Integral Transformation, Operational Calculus and Their Applications

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

Deadline for manuscript submissions: closed (31 March 2022) | Viewed by 22093

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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,

The theory and applications of integral transformations and associated 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 transformations and operational calculus as well as their multidisciplinary applications involving their symmetry properties and characteristics.

Prof. H. M. Srivastava
Guest Editor

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Keywords

  • Integral Transformations and Integral Equations as well as Other Related Operators Including Their Symmetry Properties and Characteristics
  • Applications Involving Mathematical (or Higher Transcendental) Functions Including Their Symmetry Properties and Characteristics
  • Applications Involving Fractional-Order Differential and Differintegral Equations and Their Associated Symmetry
  • Applications Involving Symmetrical Aspect of Geometric Function Theory of Complex Analysis
  • Applications Involving q-Series and q-Polynomials and Their Associated Symmetry
  • Applications Involving Special Functions of Mathematical Physics and Applied Mathematics and Their Symmetrical Aspect
  • Applications Involving Analytic Number Theory and Symmetry

Published Papers (12 papers)

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Research

13 pages, 368 KiB  
Article
Unified Integrals of Generalized Mittag–Leffler Functions and Their Graphical Numerical Investigation
by Nabiullah Khan, Mohammad Iqbal Khan, Talha Usman, Kamsing Nonlaopon and Shrideh Al-Omari
Symmetry 2022, 14(5), 869; https://doi.org/10.3390/sym14050869 - 23 Apr 2022
Cited by 1 | Viewed by 1423
Abstract
In this article, we obtain certain finite integrals concerning generalized Mittag–Leffler functions, which are evaluated in terms of the generalized Fox–Wright function. The integrals of concern are unified in nature and thereby yield some new integral formulas as special cases. Moreover, we numerically [...] Read more.
In this article, we obtain certain finite integrals concerning generalized Mittag–Leffler functions, which are evaluated in terms of the generalized Fox–Wright function. The integrals of concern are unified in nature and thereby yield some new integral formulas as special cases. Moreover, we numerically compute some integrals using the Gaussian quadrature formula and draw a comparison with the main integrals by using graphical numerical investigation. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
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21 pages, 2599 KiB  
Article
Non-Separable Linear Canonical Wavelet Transform
by Hari M. Srivastava, Firdous A. Shah, Tarun K. Garg, Waseem Z. Lone and Huzaifa L. Qadri
Symmetry 2021, 13(11), 2182; https://doi.org/10.3390/sym13112182 - 15 Nov 2021
Cited by 19 | Viewed by 1659
Abstract
This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in L2(Rn). The preliminary analysis encompasses the derivation of fundamental properties of the novel [...] Read more.
This study aims to achieve an efficient time-frequency representation of higher-dimensional signals by introducing the notion of a non-separable linear canonical wavelet transform in L2(Rn). The preliminary analysis encompasses the derivation of fundamental properties of the novel integral transform including the orthogonality relation, inversion formula, and the range theorem. To extend the scope of the study, we formulate several uncertainty inequalities, including the Heisenberg’s, logarithmic, and Nazorav’s inequalities for the proposed transform in the linear canonical domain. The obtained results are reinforced with illustrative examples. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
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8 pages, 251 KiB  
Article
A Double Logarithmic Transform Involving the Exponential and Polynomial Functions Expressed in Terms of the Hurwitz–Lerch Zeta Function
by Robert Reynolds and Allan Stauffer
Symmetry 2021, 13(11), 1983; https://doi.org/10.3390/sym13111983 - 20 Oct 2021
Cited by 2 | Viewed by 1209
Abstract
The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are [...] Read more.
The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
11 pages, 296 KiB  
Article
Double Integral of the Product of the Exponential of an Exponential Function and a Polynomial Expressed in Terms of the Lerch Function
by Robert Reynolds and Allan Stauffer
Symmetry 2021, 13(10), 1962; https://doi.org/10.3390/sym13101962 - 18 Oct 2021
Viewed by 1495
Abstract
In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by [...] Read more.
In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by emxmyexey+y(log(a)+xy)kdxdy in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. Almost all Lerch functions have an asymmetrical zero distribution. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus, a table of integral pairs is given for interested readers. All of the results in this work are new. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
24 pages, 371 KiB  
Article
Fuzzy Mixed Variational-like and Integral Inequalities for Strongly Preinvex Fuzzy Mappings
by Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed and Juan L. G. Guirao
Symmetry 2021, 13(10), 1816; https://doi.org/10.3390/sym13101816 - 29 Sep 2021
Cited by 9 | Viewed by 1441
Abstract
It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts [...] Read more.
It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite–Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
9 pages, 249 KiB  
Article
A Quadruple Definite Integral Expressed in Terms of the Lerch Function
by Robert Reynolds and Allan Stauffer
Symmetry 2021, 13(9), 1638; https://doi.org/10.3390/sym13091638 - 6 Sep 2021
Viewed by 1412
Abstract
A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority [...] Read more.
A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority of the results in this work are new. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
12 pages, 272 KiB  
Article
On New Generalized Dunkel Type Integral Inequalities with Applications
by Dong-Sheng Wang, Huan-Nan Shi, Chun-Ru Fu and Wei-Shih Du
Symmetry 2021, 13(9), 1576; https://doi.org/10.3390/sym13091576 - 27 Aug 2021
Viewed by 1580
Abstract
In this paper, by applying majorization theory, we study the Schur convexity of functions related to Dunkel integral inequality. We establish some new generalized Dunkel type integral inequalities and their applications to inequality theory. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
14 pages, 303 KiB  
Article
A Subclass of Multivalent Janowski Type q-Starlike Functions and Its Consequences
by Qiuxia Hu, Hari M. Srivastava, Bakhtiar Ahmad, Nazar Khan, Muhammad Ghaffar Khan, Wali Khan Mashwani and Bilal Khan
Symmetry 2021, 13(7), 1275; https://doi.org/10.3390/sym13071275 - 16 Jul 2021
Cited by 26 | Viewed by 1922
Abstract
In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class Ap, where class Ap is invariant (or symmetric) under rotations. The well-known class [...] Read more.
In this article, by utilizing the theory of quantum (or q-) calculus, we define a new subclass of analytic and multivalent (or p-valent) functions class Ap, where class Ap is invariant (or symmetric) under rotations. The well-known class of Janowski functions are used with the help of the principle of subordination between analytic functions in order to define this subclass of analytic and p-valent functions. This function class generalizes various other subclasses of analytic functions, not only in classical Geometric Function Theory setting, but also some q-analogue of analytic multivalent function classes. We study and investigate some interesting properties such as sufficiency criteria, coefficient bounds, distortion problem, growth theorem, radii of starlikeness and convexity for this newly-defined class. Other properties such as those involving convex combination are also discussed for these functions. In the concluding part of the article, we have finally given the well-demonstrated fact that the results presented in this article can be obtained for the (p,q)-variations, by making some straightforward simplification and will be an inconsequential exercise simply because the additional parameter p is obviously unnecessary. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
11 pages, 262 KiB  
Article
Properties of Certain Subclass of Meromorphic Multivalent Functions Associated with q-Difference Operator
by Cai-Mei Yan, Rekha Srivastava and Jin-Lin Liu
Symmetry 2021, 13(6), 1035; https://doi.org/10.3390/sym13061035 (registering DOI) - 8 Jun 2021
Cited by 4 | Viewed by 1373
Abstract
A new subclass Σp,q(α,A,B) of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient [...] Read more.
A new subclass Σp,q(α,A,B) of meromorphic multivalent functions is defined by means of a q-difference operator. Some properties of the functions in this new subclass, such as sufficient and necessary conditions, coefficient estimates, growth and distortion theorems, radius of starlikeness and convexity, partial sums and closure theorems, are investigated. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
24 pages, 351 KiB  
Article
A Spectral Calculus for Lorentz Invariant Measures on Minkowski Space
by John Mashford
Symmetry 2020, 12(10), 1696; https://doi.org/10.3390/sym12101696 - 15 Oct 2020
Cited by 4 | Viewed by 2445
Abstract
This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementary convolutions involving Feynman propagators of scalar [...] Read more.
This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementary convolutions involving Feynman propagators of scalar particles are computed. It is proved that the convolution of arbitrary causal Lorentz invariant Borel complex measures exists and the product of such measures exists in a wide class of cases. Techniques for their computation in terms of their spectral representation are presented. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
17 pages, 327 KiB  
Article
An Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions Associated with k-Fibonacci Numbers
by Muhammad Shafiq, Hari M. Srivastava, Nazar Khan, Qazi Zahoor Ahmad, Maslina Darus and Samiha Kiran
Symmetry 2020, 12(6), 1043; https://doi.org/10.3390/sym12061043 - 22 Jun 2020
Cited by 36 | Viewed by 2214
Abstract
In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class A of normalized analytic functions, where class A is invariant (or symmetric) [...] Read more.
In this paper, we use q-derivative operator to define a new class of q-starlike functions associated with k-Fibonacci numbers. This newly defined class is a subclass of class A of normalized analytic functions, where class A is invariant (or symmetric) under rotations. For this function class we obtain an upper bound of the third Hankel determinant. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
19 pages, 792 KiB  
Article
Difference of Some Positive Linear Approximation Operators for Higher-Order Derivatives
by Vijay Gupta, Ana Maria Acu and Hari Mohan Srivastava
Symmetry 2020, 12(6), 915; https://doi.org/10.3390/sym12060915 - 2 Jun 2020
Cited by 21 | Viewed by 2129
Abstract
In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between [...] Read more.
In the present paper, we deal with some general estimates for the difference of operators which are associated with different fundamental functions. In order to exemplify the theoretical results presented in (for example) Theorem 2, we provide the estimates of the differences between some of the most representative operators used in Approximation Theory in especially the difference between the Baskakov and the Szász–Mirakyan operators, the difference between the Baskakov and the Szász–Mirakyan–Baskakov operators, the difference of two genuine-Durrmeyer type operators, and the difference of the Durrmeyer operators and the Lupaş–Durrmeyer operators. By means of illustrative numerical examples, we show that, for particular cases, our result improves the estimates obtained by using the classical result of Shisha and Mond. We also provide the symmetry aspects of some of these approximations operators which we have studied in this paper. Full article
(This article belongs to the Special Issue Integral Transformation, Operational Calculus and Their Applications)
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