Special Issue "Operators of Fractional Calculus and Their Applications"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 August 2018)

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Special Issue Editor

Guest Editor
Prof. Dr. H. M. Srivastava

Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
Website | E-Mail
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 Modelling and Optimization.

Special Issue Information

Dear Colleagues,

During the past four decades or so, various operators of fractional calculus, such as those named after Riemann-Liouville, Weyl, Hadamard, Grunwald-Letnikov, Riesz, Erdelyi-Kober, Liouville-Caputo, and so on, have been found to be remarkably popular and important due mainly to their demonstrated applications in numerous seemingly diverse and widespread fields of the mathematical, physical, chemical, engineering and statistical sciences. Many of these fractional calculus operators provide several potentially useful tools for solving differential, integral, differintegral and integro-differential equations, together with the fractional-calculus analogues and extensions of each of these equation, and various other problems involving special functions of mathematical physics, as well as their extensions and generalizations in one and more variables. In this Special Issue, we invite and welcome review, expository and original research articles dealing with the recent advances in the theory of fractional calculus and its multidisciplinary applications.

Prof. Dr. Hari M. Srivastava

Guest Editor

Keywords

  • Operators of fractional calculus
  • Chaos and fractional dynamics
  • Fractional differential
  • Fractional differintegral equations
  • Fractional integro-differential equations
  • Fractional integrals
  • Fractional derivatives associated with special functions of mathematical physics
  • Applied mathematics
  • Identities and inequalities involving fractional integrals
  • Fractional derivatives

Published Papers (10 papers)

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Editorial

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Open AccessEditorial
Operators of Fractional Calculus and Their Applications
Mathematics 2018, 6(9), 157; https://doi.org/10.3390/math6090157
Received: 4 September 2018 / Revised: 4 September 2018 / Accepted: 4 September 2018 / Published: 5 September 2018
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Research

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Open AccessArticle
F-Convex Contraction via Admissible Mapping and Related Fixed Point Theorems with an Application
Mathematics 2018, 6(6), 105; https://doi.org/10.3390/math6060105
Received: 8 May 2018 / Revised: 4 June 2018 / Accepted: 5 June 2018 / Published: 20 June 2018
Cited by 2 | PDF Full-text (310 KB) | HTML Full-text | XML Full-text | Correction
Abstract
In this paper, we introduce F-convex contraction via admissible mapping in the sense of Wardowski [Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 94 (2012), 6 pages] which extends convex contraction mapping of [...] Read more.
In this paper, we introduce F-convex contraction via admissible mapping in the sense of Wardowski [Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl., 94 (2012), 6 pages] which extends convex contraction mapping of type-2 of Istrǎţescu [Some fixed point theorems for convex contraction mappings and convex non-expansive mappings (I), Libertas Mathematica, 1(1981), 151–163] and establish a fixed point theorem in the setting of metric space. Our result extends and generalizes some other similar results in the literature. As an application of our main result, we establish an existence theorem for the non-linear Fredholm integral equation and give a numerical example to validate the application of our obtained result. Full article
Open AccessArticle
Several Results of Fractional Differential and Integral Equations in Distribution
Mathematics 2018, 6(6), 97; https://doi.org/10.3390/math6060097
Received: 10 May 2018 / Revised: 3 June 2018 / Accepted: 6 June 2018 / Published: 8 June 2018
Cited by 2 | PDF Full-text (282 KB) | HTML Full-text | XML Full-text
Abstract
This paper is to study certain types of fractional differential and integral equations, such as θ(xx0)g(x)=1Γ(α)0x(xζ)α1 [...] Read more.
This paper is to study certain types of fractional differential and integral equations, such as θ ( x x 0 ) g ( x ) = 1 Γ ( α ) 0 x ( x ζ ) α 1 f ( ζ ) d ζ , y ( x ) + 0 x y ( τ ) x τ d τ = x + 2 + δ ( x ) , and x + k 0 x y ( τ ) ( x τ ) α 1 d τ = δ ( m ) ( x ) in the distributional sense by Babenko’s approach and fractional calculus. Applying convolutions and products of distributions in the Schwartz sense, we obtain generalized solutions for integral and differential equations of fractional order by using the Mittag-Leffler function, which cannot be achieved in the classical sense including numerical analysis methods, or by the Laplace transform. Full article
Open AccessArticle
Babenko’s Approach to Abel’s Integral Equations
Mathematics 2018, 6(3), 32; https://doi.org/10.3390/math6030032
Received: 25 January 2018 / Revised: 16 February 2018 / Accepted: 18 February 2018 / Published: 1 March 2018
Cited by 4 | PDF Full-text (239 KB) | HTML Full-text | XML Full-text
Abstract
The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y(t)+λΓ(α)0t(tτ)α1y(τ)d [...] Read more.
The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t ) + λ Γ ( α ) 0 t ( t τ ) α 1 y ( τ ) d τ = f ( t ) , ( t > 0 ) and its variants by fractional calculus. Applying Babenko’s approach and fractional integrals, we provide a general method for solving Abel’s integral equation and others with a demonstration of different types of examples by showing convergence of series. In particular, we extend this equation to a distributional space for any arbitrary α R by fractional operations of generalized functions for the first time and obtain several new and interesting results that cannot be realized in the classical sense or by the Laplace transform. Full article
Open AccessArticle
Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function
Mathematics 2017, 5(4), 62; https://doi.org/10.3390/math5040062
Received: 30 September 2017 / Revised: 31 October 2017 / Accepted: 6 November 2017 / Published: 10 November 2017
Cited by 2 | PDF Full-text (312 KB) | HTML Full-text | XML Full-text
Abstract
The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular [...] Read more.
The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular solution of differential equations with constant coefficients, by the Laplace transform. Full article
Open AccessFeature PaperArticle
Mixed Order Fractional Differential Equations
Mathematics 2017, 5(4), 61; https://doi.org/10.3390/math5040061
Received: 8 September 2017 / Revised: 30 October 2017 / Accepted: 31 October 2017 / Published: 7 November 2017
Cited by 1 | PDF Full-text (740 KB) | HTML Full-text | XML Full-text
Abstract
This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived. Full article
Open AccessArticle
An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations
Mathematics 2017, 5(4), 54; https://doi.org/10.3390/math5040054
Received: 2 September 2017 / Revised: 13 October 2017 / Accepted: 17 October 2017 / Published: 23 October 2017
Cited by 3 | PDF Full-text (443 KB) | HTML Full-text | XML Full-text
Abstract
The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than [...] Read more.
The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies. Full article
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Open AccessArticle
Stability of a Monomial Functional Equation on a Restricted Domain
Mathematics 2017, 5(4), 53; https://doi.org/10.3390/math5040053
Received: 24 August 2017 / Revised: 2 October 2017 / Accepted: 8 October 2017 / Published: 18 October 2017
Cited by 5 | PDF Full-text (241 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we prove the stability of the following functional equation i=0nnCi(1)nif(ix+y)n!f(x)=0 [...] Read more.
In this paper, we prove the stability of the following functional equation i = 0 n n C i ( 1 ) n i f ( i x + y ) n ! f ( x ) = 0 on a restricted domain by employing the direct method in the sense of Hyers. Full article
Open AccessArticle
New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations
Mathematics 2017, 5(4), 47; https://doi.org/10.3390/math5040047
Received: 24 August 2017 / Revised: 19 September 2017 / Accepted: 20 September 2017 / Published: 25 September 2017
Cited by 9 | PDF Full-text (1225 KB) | HTML Full-text | XML Full-text | Correction
Abstract
This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs [...] Read more.
This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods. Full article
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Other

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Open AccessCorrection
Correction: Thabet, H.; Kendre, S.; Chalishajar, D. New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations Mathematics 2017, 5, 47
Mathematics 2018, 6(2), 26; https://doi.org/10.3390/math6020026
Received: 13 February 2018 / Revised: 14 February 2018 / Accepted: 14 February 2018 / Published: 14 February 2018
Cited by 1 | PDF Full-text (144 KB) | HTML Full-text | XML Full-text
Abstract
We have found some errors in the caption of Figure 1 and Figure 2 in our paper [1], and thus would like to make the following corrections:[...] Full article
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