Editorial

7 pages, 205 KiB

Open AccessEditorial

Editorial for the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”

by Sergei Sitnik

Mathematics 2023, 11(15), 3402; https://doi.org/10.3390/math11153402 - 04 Aug 2023

Viewed by 587

Abstract

This editorial text is a short introductory guide to the book edition of the Special Issue “Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms”, which was published in the MDPI journal Mathematics in the years 2022–2023 [...] Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

Research

Jump to: Editorial

9 pages, 276 KiB

Open AccessArticle

Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

by Andrey B. Muravnik

Mathematics 2023, 11(12), 2698; https://doi.org/10.3390/math11122698 - 14 Jun 2023

Cited by 1 | Viewed by 596

Abstract

We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest [...] Read more.

We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest from researchers due to the following phenomenon: the solutions acquire qualitative properties specific for nonstationary (more exactly, parabolic) equations. In this paper, such a phenomenon is studied for nonlocal generalizations of elliptic differential equations, more exactly, for elliptic differential-difference equations with nonlocal potentials arising in various applications not covered by the classical theory. We find a Poisson-like kernel such that its convolution with the boundary-value function satisfies the investigated problem, prove that the constructed solution is infinitely smooth outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

21 pages, 996 KiB

Open AccessArticle

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

by Yongsheng Rao, Waseem Ahmad Khan, Serkan Araci and Cheon Seoung Ryoo

Mathematics 2023, 11(10), 2386; https://doi.org/10.3390/math11102386 - 20 May 2023

Cited by 2 | Viewed by 889

Abstract

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the [...] Read more.

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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16 pages, 340 KiB

Open AccessArticle

Sums Involving the Digamma Function Connected to the Incomplete Beta Function and the Bessel functions

by Juan Luis González-Santander and Fernando Sánchez Lasheras

Mathematics 2023, 11(8), 1937; https://doi.org/10.3390/math11081937 - 20 Apr 2023

Cited by 2 | Viewed by 834

Abstract

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta [...] Read more.

We calculate some infinite sums containing the digamma function in closed form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as parameter differentiation formulas for the beta incomplete function, reduction formulas of

_{3} F_{2}

hypergeometric functions, or a definite integral which does not seem to be tabulated in the most common literature. As an application of certain sums involving the digamma function, we calculated some reduction formulas for the parameter differentiation of the Mittag–Leffler function and the Wright function. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

11 pages, 319 KiB

Open AccessArticle

On Mathieu-Type Series with (p,ν)-Extended Hypergeometric Terms: Integral Representations and Upper Bounds

by Rakesh K. Parmar, Tibor K. Pogány and S. Saravanan

Mathematics 2023, 11(7), 1710; https://doi.org/10.3390/math11071710 - 03 Apr 2023

Cited by 1 | Viewed by 874

Abstract

Integral form expressions are obtained for the Mathieu-type series and for their associated alternating versions, the terms of which contain a

(p, ν)

-extended Gauss hypergeometric function. Contiguous recurrence relations are found for the Mathieu-type series with respect to two [...] Read more.

Integral form expressions are obtained for the Mathieu-type series and for their associated alternating versions, the terms of which contain a

(p, ν)

-extended Gauss hypergeometric function. Contiguous recurrence relations are found for the Mathieu-type series with respect to two parameters, and finally, particular cases and related bounding inequalities are established. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

20 pages, 380 KiB

Open AccessArticle

General Fractional Calculus in Multi-Dimensional Space: Riesz Form

by Vasily E. Tarasov

Mathematics 2023, 11(7), 1651; https://doi.org/10.3390/math11071651 - 29 Mar 2023

Cited by 10 | Viewed by 1391

Abstract

An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949. The proposed Riesz form of GFC can be considered as an extension GFC from the positive real [...] Read more.

An extension of the general fractional calculus (GFC) is proposed as a generalization of the Riesz fractional calculus, which was suggested by Marsel Riesz in 1949. The proposed Riesz form of GFC can be considered as an extension GFC from the positive real line and the Laplace convolution to the m-dimensional Euclidean space and the Fourier convolution. To formulate the general fractional calculus in the Riesz form, the Luchko approach to construction of the GFC, which was suggested by Yuri Luchko in 2021, is used. The general fractional integrals and derivatives are defined as convolution-type operators. In these definitions the Fourier convolution on m-dimensional Euclidean space is used instead of the Laplace convolution on positive semi-axis. Some properties of these general fractional operators are described. The general fractional analogs of first and second fundamental theorems of fractional calculus are proved. The fractional calculus of the Riesz potential and the fractional Laplacian of the Riesz form are special cases of proposed general fractional calculus of the Riesz form. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

19 pages, 387 KiB

Open AccessFeature PaperArticle

Inverse Sturm–Liouville Problem with Spectral Parameter in the Boundary Conditions

by Natalia P. Bondarenko and Egor E. Chitorkin

Mathematics 2023, 11(5), 1138; https://doi.org/10.3390/math11051138 - 24 Feb 2023

Cited by 7 | Viewed by 1087

Abstract

In this paper, for the first time, we study the inverse Sturm–Liouville problem with polynomials of the spectral parameter in the first boundary condition and with entire analytic functions in the second one. For the investigation of this new inverse problem, we develop [...] Read more.

In this paper, for the first time, we study the inverse Sturm–Liouville problem with polynomials of the spectral parameter in the first boundary condition and with entire analytic functions in the second one. For the investigation of this new inverse problem, we develop an approach based on the construction of a special vector functional sequence in a suitable Hilbert space. The uniqueness of recovering the potential and the polynomials of the boundary condition from a part of the spectrum is proved. Furthermore, our main results are applied to the Hochstadt–Lieberman-type problems with polynomial dependence on the spectral parameter not only in the boundary conditions but also in discontinuity (transmission) conditions inside the interval. We prove novel uniqueness theorems, which generalize and improve the previous results in this direction. Note that all the spectral problems in this paper are investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. Moreover, our method is constructive and can be developed in the future for numerical solution and for the study of solvability and stability of inverse spectral problems. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

13 pages, 283 KiB

Open AccessArticle

The General Fractional Integrals and Derivatives on a Finite Interval

by Mohammed Al-Refai and Yuri Luchko

Mathematics 2023, 11(4), 1031; https://doi.org/10.3390/math11041031 - 17 Feb 2023

Cited by 8 | Viewed by 1128

Abstract

The general fractional integrals and derivatives considered so far in the Fractional Calculus literature have been defined for the functions on the real positive semi-axis. The main contribution of this paper is in introducing the general fractional integrals and derivatives of the functions [...] Read more.

The general fractional integrals and derivatives considered so far in the Fractional Calculus literature have been defined for the functions on the real positive semi-axis. The main contribution of this paper is in introducing the general fractional integrals and derivatives of the functions on a finite interval. As in the case of the Riemann–Liouville fractional integrals and derivatives on a finite interval, we define both the left- and the right-sided operators and investigate their interconnections. The main results presented in the paper are the 1st and the 2nd fundamental theorems of Fractional Calculus formulated for the general fractional integrals and derivatives of the functions on a finite interval as well as the formulas for integration by parts that involve the general fractional integrals and derivatives. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

16 pages, 318 KiB

Open AccessArticle

Qualitative Properties of Solutions of Equations and Inequalities with KPZ-Type Nonlinearities

by Andrey B. Muravnik

Mathematics 2023, 11(4), 990; https://doi.org/10.3390/math11040990 - 15 Feb 2023

Cited by 3 | Viewed by 1003

Abstract

For quasilinear partial differential and integrodifferential equations and inequalities containing nonlinearities of the Kardar—Parisi—Zhang type, various (old and recent) results on qualitative properties of solutions (such as the stabilization of solutions, blow-up phenomena, long-time decay of solutions, and others) are presented. Descriptive examples [...] Read more.

For quasilinear partial differential and integrodifferential equations and inequalities containing nonlinearities of the Kardar—Parisi—Zhang type, various (old and recent) results on qualitative properties of solutions (such as the stabilization of solutions, blow-up phenomena, long-time decay of solutions, and others) are presented. Descriptive examples demonstrating the Bitsadze approach (the technique of monotone maps) applied in this research area are provided. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

9 pages, 289 KiB

Open AccessArticle

Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators

by Sergei M. Sitnik and Shakhobiddin T. Karimov

Mathematics 2023, 11(4), 951; https://doi.org/10.3390/math11040951 - 13 Feb 2023

Cited by 4 | Viewed by 990

Abstract

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are [...] Read more.

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

36 pages, 45866 KiB

Open AccessArticle

Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation

by Yi Ji and Yufeng Xing

Mathematics 2023, 11(3), 593; https://doi.org/10.3390/math11030593 - 23 Jan 2023

Cited by 5 | Viewed by 1519

Abstract

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear [...] Read more.

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear part and a nonlinear part. In the proposed methods of different orders, the responses of the linear part at the previous step are transferred by the generalized Padé approximations, and the nonlinear part’s responses of the previous step are approximated by the Gauss–Legendre quadrature together with the explicit Runge–Kutta method, where the explicit Runge–Kutta method is used to calculate function values at quadrature points. For reducing computations and rounding errors, the 2^m algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10⁻¹⁶ (computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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18 pages, 761 KiB

Open AccessArticle

Analytical Description of the Diffusion in a Cellular Automaton with the Margolus Neighbourhood in Terms of the Two-Dimensional Markov Chain

by Anton E. Kulagin and Alexander V. Shapovalov

Mathematics 2023, 11(3), 584; https://doi.org/10.3390/math11030584 - 22 Jan 2023

Cited by 1 | Viewed by 1219

Abstract

The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to [...] Read more.

The one-parameter two-dimensional cellular automaton with the Margolus neighbourhood is analyzed based on considering the projection of the stochastic movements of a single particle. Introducing the auxiliary random variable associated with the direction of the movement, we reduce the problem under consideration to the study of a two-dimensional Markov chain. The master equation for the probability distribution is derived and solved exactly using the probability-generating function method. The probability distribution is expressed analytically in terms of Jacobi polynomials. The moments of the obtained solution allowed us to derive the exact analytical formula for the parametric dependence of the diffusion coefficient in the two-dimensional cellular automaton with the Margolus neighbourhood. Our analytic results agree with earlier empirical results of other authors and refine them. The results are of interest for the modelling two-dimensional diffusion using cellular automata especially for the multicomponent problem. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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24 pages, 333 KiB

Open AccessArticle

Initial Problem for Two-Dimensional Hyperbolic Equation with a Nonlocal Term

by Vladimir Vasilyev and Natalya Zaitseva

Mathematics 2023, 11(1), 130; https://doi.org/10.3390/math11010130 - 27 Dec 2022

Cited by 4 | Viewed by 1009

Abstract

In this paper, we study the Cauchy problem in a strip for a two-dimensional hyperbolic equation containing the sum of a differential operator and a shift operator acting on a spatial variable that varies over the real axis. An operating scheme is used [...] Read more.

In this paper, we study the Cauchy problem in a strip for a two-dimensional hyperbolic equation containing the sum of a differential operator and a shift operator acting on a spatial variable that varies over the real axis. An operating scheme is used to construct the solutions of the equation. The solution of the problem is obtained in the form of a convolution of the function found using the operating scheme and the function from the initial conditions of the problem. It is proved that classical solutions of the considered initial problem exist if the real part of the symbol of the differential-difference operator in the equation is positive. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

12 pages, 319 KiB

Open AccessArticle

Degenerate Multi-Term Equations with Gerasimov–Caputo Derivatives in the Sectorial Case

by Vladimir E. Fedorov and Kseniya V. Boyko

Mathematics 2022, 10(24), 4699; https://doi.org/10.3390/math10244699 - 11 Dec 2022

Cited by 2 | Viewed by 601

Abstract

The unique solvability for the Cauchy problem in a class of degenerate multi-term linear equations with Gerasimov–Caputo derivatives in a Banach space is investigated. To this aim, we use the condition of sectoriality for the pair of operators at the oldest derivatives from [...] Read more.

The unique solvability for the Cauchy problem in a class of degenerate multi-term linear equations with Gerasimov–Caputo derivatives in a Banach space is investigated. To this aim, we use the condition of sectoriality for the pair of operators at the oldest derivatives from the equation and the general conditions of the other operators’ coordination with invariant subspaces, which exist due to the sectoriality. An abstract result is applied to the research of unique solvability issues for the systems of the dynamics and of the thermoconvection for some viscoelastic media. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

14 pages, 2662 KiB

Open AccessArticle

Several Types of q-Differential Equations of Higher Order and Properties of Their Solutions

by Cheon-Seoung Ryoo and Jung-Yoog Kang

Mathematics 2022, 10(23), 4469; https://doi.org/10.3390/math10234469 - 26 Nov 2022

Cited by 1 | Viewed by 630

Abstract

The purpose of this paper is to organize various types of higher order q-differential equations that are connected to q-sigmoid polynomials and obtain certain properties regarding their solutions. Using the properties of q-sigmoid polynomials, we show the symmetric properties of [...] Read more.

The purpose of this paper is to organize various types of higher order q-differential equations that are connected to q-sigmoid polynomials and obtain certain properties regarding their solutions. Using the properties of q-sigmoid polynomials, we show the symmetric properties of q-differential equations of higher order. Moreover, we derive special properties for the approximate roots of q-sigmoid polynomials which are solutions of higher order q-differential equations. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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12 pages, 381 KiB

Open AccessArticle

Recovery of Inhomogeneity from Output Boundary Data

by Vladislav V. Kravchenko, Kira V. Khmelnytskaya and Fatma Ayça Çetinkaya

Mathematics 2022, 10(22), 4349; https://doi.org/10.3390/math10224349 - 19 Nov 2022

Cited by 6 | Viewed by 1088

Abstract

We consider the Sturm–Liouville equation on a finite interval with a real-valued integrable potential and propose a method for solving the following general inverse problem. We recover the potential from a given set of the output boundary values of a solution satisfying some [...] Read more.

We consider the Sturm–Liouville equation on a finite interval with a real-valued integrable potential and propose a method for solving the following general inverse problem. We recover the potential from a given set of the output boundary values of a solution satisfying some known initial conditions for a set of values of the spectral parameter. Special cases of this problem include the recovery of the potential from the Weyl function, the inverse two-spectra Sturm–Liouville problem, as well as the recovery of the potential from the output boundary values of a plane wave that interacted with the potential. The method is based on the special Neumann series of Bessel functions representations for solutions of Sturm–Liouville equations. With their aid, the problem is reduced to the classical inverse Sturm–Liouville problem of recovering the potential from two spectra, which is solved again with the help of the same representations. The overall approach leads to an efficient numerical algorithm for solving the inverse problem. Its numerical efficiency is illustrated by several examples. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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26 pages, 438 KiB

Open AccessFeature PaperArticle

Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers

by Alexander Dyachenko and Dmitrii Karp

Mathematics 2022, 10(20), 3903; https://doi.org/10.3390/math10203903 - 20 Oct 2022

Cited by 1 | Viewed by 984

Abstract

Given real parameters

a, b, c

and integer shifts

n_{1}, n_{2}, m

, we consider the ratio [...] Read more.

Given real parameters

a, b, c

and integer shifts

n_{1}, n_{2}, m

, we consider the ratio

R (z) =_{2} F_{1} (a + n_{1}, b + n_{2}; c + m; z) /_{2} F_{1} (a, b; c; z)

of the Gauss hypergeometric functions. We find a formula for

Im R (x \pm i 0)

with

x > 1

in terms of real hypergeometric polynomial P, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for R when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

15 pages, 364 KiB

Open AccessArticle

Asymptotic Behavior of Three Connected Stochastic Delay Neoclassical Growth Systems Using Spectral Technique

by Ishtiaq Ali and Sami Ullah Khan

Mathematics 2022, 10(19), 3639; https://doi.org/10.3390/math10193639 - 05 Oct 2022

Cited by 10 | Viewed by 888

Abstract

In this study, we consider a nonlinear system of three connected delay differential neoclassical growth models along with stochastic effect and additive white noise, which is influenced by stochastic perturbation. We derived the conditions for positive equilibria, stability and positive solutions of the [...] Read more.

In this study, we consider a nonlinear system of three connected delay differential neoclassical growth models along with stochastic effect and additive white noise, which is influenced by stochastic perturbation. We derived the conditions for positive equilibria, stability and positive solutions of the stochastic system. It is observed that when a constant delay reaches a certain threshold for the steady state, the asymptotic stability is lost, and the Hopf bifurcation occurs. In the case of the finite domain, the three connected, delayed systems will not collapse to infinity but will be bounded ultimately. A Legendre spectral collocation method is used for the numerical simulations. Moreover, a comparison of a stochastic delayed system with a deterministic delayed system is also provided. Some numerical test problems are presented to illustrate the effectiveness of the theoretical results. Numerical results further illustrate the obtained stability regions and behavior of stable and unstable solutions of the proposed system. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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13 pages, 309 KiB

Open AccessArticle

Hermite-Hadamard-Type Integral Inequalities for Convex Functions and Their Applications

by Hari M. Srivastava, Sana Mehrez and Sergei M. Sitnik

Mathematics 2022, 10(17), 3127; https://doi.org/10.3390/math10173127 - 31 Aug 2022

Cited by 11 | Viewed by 1219

Abstract

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms [...] Read more.

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms of modules of certain powers of proper functions. Generalizations for convex functions are also considered. As applications, some new inequalities for the digamma function in terms of the trigamma function and some inequalities involving special means of real numbers are given. The results also include estimates via arithmetic, geometric and logarithmic means. The examples are derived in order to demonstrate that some of our results in this paper are more exact than the existing ones and some improve several known results available in the literature. The constants in the derived inequalities are calculated; some of these constants are sharp. As a visual example, graphs of some technically important functions are included in the text. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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11 pages, 293 KiB

Open AccessArticle

Bounds for Incomplete Confluent Fox–Wright Generalized Hypergeometric Functions

by Tibor K. Pogány

Mathematics 2022, 10(17), 3106; https://doi.org/10.3390/math10173106 - 29 Aug 2022

Cited by 2 | Viewed by 1117

Abstract

We establish several new functional bounds and uniform bounds (with respect to the variable) for the lower incomplete generalized Fox–Wright functions by means of the representation formulae for the McKay

I_{ν}

Bessel probability distribution’s cumulative distribution function. New cumulative distribution functions are [...] Read more.

We establish several new functional bounds and uniform bounds (with respect to the variable) for the lower incomplete generalized Fox–Wright functions by means of the representation formulae for the McKay

I_{ν}

Bessel probability distribution’s cumulative distribution function. New cumulative distribution functions are generated and expressed in terms of lower incomplete Fox–Wright functions and/or generalized hypergeometric functions, whilst in the closing part of the article, related bounding inequalities are obtained for them. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

13 pages, 273 KiB

Open AccessArticle

Fully Degenerating of Daehee Numbers and Polynomials

by Sahar Albosaily, Waseem Ahmad Khan, Serkan Araci and Azhar Iqbal

Mathematics 2022, 10(14), 2528; https://doi.org/10.3390/math10142528 - 20 Jul 2022

Cited by 2 | Viewed by 1168

Abstract

In this paper, we consider fully degenerate Daehee numbers and polynomials by using degenerate logarithm function. We investigate some properties of these numbers and polynomials. We also introduce higher-order multiple fully degenerate Daehee polynomials and numbers which can be represented in terms of [...] Read more.

In this paper, we consider fully degenerate Daehee numbers and polynomials by using degenerate logarithm function. We investigate some properties of these numbers and polynomials. We also introduce higher-order multiple fully degenerate Daehee polynomials and numbers which can be represented in terms of Riemann integrals on the interval

[0, 1]

. Finally, we derive their summation formulae. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

30 pages, 386 KiB

Open AccessArticle

On All Symmetric and Nonsymmetric Exceptional Orthogonal X₁-Polynomials Generated by a Specific Sturm–Liouville Problem

by Mohammad Masjed-Jamei, Zahra Moalemi and Nasser Saad

Mathematics 2022, 10(14), 2464; https://doi.org/10.3390/math10142464 - 15 Jul 2022

Cited by 1 | Viewed by 910

Abstract

Exceptional orthogonal X

_{1}

-polynomials of symmetric and nonsymmetric types can be considered as eigenfunctions of a Sturm–Liouville problem. In this paper, by defining a generic second-order differential equation, a unified classification of all these polynomials is presented, and 10 particular cases of [...] Read more.

Exceptional orthogonal X

_{1}

-polynomials of symmetric and nonsymmetric types can be considered as eigenfunctions of a Sturm–Liouville problem. In this paper, by defining a generic second-order differential equation, a unified classification of all these polynomials is presented, and 10 particular cases of it are then introduced and analyzed. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

14 pages, 306 KiB

Open AccessArticle

A New Parameter-Uniform Discretization of Semilinear Singularly Perturbed Problems

by Justin B. Munyakazi and Olawale O. Kehinde

Mathematics 2022, 10(13), 2254; https://doi.org/10.3390/math10132254 - 27 Jun 2022

Cited by 3 | Viewed by 1328

Abstract

In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of [...] Read more.

In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of linear singularly perturbed problems that emerges from the quasilinearization process. We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice that the method is first-order uniformly convergent. Some numerical evaluations are implemented on model examples to confirm the proposed theoretical results and to show the efficiency of the method. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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23 pages, 368 KiB

Open AccessArticle

Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

by Han Feng and Yan Ge

Mathematics 2022, 10(12), 2108; https://doi.org/10.3390/math10122108 - 17 Jun 2022

Cited by 2 | Viewed by 971

Abstract

In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on

R^{d}

. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the [...] Read more.

In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on

R^{d}

. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function

h_{κ} (x) : = \prod_{ν \in R +} {| ⟨x, ν⟩ |}^{κ_{ν}}, x \in R^{d}

on the unit sphere; the upper estimates of the Cesàro kernels and Cesàro means are obtained and used to prove the convergence of the Cesàro

(C, δ)

means in the weighted

L^{p}

space for

δ

above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

17 pages, 326 KiB

Open AccessArticle

Generalized q-Difference Equations for q-Hypergeometric Polynomials with Double q-Binomial Coefficients

by Jian Cao, Hari M. Srivastava, Hong-Li Zhou and Sama Arjika

Mathematics 2022, 10(4), 556; https://doi.org/10.3390/math10040556 - 11 Feb 2022

Cited by 11 | Viewed by 1139

Abstract

In this paper, we apply a general family of basic (or q-) polynomials with double q-binomial coefficients as well as some homogeneous q-operators in order to construct several q-difference equations involving seven variables. We derive the Rogers type and [...] Read more.

In this paper, we apply a general family of basic (or q-) polynomials with double q-binomial coefficients as well as some homogeneous q-operators in order to construct several q-difference equations involving seven variables. We derive the Rogers type and the extended Rogers type formulas as well as the Srivastava-Agarwal-type bilinear generating functions for the general q-polynomials, which generalize the generating functions for the Cigler polynomials. We also derive a class of mixed generating functions by means of the aforementioned q-difference equations. The various results, which we have derived in this paper, are new and sufficiently general in character. Moreover, the generating functions presented here are potentially applicable not only in the study of the general q-polynomials, which they have generated, but indeed also in finding solutions of the associated q-difference equations. Finally, we remark that it will be a rather trivial and inconsequential exercise to produce the so-called

(p, q)

-variations of the q-results, which we have investigated here, because the additional forced-in parameter p is obviously redundant. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

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