New Class Of Special Functions Research Articles

Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

Hypergeometric Representations and Differential-Difference Relations for Some Kernels Appearing in Mathematical Physics

We investigate the analytic properties of a new class of special functions that appear in the kernels of a class of integral operators underlying the dynamics of matter relaxation processes in attractive fields. These functions, recently introduced by the second author, generate the kernels of the principal parts of these operators and play an important role in understanding their spectral characteristics. We reveal the representations of these functions in terms of the Gauss and Clausen hypergeometric functions and present differential-difference and differential equations they satisfy. Mathematically, the results include calculation of certain trigonometric double integrals and derivation of their other properties. Furthermore, they represent a potentially useful tool in matter relaxation in an external field, the study of nanoelectronic electrolyte-based systems and dynamics of charge carriers in media with obstacles.

A new class of special functions in higher dimensions

A new class of special functions in higher dimensions

SOME RECURRENCE FORMULAS FOR A NEW CLASS OF SPECIAL POLYNOMIALS AND SPECIAL FUNCTIONS

<jats:p>In this paper we used a new class of special functions and special polynomials which are solutions different Sturm Liouvile differential equations of second order. These functions form a basis of a space of square integrable functions over set of a real numbers. We investigated some properties of these polynomials and established some recurrence formulas. Using a new class of special functions, we obtained some useful summation formulas and recurrence formulas.</jats:p>

Reducing nonlinear dynamical systems to canonical forms.

A global framework for treating nonlinear differential dynamical systems is presented. It rests on the fact that most systems can be transformed into the quasi-polynomial format. Any system in this format belongs to an infinite equivalence class characterized by two canonical forms, the Lotka-Volterra (LV) and the monomial systems. Both forms allow for finding total or partial integrability conditions, invariants and dimension reductions of the original systems. The LV form also provides Lyapunov functions and systematic tools for stability analysis. An abstract Lie algebra is shown to underlie the whole formalism. This abstract algebra can be expressed in several realizations among which are the bosonic creation-destruction operators. One of these representations allows one to obtain the analytic form of the general coefficient of the Taylor series representing the solution of the original system. This generates a new class of special functions that are solutions of these nonlinear dynamical systems. From the monomial canonical form, one can prove an equivalence relationship between urn processes and dynamical systems. This establishes a new link between nonlinear dynamics and stochastic processes.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

A Stabilized Separation of Variables Method for the Modified Biharmonic Equation

The modified biharmonic equation is encountered in a variety of application areas, including streamfunction formulations of the Navier–Stokes equations. We develop a separation of variables representation for this equation in polar coordinates, for either the interior or exterior of a disk, and derive a new class of special functions which makes the approach stable. We discuss how these functions can be used in conjunction with fast algorithms to accelerate the solution of the modified biharmonic equation or the “bi-Helmholtz” equation in more complex geometries.

Solutions of nonlinear equations of divergence type in domains having corner points

The algorithm for generation of exact solutions of the nonlinear equation in partial derivatives of a divergent type which is included in the formulation of magnetostatics, hydro-and aerodynamics, quantum mechanics (stationary Schr\"odinger equation) has been suggested. The properties of smoothness of solutions in domains with corner points (piecewise smooth boundary) have been considered. The solutions with unlimited derivatives in the corner point domain have been presented on the basis of a new class of special functions.

On a new class of special functions generated by integral-difference operators

We introduce and investigate a new class of special functions Ξ N [ k ] ( x ) , x ∈ [ 0 , 1 ) . Originally these functions naturally appeared upon spectral analysis of integral-difference operators. We discuss issues of generating functions; decompositions in series; links to Legendre polynomials, complete elliptic integrals and Gauss hypergeometric functions; functional relations for functions Ξ N [ k ] , mostly in case N = 1 .

New spectral estimations for a class of integral-difference operators and generalisation to higher dimensions

Quadratic form approach allows for new results in the analysis of a class of integral-difference operators in finite domains: non-negativity, spectral estimations, a new property of Legendre polynomials, and establishing links with weighted mean-square deviation functionals and with infinite Jacobi matrices with not-bounded coefficients. Generalisation of integral-difference operators to higher dimensions is provided and application to matter relaxation in a field is considered. A new class of special functions naturally appears.

Some new class of special functions suggested by the confluent hypergeometric function

In the present work, we introduce the function representing a rapidly convergent power series which extends the well-known confluent hypergeometric function \(_1F_1[z]\) as well as the integral function \( f(z) = \sum \nolimits _{n=1}^\infty \frac{z^n}{n!^n} \) considered by Sikkema (Differential operators and equations, P. Noordhoff N. V., Djakarta, 1953). We introduce the corresponding differential operators and obtain infinite order differential equations, for which these new special functions are the eigen functions. First we establish some properties, as the order zero of these entire (integral) functions, integral representations, differential equations involving a kind of hyper-Bessel type operators of infinite order. Then we emphasize on the special cases, especially the corresponding analogues of the exponential, circular and hyperbolic functions, called here as \({\ell }\)-H exponential function, \({\ell }\)-H circular and \({\ell }\)-H hyperbolic functions. At the end, the graphs of these functions are plotted using the Maple software.

New class of special functions based on the exact solution of a nonlinear partial differential equation of divergence type

New class of special functions based on the exact solution of a nonlinear partial differential equation of divergence type

Paul Painlevé and his contribution to science

The life and career of the great French mathematician and politician Paul Painleve is described. His contribution to the analytical theory of nonlinear differential equations was significant. The paper outlines the achievements of Paul Painleve and his students in the investigation of an interesting class of nonlinear second-order equations and new equations defining a completely new class of special functions, now called the Painleve transcendents. The contribution of Paul Painleve to the study of algebraic nonintegrability of the N-body problem, his remarkable observations in mechanics, in particular, paradoxes arising in the dynamics of systems with friction, his attempt to create the axiomatics of mechanics and his contribution to gravitation theory are discussed.

Three-variable exponential functions of the alternating group

A new class of special functions of three real variables, based on the alternating subgroup of the permutation group S3, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the E-functions of C3 is shown. Continuous interpolation of the three-dimensional data is studied and exemplified.

Bessel G-Functions. I

We introduce a new class of special functions that generalize the classical Bessel functions and prove their main properties.

Canonical Forms of Differential Equations Free from Accessory Parameters

Systems of differential equations free from accessory parameters are defined and studied by Okubo [Seminar Reports of Tokyo Metropolitan University, 1987]. They are Fuchsian on the complex projective line, and there is an algorithm of determining monodromy representations for such systems. The Gauss hypergeometric equation, the generalized hypergeometric equation, the Pochhammer equation and a one-dimensional section of the Appell hypergeometric system $F_3 $ are known to be reduced to such systems. Recently Yokoyama classified all the systems of differential equations which are irreducible and free from accessory parameters in terms of multiplicities of characteristic exponents. This paper presents canonical forms of all such systems and will define a new class of special functions.

Generalized Bessel functions in incommensurate structure analysis

The analysis of incommensurate structures is computationally more difficult than that of normal ones. This is mainly a result of the structure-factor expression, which involves numerical integrations or infinite series of Bessel functions. Both approaches have been implemented in existing computer programs. Compact analytical expressions are known for special cases only. Recently, a new theory of generalized Bessel functions has been developed. The number of theoretical results and applications is increasing rapidly. Numerical properties and algorithms are being studied. A possible application of the generalized Bessel functions for incommensurate structure analysis is proposed. These functions can be used to derive analytical expressions for structure factors and all partial derivatives for a wide class of incommensurate crystals. The existing programs can be improved by taking into account some interesting numerical and analytical properties of these new functions, like recurrence relations, analytical expressions for derivatives, generating functions and integral representations. A new class of special functions, suitable for dealing with incommensurate structures in a more analv:ical way, is emerging.

Steady infiltration in unsaturated soil from a buried circular cylinder: The separate contributions from top and bottom halves

Waechter and Philip (1985) obtained the asymptotic expansion of the mean infiltration rate for large s from a buried circular cylinder using a scattering analog. Here s(= αl/2) is defined as the ratio of the characteristic length l of the water supply surface (in fact, its radius) to the sorptive length 2α−1 of the soil and a satisfies the relationship K(ψ) = K(0) eαψ, where K is the hydraulic conductivity, and ψ is the moisture potential. This exact solution cannot be used directly to obtain the separate contributions to the mean infiltration rate from the top and the bottom halves of the cylinder; our analysis is based on a new class of special functions derived from the modified Bessel equation with a forcing term. In this paper, we obtain the separate asymptotics for the two halves for large s to make a comparison with the results of the trench problem (Waechter and Mandal, 1993). The asymptotic expansions for top and bottom halves are (2/π)(0.69553s−2/3) and (2/π)(1+0.30066s−2/3), respectively, whereas for a semicircular trench, the mean infiltration rate is given by (2/π)(1+0.30066s−2/3).