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Symmetry
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Applied Mathematics and Fractional Calculus II
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Applied Mathematics and Fractional Calculus II

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

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 28565

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editors

Dr. Francisco Martínez González

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Guest Editor
Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain
Interests: fractional calculus; real analysis; complex analysis; mathematical physics; numerical analysis; computational science; mathematical modeling; theoretical physics; signal processing
Special Issues, Collections and Topics in MDPI journals
Dr. Mohammed K. A. Kaabar

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Guest Editor
Chinese Institute of Electric Power, Samarkand International University of Technology, Samarkand 140100, Uzbekistan
Interests: mathematics; electrical engineering; computer engineering; antennas and wave propagation; modern electronics; data analysis; design project; sustainable development; new technology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. That is why the application of fractional calculus theory has become a focus of international academic research.

Dr. Francisco Martínez González
Dr. Mohammed K. A. Kaabar
Guest Editors

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Keywords

  • fractional calculus
  • fractional derivative
  • fractional integral
  • multivariable fractional calculus
  • fractional differential equations
  • fractional partial derivative equations
  • fractional physical equations

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Related Special Issue

Published Papers (19 papers)

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Research

13 pages, 299 KiB  
Article
Fractional Integrals Associated with the One-Dimensional Dunkl Operator in Generalized Lizorkin Space
by Fethi Bouzeffour
Symmetry 2023, 15(9), 1725; https://doi.org/10.3390/sym15091725 - 8 Sep 2023
Viewed by 1033
Abstract
This paper explores the realm of fractional integral calculus in connection with the one-dimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts [...] Read more.
This paper explores the realm of fractional integral calculus in connection with the one-dimensional Dunkl operator on the space of tempered functions and Lizorkin type space. The primary objective is to construct fractional integral operators within this framework. By establishing the analogous counterparts of well-known operators, including the Riesz fractional integral, Feller fractional integral, and Riemann–Liouville fractional integral operators, we demonstrate their applicability in this setting. Moreover, we show that familiar properties of fractional integrals can be derived from the obtained results, further reinforcing their significance. This investigation sheds light on the utilization of Dunkl operators in fractional calculus and provides valuable insights into the connections between different types of fractional integrals. The findings presented in this paper contribute to the broader field of fractional calculus and advance our understanding of the study of Dunkl operators in this context. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
13 pages, 695 KiB  
Article
Artificial Neural Network Solution for a Fractional-Order Human Skull Model Using a Hybrid Cuckoo Search Algorithm
by Waseem, Sabir Ali, Shahzad Khattak, Asad Ullah, Muhammad Ayaz, Fuad A. Awwad and Emad A. A. Ismail
Symmetry 2023, 15(9), 1722; https://doi.org/10.3390/sym15091722 - 8 Sep 2023
Cited by 2 | Viewed by 976
Abstract
In this study, a new fractional-order model for human skull heat conduction is tackled by using a neural network, and the results were further modified by using the hybrid cuckoo search algorithm. In order to understand the temperature distribution, we introduced memory effects [...] Read more.
In this study, a new fractional-order model for human skull heat conduction is tackled by using a neural network, and the results were further modified by using the hybrid cuckoo search algorithm. In order to understand the temperature distribution, we introduced memory effects into our model by using fractional time derivatives. The objective function was constructed in such a way that the L2−error remained at a minimum. The fractional order equation was then calculated by using the proposed biogeography-based hybrid cuckoo search (BHCS) algorithm to approximate the solution. When compared to earlier simulations based on integer-order models, this method enabled us to examine the fractional-order (FO) cases, as well as the integer order. The results are presented in the form of figures and tables for the different case studies. The results obtained for the various parameters were validated numerically against the available literature, where our proposed methodology showed better performance when compared to the least squares method (LSM). Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
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15 pages, 350 KiB  
Article
Averaging Principle for ψ-Capuo Fractional Stochastic Delay Differential Equations with Poisson Jumps
by Dandan Yang, Jingfeng Wang and Chuanzhi Bai
Symmetry 2023, 15(7), 1346; https://doi.org/10.3390/sym15071346 - 1 Jul 2023
Cited by 2 | Viewed by 1265
Abstract
In this paper, we study the averaging principle for ψ-Capuo fractional stochastic delay differential equations (FSDDEs) with Poisson jumps. Based on fractional calculus, Burkholder-Davis-Gundy’s inequality, Doob’s martingale inequality, and the Ho¨lder inequality, we prove that the solution of the [...] Read more.
In this paper, we study the averaging principle for ψ-Capuo fractional stochastic delay differential equations (FSDDEs) with Poisson jumps. Based on fractional calculus, Burkholder-Davis-Gundy’s inequality, Doob’s martingale inequality, and the Ho¨lder inequality, we prove that the solution of the averaged FSDDEs converges to that of the standard FSDDEs in the sense of Lp. Our result extends some known results in the literature. Finally, an example and simulation is performed to show the effectiveness of our result. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
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14 pages, 314 KiB  
Article
Weakly Coupled System of Semi-Linear Fractional θ-Evolution Equations with Special Cauchy Conditions
by Abdelhamid Mohammed Djaouti
Symmetry 2023, 15(7), 1341; https://doi.org/10.3390/sym15071341 - 30 Jun 2023
Cited by 4 | Viewed by 761
Abstract
In this paper, we consider a weakly system of fractional θ-evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for [...] Read more.
In this paper, we consider a weakly system of fractional θ-evolution equations. Using the fixed-point theorem, a global-in-time existence of small data solutions to the Cauchy problem is proved for one single equation. Using these results, we prove the global existence for the system under some mixed symmetrical conditions that describe the interaction between the equations of the system. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
19 pages, 2254 KiB  
Article
Numerical Simulation for a Hybrid Variable-Order Multi-Vaccination COVID-19 Mathematical Model
by Nasser Sweilam, Seham M. Al-Mekhlafi, Reem G. Salama and Tagreed A. Assiri
Symmetry 2023, 15(4), 869; https://doi.org/10.3390/sym15040869 - 6 Apr 2023
Cited by 3 | Viewed by 1554
Abstract
In this paper, a hybrid variable-order mathematical model for multi-vaccination COVID-19 is analyzed. The hybrid variable-order derivative is defined as a linear combination of the variable-order integral of Riemann–Liouville and the variable-order Caputo derivative. A symmetry parameter σ is presented in order to [...] Read more.
In this paper, a hybrid variable-order mathematical model for multi-vaccination COVID-19 is analyzed. The hybrid variable-order derivative is defined as a linear combination of the variable-order integral of Riemann–Liouville and the variable-order Caputo derivative. A symmetry parameter σ is presented in order to be consistent with the physical model problem. The existence, uniqueness, boundedness and positivity of the proposed model are given. Moreover, the stability of the proposed model is discussed. The theta finite difference method with the discretization of the hybrid variable-order operator is developed for solving numerically the model problem. This method can be explicit or fully implicit with a large stability region depending on values of the factor Θ. The convergence and stability analysis of the proposed method are proved. Moreover, the fourth order generalized Runge–Kutta method is also used to study the proposed model. Comparative studies and numerical examples are presented. We found that the proposed model is also more general than the model in the previous study; the results obtained by the proposed method are more stable than previous research in this area. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
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12 pages, 283 KiB  
Article
Studies on Special Polynomials Involving Degenerate Appell Polynomials and Fractional Derivative
by Shahid Ahmad Wani, Kinda Abuasbeh, Georgia Irina Oros and Salma Trabelsi
Symmetry 2023, 15(4), 840; https://doi.org/10.3390/sym15040840 - 31 Mar 2023
Cited by 12 | Viewed by 1377
Abstract
The focus of the research presented in this paper is on a new generalized family of degenerate three-variable Hermite–Appell polynomials defined here using a fractional derivative. The research was motivated by the investigations on the degenerate three-variable Hermite-based Appell polynomials introduced by R. [...] Read more.
The focus of the research presented in this paper is on a new generalized family of degenerate three-variable Hermite–Appell polynomials defined here using a fractional derivative. The research was motivated by the investigations on the degenerate three-variable Hermite-based Appell polynomials introduced by R. Alyosuf. We show in the paper that, for certain values, the well-known degenerate Hermite–Appell polynomials, three-variable Hermite–Appell polynomials and Appell polynomials are seen as particular cases for this new family. As new results of the investigation, the operational rule for this new generalized family is introduced and the explicit summation formula is established. Furthermore, using the determinant formulation of the Appell polynomials, the determinant form for the new generalized family is obtained and the recurrence relations are also determined considering the generating expression of the polynomials contained in the new generalized family. Certain applications of the generalized three-variable Hermite–Appell polynomials are also presented showing the connection with the equivalent results for the degenerate Hermite–Bernoulli and Hermite–Euler polynomials with three variables. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
10 pages, 308 KiB  
Article
Numerical Method for Solving Fractional Order Optimal Control Problems with Free and Non-Free Terminal Time
by Oday I. Al-Shaher, M. Mahmoudi and Mohammed S. Mechee
Symmetry 2023, 15(3), 624; https://doi.org/10.3390/sym15030624 - 2 Mar 2023
Cited by 1 | Viewed by 1277
Abstract
The optimal control theory in mathematics aims to study the finding of control for a dynamic system over time, where an objective function is optimized. It has a broad range of applications in engineering, operations research, and science. The main purpose of this [...] Read more.
The optimal control theory in mathematics aims to study the finding of control for a dynamic system over time, where an objective function is optimized. It has a broad range of applications in engineering, operations research, and science. The main purpose of this study is to provide numerical algorithms for two cases of optimal control problems of fractional order that involve fractional order derivatives with free and non-free terminal time. In addition to comparing the numerical results for three test problems with exact solutions of these problems, various computer simulations are also introduced. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
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19 pages, 329 KiB  
Article
Fractional Weighted Midpoint-Type Inequalities for s-Convex Functions
by Nassima Nasri, Fatima Aissaoui, Keltoum Bouhali, Assia Frioui, Badreddine Meftah, Khaled Zennir and Taha Radwan
Symmetry 2023, 15(3), 612; https://doi.org/10.3390/sym15030612 - 28 Feb 2023
Cited by 6 | Viewed by 1480
Abstract
In the present paper, we first prove a new integral identity. Using that identity, we establish some fractional weighted midpoint-type inequalities for functions whose first derivatives are extended s-convex. Some special cases are discussed. Finally, to prove the effectiveness of our main [...] Read more.
In the present paper, we first prove a new integral identity. Using that identity, we establish some fractional weighted midpoint-type inequalities for functions whose first derivatives are extended s-convex. Some special cases are discussed. Finally, to prove the effectiveness of our main results, we provide some applications to numerical integration as well as special means. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
19 pages, 843 KiB  
Article
Existence of Global and Local Mild Solution for the Fractional Navier–Stokes Equations
by Muath Awadalla, Azhar Hussain, Farva Hafeez and Kinda Abuasbeh
Symmetry 2023, 15(2), 343; https://doi.org/10.3390/sym15020343 - 26 Jan 2023
Cited by 2 | Viewed by 1162
Abstract
Navier–Stokes equations (NS-equations) are applied extensively for the study of various waves phenomena where the symmetries are involved. In this paper, we discuss the NS-equations with the time-fractional derivative of order β∈(0,1). In fractional media, these [...] Read more.
Navier–Stokes equations (NS-equations) are applied extensively for the study of various waves phenomena where the symmetries are involved. In this paper, we discuss the NS-equations with the time-fractional derivative of order β∈(0,1). In fractional media, these equations can be utilized to recreate anomalous diffusion equations which can be used to construct symmetries. We examine the initial value problem involving the symmetric Stokes operator and gravitational force utilizing the Caputo fractional derivative. Additionally, we demonstrate the global and local mild solutions in Hα,p. We also demonstrate the regularity of classical solutions in such circumstances. An example is presented to demonstrate the reliability of our findings. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
16 pages, 329 KiB  
Article
Analysis of Controllability of Fractional Functional Random Integroevolution Equations with Delay
by Kinda Abuasbeh, Ramsha Shafqat, Ammar Alsinai and Muath Awadalla
Symmetry 2023, 15(2), 290; https://doi.org/10.3390/sym15020290 - 20 Jan 2023
Cited by 11 | Viewed by 1260
Abstract
Various scholars have lately employed a wide range of strategies to resolve two specific types of symmetrical fractional differential equations. The evolution of a number of real-world systems in the physical and biological sciences exhibits impulsive dynamical features that can be represented via [...] Read more.
Various scholars have lately employed a wide range of strategies to resolve two specific types of symmetrical fractional differential equations. The evolution of a number of real-world systems in the physical and biological sciences exhibits impulsive dynamical features that can be represented via impulsive differential equations. In this paper, we explore some existence and controllability theories for the Caputo order q∈(1,2) of delay- and random-effect-affected fractional functional integroevolution equations (FFIEEs). In order to prove that random solutions exist, we must prove a random fixed point theorem using a stochastic domain and the mild solution. Then we demonstrate that our solutions are controllable. At the end, applications and example is illustrated which indicates the applicability of this manuscript. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
15 pages, 331 KiB  
Article
Existence and Uniqueness Results for Different Orders Coupled System of Fractional Integro-Differential Equations with Anti-Periodic Nonlocal Integral Boundary Conditions
by Ymnah Alruwaily, Shorog Aljoudi, Lamya Almaghamsi, Abdellatif Ben Makhlouf and Najla Alghamdi
Symmetry 2023, 15(1), 182; https://doi.org/10.3390/sym15010182 - 7 Jan 2023
Cited by 7 | Viewed by 1654
Abstract
This paper presents a new class of boundary value problems of integrodifferential fractional equations of different order equipped with coupled anti-periodic and nonlocal integral boundary conditions. We prove the existence and uniqueness criteria of the solutions by using the Leray-Schauder alternative and Banach [...] Read more.
This paper presents a new class of boundary value problems of integrodifferential fractional equations of different order equipped with coupled anti-periodic and nonlocal integral boundary conditions. We prove the existence and uniqueness criteria of the solutions by using the Leray-Schauder alternative and Banach contraction mapping principle. Examples are constructed for the illustration of our results. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
11 pages, 1034 KiB  
Article
The Exact Solutions of Fractional Differential Systems with n Sinusoidal Terms under Physical Conditions
by Laila F. Seddek, Essam R. El-Zahar and Abdelhalim Ebaid
Symmetry 2022, 14(12), 2539; https://doi.org/10.3390/sym14122539 - 1 Dec 2022
Cited by 1 | Viewed by 1475
Abstract
This paper considers the classes of the first-order fractional differential systems containing a finite number n of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool to solve linear fractional [...] Read more.
This paper considers the classes of the first-order fractional differential systems containing a finite number n of sinusoidal terms. The fractional derivative employs the Riemann–Liouville fractional definition. As a method of solution, the Laplace transform is an efficient tool to solve linear fractional differential equations. However, this method requires to express the initial conditions in certain fractional forms which have no physical meaning currently. This issue formulated a challenge to solve fractional systems under real/physical conditions when applying the Riemann–Liouville fractional definition. The principal incentive of this work is to overcome such difficulties via presenting a simple but effective approach. The proposed approach is successfully applied in this paper to solve linear fractional systems of an oscillatory nature. The exact solutions of the present fractional systems under physical initial conditions are derived in a straightforward manner. In addition, the obtained solutions are given in terms of the entire exponential and periodic functions with arguments of a fractional order. The symmetric/asymmetric behaviors/properties of the obtained solutions are illustrated. Moreover, the exact solutions of the classical/ordinary versions of the undertaken fractional systems are determined smoothly. In addition, the properties and the behaviors of the present solutions are discussed and interpreted. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
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11 pages, 280 KiB  
Article
Some Existence and Uniqueness Results for a Class of Fractional Stochastic Differential Equations
by Omar Kahouli, Abdellatif Ben Makhlouf, Lassaad Mchiri, Pushpendra Kumar, Naim Ben Ali and Ali Aloui
Symmetry 2022, 14(11), 2336; https://doi.org/10.3390/sym14112336 - 7 Nov 2022
Cited by 4 | Viewed by 1808
Abstract
Many techniques have been recently used by various researchers to solve some types of symmetrical fractional differential equations. In this article, we show the existence and uniqueness to the solution of ς-Caputo stochastic fractional differential equations (CSFDE) using the Banach fixed point [...] Read more.
Many techniques have been recently used by various researchers to solve some types of symmetrical fractional differential equations. In this article, we show the existence and uniqueness to the solution of ς-Caputo stochastic fractional differential equations (CSFDE) using the Banach fixed point technique (BFPT). We analyze the Hyers–Ulam stability of CSFDE using the stochastic calculus techniques. We illustrate our results with three examples. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
19 pages, 347 KiB  
Article
Mittag-Leffler Type Stability of Delay Generalized Proportional Caputo Fractional Differential Equations: Cases of Non-Instantaneous Impulses, Instantaneous Impulses and without Impulses
by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan
Symmetry 2022, 14(11), 2290; https://doi.org/10.3390/sym14112290 - 1 Nov 2022
Cited by 1 | Viewed by 1255
Abstract
In this paper, nonlinear differential equations with a generalized proportional Caputo fractional derivative and finite delay are studied in this paper. The eventual presence of impulses in the equations is considered, and the statement of initial value problems in three cases is defined: [...] Read more.
In this paper, nonlinear differential equations with a generalized proportional Caputo fractional derivative and finite delay are studied in this paper. The eventual presence of impulses in the equations is considered, and the statement of initial value problems in three cases is defined: namely non-instantaneous impulses, instantaneous impulses and no impulses. The relations between these three cases are discussed. Additionally, some stability properties are investigated. We apply the Mittag–Leffler function which plays a vital role and which gives well-known bounds on the norm of the solutions. The symmetry of this function about a line and the bounds is a property that plays an important role in stability. Several sufficient conditions are presented via appropriate new comparison results and the modified Razumikhin method. The results generalize several known results in the literature. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
22 pages, 369 KiB  
Article
On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions
by Muath Awadalla, Muthaiah Subramanian, Kinda Abuasbeh and Murugesan Manigandan
Symmetry 2022, 14(11), 2273; https://doi.org/10.3390/sym14112273 - 29 Oct 2022
Cited by 5 | Viewed by 1303
Abstract
In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii’s fixed [...] Read more.
In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii’s fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem’s concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers–Ulam stability are established in the next theorem. We establish the stability of Ulam–Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville–Caputo (ρ) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
11 pages, 255 KiB  
Article
The Fractional Hilbert Transform of Generalized Functions
by Naheed Abdullah and Saleem Iqbal
Symmetry 2022, 14(10), 2096; https://doi.org/10.3390/sym14102096 - 8 Oct 2022
Viewed by 1390
Abstract
The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmetry [...] Read more.
The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmetry to a space of generalized functions known as Boehmians. Moreover, we introduce a new fractional convolutional operator for the fractional Hilbert transform to prove a convolutional theorem similar to the classical Hilbert transform, and also to extend the fractional Hilbert transform to Boehmians. We also produce a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate the convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
20 pages, 372 KiB  
Article
On the Composition Structures of Certain Fractional Integral Operators
by Min-Jie Luo and Ravinder Krishna Raina
Symmetry 2022, 14(9), 1845; https://doi.org/10.3390/sym14091845 - 5 Sep 2022
Viewed by 1235
Abstract
This paper investigates the composition structures of certain fractional integral operators whose kernels are certain types of generalized hypergeometric functions. It is shown how composition formulas of these operators can be closely related to the various Erdélyi-type hypergeometric integrals. We also derive a [...] Read more.
This paper investigates the composition structures of certain fractional integral operators whose kernels are certain types of generalized hypergeometric functions. It is shown how composition formulas of these operators can be closely related to the various Erdélyi-type hypergeometric integrals. We also derive a derivative formula for the fractional integral operator and some applications of the operator are considered for a certain Volterra-type integral equation, which provide two generalizations to Khudozhnikov’s integral equation (see below). Some specific relationships, examples, and some future research problems are also discussed. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
10 pages, 946 KiB  
Article
Numerical Investigation of Nonlinear Shock Wave Equations with Fractional Order in Propagating Disturbance
by Jiahua Fang, Muhammad Nadeem, Mustafa Habib and Ali Akgül
Symmetry 2022, 14(6), 1179; https://doi.org/10.3390/sym14061179 - 8 Jun 2022
Cited by 24 | Viewed by 1995
Abstract
The symmetry design of the system contains integer partial differential equations and fractional-order partial differential equations with fractional derivative. In this paper, we develop a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the [...] Read more.
The symmetry design of the system contains integer partial differential equations and fractional-order partial differential equations with fractional derivative. In this paper, we develop a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the Caputo sense. This scheme is formulated using the Mohand transform (MT) and the homotopy perturbation method (HPM), altogether called Mohand homotopy perturbation transform (MHPT). Our main finding in this paper is the handling of the recurrence relation that produces the series solutions after only a few iterations. This approach presents the approximate and precise solutions in the form of convergent results with certain countable elements, without any discretization or slight perturbation theory. The numerical findings and solution graphs attained using the MHPT confirm that this approach is significant and reliable. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
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13 pages, 1638 KiB  
Article
Approximate Solution of Nonlinear Time-Fractional Klein-Gordon Equations Using Yang Transform
by Jinxing Liu, Muhammad Nadeem, Mustafa Habib and Ali Akgül
Symmetry 2022, 14(5), 907; https://doi.org/10.3390/sym14050907 - 29 Apr 2022
Cited by 18 | Viewed by 2420
Abstract
The algebras of the symmetry operators for the Klein–Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an analytical and numerical approach for [...] Read more.
The algebras of the symmetry operators for the Klein–Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an analytical and numerical approach for providing the solution to a class of linear and nonlinear fractional Klein–Gordon equations arising in classical relativistic and quantum mechanics. We study the Yang homotopy perturbation transform method (YHPTM), which is associated with the Yang transform (YT) and the homotopy perturbation method (HPM), where the fractional derivative is taken in a Caputo–Fabrizio (CF) sense. This technique provides the solution very accurately and efficiently in the form of a series with easily computable coefficients. The behavior of the approximate series solution for different fractional-order ℘ values has been shown graphically. Our numerical investigations indicate that YHPTM is a simple and powerful mathematical tool to deal with the complexity of such problems. Full article
(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus II)
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