Integral Transforms and Operational Calculus

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 January 2019) | Viewed by 103142

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Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Interests: real and complex analysis; fractional calculus and its applications; integral equations and transforms; higher transcendental functions and their applications; q-series and q-polynomials; analytic number theory; analytic and geometric Inequalities; probability and statistics; inventory modeling and optimization
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Special Issue Information

Dear Colleagues,

Investigations involving the theory and applications of integral transforms and operational calculus are remarkably wide-spread in many diverse areas of the mathematical, physical, chemical, engineering and statistical sciences. In this Special Issue, we invite and welcome review, expository and original research articles dealing with the recent advances on the topics of integral transforms and operational calculus as well as their multidisciplinary applications.

We look forward to your contributions to this Special Issue.

Kind regards,

Prof. Dr. H. M. Srivastava
Guest Editor

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Keywords

  • Integral Transforms and Integral as well as Other Related Operators
  • Applications Involving Mathematical (or Higher Transcendental) Functions
  • Applications Involving Fractional-Order Differential and Differintegral Equations
  • Applications Involving q-Series and q-Polynomials
  • Applications Involving Analytic Number Theory
  • Applications Involving Special Functions of Mathematical Physics and Applied Mathematics
  • Applications Involving Geometric Function Theory of Complex Analysis

Published Papers (36 papers)

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Research

14 pages, 334 KiB  
Article
Some Symmetric Identities for the Multiple (p, q)-Hurwitz-Euler eta Function
by Kyung-Won Hwang and Cheon Seoung Ryoo
Symmetry 2019, 11(5), 645; https://doi.org/10.3390/sym11050645 - 8 May 2019
Cited by 5 | Viewed by 2295
Abstract
The main purpose of this paper is to find some interesting symmetric identities for the ( p , q ) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple ( p , q ) -Hurwitz-Euler eta function by generalizing the [...] Read more.
The main purpose of this paper is to find some interesting symmetric identities for the ( p , q ) -Hurwitz-Euler eta function in a complex field. Firstly, we define the multiple ( p , q ) -Hurwitz-Euler eta function by generalizing the Carlitz’s form ( p , q ) -Euler numbers and polynomials. We find some formulas and properties involved in Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order. We find new symmetric identities for multiple ( p , q ) -Hurwitz-Euler eta functions. We also obtain symmetric identities for Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order by using symmetry about multiple ( p , q ) -Hurwitz-Euler eta functions. Finally, we study the distribution and symmetric properties of the zero of Carlitz’s form ( p , q ) -Euler numbers and polynomials with higher order. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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20 pages, 314 KiB  
Article
A Note on the Truncated-Exponential Based Apostol-Type Polynomials
by H. M. Srivastava, Serkan Araci, Waseem A. Khan and Mehmet Acikgöz
Symmetry 2019, 11(4), 538; https://doi.org/10.3390/sym11040538 - 15 Apr 2019
Cited by 18 | Viewed by 3343
Abstract
In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric [...] Read more.
In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
16 pages, 276 KiB  
Article
A Certain Family of Integral Operators Associated with the Struve Functions
by Shahid Mahmood, H.M. Srivastava, Sarfraz Nawaz Malik, Mohsan Raza, Neelam Shahzadi and Saira Zainab
Symmetry 2019, 11(4), 463; https://doi.org/10.3390/sym11040463 - 2 Apr 2019
Cited by 4 | Viewed by 2350
Abstract
This article presents the study of Struve functions and certain integral operators associated with the Struve functions. It contains the investigation of certain geometric properties like the strong starlikeness and strong convexity of the Struve functions. It also includes the criteria of univalence [...] Read more.
This article presents the study of Struve functions and certain integral operators associated with the Struve functions. It contains the investigation of certain geometric properties like the strong starlikeness and strong convexity of the Struve functions. It also includes the criteria of univalence for a family of certain integral operators associated with the generalized Struve functions. The starlikeness and uniform convexity of the said integral operators are also part of this research. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
19 pages, 5680 KiB  
Article
Exploiting the Symmetry of Integral Transforms for Featuring Anuran Calls
by Amalia Luque, Jesús Gómez-Bellido, Alejandro Carrasco and Julio Barbancho
Symmetry 2019, 11(3), 405; https://doi.org/10.3390/sym11030405 - 20 Mar 2019
Cited by 3 | Viewed by 2747
Abstract
The application of machine learning techniques to sound signals requires the previous characterization of said signals. In many cases, their description is made using cepstral coefficients that represent the sound spectra. In this paper, the performance in obtaining cepstral coefficients by two integral [...] Read more.
The application of machine learning techniques to sound signals requires the previous characterization of said signals. In many cases, their description is made using cepstral coefficients that represent the sound spectra. In this paper, the performance in obtaining cepstral coefficients by two integral transforms, Discrete Fourier Transform (DFT) and Discrete Cosine Transform (DCT), are compared in the context of processing anuran calls. Due to the symmetry of sound spectra, it is shown that DCT clearly outperforms DFT, and decreases the error representing the spectrum by more than 30%. Additionally, it is demonstrated that DCT-based cepstral coefficients are less correlated than their DFT-based counterparts, which leads to a significant advantage for DCT-based cepstral coefficients if these features are later used in classification algorithms. Since the DCT superiority is based on the symmetry of sound spectra and not on any intrinsic advantage of the algorithm, the conclusions of this research can definitely be extrapolated to include any sound signal. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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18 pages, 841 KiB  
Article
Fuzzy Volterra Integro-Differential Equations Using General Linear Method
by Zanariah Abdul Majid, Faranak Rabiei, Fatin Abd Hamid and Fudziah Ismail
Symmetry 2019, 11(3), 381; https://doi.org/10.3390/sym11030381 - 15 Mar 2019
Cited by 5 | Viewed by 2818
Abstract
In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The [...] Read more.
In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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13 pages, 258 KiB  
Article
Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions
by Shahid Mahmood, Hari M. Srivastava, Nazar Khan, Qazi Zahoor Ahmad, Bilal Khan and Irfan Ali
Symmetry 2019, 11(3), 347; https://doi.org/10.3390/sym11030347 - 7 Mar 2019
Cited by 85 | Viewed by 3093
Abstract
The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences [...] Read more.
The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
14 pages, 300 KiB  
Article
Fractional Telegraph Equation and Its Solution by Natural Transform Decomposition Method
by Hassan Eltayeb, Yahya T. Abdalla, Imed Bachar and Mohamed H. Khabir
Symmetry 2019, 11(3), 334; https://doi.org/10.3390/sym11030334 - 6 Mar 2019
Cited by 48 | Viewed by 3746
Abstract
In this work, the natural transform decomposition method (NTDM) is applied to solve the linear and nonlinear fractional telegraph equations. This method is a combined form of the natural transform and the Adomian decomposition methods. In addition, we prove the convergence of our [...] Read more.
In this work, the natural transform decomposition method (NTDM) is applied to solve the linear and nonlinear fractional telegraph equations. This method is a combined form of the natural transform and the Adomian decomposition methods. In addition, we prove the convergence of our method. Finally, three examples have been employed to illustrate the preciseness and effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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16 pages, 347 KiB  
Article
Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications
by Asifa Tassaddiq
Symmetry 2019, 11(3), 311; https://doi.org/10.3390/sym11030311 - 1 Mar 2019
Cited by 6 | Viewed by 2130
Abstract
In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but [...] Read more.
In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter s ; nevertheless, using our new difference equations, we can evaluate these functions for the large values of s . It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
14 pages, 273 KiB  
Article
Some General Classes of q-Starlike Functions Associated with the Janowski Functions
by Hari M. Srivastava, Muhammad Tahir, Bilal Khan, Qazi Zahoor Ahmad and Nazar Khan
Symmetry 2019, 11(2), 292; https://doi.org/10.3390/sym11020292 - 23 Feb 2019
Cited by 102 | Viewed by 3860
Abstract
By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we [...] Read more.
By making use of the concept of basic (or q-) calculus, various families of q-extensions of starlike functions of order α in the open unit disk U were introduced and studied from many different viewpoints and perspectives. In this paper, we first investigate the relationship between various known classes of q-starlike functions that are associated with the Janowski functions. We then introduce and study a new subclass of q-starlike functions that involves the Janowski functions. We also derive several properties of such families of q-starlike functions with negative coefficients including (for example) distortion theorems. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
14 pages, 273 KiB  
Article
Some Subclasses of Uniformly Univalent Functions with Respect to Symmetric Points
by Shahid Mahmood, Hari M. Srivastava and Sarfraz Nawaz Malik
Symmetry 2019, 11(2), 287; https://doi.org/10.3390/sym11020287 - 22 Feb 2019
Cited by 8 | Viewed by 2392
Abstract
This article presents the study of certain analytic functions defined by bounded radius rotations associated with conic domain. Many geometric properties like coefficient estimate, radii problems, arc length, integral representation, inclusion results and growth rate of coefficients of Taylor’s series representation are investigated. [...] Read more.
This article presents the study of certain analytic functions defined by bounded radius rotations associated with conic domain. Many geometric properties like coefficient estimate, radii problems, arc length, integral representation, inclusion results and growth rate of coefficients of Taylor’s series representation are investigated. By varying the parameters in results, several well-known results in literature are obtained as special cases. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
18 pages, 339 KiB  
Article
Existence Theory for Nonlinear Third-Order Ordinary Differential Equations with Nonlocal Multi-Point and Multi-Strip Boundary Conditions
by Ahmed Alsaedi, Mona Alsulami, Hari M. Srivastava, Bashir Ahmad and Sotiris K. Ntouyas
Symmetry 2019, 11(2), 281; https://doi.org/10.3390/sym11020281 - 22 Feb 2019
Cited by 10 | Viewed by 2632
Abstract
We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely [...] Read more.
We investigate the solvability and Ulam stability for a nonlocal nonlinear third-order integro-multi-point boundary value problem on an arbitrary domain. The nonlinearity in the third-order ordinary differential equation involves the unknown function together with its first- and second-order derivatives. Our main results rely on the modern tools of functional analysis and are well illustrated with the aid of examples. An analogue problem involving non-separated integro-multi-point boundary conditions is also discussed. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
8 pages, 218 KiB  
Article
Geometric Properties of Certain Analytic Functions Associated with the Dziok-Srivastava Operator
by Cai-Mei Yan and Jin-Lin Liu
Symmetry 2019, 11(2), 259; https://doi.org/10.3390/sym11020259 - 19 Feb 2019
Cited by 2 | Viewed by 1615
Abstract
The objective of the present paper is to derive certain geometric properties of analytic functions associated with the Dziok–Srivastava operator. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
8 pages, 273 KiB  
Article
Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )
by Jagdish Narayan Pandey, Jay Singh Maurya, Santosh Kumar Upadhyay and Hari Mohan Srivastava
Symmetry 2019, 11(2), 235; https://doi.org/10.3390/sym11020235 - 15 Feb 2019
Cited by 17 | Viewed by 2741
Abstract
In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f S ( R n ) with wavelet kernel ψ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the [...] Read more.
In this paper, we define a continuous wavelet transform of a Schwartz tempered distribution f S ( R n ) with wavelet kernel ψ S ( R n ) and derive the corresponding wavelet inversion formula interpreting convergence in the weak topology of S ( R n ) . It turns out that the wavelet transform of a constant distribution is zero and our wavelet inversion formula is not true for constant distribution, but it is true for a non-constant distribution which is not equal to the sum of a non-constant distribution with a non-zero constant distribution. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
11 pages, 268 KiB  
Article
A Dunkl–Type Generalization of Szász–Kantorovich Operators via Post–Quantum Calculus
by Md. Nasiruzzaman, Aiman Mukheimer and M. Mursaleen
Symmetry 2019, 11(2), 232; https://doi.org/10.3390/sym11020232 - 15 Feb 2019
Cited by 13 | Viewed by 2201
Abstract
In this paper, we define the ( p , q ) -variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of [...] Read more.
In this paper, we define the ( p , q ) -variant of Szász–Kantorovich operators via Dunkl-type generalization generated by an exponential function and study the Korovkin-type results. We also obtain the convergence of our operators in weighted space by the modulus of continuity, Lipschitz class, and Peetre’s K-functionals. The extra parameter p provides more flexibility in approximation and plays an important role in symmetrizing these newly-defined operators. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
17 pages, 315 KiB  
Article
Starlike Functions Related to the Bell Numbers
by Nak Eun Cho, Sushil Kumar, Virendra Kumar, V. Ravichandran and H. M. Srivastava
Symmetry 2019, 11(2), 219; https://doi.org/10.3390/sym11020219 - 13 Feb 2019
Cited by 44 | Viewed by 3070
Abstract
The present paper aims to establish the first order differential subordination relations between functions with a positive real part and starlike functions related to the Bell numbers. In addition, several sharp radii estimates for functions in the class of starlike functions associated with [...] Read more.
The present paper aims to establish the first order differential subordination relations between functions with a positive real part and starlike functions related to the Bell numbers. In addition, several sharp radii estimates for functions in the class of starlike functions associated with the Bell numbers are determined. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
11 pages, 774 KiB  
Article
On Meromorphic Functions Defined by a New Operator Containing the Mittag–Leffler Function
by Suhila Elhaddad and Maslina Darus
Symmetry 2019, 11(2), 210; https://doi.org/10.3390/sym11020210 - 12 Feb 2019
Cited by 16 | Viewed by 2965
Abstract
This study defines a new linear differential operator via the Hadamard product between a q-hypergeometric function and Mittag–Leffler function. The application of the linear differential operator generates a new subclass of meromorphic function. Additionally, the study explores various properties and features, such [...] Read more.
This study defines a new linear differential operator via the Hadamard product between a q-hypergeometric function and Mittag–Leffler function. The application of the linear differential operator generates a new subclass of meromorphic function. Additionally, the study explores various properties and features, such as convex properties, distortion, growth, coefficient inequality and radii of starlikeness. Finally, the work discusses closure theorems and extreme points. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
17 pages, 551 KiB  
Article
Hermite-Type Collocation Methods to Solve Volterra Integral Equations with Highly Oscillatory Bessel Kernels
by Chunhua Fang, Guo He and Shuhuang Xiang
Symmetry 2019, 11(2), 168; https://doi.org/10.3390/sym11020168 - 1 Feb 2019
Cited by 12 | Viewed by 2857
Abstract
In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The [...] Read more.
In this paper, we present two kinds of Hermite-type collocation methods for linear Volterra integral equations of the second kind with highly oscillatory Bessel kernels. One method is direct Hermite collocation method, which used direct two-points Hermite interpolation in the whole interval. The other one is piecewise Hermite collocation method, which used a two-points Hermite interpolation in each subinterval. These two methods can calculate the approximate value of function value and derivative value simultaneously. Both methods are constructed easily and implemented well by the fast computation of highly oscillatory integrals involving Bessel functions. Under some conditions, the asymptotic convergence order with respect to oscillatory factor of these two methods are established, which are higher than the existing results. Some numerical experiments are included to show efficiency of these two methods. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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19 pages, 558 KiB  
Article
Certain Results of q -Sheffer–Appell Polynomials
by Ghazala Yasmin, Abdulghani Muhyi and Serkan Araci
Symmetry 2019, 11(2), 159; https://doi.org/10.3390/sym11020159 - 1 Feb 2019
Cited by 12 | Viewed by 2973
Abstract
In this paper, the class of q -Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of q -Sheffer–Appell polynomials are investigated and some properties of these members are derived. [...] Read more.
In this paper, the class of q -Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of q -Sheffer–Appell polynomials are investigated and some properties of these members are derived. In addition, the class of 2D q -Sheffer–Appell polynomials is introduced. Further, the graphs of some members of q -Sheffer–Appell polynomials and 2D q -Sheffer–Appell polynomials are plotted for different values of indices by using Matlab. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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9 pages, 224 KiB  
Article
The Order of Strongly Starlikeness of the Generalized α-Convex Functions
by Yuan Yuan, Rekha Srivastava and Jin-Lin Liu
Symmetry 2019, 11(1), 76; https://doi.org/10.3390/sym11010076 - 11 Jan 2019
Cited by 1 | Viewed by 2369
Abstract
We consider the order of the strongly-starlikeness of the generalized α -convex functions. Some sufficient conditions for functions to be p-valently strongly-starlike are given. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
10 pages, 243 KiB  
Article
On a Generalization of the Initial-Boundary Problem for the Vibrating String Equation
by Djumaklich Amanov, Gafurjan Ibragimov and Adem Kılıçman
Symmetry 2019, 11(1), 73; https://doi.org/10.3390/sym11010073 - 10 Jan 2019
Cited by 4 | Viewed by 2383
Abstract
In the present paper, we study a generalization of the initial-boundary problem for the inhomogeneous vibrating string equation. The initial conditions include the higher order derivatives of the unknown function. The problem is studied under homogeneous boundary conditions of the first kind. The [...] Read more.
In the present paper, we study a generalization of the initial-boundary problem for the inhomogeneous vibrating string equation. The initial conditions include the higher order derivatives of the unknown function. The problem is studied under homogeneous boundary conditions of the first kind. The uniqueness and existence of a regular solution of the problem are proved. To prove the main result we use the spectral decomposition method. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
13 pages, 262 KiB  
Article
Geometric Properties of Normalized Mittag–Leffler Functions
by Saddaf Noreen, Mohsan Raza, Jin-Lin Liu and Muhammad Arif
Symmetry 2019, 11(1), 45; https://doi.org/10.3390/sym11010045 - 3 Jan 2019
Cited by 15 | Viewed by 2741
Abstract
The aim of this paper is to investigate certain properties such as convexity of order μ , close-to-convexity of order 1 + μ /2 and starlikeness of normalized Mittag–Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity [...] Read more.
The aim of this paper is to investigate certain properties such as convexity of order μ , close-to-convexity of order 1 + μ /2 and starlikeness of normalized Mittag–Leffler function. We use some inequalities to prove our results. We also discuss the close-to-convexity of Mittag–Leffler functions with respect to certain starlike functions. Furthermore, we find the conditions for the above-mentioned function to belong to the Hardy space H p . Some of our results improve the results in the literature. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
12 pages, 289 KiB  
Article
Some Generating Functions for q-Polynomials
by Howard S. Cohl, Roberto S. Costas-Santos and Tanay V. Wakhare
Symmetry 2018, 10(12), 758; https://doi.org/10.3390/sym10120758 - 16 Dec 2018
Cited by 1 | Viewed by 3028
Abstract
Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of [...] Read more.
Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of the basic hypergeometric series 4 ϕ 5 , 5 ϕ 5 , 4 ϕ 3 , 3 ϕ 2 , 2 ϕ 1 , and q-Pochhammer symbols. Starting with our q-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
20 pages, 362 KiB  
Article
A New Representation for Srivastava’s λ-Generalized Hurwitz-Lerch Zeta Functions
by Asifa Tassaddiq
Symmetry 2018, 10(12), 733; https://doi.org/10.3390/sym10120733 - 8 Dec 2018
Cited by 12 | Viewed by 2043
Abstract
Taking inspiration principally from some of the latest research, we develop a new series representation for the λ-generalized Hurwitz-Lerch zeta functions. This representation led to important new results. The Fourier transform played a foundational role in this work. The duality property of [...] Read more.
Taking inspiration principally from some of the latest research, we develop a new series representation for the λ-generalized Hurwitz-Lerch zeta functions. This representation led to important new results. The Fourier transform played a foundational role in this work. The duality property of the Fourier transform became significant for checking the consistency of the results. Some known data has been verified as special cases of the results obtained in this investigation. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
15 pages, 324 KiB  
Article
On Some Statistical Approximation by (p,q)-Bleimann, Butzer and Hahn Operators
by Khursheed J. Ansari, Ishfaq Ahmad, M. Mursaleen and Iqtadar Hussain
Symmetry 2018, 10(12), 731; https://doi.org/10.3390/sym10120731 - 7 Dec 2018
Cited by 12 | Viewed by 2403
Abstract
In this article, we propose a different generalization of ( p , q ) -BBH operators and carry statistical approximation properties of the introduced operators towards a function which has to be approximated where ( p , q ) -integers contains symmetric property. [...] Read more.
In this article, we propose a different generalization of ( p , q ) -BBH operators and carry statistical approximation properties of the introduced operators towards a function which has to be approximated where ( p , q ) -integers contains symmetric property. We establish a Korovkin approximation theorem in the statistical sense and obtain the statistical rates of convergence. Furthermore, we also introduce a bivariate extension of proposed operators and carry many statistical approximation results. The extra parameter p plays an important role to symmetrize the q-BBH operators. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
11 pages, 706 KiB  
Article
Optimizing a Password Hashing Function with Hardware-Accelerated Symmetric Encryption
by Rafael Álvarez, Alicia Andrade and Antonio Zamora
Symmetry 2018, 10(12), 705; https://doi.org/10.3390/sym10120705 - 3 Dec 2018
Cited by 5 | Viewed by 3647
Abstract
Password-based key derivation functions (PBKDFs) are commonly used to transform user passwords into keys for symmetric encryption, as well as for user authentication, password hashing, and preventing attacks based on custom hardware. We propose two optimized alternatives that enhance the performance of a [...] Read more.
Password-based key derivation functions (PBKDFs) are commonly used to transform user passwords into keys for symmetric encryption, as well as for user authentication, password hashing, and preventing attacks based on custom hardware. We propose two optimized alternatives that enhance the performance of a previously published PBKDF. This design is based on (1) employing a symmetric cipher, the Advanced Encryption Standard (AES), as a pseudo-random generator and (2) taking advantage of the support for the hardware acceleration for AES that is available on many common platforms in order to mitigate common attacks to password-based user authentication systems. We also analyze their security characteristics, establishing that they are equivalent to the security of the core primitive (AES), and we compare their performance with well-known PBKDF algorithms, such as Scrypt and Argon2, with favorable results. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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10 pages, 1382 KiB  
Article
Complex Spirals and Pseudo-Chebyshev Polynomials of Fractional Degree
by Paolo Emilio Ricci
Symmetry 2018, 10(12), 671; https://doi.org/10.3390/sym10120671 - 28 Nov 2018
Cited by 10 | Viewed by 4659
Abstract
The complex Bernoulli spiral is connected to Grandi curves and Chebyshev polynomials. In this framework, pseudo-Chebyshev polynomials are introduced, and some of their properties are borrowed to form classical trigonometric identities; in particular, a set of orthogonal pseudo-Chebyshev polynomials of half-integer degree is [...] Read more.
The complex Bernoulli spiral is connected to Grandi curves and Chebyshev polynomials. In this framework, pseudo-Chebyshev polynomials are introduced, and some of their properties are borrowed to form classical trigonometric identities; in particular, a set of orthogonal pseudo-Chebyshev polynomials of half-integer degree is derived. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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20 pages, 358 KiB  
Article
Generalized Liouville–Caputo Fractional Differential Equations and Inclusions with Nonlocal Generalized Fractional Integral and Multipoint Boundary Conditions
by Ahmed Alsaedi, Madeaha Alghanmi, Bashir Ahmad and Sotiris K. Ntouyas
Symmetry 2018, 10(12), 667; https://doi.org/10.3390/sym10120667 - 26 Nov 2018
Cited by 10 | Viewed by 2360
Abstract
We develop the existence criteria for solutions of Liouville–Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are [...] Read more.
We develop the existence criteria for solutions of Liouville–Caputo-type generalized fractional differential equations and inclusions equipped with nonlocal generalized fractional integral and multipoint boundary conditions. Modern techniques of functional analysis are employed to derive the main results. Examples illustrating the main results are also presented. It is imperative to mention that our results correspond to the ones for a symmetric second-order nonlocal multipoint integral boundary value problem under suitable conditions (see the last section). Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
8 pages, 243 KiB  
Article
Convolution and Partial Sums of Certain Multivalent Analytic Functions Involving Srivastava–Tomovski Generalization of the Mittag–Leffler Function
by Yi-Hui Xu and Jin-Lin Liu
Symmetry 2018, 10(11), 597; https://doi.org/10.3390/sym10110597 - 5 Nov 2018
Cited by 1 | Viewed by 2412
Abstract
We derive several properties such as convolution and partial sums of multivalent analytic functions associated with an operator involving Srivastava–Tomovski generalization of the Mittag–Leffler function. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
15 pages, 630 KiB  
Article
Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation
by Adem Kilicman and Rathinavel Silambarasan
Symmetry 2018, 10(10), 527; https://doi.org/10.3390/sym10100527 - 21 Oct 2018
Cited by 27 | Viewed by 3908
Abstract
The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems [...] Read more.
The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems yields the analytical solution. By this way, various exact solutions including complex structures are found, and their behavior is drawn in the 2D plane by Maple to compare the uniqueness and wave traveling of the solutions. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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13 pages, 257 KiB  
Article
On the Existence of the Solutions of a Fredholm Integral Equation with a Modified Argument in Hölder Spaces
by Merve Temizer Ersoy and Hasan Furkan
Symmetry 2018, 10(10), 522; https://doi.org/10.3390/sym10100522 - 19 Oct 2018
Cited by 9 | Viewed by 2292
Abstract
This article concerns the entity of solutions of a quadratic integral equation of the Fredholm type with an altered argument, [...] Read more.
This article concerns the entity of solutions of a quadratic integral equation of the Fredholm type with an altered argument, x ( t ) = p ( t ) + x ( t ) 0 1 k ( t , τ ) ( T x ) ( τ ) d τ , where p , k are given functions, T is the given operator satisfying conditions specified later and x is an unknown function. Through the classical Schauder fixed point theorem and a new conclusion about the relative compactness in Hölder spaces, we obtain the existence of solutions under certain assumptions. Our work is more general than the previous works in the Conclusion section. At the end, we introduce several tangible examples where our entity result can be adopted. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
13 pages, 278 KiB  
Article
A Class of Nonlinear Boundary Value Problems for an Arbitrary Fractional-Order Differential Equation with the Riemann-Stieltjes Functional Integral and Infinite-Point Boundary Conditions
by Hari M. Srivastava, Ahmed M. A. El-Sayed and Fatma M. Gaafar
Symmetry 2018, 10(10), 508; https://doi.org/10.3390/sym10100508 - 16 Oct 2018
Cited by 27 | Viewed by 2831
Abstract
In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the [...] Read more.
In this paper, we investigate the existence of an absolute continuous solution to a class of first-order nonlinear differential equation with integral boundary conditions (BCs) or with infinite-point BCs. The Liouville-Caputo fractional derivative is involved in the nonlinear function. We first consider the existence of a solution for the first-order nonlinear differential equation with m-point nonlocal BCs. The existence of solutions of our problems is investigated by applying the properties of the Riemann sum for continuous functions. Several examples are given in order to illustrate our results. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
8 pages, 280 KiB  
Article
Third-Order Hankel Determinant for Certain Class of Analytic Functions Related with Exponential Function
by Hai-Yan Zhang, Huo Tang and Xiao-Meng Niu
Symmetry 2018, 10(10), 501; https://doi.org/10.3390/sym10100501 - 15 Oct 2018
Cited by 19 | Viewed by 3241
Abstract
Let S l * denote the class of analytic functions f in the open unit disk D = { z : | z | < 1 } normalized by [...] Read more.
Let S l * denote the class of analytic functions f in the open unit disk D = { z : | z | < 1 } normalized by f ( 0 ) = f ( 0 ) 1 = 0 , which is subordinate to exponential function, z f ( z ) f ( z ) e z ( z D ) . In this paper, we aim to investigate the third-order Hankel determinant H 3 ( 1 ) for this function class S l * associated with exponential function and obtain the upper bound of the determinant H 3 ( 1 ) . Meanwhile, we give two examples to illustrate the results obtained. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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11 pages, 281 KiB  
Article
Geometric Properties of Lommel Functions of the First Kind
by Young Jae Sim, Oh Sang Kwon and Nak Eun Cho
Symmetry 2018, 10(10), 455; https://doi.org/10.3390/sym10100455 - 1 Oct 2018
Cited by 4 | Viewed by 2164
Abstract
In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to [...] Read more.
In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
12 pages, 300 KiB  
Article
Symmetric Identities for (P,Q)-Analogue of Tangent Zeta Function
by Cheon Seoung Ryoo
Symmetry 2018, 10(9), 395; https://doi.org/10.3390/sym10090395 - 11 Sep 2018
Cited by 4 | Viewed by 2710
Abstract
The goal of this paper is to define the ( p , q ) -analogue of tangent numbers and polynomials by generalizing the tangent numbers and polynomials and Carlitz-type q-tangent numbers and polynomials. We get some explicit formulas and properties in conjunction [...] Read more.
The goal of this paper is to define the ( p , q ) -analogue of tangent numbers and polynomials by generalizing the tangent numbers and polynomials and Carlitz-type q-tangent numbers and polynomials. We get some explicit formulas and properties in conjunction with ( p , q ) -analogue of tangent numbers and polynomials. We give some new symmetric identities for ( p , q ) -analogue of tangent polynomials by using ( p , q ) -tangent zeta function. Finally, we investigate the distribution and symmetry of the zero of ( p , q ) -analogue of tangent polynomials with numerical methods. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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13 pages, 751 KiB  
Article
Sharp Bounds on the Higher Order Schwarzian Derivatives for Janowski Classes
by Nak Eun Cho, Virendra Kumar and V. Ravichandran
Symmetry 2018, 10(8), 348; https://doi.org/10.3390/sym10080348 - 18 Aug 2018
Cited by 6 | Viewed by 2599
Abstract
Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp [...] Read more.
Higher order Schwarzian derivatives for normalized univalent functions were first considered by Schippers, and those of convex functions were considered by Dorff and Szynal. In the present investigation, higher order Schwarzian derivatives for the Janowski star-like and convex functions are considered, and sharp bounds for the first three consecutive derivatives are investigated. The results obtained in this paper generalize several existing results in this direction. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
15 pages, 484 KiB  
Article
On the Convolution Quadrature Rule for Integral Transforms with Oscillatory Bessel Kernels
by Junjie Ma and Huilan Liu
Symmetry 2018, 10(7), 239; https://doi.org/10.3390/sym10070239 - 25 Jun 2018
Cited by 8 | Viewed by 3500
Abstract
Lubich’s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the [...] Read more.
Lubich’s convolution quadrature rule provides efficient approximations to integrals with special kernels. Particularly, when it is applied to computing highly oscillatory integrals, numerical tests show it does not suffer from fast oscillation. This paper is devoted to studying the convergence property of the convolution quadrature rule for highly oscillatory problems. With the help of operational calculus, the convergence rate of the convolution quadrature rule with respect to the frequency is derived. Furthermore, its application to highly oscillatory integral equations is also investigated. Numerical results are presented to verify the effectiveness of the convolution quadrature rule in solving highly oscillatory problems. It is found from theoretical and numerical results that the convolution quadrature rule for solving highly oscillatory problems is efficient and high-potential. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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