Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A₂(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).     

Mellin transform of quartic products of shifted Airy functions

The Mellin transform of quartic products of shifted Airy functions is evaluated in a closed form. Some particular cases expressed in terms of the logarithm function and complete elliptic integrals special values are presented.     

Fractional derivatives of products of Airy functions

Fractional derivatives of the products of Airy functions are investigated, and Dα{Ai(x)×Bi(x)}, where Ai(x) and Bi(x) are the Airy functions of the first and second type, respectively. They turn out to be linear combinations of Dα{Ai(x)} and Dα{Gi(x)}, where Gi(x) is the Scorer function. It is also proved that the Wronskian W(x) of the system of half integrals {D−1/2Ai(x),D−1/2Gi(x)} and its Hilbert transform can be considered special functions in their own right since they are expressed in terms of and Ai(x)Bi(x), respectively. Various integral relations are established. Integral representations for Dα{Ai(x−a)Ai(x+a)} and its Hilbert transform −HDα{Ai(x−a)Ai(x+a)} are derived.     

On solvability of differential equations with the Riesz fractional derivative

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy-type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well-known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.     

Differential and integral relations involving fractional derivatives of Airy functions and applications

Various differential and integral relations are deduced that involve fractional derivatives of the Airy function Ai(x) and the Scorer function Gi(x). Several new Wronskian relations are obtained that lead to the calculation of a number of indefinite integrals containing fractional derivatives of the Airy functions. New fractional derivative conservation laws are derived for equations of the Korteweg-de Vries type.     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

Solution to system of partial fractional differential equations using the fractional exponential operators

In this article, fractional exponential operator is considered as a general approach for solving partial fractional differential equations. An integral representation for this operator is derived from the Bromwich integral for the inverse Mellin transform. Also, effectiveness of this operator for obtaining the formal solution of system of diffusion equations is discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order algorithms for Riesz derivative and their applications (V)

In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even‐order numerical differential formulas (named the fractional‐compact formulas) are established for the Riesz derivatives with order . Properties of coefficients in the derived formulas are studied in details. Then applying the constructed fourth‐order formula, a difference scheme is proposed to solve the Riesz spatial telegraph equation. By the energy method, the constructed numerical algorithm is proved to be stable and convergent with order , where τ and h are the temporal and spatial stepsizes, respectively. Finally, several numerical examples are presented to verify the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1754–1794, 2017     

Cauchy-type problem for an abstract differential equation with fractional derivatives

The uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator A is proved. For an unbounded operator A we present a test for the uniform well-posedness of the problem under consideration consistent with the test for the uniform well-posedness of the Cauchy problem for an equation of second order.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 28–41.Original Russian Text Copyright © 2005 by A. V. Glushak.This revised version was published online in April 2005 with a corrected issue number.     

A novel attempt for finding comparatively accurate solution for sine‐Gordon equation comprising Riesz space fractional derivative

In this paper, a numerical procedure involving Chebyshev wavelet method has been implemented for computing the approximate solution of Riesz space fractional sine‐Gordon equation (SGE). Two‐dimensional Chebyshev wavelet method is implemented to calculate the numerical solution of space fractional SGE. The fractional SGE is considered as an interpolation between the classical SGE (corresponding to α = 2) and nonlocal SGE (corresponding to α = 1). As a consequence, the approximate solutions of fractional SGE obtained by using Chebyshev wavelet approach were compared with those derived by using modified homotopy analysis method with Fourier transform. Copyright © 2015 John Wiley & Sons, Ltd.     

Limiting weak-type behaviors for Riesz transforms and maximal operators in Bessel setting

We establish the limiting weak type behaviors of Riesz transforms associated to the Bessel operators on IR+. which are closely related to the best constants of the weak type (1,1) estimates for such operators. Meanwhile, the corresponding results for Hardy-Littlewood maximal operator and fractional maximal operator in Bessel setting are also obtained.     

On nonoscillating integrals for computing inhomogeneous Airy functions

Integral representations are considered of solutions of the inhomogeneous Airy differential equation . The solutions of these equations are also known as Scorer functions. Certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane. By using steepest descent methods from asymptotics, the standard integral representations of the Scorer functions are modified in order to obtain nonoscillating integrals for complex values of . In this way stable representations for numerical evaluations of the functions are obtained. The methods are illustrated with numerical results.

     

Riesz Transforms and Harmonic Lip1-Capacity in Cantor Sets

We estimate the L²-norm of the s-dimensional Riesz transformson some Cantor sets in R^d. Towards this end, we show that theRiesz transforms truncated at different scales behave in a quasiorthogonalway. As an application, we obtain some precise numerical estimatesfor the Lipschitz harmonic capacity of these sets. 2000 MathematicsSubject Classification 42B20, 42B25.     

Integrals involving products of Airy functions, their derivatives and Bessel functions

A new integral representation of the Hankel transform type is deduced for the function Fn(x,Z)=Zn−1Ai(x−Z)Ai(x+Z) with x∈R, Z>0 and n∈N. This formula involves the product of Airy functions, their derivatives and Bessel functions. The presence of the latter allows one to perform various transformations with respect to Z and obtain new integral formulae of the type of the Mellin transform, K-transform, Laplace and Fourier transform. Some integrals containing Airy functions, their derivatives and Chebyshev polynomials of the first and second kind are computed explicitly. A new representation is given for the function ²|Ai(z)| with z∈C.     

Characterizations of the boundedness of generalized fractional maximal functions and related operators in Orlicz spaces

Given and a Young function η, we consider the generalized fractional maximal operator defined by where the supremum is taken over every ball B contained in . In this article, we give necessary and sufficient Dini type conditions on the functions , and η such that is bounded from the Orlicz space into the Orlicz space . We also present a version of this result for open subsets of with finite measure. Both results generalize those contained in 6 and 14 when , respectively. As a consequence, we obtain a characterization of the functions involved in the boundedness of the higher order commutators of the fractional integral operator with BMO symbols. Moreover, we give sufficient conditions that guarantee the continuity in Orlicz spaces of a large class of fractional integral operators of convolution type with less regular kernels and their commutators, which are controlled by .     

On the Wright hypergeometric matrix functions and their fractional calculus

In this paper we focus on the Wright hypergeometric matrix functions and incomplete Wright Gauss hypergeometric matrix functions by using Pochhammer matrix symbol. We first introduce the Wright hypergeometric functions of a matrix argument and examine the convergence of these matrix functions in the unit circle, then we discuss the integral representations and differential formulas of the Wright hypergeometric matrix functions. We have also carried out a similar study process for incomplete Wright Gauss hypergeometric matrix functions. Finally, we obtain some results on the transform and fractional calculus of these Wright hypergeometric matrix functions.     

Space-time fractional derivative operators

Evolution equations for anomalous diffusion employ fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. This paper develops the mathematical foundations of those operators.

     

Integrals containing products of Airy functions and planar photon propagation in a magnetic field

We derive a number of new results on integrals of products of Airy functions, using various integral transform techniques. As an application, we deduce a compact integral representation, suitable for numerical integration, of the one‐loop photon propagator in a magnetic field in 2 + 1 dimensional scalar quantum electrodynamics. Copyright © 2011 John Wiley & Sons, Ltd.