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New Challenges Arising in Engineering Problems with Fractional and Integer...
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New Challenges Arising in Engineering Problems with Fractional and Integer Order II

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (1 September 2022) | Viewed by 28888

Special Issue Editors

Prof. Dr. Haci Mehmet Baskonus

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Guest Editor
Department of Mathematics, Faculty of Education, Harran University, Sanliurfa 63100, Turkey
Interests: partial differential equations; fractional calculus; analytical methods; numerical methods; mathematical physics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Luis Manuel Sánchez Ruiz

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Guest Editor
ETSID-Department of Applied Mathematics, Universitat Politecnica de Valencia, 46022 Valencia, Spain
Interests: fractional calculus; analytical and computational methods; differential and difference equations; real and complex analysis; applied and computational mathematics; mathematical physics
Special Issues, Collections and Topics in MDPI journals
Dr. Armando Ciancio

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Guest Editor
Department of Biomedical and Dental Sciences and Morphofunctional Imaging, University of Messina, 98125 Messina, Italy
Interests: time series based on wavelets; analysis of solutions in the field of physical-mathematical models of rheological media; fractional calculus; mathematical models in economics and finance; physical-mathematical models for biological media and applications to biotechnological and medical sciences
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Recently, many new models have been developed that deal with real-world problems that are seen as serious threats to the future of humankind. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research conducted on fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world.

The focus of this Special Issue will be on reviewing new developments based on fractional differentiation and integration, both with respect to theoretical and numerical aspects.

This Special Issue is a place for experts to share new ideas on theories, applications, and numerical and analytical methods and simulations of fractional calculus and fractional differential equations, as well as integer order. Topics of interest are defined below, and submissions relating to relevant fields are welcome.

  • New analytical and numerical methods to solve partial differential equations
  • Computational methods for fractional differential equations
  • Analysis, modeling, and control of phenomena in the following:
    • Electrical engineering
    • Fluids dynamics and thermal engineering
    • Mechanics
    • Biology
    • Physics
    • Applied sciences
    • Computer science
  • Engineering problems
  • Deterministic and stochastic fractional order models

This Special Issue is organized together with the 6th International Conference on Computational Mathematics and Engineering Sciences (CMES-2022) (20–22 May 2022, Ordu, Turkey); hence, participants in CMES-2022 are especially welcome to submit their contributions. However, this Special Issue will accept contributions from all authors, not just conference participants.

Prof. Dr. Haci Mehmet Baskonus
Prof. Dr. Luis Manuel Sánchez Ruiz
Prof. Dr. Armando Ciancio
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Editorial

Jump to: Research

3 pages, 211 KiB  
Editorial
New Challenges Arising in Engineering Problems with Fractional and Integer Order-II
by Haci Mehmet Baskonus, Luis Manuel Sánchez Ruiz and Armando Ciancio
Fractal Fract. 2022, 6(11), 665; https://doi.org/10.3390/fractalfract6110665 - 11 Nov 2022
Viewed by 1008
Abstract
Modern science is one of the most-used commodities globally, and it is especially important in determining the sources of various threats faced by the world [...] Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)

Research

Jump to: Editorial

12 pages, 390 KiB  
Article
Fractional-Order PD Attitude Control for a Type of Spacecraft with Flexible Appendages
by Shuo Zhang, Yukang Zhou and Suting Cai
Fractal Fract. 2022, 6(10), 601; https://doi.org/10.3390/fractalfract6100601 - 16 Oct 2022
Cited by 4 | Viewed by 1169
Abstract
As large-sized spacecraft have been developed, they have been equipped with flexible appendages, such as solar cell plates and mechanical flexible arms. The attitude control of spacecraft with flexible appendages has become more complex, with higher requirements. In this paper, a fractional-order PD [...] Read more.
As large-sized spacecraft have been developed, they have been equipped with flexible appendages, such as solar cell plates and mechanical flexible arms. The attitude control of spacecraft with flexible appendages has become more complex, with higher requirements. In this paper, a fractional-order PD attitude control method for a type of spacecraft with flexible appendages is presented. Firstly, a lumped parameter model of a spacecraft with flexible appendages is constructed, which provides the transfer function of the attitude angle and external moment. Then, a design method for the fractional-order PD controller for the attitude control of a spacecraft with flexible appendages is provided. Based on the designed steps, a numerical example is provided to compare the control performances between the fractional-order and integer-order PD controllers. Finally, the obtained numerical results are presented to verify the effectiveness of the proposed control method. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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29 pages, 4156 KiB  
Article
Optimal Frequency Control of Multi-Area Hybrid Power System Using New Cascaded TID-PIλDμN Controller Incorporating Electric Vehicles
by Amira Hassan, Mokhtar Aly, Ahmed Elmelegi, Loai Nasrat, Masayuki Watanabe and Emad A. Mohamed
Fractal Fract. 2022, 6(10), 548; https://doi.org/10.3390/fractalfract6100548 - 28 Sep 2022
Cited by 13 | Viewed by 1536
Abstract
Modern structures of electrical power systems are expected to have more domination of renewable energy sources. However, renewable energy-based generation systems suffer from their lack of or reduced rotating masses, which is the main source of power system inertia. Therefore, the frequency of [...] Read more.
Modern structures of electrical power systems are expected to have more domination of renewable energy sources. However, renewable energy-based generation systems suffer from their lack of or reduced rotating masses, which is the main source of power system inertia. Therefore, the frequency of modern power systems represents an important indicator of their proper and safe operation. In addition, the uncertainties and randomness of the renewable energy sources and the load variations can result in frequency undulation problems. In this context, this paper presents an improved cascaded fractional order-based frequency regulation controller for a two-area interconnected power system. The proposed controller uses the cascade structure of the tilt integral derivative (TID) with the fractional order proportional integral derivative with a filter (FOPIDN or PIλDμN) controller (namely the cascaded TID-FOPIDN or TID-PIλDμN controller). Moreover, an optimized TID control method is presented for the electric vehicles (EVs) to maximize their benefits and contribution to the frequency regulation of power systems. The recent widely employed marine predators optimization algorithm (MPA) is utilized to design the new proposed controllers. The proposed controller and design method are tested and validated at various load and renewable source variations, as is their robustness against parameter uncertainties of power systems. Performance comparisons of the proposed controller with featured frequency regulation controllers in the literature are provided to verify the superiority of the new proposed controller. The obtained results confirm the stable operation and the frequency regulation performance of the new proposed controller with optimized controller parameters and without the need for complex design methods. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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19 pages, 1911 KiB  
Article
Computational Analysis of Local Fractional LWR Model Occurring in a Fractal Vehicular Traffic Flow
by Ved Prakash Dubey, Devendra Kumar, Hashim M. Alshehri, Sarvesh Dubey and Jagdev Singh
Fractal Fract. 2022, 6(8), 426; https://doi.org/10.3390/fractalfract6080426 - 31 Jul 2022
Cited by 12 | Viewed by 1612
Abstract
In this paper, we implement computational methods, namely the local fractional natural homotopy analysis method (LFNHAM) and local fractional natural decomposition method (LFNDM), to examine the solution for the local fractional Lighthill–Whitham–Richards (LFLWR) model occurring in a fractal vehicular traffic flow. The LWR [...] Read more.
In this paper, we implement computational methods, namely the local fractional natural homotopy analysis method (LFNHAM) and local fractional natural decomposition method (LFNDM), to examine the solution for the local fractional Lighthill–Whitham–Richards (LFLWR) model occurring in a fractal vehicular traffic flow. The LWR approach preferably models the traffic flow and represents the traffic patterns via the supposition of speed–density equilibrium relationship and continuity equation. This model is mostly preferred for modeling of traffic flow because of its simple approach and interpretive ability to examine the qualitative patterns of traffic flow. The methods applied here incorporate the local fractional natural transform (LFNT) and derive the solutions for the LFLWR model in a closed form. Two examples are provided to demonstrate the accuracy and efficiency of the suggested methods. Furthermore, the numerical simulations have also been presented for each of the examples in the fractal domain. Additionally, the explored solutions for both examples have also been compared and are in good match with already existing solutions in literature. The methods applied in this work make the computational process easier as compared to other iterative methods and still provide precise solutions. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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14 pages, 2046 KiB  
Article
An Efficient Numerical Simulation for the Fractional COVID-19 Model Using the GRK4M Together with the Fractional FDM
by Yasser Ibrahim, Mohamed Khader, Ahmed Megahed, Fawzy Abd El-Salam and Mohamed Adel
Fractal Fract. 2022, 6(6), 304; https://doi.org/10.3390/fractalfract6060304 - 31 May 2022
Cited by 8 | Viewed by 1543
Abstract
In this research, we studied a mathematical model formulated with six fractional differential equations to characterize a COVID-19 outbreak. For the past two years, the disease transmission has been increasing all over the world. We included the considerations of people with infections who [...] Read more.
In this research, we studied a mathematical model formulated with six fractional differential equations to characterize a COVID-19 outbreak. For the past two years, the disease transmission has been increasing all over the world. We included the considerations of people with infections who were both asymptomatic and symptomatic as well as the fact that an individual who has been exposed is either quarantined or moved to one of the diseased classes, with the chance that a susceptible individual could also migrate to the quarantined class. The suggested model is solved numerically by implementing the generalized Runge–Kutta method of the fourth order (GRK4M). We discuss the stability analysis of the GRK4M as a general study. The acquired findings are compared with those obtained using the fractional finite difference method (FDM), where we used the Grünwald–Letnikov approach to discretize the fractional differentiation operator. The FDM is mostly reliant on correctly converting the suggested model into a system of algebraic equations. By applying the proposed methods, the numerical results reveal that these methods are straightforward to apply and computationally very effective at presenting a numerical simulation of the behavior of all components of the model under study. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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12 pages, 907 KiB  
Article
A New Fifth-Order Finite Difference Compact Reconstruction Unequal-Sized WENO Scheme for Fractional Differential Equations
by Yan Zhang and Jun Zhu
Fractal Fract. 2022, 6(6), 294; https://doi.org/10.3390/fractalfract6060294 - 27 May 2022
Cited by 2 | Viewed by 1408
Abstract
This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain [...] Read more.
This paper designs a new finite difference compact reconstruction unequal-sized weighted essentially nonoscillatory scheme (CRUS-WENO) for solving fractional differential equations containing the fractional Laplacian operator. This new CRUS-WENO scheme uses stencils of different sizes to achieve fifth-order accuracy in smooth regions and maintain nonoscillatory properties near discontinuities. The fractional Laplacian operator of order β(0<β<1) is split into the integral part and the first derivative term. Using the Gauss–Jacobi quadrature method to solve the integral part of the fractional Laplacian operators, a new finite difference CRUS-WENO scheme is presented to discretize the first derivative term of the fractional equation. This new CRUS-WENO scheme has the advantages of a narrower large stencil and high spectral resolution. In addition, the linear weights of the new CRUS-WENO scheme can be any positive numbers whose sum is one, which greatly reduces the calculation cost. Some numerical examples are given to show the effectiveness and feasibility of this new CRUS-WENO scheme in solving fractional equations containing the fractional Laplacian operator. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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16 pages, 3057 KiB  
Article
The Tracking Control of the Variable-Order Fractional Differential Systems by Time-Varying Sliding-Mode Control Approach
by Jingfei Jiang, Xin Xu, Kun Zhao, Juan L. G. Guirao, Tareq Saeed and Huatao Chen
Fractal Fract. 2022, 6(5), 231; https://doi.org/10.3390/fractalfract6050231 - 22 Apr 2022
Cited by 3 | Viewed by 1570
Abstract
This paper is concerned with the problem of tracking control for a class of variable-order fractional uncertain system. In order to realize the global robustness of systems, two types of controllers are designed by the global sliding-mode control method. The first one is [...] Read more.
This paper is concerned with the problem of tracking control for a class of variable-order fractional uncertain system. In order to realize the global robustness of systems, two types of controllers are designed by the global sliding-mode control method. The first one is based on a full-order global sliding-mode surface with variable-order fractional type, and the control law is continuous, which is free of chattering. The other one is a novel time-varying control law, which drives the error signals to stay on the proposed reduced-order sliding-mode surface and then converges to the origin. The stability of the controllers proposed is proved by the use of the variable-order fractional type Lyapunov stability theorem and the numerical simulation is given to validate the effectiveness of the theoretical results. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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28 pages, 8852 KiB  
Article
Different Stochastic Resonances Induced by Multiplicative Polynomial Trichotomous Noise in a Fractional Order Oscillator with Time Delay and Fractional Gaussian Noise
by Zhi Yan, Juan L. G. Guirao, Tareq Saeed, Huatao Chen and Xianbin Liu
Fractal Fract. 2022, 6(4), 191; https://doi.org/10.3390/fractalfract6040191 - 30 Mar 2022
Cited by 9 | Viewed by 1679
Abstract
A general investigation on the mechanism of stochastic resonance is reported in a time-delay fractional Langevin system, which endues a nonlinear form multiplicative colored noise and fractional Gaussian noise. In terms of theoretical analysis, both the expressions of output steady-state amplitude and that [...] Read more.
A general investigation on the mechanism of stochastic resonance is reported in a time-delay fractional Langevin system, which endues a nonlinear form multiplicative colored noise and fractional Gaussian noise. In terms of theoretical analysis, both the expressions of output steady-state amplitude and that of the first moment of system response are obtained by utilizing stochastic averaging method, fractional Shapiro and Laplace methods. Due to the presence of trichotomous colored noise, the excitation frequency can induce fresh multimodal Bona fide stochastic resonance, exhibiting much more novel dynamical behaviors than the non-disturbance case. It is verified that multimodal pattern only appears with small noise switching rate and memory damping order. The explicit expressions of critical noise intensity corresponding to the generalized stochastic resonance are given for the first time, by which it is determined that nonlinear form colored noise induces much more of a comprehensive resonant phenomena than the linear form. In the case of slow transfer rate noise, a newfangled phenomenon of double hypersensitive response induced by a variation in noise intensity is discovered and verified for the first time, with the necessary range of parameters for this phenomenon given. In terms of numerical scheme, an efficient and feasible algorithm for generating trichotomous noise is proposed, by which an algorithm based on the Caputo fractional derivative are applied. The numerical results match well with the analytical ones. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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13 pages, 2469 KiB  
Article
Modified Predictor–Corrector Method for the Numerical Solution of a Fractional-Order SIR Model with 2019-nCoV
by Wei Gao, Pundikala Veeresha, Carlo Cattani, Chandrali Baishya and Haci Mehmet Baskonus
Fractal Fract. 2022, 6(2), 92; https://doi.org/10.3390/fractalfract6020092 - 06 Feb 2022
Cited by 48 | Viewed by 1957
Abstract
In this paper, we analyzed and found the solution for a suitable nonlinear fractional dynamical system that describes coronavirus (2019-nCoV) using a novel computational method. A compartmental model with four compartments, namely, susceptible, infected, reported and unreported, was adopted and modified to a [...] Read more.
In this paper, we analyzed and found the solution for a suitable nonlinear fractional dynamical system that describes coronavirus (2019-nCoV) using a novel computational method. A compartmental model with four compartments, namely, susceptible, infected, reported and unreported, was adopted and modified to a new model incorporating fractional operators. In particular, by using a modified predictor–corrector method, we captured the nature of the obtained solution for different arbitrary orders. We investigated the influence of the fractional operator to present and discuss some interesting properties of the novel coronavirus infection. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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21 pages, 415 KiB  
Article
Exact Controllability Results for Sobolev-Type Hilfer Fractional Neutral Delay Volterra-Fredholm Integro-Differential Systems
by Velusamy Vijayakumar, Saud Fahad Aldosary and Kottakkaran Sooppy Nisar
Fractal Fract. 2022, 6(2), 81; https://doi.org/10.3390/fractalfract6020081 - 01 Feb 2022
Cited by 6 | Viewed by 1495
Abstract
This manuscript mainly focuses on the exact controllability of Sobolev-type Hilfer fractional neutral delay Volterra-Fredholm integro-differential systems. The principal findings of this discussion are established by using the theories on fractional calculus, the measure of noncompactness and Mönch fixed point technique. Initially, the [...] Read more.
This manuscript mainly focuses on the exact controllability of Sobolev-type Hilfer fractional neutral delay Volterra-Fredholm integro-differential systems. The principal findings of this discussion are established by using the theories on fractional calculus, the measure of noncompactness and Mönch fixed point technique. Initially, the exact controllability of the system is presented and then we improve the discussion to the system with nonlocal conditions. Finally, abstract and filter systems are provided for the illustration. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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14 pages, 343 KiB  
Article
A Note on Approximate Controllability of Fractional Semilinear Integrodifferential Control Systems via Resolvent Operators
by Velusamy Vijayakumar, Kottakkaran Sooppy Nisar, Dimplekumar Chalishajar, Anurag Shukla, Muslim Malik, Ateq Alsaadi and Saud Fahad Aldosary
Fractal Fract. 2022, 6(2), 73; https://doi.org/10.3390/fractalfract6020073 - 29 Jan 2022
Cited by 41 | Viewed by 2500
Abstract
This article primarily focuses on the approximate controllability of fractional semilinear integrodifferential equations using resolvent operators. Two alternative sets of necessary requirements have been studied. In the first set, we use theories from functional analysis, the compactness of an associated resolvent operator, for [...] Read more.
This article primarily focuses on the approximate controllability of fractional semilinear integrodifferential equations using resolvent operators. Two alternative sets of necessary requirements have been studied. In the first set, we use theories from functional analysis, the compactness of an associated resolvent operator, for our discussion. The primary discussion is proved in the second set by employing Gronwall’s inequality, which prevents the need for compactness of the resolvent operator and the standard fixed point theorems. Then, we extend the discussions to the fractional Sobolev-type semilinear integrodifferential systems. Finally, some theoretical and practical examples are provided to illustrate the obtained theoretical results. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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14 pages, 1417 KiB  
Article
A Neuro-Evolution Heuristic Using Active-Set Techniques to Solve a Novel Nonlinear Singular Prediction Differential Model
by Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Thongchai Botmart and Wajaree Weera
Fractal Fract. 2022, 6(1), 29; https://doi.org/10.3390/fractalfract6010029 - 04 Jan 2022
Cited by 19 | Viewed by 1277
Abstract
In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and [...] Read more.
In this study, a novel design of a second kind of nonlinear Lane–Emden prediction differential singular model (NLE-PDSM) is presented. The numerical solutions of this model were investigated via a neuro-evolution computing intelligent solver using artificial neural networks (ANNs) optimized by global and local search genetic algorithms (GAs) and the active-set method (ASM), i.e., ANN-GAASM. The novel NLE-PDSM was derived from the standard LE and the PDSM along with the details of singular points, prediction terms and shape factors. The modeling strength of ANN was implemented to create a merit function based on the second kind of NLE-PDSM using the mean squared error, and optimization was performed through the GAASM. The corroboration, validation and excellence of the ANN-GAASM for three distinct problems were established through relative studies from exact solutions on the basis of stability, convergence and robustness. Furthermore, explanations through statistical investigations confirmed the worth of the proposed scheme. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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14 pages, 4638 KiB  
Article
A Comprehensive Mathematical Model for SARS-CoV-2 in Caputo Derivative
by Yu Gu, Muhammad Altaf Khan, Y. S. Hamed and Bassem F. Felemban
Fractal Fract. 2021, 5(4), 271; https://doi.org/10.3390/fractalfract5040271 - 14 Dec 2021
Cited by 13 | Viewed by 1889
Abstract
In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical [...] Read more.
In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical differential equations and then extend it into fractional differential equations using the definition of the Caputo derivative. We explore in detail the stability results for the model of the disease-free case when R0<1. We show that the model is stable locally when R0<1. We give the result that the model is globally asymptotically stable whenever R0≤1. Further, to estimate the model parameters, we consider the real data of the fourth wave from Pakistan and provide a reasonable fitting to the data. We estimate the basic reproduction number for the proposed data to be R0=1.0779. Moreover, using the real parameters, we present the numerical solution by first giving a reliable scheme that can numerically handle the solution of the model. In our simulation, we give the graphical results for some sensitive parameters that have a large impact on disease elimination. Our results show that taking into consideration all the possible interactions can describe COVID-19 infection. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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15 pages, 1260 KiB  
Article
Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative
by Li Yan, Gulnur Yel, Ajay Kumar, Haci Mehmet Baskonus and Wei Gao
Fractal Fract. 2021, 5(4), 238; https://doi.org/10.3390/fractalfract5040238 - 23 Nov 2021
Cited by 12 | Viewed by 1493
Abstract
This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these [...] Read more.
This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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14 pages, 814 KiB  
Article
More General Weighted-Type Fractional Integral Inequalities via Chebyshev Functionals
by Gauhar Rahman, Arshad Hussain, Asad Ali, Kottakkaran Sooppy Nisar and Roshan Noor Mohamed
Fractal Fract. 2021, 5(4), 232; https://doi.org/10.3390/fractalfract5040232 - 18 Nov 2021
Cited by 7 | Viewed by 1315
Abstract
The purpose of this research paper is first to propose the generalized weighted-type fractional integrals. Then, we investigate some novel inequalities for a class of differentiable functions related to Chebyshev’s functionals by utilizing the proposed modified weighted-type fractional integral incorporating another function in [...] Read more.
The purpose of this research paper is first to propose the generalized weighted-type fractional integrals. Then, we investigate some novel inequalities for a class of differentiable functions related to Chebyshev’s functionals by utilizing the proposed modified weighted-type fractional integral incorporating another function in the kernel F(θ). For the weighted and extended Chebyshev’s functionals, we also propose weighted fractional integral inequalities. With specific choices of ϖ(θ) and F(θ) as stated in the literature, one may easily study certain new inequalities involving all other types of weighted fractional integrals related to Chebyshev’s functionals. Furthermore, the inequalities for all other type of fractional integrals associated with Chebyshev’s functionals with certain choices of ϖ(θ) and F(θ) are covered from the obtained generalized weighted-type fractional integral inequalities. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
32 pages, 1730 KiB  
Article
A Novel Treatment of Fuzzy Fractional Swift–Hohenberg Equation for a Hybrid Transform within the Fractional Derivative Operator
by Saima Rashid, Rehana Ashraf and Fatimah S. Bayones
Fractal Fract. 2021, 5(4), 209; https://doi.org/10.3390/fractalfract5040209 - 11 Nov 2021
Cited by 8 | Viewed by 1646
Abstract
This article investigates the semi-analytical method coupled with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). In addition, we apply this method to the time-fractional Swift–Hohenberg [...] Read more.
This article investigates the semi-analytical method coupled with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). In addition, we apply this method to the time-fractional Swift–Hohenberg equation (SHe) with various initial conditions (IC) under gH-differentiability. Some aspects of the fuzzy Caputo fractional derivative (CFD) with the Elzaki transform are presented. Moreover, we established the general formulation and approximate findings by testing examples in series form of the models under investigation with success. With the aid of the projected method, we establish the approximate analytical results of SHe with graphical representations of initial value problems by inserting the uncertainty parameter 0≤℘≤1 with different fractional orders. It is expected that fuzzy EADM will be powerful and accurate in configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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18 pages, 380 KiB  
Article
An Approximate Solution of the Time-Fractional Two-Mode Coupled Burgers Equation
by Rachana Shokhanda, Pranay Goswami, Ji-Huan He and Ali Althobaiti
Fractal Fract. 2021, 5(4), 196; https://doi.org/10.3390/fractalfract5040196 - 04 Nov 2021
Cited by 18 | Viewed by 1414
Abstract
In this paper, we consider the time-fractional two-mode coupled Burgers equation with the Caputo fractional derivative. A modified homotopy perturbation method coupled with Laplace transform (He-Laplace method) is applied to find its approximate analytical solution. The method is to decompose the equation into [...] Read more.
In this paper, we consider the time-fractional two-mode coupled Burgers equation with the Caputo fractional derivative. A modified homotopy perturbation method coupled with Laplace transform (He-Laplace method) is applied to find its approximate analytical solution. The method is to decompose the equation into a series of linear equations, which can be effectively and easily solved by the Laplace transform. The solution process is illustrated step by step, and the results show that the present method is extremely powerful for fractional differential equations. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)
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