Editorial

4 pages, 172 KiB

Open AccessEditorial

Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of His 80th Birthday

by Sotiris K. Ntouyas

Axioms 2020, 9(4), 135; https://doi.org/10.3390/axioms9040135 - 18 Nov 2020

Viewed by 1441

Abstract

Differential and difference equations play an important role in many branches of mathematics [...] Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

Research

Jump to: Editorial

15 pages, 648 KiB

Open AccessArticle

On the Triple Lauricella–Horn–Karlsson q-Hypergeometric Functions

by Thomas Ernst

Axioms 2020, 9(3), 93; https://doi.org/10.3390/axioms9030093 - 31 Jul 2020

Cited by 3 | Viewed by 2120

Abstract

The Horn–Karlsson approach to find convergence regions is applied to find convergence regions for triple q-hypergeometric functions. It turns out that the convergence regions are significantly increased in the q-case; just as for q-Appell and q-Lauricella functions, additions are [...] Read more.

The Horn–Karlsson approach to find convergence regions is applied to find convergence regions for triple q-hypergeometric functions. It turns out that the convergence regions are significantly increased in the q-case; just as for q-Appell and q-Lauricella functions, additions are replaced by Ward q-additions. Mostly referring to Krishna Srivastava 1956, we give q-integral representations for these functions. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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18 pages, 2600 KiB

Open AccessEditor’s ChoiceArticle

The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem

by Konstantinos Kalimeris and Athanassios S. Fokas

Axioms 2020, 9(3), 89; https://doi.org/10.3390/axioms9030089 - 28 Jul 2020

Cited by 1 | Viewed by 2723

Abstract

Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on [...] Read more.

Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on each side of the hexagon. We show that if this function is odd, then this problem can be solved in closed form; numerical verification is also provided. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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17 pages, 403 KiB

Open AccessArticle

On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator

by Alexander Yeliseev

Axioms 2020, 9(3), 86; https://doi.org/10.3390/axioms9030086 - 23 Jul 2020

Cited by 8 | Viewed by 1584

Abstract

An asymptotic solution of the linear Cauchy problem in the presence of a “weak” turning point for the limit operator is constructed by the method of S. A. Lomov regularization. The main singularities of this problem are written out explicitly. Estimates are given [...] Read more.

An asymptotic solution of the linear Cauchy problem in the presence of a “weak” turning point for the limit operator is constructed by the method of S. A. Lomov regularization. The main singularities of this problem are written out explicitly. Estimates are given for

ε

that characterize the behavior of singularities for

ϵ \to 0

. The asymptotic convergence of a regularized series is proven. The results are illustrated by an example. Bibliography: six titles. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

28 pages, 464 KiB

Open AccessArticle

Eigenfunction Families and Solution Bounds for Multiplicatively Advanced Differential Equations

by David W. Pravica, Njinasoa Randriampiry and Michael J. Spurr

Axioms 2020, 9(3), 83; https://doi.org/10.3390/axioms9030083 - 21 Jul 2020

Cited by 3 | Viewed by 2267

Abstract

A family of Schwartz functions

W (t)

are interpreted as eigensolutions of MADEs in the sense that

W^{(δ)} (t) = E W (q^{γ} t)

where the eigenvalue

E \in R

is independent of [...] Read more.

A family of Schwartz functions

W (t)

are interpreted as eigensolutions of MADEs in the sense that

W^{(δ)} (t) = E W (q^{γ} t)

where the eigenvalue

E \in R

is independent of the advancing parameter

q > 1

. The parameters

δ, γ \in N

are characteristics of the MADE. Some issues, which are related to corresponding q-advanced PDEs, are also explored. In the limit that

q \to 1^{+}

we show convergence of MADE eigenfunctions to solutions of ODEs, which involve only simple exponentials and trigonometric functions. The limit eigenfunctions (

q = 1^{+}

) are not Schwartz, thus convergence is only uniform in

t \in R

on compact sets. An asymptotic analysis is provided for MADEs which indicates how to extend solutions in a neighborhood of the origin

t = 0

. Finally, an expanded table of Fourier transforms is provided that includes Schwartz solutions to MADEs. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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11 pages, 231 KiB

Open AccessArticle

On the Periodicity of General Class of Difference Equations

by Osama Moaaz, Hamida Mahjoub and Ali Muhib

Axioms 2020, 9(3), 75; https://doi.org/10.3390/axioms9030075 - 01 Jul 2020

Cited by 4 | Viewed by 1600

Abstract

In this paper, we are interested in studying the periodic behavior of solutions of nonlinear difference equations. We used a new method to find the necessary and sufficient conditions for the existence of periodic solutions. Through examples, we compare the results of this [...] Read more.

In this paper, we are interested in studying the periodic behavior of solutions of nonlinear difference equations. We used a new method to find the necessary and sufficient conditions for the existence of periodic solutions. Through examples, we compare the results of this method with the usual method. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

18 pages, 634 KiB

Open AccessArticle

Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations

by Ilya Boykov, Vladimir Roudnev and Alla Boykova

Axioms 2020, 9(3), 74; https://doi.org/10.3390/axioms9030074 - 01 Jul 2020

Cited by 8 | Viewed by 1962

Abstract

We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today, hypersingular integral equations of this type are widely used in physics [...] Read more.

We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today, hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approximate equations system in terms of the operator logarithmic norm. This makes it possible to estimate the norm of the inverse matrix for an approximating system. The advantage of the method for nonlinear equations is that neither the existence or reversibility of the nonlinear operator derivative is required. Examples are given illustrating the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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21 pages, 384 KiB

Open AccessArticle

Existence Results for Nonlocal Multi-Point and Multi-Term Fractional Order Boundary Value Problems

by Bashir Ahmad, Najla Alghamdi, Ahmed Alsaedi and Sotiris K. Ntouyas

Axioms 2020, 9(2), 70; https://doi.org/10.3390/axioms9020070 - 24 Jun 2020

Cited by 1 | Viewed by 2022

Abstract

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point and integral boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with [...] Read more.

In this paper, we discuss the existence and uniqueness of solutions for a new class of multi-point and integral boundary value problems of multi-term fractional differential equations by using standard fixed point theorems. We also demonstrate the application of the obtained results with the aid of examples. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

23 pages, 840 KiB

Open AccessArticle

Generalized Nabla Differentiability and Integrability for Fuzzy Functions on Time Scales

by R. Leelavathi, G. Suresh Kumar, Ravi P. Agarwal, Chao Wang and M.S.N. Murty

Axioms 2020, 9(2), 65; https://doi.org/10.3390/axioms9020065 - 08 Jun 2020

Cited by 3 | Viewed by 2179

Abstract

This paper mainly deals with introducing and studying the properties of generalized nabla differentiability for fuzzy functions on time scales via Hukuhara difference. Further, we obtain embedding results on

E_{n}

for generalized nabla differentiable fuzzy functions. Finally, we prove a fundamental theorem [...] Read more.

This paper mainly deals with introducing and studying the properties of generalized nabla differentiability for fuzzy functions on time scales via Hukuhara difference. Further, we obtain embedding results on

E_{n}

for generalized nabla differentiable fuzzy functions. Finally, we prove a fundamental theorem of a nabla integral calculus for fuzzy functions on time scales under generalized nabla differentiability. The obtained results are illustrated with suitable examples. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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16 pages, 819 KiB

Open AccessArticle

Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals

by Ahmed Alsaedi, Abrar Broom, Sotiris K. Ntouyas and Bashir Ahmad

Axioms 2020, 9(2), 50; https://doi.org/10.3390/axioms9020050 - 01 May 2020

Cited by 4 | Viewed by 2042

Abstract

In this paper, we study the existence of solutions for nonlocal single and multi-valued boundary value problems involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals. The existence of solutions for the single-valued case relies on Sadovskii’s fixed [...] Read more.

In this paper, we study the existence of solutions for nonlocal single and multi-valued boundary value problems involving right-Caputo and left-Riemann–Liouville fractional derivatives of different orders and right-left Riemann–Liouville fractional integrals. The existence of solutions for the single-valued case relies on Sadovskii’s fixed point theorem. The first existence results for the multi-valued case are proved by applying Bohnenblust-Karlin’s fixed point theorem, while the second one is based on Martelli’s fixed point theorem. We also demonstrate the applications of the obtained results. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

12 pages, 279 KiB

Open AccessArticle

Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time

by Davor Dragičević and Ciprian Preda

Axioms 2020, 9(2), 47; https://doi.org/10.3390/axioms9020047 - 27 Apr 2020

Cited by 5 | Viewed by 2035

Abstract

For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion [...] Read more.

For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

21 pages, 290 KiB

Open AccessArticle

Nonlocal Inverse Problem for a Pseudohyperbolic- Pseudoelliptic Type Integro-Differential Equations

by Tursun K. Yuldashev

Axioms 2020, 9(2), 45; https://doi.org/10.3390/axioms9020045 - 27 Apr 2020

Cited by 29 | Viewed by 2997

Abstract

The questions of solvability of a nonlocal inverse boundary value problem for a mixed pseudohyperbolic-pseudoelliptic integro-differential equation with spectral parameters are considered. Using the method of the Fourier series, a system of countable systems of ordinary integro-differential equations is obtained. To determine arbitrary [...] Read more.

The questions of solvability of a nonlocal inverse boundary value problem for a mixed pseudohyperbolic-pseudoelliptic integro-differential equation with spectral parameters are considered. Using the method of the Fourier series, a system of countable systems of ordinary integro-differential equations is obtained. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system regular and irregular values of the spectral parameters were calculated. The unique solvability of the inverse boundary value problem for regular values of spectral parameters is proved. For irregular values of spectral parameters is established a criterion of existence of an infinite set of solutions of the inverse boundary value problem. The results are formulated as a theorem. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

13 pages, 318 KiB

Open AccessArticle

Generalized Briot-Bouquet Differential Equation Based on New Differential Operator with Complex Connections

by Rabha W. Ibrahim, Rafida M. Elobaid and Suzan J. Obaiys

Axioms 2020, 9(2), 42; https://doi.org/10.3390/axioms9020042 - 21 Apr 2020

Cited by 8 | Viewed by 2217

Abstract

A class of Briot–Bouquet differential equations is a magnificent part of investigating the geometric behaviors of analytic functions, using the subordination and superordination concepts. In this work, we aim to formulate a new differential operator with complex connections (coefficients) in the open unit [...] Read more.

A class of Briot–Bouquet differential equations is a magnificent part of investigating the geometric behaviors of analytic functions, using the subordination and superordination concepts. In this work, we aim to formulate a new differential operator with complex connections (coefficients) in the open unit disk and generalize a class of Briot–Bouquet differential equations (BBDEs). We study and generalize new classes of analytic functions based on the new differential operator. Consequently, we define a linear operator with applications. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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7 pages, 237 KiB

Open AccessArticle

Coincidence Continuation Theory for Multivalued Maps with Selections in a Given Class

by Donal O’Regan

Axioms 2020, 9(2), 37; https://doi.org/10.3390/axioms9020037 - 10 Apr 2020

Cited by 1 | Viewed by 1485

Abstract

This paper considers the topological transversality theorem for general multivalued maps which have selections in a given class of maps. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

10 pages, 252 KiB

Open AccessArticle

On a Harmonic Univalent Subclass of Functions Involving a Generalized Linear Operator

by Abdeljabbar Talal Yousef and Zabidin Salleh

Axioms 2020, 9(1), 32; https://doi.org/10.3390/axioms9010032 - 24 Mar 2020

Cited by 8 | Viewed by 2456

Abstract

In this paper, a subclass of complex-valued harmonic univalent functions defined by a generalized linear operator is introduced. Some interesting results such as coefficient bounds, compactness, and other properties of this class are obtained. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

14 pages, 4698 KiB

Open AccessArticle

Dynamics of HIV-TB Co-Infection Model

by Nita H Shah, Nisha Sheoran and Yash Shah

Axioms 2020, 9(1), 29; https://doi.org/10.3390/axioms9010029 - 11 Mar 2020

Cited by 10 | Viewed by 3588

Abstract

According to World Health Organization (WHO), the population suffering from human immunodeficiency virus (HIV) infection over a period of time may suffer from TB infection which increases the death rate. There is no cure for acquired immunodeficiency syndrome (AIDS) to date but antiretrovirals [...] Read more.

According to World Health Organization (WHO), the population suffering from human immunodeficiency virus (HIV) infection over a period of time may suffer from TB infection which increases the death rate. There is no cure for acquired immunodeficiency syndrome (AIDS) to date but antiretrovirals (ARVs) can slow down the progression of disease as well as prevent secondary infections or complications. This is considered as a medication in this paper. This scenario of HIV-TB co-infection is modeled using a system of non-linear differential equations. This model considers HIV-infected individual as the initial stage. Four equilibrium points are found. Reproduction number R₀ is calculated. If R₀ >1 disease persists uniformly, with reference to the reproduction number, backward bifurcation is computed for pre-AIDS (latent) stage. Global stability is established for the equilibrium points where there is no Pre-AIDS TB class, point without co-infection and for the endemic point. Numerical simulation is carried out to validate the data. Sensitivity analysis is carried out to determine the importance of model parameters in the disease dynamics. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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16 pages, 387 KiB

Open AccessArticle

Stability of Equilibria of Rumor Spreading Model under Stochastic Perturbations

by Leonid Shaikhet

Axioms 2020, 9(1), 24; https://doi.org/10.3390/axioms9010024 - 18 Feb 2020

Cited by 3 | Viewed by 2391

Abstract

The known mathematical model of rumor spreading, which is described by a system of four nonlinear differential equations and is very popular in research, is considered. It is supposed that the considered model is influenced by stochastic perturbations that are of the type [...] Read more.

The known mathematical model of rumor spreading, which is described by a system of four nonlinear differential equations and is very popular in research, is considered. It is supposed that the considered model is influenced by stochastic perturbations that are of the type of white noise and are proportional to the deviation of the system state from its equilibrium point. Sufficient conditions of stability in probability for each from the five equilibria of the considered model are obtained by virtue of the Routh–Hurwitz criterion and the method of linear matrix inequalities (LMIs). The obtained results are illustrated by numerical analysis of appropriate LMIs and numerical simulations of solutions of the considered system of stochastic differential equations. The research method can also be used in other applications for similar nonlinear models with the order of nonlinearity higher than one. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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11 pages, 774 KiB

Open AccessArticle

Oscillation Results for Higher Order Differential Equations

by Choonkil Park, Osama Moaaz and Omar Bazighifan

Axioms 2020, 9(1), 14; https://doi.org/10.3390/axioms9010014 - 03 Feb 2020

Cited by 32 | Viewed by 2548

Abstract

The objective of our research was to study asymptotic properties of the class of higher order differential equations with a p-Laplacian-like operator. Our results supplement and improve some known results obtained in the literature. An illustrative example is provided. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

11 pages, 782 KiB

Open AccessArticle

Harmonic Starlike Functions with Respect to Symmetric Points

by Nak Eun Cho and Jacek Dziok

Axioms 2020, 9(1), 3; https://doi.org/10.3390/axioms9010003 - 22 Dec 2019

Cited by 4 | Viewed by 2290

Abstract

In the paper we define classes of harmonic starlike functions with respect to symmetric points and obtain some analytic conditions for these classes of functions. Some results connected to subordination properties, coefficient estimates, integral representation, and distortion theorems are also obtained. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

9 pages, 236 KiB

Open AccessArticle

Solutions of the Generalized Abel’s Integral Equations of the Second Kind with Variable Coefficients

by Chenkuan Li and Hunter Plowman

Axioms 2019, 8(4), 137; https://doi.org/10.3390/axioms8040137 - 05 Dec 2019

Cited by 12 | Viewed by 2464

Abstract

Applying Babenko’s approach, we construct solutions for the generalized Abel’s integral equations of the second kind with variable coefficients on R and

R^{n}

, and show their convergence and stability in the spaces of Lebesgue integrable functions, with several illustrative examples. [...] Read more.

Applying Babenko’s approach, we construct solutions for the generalized Abel’s integral equations of the second kind with variable coefficients on R and

R^{n}

, and show their convergence and stability in the spaces of Lebesgue integrable functions, with several illustrative examples. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

11 pages, 258 KiB

Open AccessArticle

General Linear Recurrence Sequences and Their Convolution Formulas

by Paolo Emilio Ricci and Pierpaolo Natalini

Axioms 2019, 8(4), 132; https://doi.org/10.3390/axioms8040132 - 19 Nov 2019

Cited by 3 | Viewed by 2095

Abstract

We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to find convolution formulas for second order linear recurrence polynomials generated by

{(\frac{1}{1 + a t + b t^{2}})}^{x} .

The case of generating functions containing [...] Read more.

We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to find convolution formulas for second order linear recurrence polynomials generated by

{(\frac{1}{1 + a t + b t^{2}})}^{x} .

The case of generating functions containing parameters, even in the numerator is considered. Convolution formulas and general recurrence relations are derived. Many illustrative examples and a straightforward extension to the case of matrix polynomials are shown. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

21 pages, 1835 KiB

Open AccessArticle

Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches

by Rasool Shah, Hassan Khan and Dumitru Baleanu

Axioms 2019, 8(4), 125; https://doi.org/10.3390/axioms8040125 - 07 Nov 2019

Cited by 33 | Viewed by 3286

Abstract

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are [...] Read more.

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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7 pages, 716 KiB

Open AccessArticle

On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order

by Temirkhan Aleroev

Axioms 2019, 8(4), 117; https://doi.org/10.3390/axioms8040117 - 18 Oct 2019

Cited by 4 | Viewed by 2301

Abstract

The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, [...] Read more.

The present paper is devoted to the spectral analysis of operators induced by fractional differential equations and boundary conditions of Sturm-Liouville type. It should be noted that these operators are non-self-adjoint. The spectral structure of such operators has been insufficiently explored. In particular, a study of the completeness of systems of eigenfunctions and associated functions has begun relatively recently. In this paper, the completeness of the system of eigenfunctions and associated functions of one class of non-self-adjoint integral operators corresponding boundary value problems for fractional differential equations is established. The proof is based on the well-known Theorem of M.S. Livshits on the spectral decomposition of linear non-self-adjoint operators, as well as on the sectoriality of the fractional differentiation operator. The results of Dzhrbashian-Nersesian on the asymptotics of the zeros of the Mittag-Leffler function are used. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

16 pages, 272 KiB

Open AccessArticle

Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

by Irem Kucukoglu, Burcin Simsek and Yilmaz Simsek

Axioms 2019, 8(4), 112; https://doi.org/10.3390/axioms8040112 - 11 Oct 2019

Cited by 18 | Viewed by 2849

Abstract

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new [...] Read more.

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. Full article

(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)

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