extended and reinterpreted to cover cases when kernel distributions are either nonintegrable or of nonconstant rank, and constraint sets need not be closed.     

A posteriori error estimates with point sources in fractional sobolev spaces

We consider Poisson's equation with a finite number of weighted Dirac masses as a source term, together with its discretization by means of conforming finite elements. For the error in fractional Sobolev spaces, we propose residual‐type a posteriori estimators with a specifically tailored oscillation and show that, on two‐dimensional polygonal domains, they are reliable and locally efficient. In numerical tests, their use in an adaptive algorithm leads to optimal error decay rates. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1018–1042, 2017     

A FORMULATION OF HAMILTONIAN MECHANICS USING GEOMETRIC ALGEBRA

Geometric Algebra is introduced. The basic concepts of Hamilton mechanics are derived using the Geometric Algebra. Standard Poisson bracket and standard Lagrange bracket is introduced in terms of the Geometric Algebra. The Dirac quantization map is derived in a new view, which is used to derive the plus Poisson bracket, which seems different from the previous forms by other authors.     

Twistor Operators and Eigenvalues of the Dirac Operator on Compact Quaternion-Kähler Spin Manifolds

Quaternion-Kähler twistor operators are introduced. Using these operators with the Lichnerowicz formula, we get lower bounds for the square of the eigenvalues of the Dirac operator in terms of the eigenvalues of the fundamental 4-form.     

Factorization of the Non-Stationary Schrödinger Operator

We consider a factorization of the non-stationary Schr?dinger operator based on the parabolic Dirac operator introduced by Cerejeiras, K?hler and Sommen. Based on the fundamental solution for the parabolic Dirac operators, we shall construct appropriated Teodurescu and Cauchy-Bitsadze operators. Afterwards we will describe how to solve the nonlinear Schr?dinger equation using Banach fixed point theorem.     

Scaling limit solution of a fractional Burgers equation

A fractional version of the heat equation, involving fractional powers of the negative Laplacian operator, with random initial conditions of exponential type, is introduced. Two cases are considered, depending on whether the Hopf–Cole transformation of such random initial conditions coincides, in the mean-square sense, with the gradient of the fractional Riesz–Bessel motion introduced in Anh et al. (J. Statist. Plann. Inference 80 (1999) 95–110), or with a quadratic function of such a random field. The scaling limits of the random fields defined by the Hopf–Cole transformation of the solutions to the fractional heat equation introduced in the two cases considered are then calculated via their spectral representations.     

GENERALIZED FRACTIONAL TRACE VARIATIONAL IDENTITY AND A NEW FRACTIONAL INTEGRABLE COUPLINGS OF SOLITON HIERARCHY

Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.