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2 pages, 141 KiB

Open AccessEditorial

Operators of Fractional Calculus and Their Applications

by Hari Mohan Srivastava

Mathematics 2018, 6(9), 157; https://doi.org/10.3390/math6090157 - 5 Sep 2018

Cited by 2 | Viewed by 3034

Abstract

n/a Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

Research

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15 pages, 310 KiB

Open AccessArticle

F-Convex Contraction via Admissible Mapping and Related Fixed Point Theorems with an Application

by Y. Mahendra Singh, Mohammad Saeed Khan and Shin Min Kang

Mathematics 2018, 6(6), 105; https://doi.org/10.3390/math6060105 - 20 Jun 2018

Cited by 14 | Viewed by 3626 | 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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

19 pages, 282 KiB

Open AccessArticle

Several Results of Fractional Differential and Integral Equations in Distribution

by Chenkuan Li, Changpin Li and Kyle Clarkson

Mathematics 2018, 6(6), 97; https://doi.org/10.3390/math6060097 - 8 Jun 2018

Cited by 13 | Viewed by 2928

Abstract

This paper is to study certain types of fractional differential and integral equations, such as [...] Read more.

This paper is to study certain types of fractional differential and integral equations, such as

θ (x - x_{0}) g (x) = \frac{1}{Γ (α)} \int_{0}^{x} {(x - ζ)}^{α - 1} f (ζ) d ζ

,

y (x) + \int_{0}^{x} \frac{y (τ)}{\sqrt{x - τ}} d τ = x_{+}^{- \sqrt{2}} + δ (x)

, and

x_{+}^{k} \int_{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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

15 pages, 239 KiB

Open AccessArticle

Babenko’s Approach to Abel’s Integral Equations

by Chenkuan Li and Kyle Clarkson

Mathematics 2018, 6(3), 32; https://doi.org/10.3390/math6030032 - 1 Mar 2018

Cited by 20 | Viewed by 3999

Abstract

The goal of this paper is to investigate the following Abel’s integral equation of the second kind: [...] Read more.

The goal of this paper is to investigate the following Abel’s integral equation of the second kind:

y (t) + \frac{λ}{Γ (α)} \int_{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 \in 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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

312 KiB

Open AccessArticle

Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function

by Tohru Morita and Ken-ichi Sato

Mathematics 2017, 5(4), 62; https://doi.org/10.3390/math5040062 - 10 Nov 2017

Cited by 4 | Viewed by 3708

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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

740 KiB

Open AccessFeature PaperArticle

Mixed Order Fractional Differential Equations

by Michal Fečkan and JinRong Wang

Mathematics 2017, 5(4), 61; https://doi.org/10.3390/math5040061 - 7 Nov 2017

Cited by 4 | Viewed by 3549

Abstract

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived. Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

443 KiB

Open AccessArticle

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

by Suranon Yensiri and Ruth J. Skulkhu

Mathematics 2017, 5(4), 54; https://doi.org/10.3390/math5040054 - 23 Oct 2017

Cited by 14 | Viewed by 6020

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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

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241 KiB

Open AccessArticle

Stability of a Monomial Functional Equation on a Restricted Domain

by Yang-Hi Lee

Mathematics 2017, 5(4), 53; https://doi.org/10.3390/math5040053 - 18 Oct 2017

Cited by 10 | Viewed by 3255

Abstract

In this paper, we prove the stability of the following functional equation [...] Read more.

In this paper, we prove the stability of the following functional equation

\sum_{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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

1225 KiB

Open AccessArticle

New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations

by Hayman Thabet, Subhash Kendre and Dimplekumar Chalishajar

Mathematics 2017, 5(4), 47; https://doi.org/10.3390/math5040047 - 25 Sep 2017

Cited by 32 | Viewed by 5764 | 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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

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1 pages, 144 KiB

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

by Hayman Thabet, Subhash Kendre and Dimplekumar Chalishajar

Mathematics 2018, 6(2), 26; https://doi.org/10.3390/math6020026 - 14 Feb 2018

Cited by 2 | Viewed by 2328

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

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)

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