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4 pages, 174 KiB

Open AccessEditorial

Fractional Calculus—Theory and Applications

by Jorge E. Macías-Díaz

Axioms 2022, 11(2), 43; https://doi.org/10.3390/axioms11020043 - 22 Jan 2022

Cited by 3 | Viewed by 2438

Abstract

In recent years, fractional calculus has witnessed tremendous progress in various areas of sciences and mathematics [...] Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

Research

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13 pages, 717 KiB

Open AccessArticle

Design, Analysis and Comparison of a Nonstandard Computational Method for the Solution of a General Stochastic Fractional Epidemic Model

by Nauman Ahmed, Jorge E. Macías-Díaz, Ali Raza, Dumitru Baleanu, Muhammad Rafiq, Zafar Iqbal and Muhammad Ozair Ahmad

Axioms 2022, 11(1), 10; https://doi.org/10.3390/axioms11010010 - 24 Dec 2021

Cited by 10 | Viewed by 2874

Abstract

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the [...] Read more.

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number,

R_{0}

, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when

R_{0} < 1

. A similar result is obtained for the endemic equilibrium when

R_{0} > 1

. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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25 pages, 3396 KiB

Open AccessArticle

GPU Based Modelling and Analysis for Parallel Fractional Order Derivative Model of the Spiral-Plate Heat Exchanger

by Guanqiang Dong and Mingcong Deng

Axioms 2021, 10(4), 344; https://doi.org/10.3390/axioms10040344 - 16 Dec 2021

Cited by 7 | Viewed by 2407

Abstract

Heat exchangers are commonly used in various industries. A spiral-plate heat exchanger with two fluids is a compact plant that only requires a small space and is excellent in high heat transfer efficiency. However, the spiral-plate heat exchanger is a nonlinear plant with [...] Read more.

Heat exchangers are commonly used in various industries. A spiral-plate heat exchanger with two fluids is a compact plant that only requires a small space and is excellent in high heat transfer efficiency. However, the spiral-plate heat exchanger is a nonlinear plant with uncertainties, considering the difference between the heat fluid, the heated fluid, and other complex factors. The fractional order derivation model is more accurate than the traditional integer order model. In this paper, a parallel fractional order derivation model is proposed by considering the merit of the graphics processing unit (GPU). Then, the parallel fractional order derivation model for the spiral-plate heat exchanger is constructed. Simulations show the relationships between the output temperature of heated fluid and the orders of fractional order derivatives with two directional fluids impacted by complex factors, namely, the volume flow rate in hot fluid, and the volume flow rate in cold fluid, respectively. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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9 pages, 920 KiB

Open AccessArticle

Forecasting Economic Growth of the Group of Seven via Fractional-Order Gradient Descent Approach

by Xiaoling Wang, Michal Fečkan and JinRong Wang

Axioms 2021, 10(4), 257; https://doi.org/10.3390/axioms10040257 - 15 Oct 2021

Cited by 7 | Viewed by 1695

Abstract

This paper establishes a model of economic growth for all the G7 countries from 1973 to 2016, in which the gross domestic product (GDP) is related to land area, arable land, population, school attendance, gross capital formation, exports of goods and services, general [...] Read more.

This paper establishes a model of economic growth for all the G7 countries from 1973 to 2016, in which the gross domestic product (GDP) is related to land area, arable land, population, school attendance, gross capital formation, exports of goods and services, general government, final consumer spending and broad money. The fractional-order gradient descent and integer-order gradient descent are used to estimate the model parameters to fit the GDP and forecast GDP from 2017 to 2019. The results show that the convergence rate of the fractional-order gradient descent is faster and has a better fitting accuracy and prediction effect. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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21 pages, 1181 KiB

Open AccessArticle

The Approximate and Analytic Solutions of the Time-Fractional Intermediate Diffusion Wave Equation Associated with the Fokker–Planck Operator and Applications

by Entsar A. Abdel-Rehim

Axioms 2021, 10(3), 230; https://doi.org/10.3390/axioms10030230 - 17 Sep 2021

Cited by 4 | Viewed by 2016

Abstract

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the [...] Read more.

In this paper, the time-fractional wave equation associated with the space-fractional Fokker–Planck operator and with the time-fractional-damped term is studied. The concept of the Green function is implemented to drive the analytic solution of the three-term time-fractional equation. The explicit expressions for the Green function

G_{3} (t)

of the three-term time-fractional wave equation with constant coefficients is also studied for two physical and biological models. The explicit analytic solutions, for the two studied models, are expressed in terms of the Weber, hypergeometric, exponential, and Mittag–Leffler functions. The relation to the diffusion equation is given. The asymptotic behaviors of the Mittag–Leffler function, the hypergeometric function

1 F 1

, and the exponential functions are compared numerically. The Grünwald–Letnikov scheme is used to derive the approximate difference schemes of the Caputo time-fractional operator and the Feller–Riesz space-fractional operator. The explicit difference scheme is numerically studied, and the simulations of the approximate solutions are plotted for different values of the fractional orders. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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18 pages, 370 KiB

Open AccessArticle

Some New Fractional Estimates of Inequalities for LR-

p

-Convex Interval-Valued Functions by Means of Pseudo Order Relation

by Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor, Dumitru Baleanu and Juan Luis García Guirao

Axioms 2021, 10(3), 175; https://doi.org/10.3390/axioms10030175 - 31 Jul 2021

Cited by 27 | Viewed by 2746

Abstract

It is a familiar fact that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval [...] Read more.

It is a familiar fact that interval analysis provides tools to deal with data uncertainty. In general, interval analysis is typically used to deal with the models whose data are composed of inaccuracies that may occur from certain kinds of measurements. In interval analysis, both the inclusion relation

(\subseteq)

and pseudo order relation

(\leq_{p})

are two different concepts. In this article, by using pseudo order relation, we introduce the new class of nonconvex functions known as LR-

p

-convex interval-valued functions (LR-

p

-convex-IVFs). With the help of this relation, we establish a strong relationship between LR-

p

-convex-IVFs and Hermite-Hadamard type inequalities (HH-type inequalities) via Katugampola fractional integral operator. Moreover, we have shown that our results include a wide class of new and known inequalities for LR-

p

-convex-IVFs and their variant forms as special cases. Useful examples that demonstrate the applicability of the theory proposed in this study are given. The concepts and techniques of this paper may be a starting point for further research in this area. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

15 pages, 298 KiB

Open AccessArticle

Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Systems with Nonlocal Coupled Fractional Integral Boundary Conditions

by Chanakarn Kiataramkul, Weera Yukunthorn, Sotiris K. Ntouyas and Jessada Tariboon

Axioms 2021, 10(3), 174; https://doi.org/10.3390/axioms10030174 - 31 Jul 2021

Cited by 16 | Viewed by 2022

Abstract

In this paper, we initiate the study of existence of solutions for a fractional differential system which contains mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives, complemented with nonlocal coupled fractional integral boundary conditions. We derive necessary conditions for the existence and uniqueness of solutions [...] Read more.

In this paper, we initiate the study of existence of solutions for a fractional differential system which contains mixed Riemann–Liouville and Hadamard–Caputo fractional derivatives, complemented with nonlocal coupled fractional integral boundary conditions. We derive necessary conditions for the existence and uniqueness of solutions of the considered system, by using standard fixed point theorems, such as Banach contraction mapping principle and Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

16 pages, 1805 KiB

Open AccessArticle

A Novel Numerical Method for Solving Fractional Diffusion-Wave and Nonlinear Fredholm and Volterra Integral Equations with Zero Absolute Error

by Mutaz Mohammad, Alexandre Trounev and Mohammed Alshbool

Axioms 2021, 10(3), 165; https://doi.org/10.3390/axioms10030165 - 28 Jul 2021

Cited by 13 | Viewed by 2813

Abstract

In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an

M \times M

collocation points. The proposed equations are [...] Read more.

In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an

M \times M

collocation points. The proposed equations are presented based on Caputo fractional derivative where we reduce the resulting system to a system of algebraic equations by implementing the Gaussian quadrature discretization. The reduced system is generated via the truncated Euler wavelet expansion. Several examples with known exact solutions have been solved with zero absolute error. This method is also applied to the Fredholm and Volterra nonlinear integral equations and achieves the desired absolute error of

0 \times 10^{- 31}

for all tested examples. The new numerical scheme is exceptional in terms of its novelty, efficiency and accuracy in the field of numerical approximation. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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15 pages, 309 KiB

Open AccessFeature PaperArticle

Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

by Suphawat Asawasamrit, Yasintorn Thadang, Sotiris K. Ntouyas and Jessada Tariboon

Axioms 2021, 10(3), 130; https://doi.org/10.3390/axioms10030130 - 23 Jun 2021

Cited by 16 | Viewed by 1675

Abstract

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence [...] Read more.

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

20 pages, 404 KiB

Open AccessArticle

A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

by Hristo Kiskinov, Magdalena Veselinova, Ekaterina Madamlieva and Andrey Zahariev

Axioms 2021, 10(2), 75; https://doi.org/10.3390/axioms10020075 - 27 Apr 2021

Cited by 2 | Viewed by 2026

Abstract

In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and [...] Read more.

In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

26 pages, 342 KiB

Open AccessArticle

Fractional Coupled Hybrid Sturm–Liouville Differential Equation with Multi-Point Boundary Coupled Hybrid Condition

by Mohadeseh Paknazar and Manuel De La Sen

Axioms 2021, 10(2), 65; https://doi.org/10.3390/axioms10020065 - 16 Apr 2021

Cited by 3 | Viewed by 1725

Abstract

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville [...] Read more.

The Sturm–Liouville differential equation is an important tool for physics, applied mathematics, and other fields of engineering and science and has wide applications in quantum mechanics, classical mechanics, and wave phenomena. In this paper, we investigate the coupled hybrid version of the Sturm–Liouville differential equation. Indeed, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with multi-point boundary coupled hybrid condition. Furthermore, we study the existence of solutions for the coupled hybrid Sturm–Liouville differential equation with an integral boundary coupled hybrid condition. We give an application and some examples to illustrate our results. Full article

(This article belongs to the Special Issue Fractional Calculus - Theory and Applications)

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