Search Results (117)

In this study, we consider and analyze an inhomogeneous Whittaker-type differential equation of the form ${\displaystyle \frac{d^{2}y\left(x\right)}{d{x}^2}}+{\displaystyle \frac{1}{x}}{\displaystyle \frac{dy\left(x\right)}{dx}}-{\alpha }^2\left(x^2-{\beta }^2\right)y\left(x\right)=g\left(x\right)$, where $\alpha$ and $\beta$ are given parameters. We investigate the analytical structure of its solution through the application of the Whittaker integral representation. The analysis encompasses both initial value problems (IVPs) and boundary value problems (BVPs), wherein appropriate conditions are imposed within a unified analytical framework. Furthermore, a systematic methodology is developed for constructing explicit solutions within the framework of Whittaker function theory. This approach not only elucidates the functional behaviour of the solutions but also provides a foundation for extending the analysis to more general classes of second-order linear differential equations. Full article

The second virial coefficient (SVC) of the Lennard-Jones fluid is a cornerstone of molecular theory, yet its calculation has traditionally relied on the complex integration of the pair potential. This work introduces a fundamentally different approach by reformulating the problem in terms of ordinary differential equations (ODEs). For the classical component of the SVC, we generalize the confluent hypergeometric and Weber–Hermite equations. For the first quantum correction, we present entirely new ODEs and their corresponding exact-analytical solutions. The most striking result of this framework is the discovery that these ODEs can be transformed into Schrödinger-like equations. The classical term corresponds to a harmonic oscillator, while the quantum correction includes additional inverse-power potential terms. This formulation not only provides a versatile method for expressing the virial coefficient through a linear combination of functions (including Kummer, Weber, and Whittaker functions) but also reveals a profound and previously unknown mathematical structure underlying a classical thermodynamic property. Full article

This work addresses the Cauchy problem for a linear equation with a first-order time derivative t and an arbitrary-order spatial derivative x. This equation is a generalization of the linear heat equation of the second order in the case of arbitrary order with respect to spatial variable. The considered linear equation arises from the linearization of the Burgers hierarchy of equations. The Cauchy problem to a linear equation can be solved using the Green function method. The Green function is explicitly constructed for the case of dissipative and dispersive equations and is expressed in terms of generalized hypergeometric functions. The general formulas obtained for representing Green’s function are new. A discussion of specific cases of the equation is also provided. Full article

The aim of this work is to model the growth of an economic system and, in particular, the evolution of capital accumulation over time, analysing the feasibility of a closed-form solution to the initial value problem that governs the capital-per-capita dynamics. The latter are related to the labour-force dynamics, which are assumed to follow a von Bertalanffy model, studied in the literature in its simplest form and for which the existence of an exact solution, in terms of hypergeometric functions, is known. Here, we consider an extended form of the von Bertalanffy equation, which we make dependent on two parameters, rather than the single-parameter model known in the literature, to better capture the features that a reliable economic growth model should possess. Furthermore, we allow one of the two parameters to vary over time, making it dependent on a periodic function to account for seasonality. We prove that the two-parameter model admits an exact solution, in terms of hypergeometric functions, when both parameters are constant. In the time-varying case, although it is not possible to obtain a closed-form solution, we are able to find two exact solutions that closely bound, from below and from above, the desired one, as well as its numerical approximation. The presented models are implemented in the Mathematica environment, where simulations, parameter sensitivity analyses and comparisons with the known single-parameter model are also performed, validating our findings. Full article

In this paper, we study a generalized Duffin–Kemmer equation for a spin-1 particle with two characteristics, anomalous magnetic moment and polarizability in the presence of external uniform magnetic and electric fields. After separating the variables, we obtained a system of 10 first-order partial differential equations for 10 functions $f_A(r,z)$. To resolve this complicated problem, we first took into account existing symmetry in the structure of the derived system. The main step consisted of applying a special method for fixing the r-dependence of ten functions $f_A(r,z),A=1,\dots ,10$. We used the approach of Fedorov–Gronskiy, according to which the complete 10-component wave function is decomposed into the sum of three projective constituents. The dependence of each component on the polar coordinate r is determined by only one corresponding function, $F_i\left(r\right),i=1,2,3$. These three basic functions are constructed in terms of confluent hypergeometric functions, and in this process a quantization rule arises due to the presence of a magnetic field.In fact, this approach is a step-by-step algebraization of the systems of equations in partial derivatives. After that, we derived a system of 10 ordinary differential equations for 10 functions $f_A\left(z\right)$. This system was solved using the elimination method and with the help of special linear combinined with the involved functions. As a result, we found three separated second-order differential equations, and their solutions were constructed in the terms of the confluent hypergeometric functions. Thus, in this paper, the three types of solutions for a vector particle with two additional electromagnetic characteristics in the presence of both external uniform magnetic and electric fields. Full article

In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$, along with their derivatives, is introduced to facilitate the construction of the solution. The analytical properties of ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$ are rigorously investigated, and explicit closed-form expressions for ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$ and its derivatives are derived in terms of Whittaker functions ${\mathrm{M}}_{\kappa ,\mu }\left(z\right)$ and ${\mathrm{W}}_{\kappa ,\mu }\left(z\right)$, confluent hypergeometric functions, and other special functions including Bessel functions, modified Bessel functions, and the incomplete gamma functions, along with their respective derivatives. These expressions are obtained for specific parameter values using symbolic computation in Maple. The results contribute to the broader analytical framework for solving inhomogeneous linear differential equations with applications in engineering, mathematical physics, and biological modeling. Full article

The current paper presents a novel numerical technique to handle variable-order multiterm fractional differential equations (VO-MTFDEs) supplemented with initial conditions (ICs) by introducing generalized fractional Jacobi functions (GFJFs). These GFJFs satisfy the associated ICs. A crucial part of this approach is using the spectral collocation method (SCM) and building operational matrices (OMs) for both integer-order and variable-order fractional derivatives in the context of GFJFs. These lead to efficient and accurate computations. The suggested algorithm’s convergence and error analysis are proved. The feasibility of the suggested procedure is confirmed via five numerical test examples. Full article

We investigate the influence of the higher-order curvature corrections on a static configuration with constant density in Einstein–Gauss–Bonnet (EGB) gravity. This analysis is applied to both neutral and charged fluid distributions in arbitrary spacetime dimensions. The EGB field equations are generated, and the condition of pressure isotropy is shown to generalise the general relativity equation. The gravitational potentials are unique in all spacetime dimensions for neutral gravitating spheres. Charged gravitating spheres are not unique and depend on the form of the electric field. Our treatment is extended to the particular case of a charged fluid distribution with a constant energy density and constant electric field intensity. The charged EGB field equations are integrated to give exact models in terms of hypergeometric functions which can also be written as a series. Full article

In this study, we aim to construct a finite set of orthogonal matrix polynomials for the first time, along with their finite orthogonality, matrix differential equation, Rodrigues’ formula, several recurrence relations including three-term relation, forward and backward shift operators, generating functions, integral representation and their relation with Jacobi matrix polynomials. Thus, the concept of “finite”, which is used to impose parametric constraints for orthogonal polynomials, is transferred to the theory of matrix polynomials for the first time in the literature. Moreover, this family reduces to the finite orthogonal M polynomials in the scalar case when the degree is 1, thereby providing a matrix generalization of finite orthogonal M polynomials in one variable. Full article

The present review discusses the solution of the Heun, confluent, biconfluent, double confluent, and triconfluent equations in terms of polynomial expansions, and applies the results to generate exactly solvable Schrödinger potentials. Although there are more general approaches to solve these differential equations in terms of the expansions of certain special functions, the importance of polynomial solutions is unquestionable, as most of the known potentials are solvable in terms of the hypergeometric and confluent hypergeometric functions; i.e., Natanzon-class potentials possess bound-state solutions in terms of classical orthogonal polynomials, to which the (confluent) hypergeometric functions can be reduced. Since some of the Heun-type equations contain the hypergeometric and/or confluent hypergeometric differential equations as special limits, the potentials generated from them may also contain Natanzon-class potentials as special cases. A power series expansion is assumed around one of the singular points of each differential equation, and recurrence relations are obtained for the expansion coefficients. With the exception of the triconfluent Heun equations, these are three-term recurrence relations, the termination of which is achieved by prescribing certain conditions. In the case of the biconfluent and double confluent Heun equations, the expansion coefficients can be obtained in the standard way, i.e., after finding the roots of an (N + 1)th-order polynomial in one of the parameters, which, in turn, follows from requiring the vanishing of an (N + 1) × (N + 1) determinant. However, in the case of the Heun and confluent Heun equations, the recurrence relation can be solved directly, and the solutions are obtained in terms of rationally extended X₁-type Jacobi and Laguerre polynomials, respectively. Examples for solvable potentials are presented for the Heun, confluent, biconfluent, and double confluent Heun equations, and alternative methods for obtaining the same potentials are also discussed. These are the schemes based on the rational extension of Bochner-type differential equations (for the Heun and confluent Heun equation) and solutions based on quasi-exact solvability (QES) and on continued fractions (for the biconfluent and double confluent equation). Possible further lines of investigations are also outlined concerning physical problems that require the solution of second-order differential equations, i.e., the Schrödinger equation with position-dependent mass and relativistic wave equations. Full article

We construct the Green functions (or Feynman’s propagators) for the Schrödinger equations of the form $i{\psi }_t+{\displaystyle \frac{1}{4}}{\psi }_{xx}\pm t{x}^2\psi =0$ (for the wave function $\psi$ and its time (t) and x-space derivatives) in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schrödinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of the quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann’s functions is established. The relevant bibliography, to the best of our knowledge, is addressed. Full article

This paper, we study systems of difference equations numerically and theoretically. These systems have been examined by many researchers. We focus on their general forms and limits. We consider different orders of difference systems. In certain cases, we use a computer in order to verify the summation laws that we encountered. We consider a system with two parameters. For certain values of the parameters, we determine the explicit form of the solution and show that the limit of this sequence tends to zero. We show that for the values of a parameter between 0 and 1, the limit of the sequence is a nonzero value, while for the values greater than 1, the limit of the sequence becomes zero. Full article

The primary objective of this study is to explore sufficient conditions for the existence, uniqueness, and optimal stability of positive solutions to a finite system of Hilfer–Hadamard fractional differential equations with two-point boundary conditions. Our analysis centers around transforming fractional differential equations into fractional integral equations under minimal requirements. This investigation employs several well-known special control functions, including the Mittag–Leffler function, the Wright function, and the hypergeometric function. The results are obtained by constructing upper and lower control functions for nonlinear expressions without any monotonous conditions, utilizing fixed point theorems, such as Banach and Schauder, and applying techniques from nonlinear functional analysis. To demonstrate the practical implications of the theoretical findings, a pertinent example is provided, which validates the results obtained. Full article

The evaluation of low-degree hypergeometric polynomials to zero defines algebraic hypersurfaces in the affine space of the free parameters and the argument of the hypergeometric function. This article investigates the algebraic surfaces defined by the hypergeometric equation ${}_2\mathrm{F}_1(-N,b;c;z)=0$ with $N=3$ or $N=4$. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps. Thereby, this article contributes to the systematic enumeration of Belyi maps. Full article

The present study investigates the entropy generation of chemically reactive micropolar hybrid nanoparticle motion with mass transfer. Magnetic oxide (Fe₃O₄) and copper oxide (CuO) nanoparticles were mixed in water to form a hybrid nanofluid. The governing equations for velocity, concentration, and temperature are transformed into ordinary differential equations along with the boundary conditions. In the fluid region, the heat balance is kept conservative with a source/sink that relies on the temperature. In the case of radiation, there is a differential equation along with several characteristic coefficients that transform hypergeometric and Kummer’s differential equations by a new variable. Furthermore, the results of the current problem can be discussed by implementing a graphical representation with different factors, namely the Brinkman number, porosity parameter, magnetic field, micropolar parameter, thermal radiation, Schmidt number, heat source/sink parameter, and mass transpiration. The results of this study are presented through graphical representations that depict various factors influencing the flow profiles and physical characteristics. The results reveal that an increase in the magnetic field leads to a reduction in velocity and entropy production. Furthermore, temperature and entropy generation rise with a stronger radiation parameter, whereas the Nusselt number experiences a decline. This study has several industrial applications in technology and manufacturing processes, including paper production, polymer extrusion, and the development of specialized materials. Full article