Special Functions Of Mathematical Physics Research Articles

Fox’s H-Functions: A Gentle Introduction to Astrophysical Thermonuclear Functions

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper.

Numerical integration of the Cauchy problem with non-singular special points

Solutions of many applied Cauchy problems for ordinary differential equations have one or more multiple zeros on the integration segment. Examples are the equations of special functions of mathematical physics. The presence of multiples of zeros significantly complicates the numerical calculation, since such problems are ill-conditioned. Round-off errors may corrupt all decimal digits of the solution. Therefore, multiple zeros should be treated as special points of the differential equations. In the present paper, a local solution transformation is proposed, which converts the multiple zero into a simple one. The calculation of the latter is not difficult. This makes it possible to dramatically improve the accuracy and reliability of the calculation. Illustrative examples have been carried out, which confirm the advantages of the proposed method.

Calculation of special functions arising in the problem of diffraction by a dielectric ball

To apply the incomplete Galerkin method to the problem of the scattering of electromagnetic waves by lenses, it is necessary to study the differential equations for the field amplitudes. These equations belong to the class of linear ordinary differential equations with Fuchsian singularities and, in the case of the Lneburg lens, are integrated in special functions of mathematical physics, namely, the Whittaker and Heun functions. The Maple computer algebra system has tools for working with Whittaker and Heun functions, but in some cases this system gives very large values for these functions, and their plots contain various kinds of artifacts. Therefore, the results of calculations in the Maple11 and Maple2019 systems of special functions related to the problem of scattering by a Lneburg lens need additional verification. For this purpose, an algorithm for finding solutions to linear ordinary differential equations with Fuchsian singular points by the method of Frobenius series was implemented, designed as a software package Fucsh for Sage. The problem of scattering by a Lneburg lens is used as a test case. The calculation results are compared with similar results obtained in different versions of CAS Maple. Fuchs for Sage allows computing solutions to other linear differential equations that cannot be expressed in terms of known special functions.

THE ANALYTIC T-MATRIX FOR THE HULTHÉN POTENTIAL IN ALL PARTIAL WAVES

Analytical expression for the Hulthén off-shell T -matrix in all partial waves is constructed in its maximal reduced form. As a basic requirement the off-shell solutions and Jost function are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics . The off-shell T -matrix is found to be in order with respect to its limiting behaviours and off-shell unitarity.

Asymptotically accurate error estimates of exponential convergence for the trapezoidal rule

In many applied problems, efficient calculation of quadratures with high accuracy is required. The examples are: calculation of special functions of mathematical physics, calculation of Fourier coefficients of a given function, Fourier and Laplace transformations, numerical solution of integral equations, solution of boundary value problems for partial differential equations in integral form, etc. For grid calculation of quadratures, the trapezoidal, the mean and the Simpson methods are usually used. Commonly, the error of these methods depends quadratically on the grid step, and a large number of steps are required to obtain good accuracy. However, there are some cases when the error of the trapezoidal method depends on the step value not quadratically, but exponentially. Such cases are integral of a periodic function over the full period and the integral over the entire real axis of a function that decreases rapidly enough at infinity. If the integrand has poles of the first order on the complex plane, then the Trefethen-Weidemann majorant accuracy estimates are valid for such quadratures. In the present paper, new error estimates of exponentially converging quadratures from periodic functions over the full period are constructed. The integrand function can have an arbitrary number of poles of an integer order on the complex plane. If the grid is sufficiently detailed, i.e., it resolves the profile of the integrand function, then the proposed estimates are not majorant, but asymptotically sharp. Extrapolating, i.e., excluding this error from the numerical quadrature, it is possible to calculate the integrals of these classes with the accuracy of rounding errors already on extremely coarse grids containing only 10 steps.

Hulthén Half-off-Shell T Matrix-Application to n-p and n-d Systems

In a previous paper, one of us (UL) constructed an exact analytical expression for the irregular (Jost) solution, off the energy shell, for the Hulthen potential by using the method of ordinary differential equation along with the properties of certain special functions of mathematical physics. In the present text, closed form expressions for the off-shell irregular solution and half-off-shell Tmatrix for motion in the Hulthen potential are re-derived by using the direct integration approach to the problem under consideration. We demonstrate the usefulness of our expressions through some model calculations and achieve excellent agreement between bound-and scattering-state observables and standard results.

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

Off-shell Jost function for the Hulthén potential in all partial waves

A new expression for the Hulthén off-shell Jost function in all partial waves is constructed in its maximal reduced form. As a basic requirement the on-shell solutions are first developed by following the differential equation approach to the problem together with judicious exploitation of the properties of certain special functions of mathematical physics. Utilizing the properties of the Jost function, the binding energies and phase shifts for N-N and n-d systems are computed and found excellent agreement with standard data.

Two-sided inequalities for the Struve and Lommel functions

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial differential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function 1F2(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function 1F2(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.

AN ELEMENTARY AND INTRODUCTORY APPROACH TO FRACTIONAL CALCULUS AND ITS APPLICATIONS

A b s t r a c t: The subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It does indeed provide several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables. The main object of this paper* is to present a brief elementary and introductory approach to the theory of fractional calculus and its applications especially in developing solutions of certain interesting families of ordinary and partial fractional differintegral equations. Relevant connections of some of the results presented in this lecture with those obtained in many other earlier works on this subject will also be indicated.

A unified study of orthogonal polynomials via Lie algebra

In this paper, we discuss some operators defined on Lie algebras for the purpose of deriving properties of some special functions. The method developed in this paper can also be used to study some other special functions of mathematical physics. We have established a general theorem concerning eigenvectors for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. Further, using this result, we have obtained differential recurrence relations and differential equations for the extended Jacobi polynomials and the Gegenbauer polynomials. Results of many researchers; see for example Radulescu (1991), Mandal (1991), Pathan and Khan (2003), Humi, and the references therein, follow as special cases of our results.

Similarity Solutions to Burgers\textquoteright \ Equation in Terms of Special Functions of Mathematical Physics

Similarity Solutions to Burgers\textquoteright \ Equation in Terms of Special Functions of Mathematical Physics

Störmer problem restricted to a spherical surface and the Euler and Lagrange tops

In a recent work, Cortés and Poza (2015 Eur. J. Phys. 36 055009) analysed, in full, the dynamics of a charged particle in the field of a magnetic dipole restricted to a spherical surface with the dipole at its centre. This model can be considered as the classical non-relativistic Störmer problem on a sphere. Here, we started from a Lagrangian approach: we derived the Hamilton equations of motion and observed that in this restricted case the equations can be reduced to quadratures, and they were integrated numerically. From the Hamiltonian function we found, for the polar angle, an equivalent one-dimensional system of a particle in the presence of an effective potential. In the present work we start from a change of variable to the cosine of the polar angle. In terms of this variable we obtain an equation that turns out to be the same as the one of a particle in a quartic potential. Then, we can actually solve the equations of motion for the polar angle using Jacobi elliptic functions, and for the azimuthal angle we use the same integrals which were expressed by Jacobi in terms of theta functions, both in the Euler and Lagrange tops. In this restricted Störmer problem, the student at undergraduate or graduate level will have a good example of an integrable nonlinear physical system in which, after analysis of its complex dynamics, one can obtain an analytical solution by means of some special functions of mathematical physics. Additionally, one discovers that the equations of motion of this restricted case of a charge in a magnetic dipole field have the same mathematical structure as the corresponding equations of other well known integrable classical dynamical systems.

Certain properties of Gegenbauer polynomials via Lie algebra

We establish a result for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. Then we use this result to derive some properties for Gegenbauer polynomials, for example, Rodrigues formula. The method developed here is potentially useful to investigate some other special functions of mathematical physics.

On- and off-shell Jost functions and their integral representations

By judicious exploitation of the transpose operator relation in conjunction with the differential equations of special functions of mathematical physics, integral representations of the on- and off-shell Jost functions are derived from the particular integrals of the inhomogeneous Schrodinger equation. Using the particular integral of the inhomogeneous Schrodinger equation, exact analytical expressions for the Coulomb and Coulomb plus Yamaguchi off-shell Jost solutions are constructed in the maximal reduced form. As a case study, the limiting behaviours and the on-shell discontinuities of the Coulomb plus Yamaguchi Jost solutions are verified numerically.

Some new inequalities involving generalized Erdelyi-Kober fractional q-integral operator

During the past four decades and longer, the subject of fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has provided several potentially useful tools for solving differential, integral and integro-differential equations, and various other problems involving special functions of mathematical physics as well as their extensions (q-extensions) and generalizations in one and more variables. Here, in this paper, we aim to establish some new and potentially useful inequalities involving generalized Erdelyi-Kober fractional q-integral operator of the two parameters of deformation q1 and 3578 Junesang Choi, Daniele Ritelli and Praveen Agarwal q2 due to Gaulue [12], by following the similar process used by Gaulue [13] and Dumitru and Agarwal [5]. Relevant connections of the results presented here with those earlier ones are also pointed out. Mathematics Subject Classification: Primary 26D10, 26D15; Secondary 26A33, 05A30

A generalized Cole–Hopf transformation for nonlinear ODEs

We introduce a ‘hybrid’ Cole–Hopf–Darboux transformation to relate the solutions of nonlinear and linear second-order differential equations and derive classification and sufficient condition for this correspondence. We explore physical applications of this correspondence to nonlinear oscillations, the Duffing equation and a nonlinear form of the Schrödinger equation for the nonrelativistic hydrogen atom. In addition, we show that solutions of some nonlinear second-order equations are related to the special functions of mathematical physics through this transformation. These nonlinear equations can be viewed as the ‘class of special nonlinear equations’ which correspond to the linear differential equations which define the special functions of mathematical physics.

Erratum to “Formulas and Theorems for the Special Functions of Mathematical Physics” by W. Magnus, F. Oberhettinger, R. P. Soni

Erratum to “Formulas and Theorems for the Special Functions of Mathematical Physics” by W. Magnus, F. Oberhettinger, R. P. Soni

MORE SPECIAL FUNCTIONS TRAPPED

We extend the technique of using the trapezoidal rule for efficient evaluation of the special functions of mathematical physics given by integral representations. This technique was recently used for Bessel functions, and here we treat incomplete gamma functions and the general confluent hypergeometric function.

NUMERICAL CALCULATION OF BESSEL FUNCTIONS

A new computational procedure is offered to provide simple, accurate and flexible methods for using modern computers to give numerical evaluations of the various Bessel functions. The trapezoidal rule, applied to suitable integral representations, may become the method of choice for evaluation of the many special functions of mathematical physics.