Research

13 pages, 2539 KiB

Open AccessArticle

Energy-Preserving AVF Methods for Riesz Space-Fractional Nonlinear KGZ and KGS Equations

by Jianqiang Sun, Siqi Yang and Lijuan Zhang

Fractal Fract. 2023, 7(10), 711; https://doi.org/10.3390/fractalfract7100711 - 27 Sep 2023

Viewed by 753

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained [...] Read more.

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained and solved by the second-order average vector field (AVF) method. The energy conservation property of these new discrete schemes of the fractional KGZ and KGS equations is proven. These schemes are applied to simulate the evolution of two fractional differential equations. Numerical results show that these schemes can simulate the evolution of these fractional differential equations well and maintain the energy-preserving property. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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21 pages, 6876 KiB

Open AccessArticle

Fractional-Order Total Variation Geiger-Mode Avalanche Photodiode Lidar Range-Image Denoising Algorithm Based on Spatial Kernel Function and Range Kernel Function

by Xuyang Wei, Chunyang Wang, Da Xie, Kai Yuan, Xuelian Liu, Zihao Wang, Xinjian Wang and Tingsheng Huang

Fractal Fract. 2023, 7(9), 674; https://doi.org/10.3390/fractalfract7090674 - 07 Sep 2023

Viewed by 775

Abstract

A Geiger-mode avalanche photodiode (GM-APD) laser radar range image has much noise when the signal-to-background ratios (SBRs) are low, making it difficult to recover the real target scene. In this paper, based on the GM-APD lidar denoising model of fractional-order total variation (FOTV), [...] Read more.

A Geiger-mode avalanche photodiode (GM-APD) laser radar range image has much noise when the signal-to-background ratios (SBRs) are low, making it difficult to recover the real target scene. In this paper, based on the GM-APD lidar denoising model of fractional-order total variation (FOTV), the spatial relationship and similarity relationship between pixels are obtained by using a spatial kernel function and range kernel function to optimize the fractional differential operator, and a new FOTV GM-APD lidar range-image denoising algorithm is designed. The lost information and range anomalous noise are suppressed while the target details and contour information are preserved. The Monte Carlo simulation and experimental results show that, under the same SBRs and statistical frame number, the proposed algorithm improves the target restoration degree by at least 5.11% and the peak signal-to-noise ratio (PSNR) by at least 24.6%. The proposed approach can accomplish the denoising of GM-APD lidar range images when SBRs are low. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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23 pages, 1566 KiB

Open AccessArticle

Eighth-Kind Chebyshev Polynomials Collocation Algorithm for the Nonlinear Time-Fractional Generalized Kawahara Equation

by Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri, Amr Kamel Amin and Ahmed Gamal Atta

Fractal Fract. 2023, 7(9), 652; https://doi.org/10.3390/fractalfract7090652 - 29 Aug 2023

Cited by 11 | Viewed by 899

Abstract

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These [...] Read more.

In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through a series of numerical experiments accompanied by comparative assessments. By undertaking these steps, we seek to communicate our findings comprehensively while ensuring the method’s applicability and precision are demonstrated. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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17 pages, 1625 KiB

Open AccessArticle

A Fast θ Scheme Combined with the Legendre Spectral Method for Solving a Fractional Klein–Gordon Equation

by Yanan Li, Yibin Xu, Yanqin Liu and Yanfeng Shen

Fractal Fract. 2023, 7(8), 635; https://doi.org/10.3390/fractalfract7080635 - 20 Aug 2023

Viewed by 697

Abstract

In the current work, a fast

θ

scheme combined with the Legendre spectral method was developed for solving a fractional Klein–Gordon equation (FKGE). The numerical scheme was provided by the Legendre spectral method in the spatial direction, and for the temporal direction, a [...] Read more.

In the current work, a fast

θ

scheme combined with the Legendre spectral method was developed for solving a fractional Klein–Gordon equation (FKGE). The numerical scheme was provided by the Legendre spectral method in the spatial direction, and for the temporal direction, a

θ

scheme of order

O (τ^{2})

with a fast algorithm was taken into account. The fast algorithm could decrease the computational cost from

O (M^{2})

to

O (M log M)

, where M denotes the number of time levels. In addition, correction terms could be employed to improve the convergence rate when the solutions have weak regularity. We proved theoretically that the scheme is unconditionally stable and obtained an error estimate. The numerical experiments demonstrated that our numerical scheme is accurate and efficient. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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20 pages, 1210 KiB

Open AccessArticle

Asymptotic and Pinning Synchronization of Fractional-Order Nonidentical Complex Dynamical Networks with Uncertain Parameters

by Yu Wang, Xiliang He and Tianzeng Li

Fractal Fract. 2023, 7(8), 571; https://doi.org/10.3390/fractalfract7080571 - 25 Jul 2023

Cited by 1 | Viewed by 701

Abstract

This paper is concerned with the asymptotic and pinning synchronization of fractional-order nonidentical complex dynamical networks with uncertain parameters (FONCDNUP). First of all, some synchronization criteria of FONCDNUP are proposed by using the stability of fractional-order dynamical systems and inequality theory. Moreover, a [...] Read more.

This paper is concerned with the asymptotic and pinning synchronization of fractional-order nonidentical complex dynamical networks with uncertain parameters (FONCDNUP). First of all, some synchronization criteria of FONCDNUP are proposed by using the stability of fractional-order dynamical systems and inequality theory. Moreover, a novel controller is derived by using the Lyapunov direct method and the differential inclusion theory. Next, based on the Lyapunov stability theory and pinning control techniques, a new group of sufficient conditions to assure the synchronization for FONCDNUP are obtained by adding controllers to the sub-nodes of networks. At last, two numerical simulations are utilized to illustrate the validity and rationality of the acquired results. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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29 pages, 591 KiB

Open AccessArticle

Probing Families of Optical Soliton Solutions in Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model with Improved Versions of Extended Direct Algebraic Method

by Humaira Yasmin, Noufe H. Aljahdaly, Abdulkafi Mohammed Saeed and Rasool Shah

Fractal Fract. 2023, 7(7), 512; https://doi.org/10.3390/fractalfract7070512 - 28 Jun 2023

Cited by 41 | Viewed by 829

Abstract

In this investigation, we utilize advanced versions of the Extended Direct Algebraic Method (EDAM), namely the modified EDAM (mEDAM) and r+mEDAM, to explore families of optical soliton solutions in the Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model (FPRKLM). Our study stands out due to its [...] Read more.

In this investigation, we utilize advanced versions of the Extended Direct Algebraic Method (EDAM), namely the modified EDAM (mEDAM) and r+mEDAM, to explore families of optical soliton solutions in the Fractional Perturbed Radhakrishnan–Kundu–Lakshmanan Model (FPRKLM). Our study stands out due to its in-depth investigation and the identification of multiple localized and stable soliton families, illuminating their complex behavior. We offer visual validation via carefully designed 3D graphics that capture the complex behaviors of these solitons. The implications of our research extend to fiber optics, communication systems, and nonlinear optics, with the potential for driving developments in optical devices and information processing technologies. This study conveys an important contribution to the field of nonlinear optics, paving the way for future advancements and a greater comprehension of optical solitons and their applications. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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13 pages, 310 KiB

Open AccessArticle

Local Error Estimate of an L1-Finite Difference Scheme for the Multiterm Two-Dimensional Time-Fractional Reaction–Diffusion Equation with Robin Boundary Conditions

by Jian Hou, Xiangyun Meng, Jingjia Wang, Yongsheng Han and Yongguang Yu

Fractal Fract. 2023, 7(6), 453; https://doi.org/10.3390/fractalfract7060453 - 01 Jun 2023

Viewed by 1018

Abstract

In this paper, the numerical method for a multiterm time-fractional reaction–diffusion equation with classical Robin boundary conditions is considered. The full discrete scheme is constructed with the L1-finite difference method, which entails using the L1 scheme on graded meshes for the temporal discretisation [...] Read more.

In this paper, the numerical method for a multiterm time-fractional reaction–diffusion equation with classical Robin boundary conditions is considered. The full discrete scheme is constructed with the L1-finite difference method, which entails using the L1 scheme on graded meshes for the temporal discretisation of each Caputo fractional derivative and using the finite difference method on uniform meshes for spatial discretisation. By dealing with the discretisation of Robin boundary conditions carefully, sharp error analysis at each time level is proven. Additionally, numerical results that can confirm the sharpness of the error estimates are presented. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

18 pages, 494 KiB

Open AccessArticle

Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

by Bo Yu, Xiang Li and Ning Dong

Fractal Fract. 2023, 7(5), 380; https://doi.org/10.3390/fractalfract7050380 - 01 May 2023

Viewed by 984

Abstract

The implicit difference approach is used to discretize a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and [...] Read more.

The implicit difference approach is used to discretize a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and of nonsingular M-matrix structure, respectively. A low-ranked doubling Smith method with determined optimally iterative parameters is presented for solving the corresponding matrix equation. In comparison to the existing Krylov solver with Fast Fourier Transform (FFT) for the sequence Toeplitz linear system, numerical examples demonstrate that the proposed method is more effective on CPU time for solving large-scale problems. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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14 pages, 462 KiB

Open AccessArticle

Fractional-Order Windkessel Boundary Conditions in a One-Dimensional Blood Flow Model for Fractional Flow Reserve (FFR) Estimation

by Timur Gamilov and Ruslan Yanbarisov

Fractal Fract. 2023, 7(5), 373; https://doi.org/10.3390/fractalfract7050373 - 30 Apr 2023

Cited by 1 | Viewed by 1136

Abstract

Recent studies have demonstrated the benefits of using fractional derivatives to simulate a blood pressure profile. In this work we propose to combine a one-dimensional model of coronary blood flow with fractional-order Windkessel boundary conditions. This allows us to obtain a greater variety [...] Read more.

Recent studies have demonstrated the benefits of using fractional derivatives to simulate a blood pressure profile. In this work we propose to combine a one-dimensional model of coronary blood flow with fractional-order Windkessel boundary conditions. This allows us to obtain a greater variety of blood pressure profiles for better model personalization An algorithm of parameter identification is described, which is used to fit the measured mean value of arterial pressure and estimate the fractional flow reserve (FFR) for a given patient. The proposed framework is used to investigate sensitivity of mean blood pressure and fractional flow reserve to fractional order. We demonstrate that the fractional derivative order significantly affects the fractional flow reserve (FFR), which is used as an indicator of stenosis significance. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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13 pages, 2268 KiB

Open AccessArticle

On the Solutions of a Class of Discrete PWC Systems Modeled with Caputo-Type Delta Fractional Difference Equations

by Marius-F. Danca and Jagan Mohan Jonnalagadda

Fractal Fract. 2023, 7(4), 304; https://doi.org/10.3390/fractalfract7040304 - 30 Mar 2023

Viewed by 831

Abstract

In this paper, it is shown that a class of discrete Piece Wise Continuous (PWC) systems with Caputo-type delta fractional difference may not have solutions. To overcome this obstacle, the discontinuous problem is restarted as a continuous fractional problem. First, the single-valued PWC [...] Read more.

In this paper, it is shown that a class of discrete Piece Wise Continuous (PWC) systems with Caputo-type delta fractional difference may not have solutions. To overcome this obstacle, the discontinuous problem is restarted as a continuous fractional problem. First, the single-valued PWC problem is transformed into a set-valued one via Filippov’s theory, after which Cellina’s theorem allows the restart of the problem into a single-valued continuous one. A numerical example is proposed and analyzed. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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24 pages, 530 KiB

Open AccessArticle

Fractional Order Runge–Kutta Methods

by Farideh Ghoreishi, Rezvan Ghaffari and Nasser Saad

Fractal Fract. 2023, 7(3), 245; https://doi.org/10.3390/fractalfract7030245 - 08 Mar 2023

Cited by 2 | Viewed by 1390

Abstract

This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution of fractional differential equations (FDEs). We construct explicit and implicit FORK methods for FDEs by using the Caputo generalized Taylor series formula. Due to the dependence [...] Read more.

This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution of fractional differential equations (FDEs). We construct explicit and implicit FORK methods for FDEs by using the Caputo generalized Taylor series formula. Due to the dependence of fractional derivatives on a fixed base point, in the proposed method, we had to modify the right-hand side of the given equation in all steps of the FORK methods. Some coefficients for explicit and implicit FORK schemes are presented. The convergence analysis of the proposed method is also discussed. Numerical experiments are presented to clarify the effectiveness and robustness of the method. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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19 pages, 889 KiB

Open AccessArticle

An Application of the Distributed-Order Time- and Space-Fractional Diffusion-Wave Equation for Studying Anomalous Transport in Comb Structures

by Lin Liu, Sen Zhang, Siyu Chen, Fawang Liu, Libo Feng, Ian Turner, Liancun Zheng and Jing Zhu

Fractal Fract. 2023, 7(3), 239; https://doi.org/10.3390/fractalfract7030239 - 07 Mar 2023

Cited by 3 | Viewed by 1067

Abstract

A comb structure consists of a one-dimensional backbone with lateral branches. These structures have widespread application in medicine and biology. Such a structure promotes an anomalous diffusion process along the backbone (x-direction), along with classical diffusion along the branches (y [...] Read more.

A comb structure consists of a one-dimensional backbone with lateral branches. These structures have widespread application in medicine and biology. Such a structure promotes an anomalous diffusion process along the backbone (x-direction), along with classical diffusion along the branches (y-direction). In this work, we propose a distributed-order time- and space-fractional diffusion-wave equation to model a comb structure in the more general setting. The distributed-order time- and space-fractional diffusion-wave equation is firstly formulated to study the anomalous diffusion in the comb model subject to an irregular convex domain with the motivation that the time-fractional derivative considers the memory characteristic and the space one with the variable diffusion coefficient possesses the nonlocal characteristic. The finite element method is applied to obtain the numerical solution. The stability and convergence of the numerical discretization scheme are derived and analyzed. Two numerical examples of relevance to the comb model are given to verify the correctness of the numerical method. Moreover, the influence of the involved parameters on the three-dimensional and axial projection drawing particle distribution subject to an elliptical domain are analyzed, and the physical meanings are interpreted in detail. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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19 pages, 659 KiB

Open AccessArticle

A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a General Discontinuous Kernel

by Sharifah E. Alhazmi and Mohamed A. Abdou

Fractal Fract. 2023, 7(2), 173; https://doi.org/10.3390/fractalfract7020173 - 09 Feb 2023

Cited by 4 | Viewed by 1173

Abstract

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, [...] Read more.

In this study, a fractional nonlinear mixed integro-differential equation (Fr-NMIDE) is presented and has a general discontinuous kernel based on position and time space. Conditions of the existence and uniqueness of the solution is provided through the principal form of the integral equation, based on the Banach fixed point theorem. After applying the properties of a fractional integral, the Fr-NMIDE conformed to the Volterra–Hammerstein integral equation (V-HIE) of the second kind, with a general discontinuous kernel in position with the Hammerstein integral term and a continuous kernel in time to the Volterra term. Then, using a technique of the separating method, we obtained HIE, where its physical coefficients were variable in time. The Toeplitz matrix method (TMM) and its schemes were used to obtain a nonlinear algebraic system by studying the convergence of the system. The Maple 18 program was implemented to present the numerical results, along with corresponding errors. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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18 pages, 2366 KiB

Open AccessArticle

Fourth-Order Numerical Solutions for a Fuzzy Time-Fractional Convection–Diffusion Equation under Caputo Generalized Hukuhara Derivative

by Hamzeh Zureigat, Mohammed Al-Smadi, Areen Al-Khateeb, Shrideh Al-Omari and Sharifah E. Alhazmi

Fractal Fract. 2023, 7(1), 47; https://doi.org/10.3390/fractalfract7010047 - 30 Dec 2022

Cited by 10 | Viewed by 1307

Abstract

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting [...] Read more.

The fuzzy fractional differential equation explains more complex real-world phenomena than the fractional differential equation does. Therefore, numerous techniques have been timely derived to solve various fractional time-dependent models. In this paper, we develop two compact finite difference schemes and employ the resulting schemes to obtain a certain solution for the fuzzy time-fractional convection–diffusion equation. Then, by making use of the Caputo fractional derivative, we provide new fuzzy analysis relying on the concept of fuzzy numbers. Further, we approximate the time-fractional derivative by using a fuzzy Caputo generalized Hukuhara derivative under the double-parametric form of fuzzy numbers. Furthermore, we introduce new computational techniques, based on fuzzy double-parametric form, to shift the given problem from one fuzzy domain to another crisp domain. Moreover, we discuss some stability and error analysis for the proposed techniques by using the Fourier method. Over and above, we derive several numerical experiments to illustrate reliability and feasibility of our proposed approach. It was found that the fuzzy fourth-order compact implicit scheme produces slightly better results than the fourth-order compact FTCS scheme. Furthermore, the proposed methods were found to be feasible, appropriate, and accurate, as demonstrated by a comparison of analytical and numerical solutions at various fuzzy values. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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20 pages, 1223 KiB

Open AccessArticle

A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

by Yusuf O. Afolabi, Toheeb A. Biala, Olaniyi S. Iyiola, Abdul Q. M. Khaliq and Bruce A. Wade

Fractal Fract. 2023, 7(1), 40; https://doi.org/10.3390/fractalfract7010040 - 30 Dec 2022

Cited by 1 | Viewed by 1495

Abstract

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order

α,

), where time accuracy is of the order [...] Read more.

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order

α,

), where time accuracy is of the order

(2 - α)

or

(1 + α)

. To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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9 pages, 307 KiB

Open AccessArticle

Sequential Caputo–Hadamard Fractional Differential Equations with Boundary Conditions in Banach Spaces

by Ramasamy Arul, Panjayan Karthikeyan, Kulandhaivel Karthikeyan, Ymnah Alruwaily, Lamya Almaghamsi and El-sayed El-hady

Fractal Fract. 2022, 6(12), 730; https://doi.org/10.3390/fractalfract6120730 - 10 Dec 2022

Cited by 4 | Viewed by 1155

Abstract

We present the existence of solutions for sequential Caputo–Hadamard fractional differential equations (SC-HFDE) with fractional boundary conditions (FBCs). Known fixed-point techniques are used to analyze the existence of the problem. In particular, the contraction mapping principle is used to investigate the uniqueness results. [...] Read more.

We present the existence of solutions for sequential Caputo–Hadamard fractional differential equations (SC-HFDE) with fractional boundary conditions (FBCs). Known fixed-point techniques are used to analyze the existence of the problem. In particular, the contraction mapping principle is used to investigate the uniqueness results. Existence results are obtained via Krasnoselkii’s theorem. An example is used to illustrate the results. In this way, our work generalizes several recent interesting results. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

17 pages, 3516 KiB

Open AccessArticle

Analytical Solution of Coupled Hirota–Satsuma and KdV Equations

by Rania Saadeh, Osama Ala’yed and Ahmad Qazza

Fractal Fract. 2022, 6(12), 694; https://doi.org/10.3390/fractalfract6120694 - 23 Nov 2022

Cited by 12 | Viewed by 1447

Abstract

In this study, we applied the Laplace residual power series method (LRPSM) to expand the solution of the nonlinear time-fractional coupled Hirota–Satsuma and KdV equations in the form of a rapidly convergent series while considering Caputo fractional derivatives. We demonstrate the applicability and [...] Read more.

In this study, we applied the Laplace residual power series method (LRPSM) to expand the solution of the nonlinear time-fractional coupled Hirota–Satsuma and KdV equations in the form of a rapidly convergent series while considering Caputo fractional derivatives. We demonstrate the applicability and accuracy of the proposed method with some examples. The numerical results and the graphical representations reveal that the proposed method performs extremely well in terms of efficiency and simplicity. Therefore, it can be utilized to solve more problems in the field of non-linear fractional differential equations. To show the validity of the proposed method, we present a numerical application, compute two kinds of errors, and sketch figures of the obtained results. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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23 pages, 5583 KiB

Open AccessArticle

Numerical Simulations of the Oscillating Second-Grade Fluid through a Rectangular Cross Duct with Fractional Constitution Relationship

by Bo Zhang, Lin Liu, Siyu Chen, Sen Zhang, Lang Liu, Libo Feng, Jing Zhu, Jiangshan Zhang and Liancun Zheng

Fractal Fract. 2022, 6(11), 666; https://doi.org/10.3390/fractalfract6110666 - 11 Nov 2022

Viewed by 1044

Abstract

An oscillating second-grade fluid through a rectangular cross duct is studied. A traditional integer time derivative in the kinematic tensors is substituted by a fractional operator that considers the memory characteristics. To treat the fractional governing equation, an analytical method was obtained. To [...] Read more.

An oscillating second-grade fluid through a rectangular cross duct is studied. A traditional integer time derivative in the kinematic tensors is substituted by a fractional operator that considers the memory characteristics. To treat the fractional governing equation, an analytical method was obtained. To analyze the impact of the parameters more intuitively, the difference method was applied to determine the numerical expression and draw with the help of computer simulation. To reduce the cost of the amount of computation and storage, a fast scheme was proposed, one which can greatly improve the calculation speed. To verify the correctness of the difference scheme, the contrast between the numerical expression and the exact expression—constructed by introducing a source term—was given and the superiority of the fast scheme is discussed. Furthermore, the influences of the involved parameters, including the parameter of retardation time, fractional parameter, magnetic parameter, and oscillatory frequency parameter, on the distributions of velocity and shear force at the wall surface with oscillatory flow are analyzed in detail. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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27 pages, 7020 KiB

Open AccessArticle

A Novel Analytical LRPSM for Solving Nonlinear Systems of FPDEs

by Hussam Aljarrah, Mohammad Alaroud, Anuar Ishak and Maslina Darus

Fractal Fract. 2022, 6(11), 650; https://doi.org/10.3390/fractalfract6110650 - 04 Nov 2022

Cited by 2 | Viewed by 986

Abstract

This article employs the Laplace residual power series approach to study nonlinear systems of time-fractional partial differential equations with time-fractional Caputo derivative. The proposed technique is based on a new fractional expansion of the Maclurian series, which provides a rapid convergence series solution [...] Read more.

This article employs the Laplace residual power series approach to study nonlinear systems of time-fractional partial differential equations with time-fractional Caputo derivative. The proposed technique is based on a new fractional expansion of the Maclurian series, which provides a rapid convergence series solution where the coefficients of the proposed fractional expansion are computed with the limit concept. The nonlinear systems studied in this work are the Broer-Kaup system, the Burgers’ system of two variables, and the Burgers’ system of three variables, which are used in modeling various nonlinear physical applications such as shock waves, processes of the wave, transportation of vorticity, dispersion in porous media, and hydrodynamic turbulence. The results obtained are reliable, efficient, and accurate with minimal computations. The proposed technique is analyzed by applying it to three attractive problems where the approximate analytical solutions are formulated in rapid convergent fractional Maclurian formulas. The results are studied numerically and graphically to show the performance and validity of the technique, as well as the fractional order impact on the behavior of the solutions. Moreover, numerical comparisons are made with other well-known methods, proving that the results obtained in the proposed technique are much better and the most accurate. Finally, the obtained outcomes and simulation data show that the present method provides a sound methodology and suitable tool for solving such nonlinear systems of time-fractional partial differential equations. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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11 pages, 1477 KiB

Open AccessArticle

Analytical Solutions of the Nonlinear Time-Fractional Coupled Boussinesq-Burger Equations Using Laplace Residual Power Series Technique

by Aref Sarhan, Aliaa Burqan, Rania Saadeh and Zeyad Al-Zhour

Fractal Fract. 2022, 6(11), 631; https://doi.org/10.3390/fractalfract6110631 - 29 Oct 2022

Cited by 8 | Viewed by 1646

Abstract

In this paper, we present the series solutions of the nonlinear time-fractional coupled Boussinesq-Burger equations (T-FCB-BEs) using Laplace-residual power series (L-RPS) technique in the sense of Caputo fractional derivative (C-FD). To assert the efficiency, simplicity, performance, and reliability of our proposed method, an [...] Read more.

In this paper, we present the series solutions of the nonlinear time-fractional coupled Boussinesq-Burger equations (T-FCB-BEs) using Laplace-residual power series (L-RPS) technique in the sense of Caputo fractional derivative (C-FD). To assert the efficiency, simplicity, performance, and reliability of our proposed method, an attractive and interesting numerical example is tested analytically and graphically. In addition, our obtained results show that this algorithm is compatible and accurate for investigating the fractional-order solutions of engineering and physical applications. Finally, Mathematica software 14 is applied to compute the numerical and graphical results. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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17 pages, 320 KiB

Open AccessArticle

A Finite-State Stationary Process with Long-Range Dependence and Fractional Multinomial Distribution

by Jeonghwa Lee

Fractal Fract. 2022, 6(10), 596; https://doi.org/10.3390/fractalfract6100596 - 14 Oct 2022

Viewed by 906

Abstract

We propose a discrete-time, finite-state stationary process that can possess long-range dependence. Among the interesting features of this process is that each state can have different long-term dependency, i.e., the indicator sequence can have a different Hurst index for different states. Furthermore, inter-arrival [...] Read more.

We propose a discrete-time, finite-state stationary process that can possess long-range dependence. Among the interesting features of this process is that each state can have different long-term dependency, i.e., the indicator sequence can have a different Hurst index for different states. Furthermore, inter-arrival time for each state follows heavy tail distribution, with different states showing different tail behavior. A possible application of this process is to model over-dispersed multinomial distribution. In particular, we define a fractional multinomial distribution from our model. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

13 pages, 2181 KiB

Open AccessArticle

A Fourth-Order Time-Stepping Method for Two-Dimensional, Distributed-Order, Space-Fractional, Inhomogeneous Parabolic Equations

by Muhammad Yousuf, Khaled M. Furati and Abdul Q. M. Khaliq

Fractal Fract. 2022, 6(10), 592; https://doi.org/10.3390/fractalfract6100592 - 13 Oct 2022

Viewed by 1076

Abstract

Distributed-order, space-fractional diffusion equations are used to describe physical processes that lack power-law scaling. A fourth-order-accurate, A-stable time-stepping method was developed, analyzed, and implemented to solve inhomogeneous parabolic problems having Riesz-space-fractional, distributed-order derivatives. The considered problem was transformed into a multi-term, space-fractional [...] Read more.

Distributed-order, space-fractional diffusion equations are used to describe physical processes that lack power-law scaling. A fourth-order-accurate, A-stable time-stepping method was developed, analyzed, and implemented to solve inhomogeneous parabolic problems having Riesz-space-fractional, distributed-order derivatives. The considered problem was transformed into a multi-term, space-fractional problem using Simpson’s three-eighths rule. The method is based on an approximation of matrix exponential functions using fourth-order diagonal Padé approximation. The Gaussian quadrature approach is used to approximate the integral matrix exponential function, along with the inhomogeneous term. Partial fraction splitting is used to address the issues regarding stability and computational efficiency. Convergence of the method was proved analytically and demonstrated through numerical experiments. CPU time was recorded in these experiments to show the computational efficiency of the method. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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13 pages, 2326 KiB

Open AccessArticle

Mixed Convection of Fractional Nanofluids Considering Brownian Motion and Thermophoresis

by Mingwen Chen, Yefan Tian, Weidong Yang and Xuehui Chen

Fractal Fract. 2022, 6(10), 584; https://doi.org/10.3390/fractalfract6100584 - 12 Oct 2022

Viewed by 1305

Abstract

In this paper, the mixed convective heat transfer mechanism of nanofluids is investigated. Based on the Buongiorno model, we develop a novel Cattaneo–Buongiorno model that reflects the non-local properties as well as Brownian motion and thermophoresis diffusion. Due to the highly non-linear character [...] Read more.

In this paper, the mixed convective heat transfer mechanism of nanofluids is investigated. Based on the Buongiorno model, we develop a novel Cattaneo–Buongiorno model that reflects the non-local properties as well as Brownian motion and thermophoresis diffusion. Due to the highly non-linear character of the equations, the finite difference method is employed to numerically solve the governing equations. The effectiveness of the numerical method and the convergence order are presented. The results show that the rise in the fractional parameter

δ

enhances the energy transfer process of nanofluids, while the fractional parameter

γ

has the opposite effect. In addition, the effects of Brownian motion and thermophoresis diffusion parameters are also discussed. We infer that the flow and heat transfer mechanism of the viscoelastic nanofluids can be more clearly revealed by controlling the parameters in the Cattaneo–Buongiorno model. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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13 pages, 321 KiB

Open AccessArticle

Initial Boundary Value Problem for a Fractional Viscoelastic Equation of the Kirchhoff Type

by Yang Liu and Li Zhang

Fractal Fract. 2022, 6(10), 581; https://doi.org/10.3390/fractalfract6100581 - 11 Oct 2022

Cited by 1 | Viewed by 1046

Abstract

In this paper, we study the initial boundary value problem for a fractional viscoelastic equation of the Kirchhoff type. In suitable functional spaces, we define a potential well. In the framework of the potential well theory, we obtain the global existence of solutions [...] Read more.

In this paper, we study the initial boundary value problem for a fractional viscoelastic equation of the Kirchhoff type. In suitable functional spaces, we define a potential well. In the framework of the potential well theory, we obtain the global existence of solutions by using the Galerkin approximations. Moreover, we derive the asymptotic behavior of solutions by means of the perturbed energy method. Our main results provide sufficient conditions for the qualitative properties of solutions in time. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

14 pages, 295 KiB

Open AccessArticle

Existence of Solutions to a System of Riemann-Liouville Fractional Differential Equations with Coupled Riemann-Stieltjes Integrals Boundary Conditions

by Yuan Ma and Dehong Ji

Fractal Fract. 2022, 6(10), 543; https://doi.org/10.3390/fractalfract6100543 - 26 Sep 2022

Cited by 2 | Viewed by 908

Abstract

A general system of fractional differential equations with coupled fractional Stieltjes integrals and a Riemann–Liouville fractional integral in boundary conditions is studied in the context of pattern formation. We need to transform the fractional differential system into the corresponding integral operator to obtain [...] Read more.

A general system of fractional differential equations with coupled fractional Stieltjes integrals and a Riemann–Liouville fractional integral in boundary conditions is studied in the context of pattern formation. We need to transform the fractional differential system into the corresponding integral operator to obtain the existence and uniqueness of solutions for the system. The contraction mapping principle in Banach space and the alternative theorem of Leray–Schauder are applied. Finally, we give two applications to illustrate our theoretical results. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

23 pages, 362 KiB

Open AccessArticle

An Implicit Difference Scheme for the Fourth-Order Nonlinear Evolution Equation with Multi-Term Riemann–Liouvile Fractional Integral Kernels

by Xiaoxuan Jiang, Xuehua Yang, Haixiang Zhang and Qingqing Tian

Fractal Fract. 2022, 6(8), 443; https://doi.org/10.3390/fractalfract6080443 - 15 Aug 2022

Cited by 1 | Viewed by 1109

Abstract

In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by [...] Read more.

In this paper, an implicit difference scheme is proposed and analyzed for a class of nonlinear fourth-order equations with the multi-term Riemann–Liouvile (R–L) fractional integral kernels. For the nonlinear convection term, we handle implicitly and attain a system of nonlinear algebraic equations by using the Galerkin method based on piecewise linear test functions. The Riemann–Liouvile fractional integral terms are treated by convolution quadrature. In order to obtain a fully discrete method, the standard central difference approximation is used to discretize the spatial derivative. The stability and convergence are rigorously proved by the discrete energy method. In addition, the existence and uniqueness of numerical solutions for nonlinear systems are proved strictly. Additionally, we introduce and compare the Besse relaxation algorithm, the Newton iterative method, and the linearized iterative algorithm for solving the nonlinear systems. Numerical results confirm the theoretical analysis and show the effectiveness of the method. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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16 pages, 614 KiB

Open AccessArticle

The Construction of High-Order Robust Theta Methods with Applications in Subdiffusion Models

by Baoli Yin, Guoyu Zhang, Yang Liu and Hong Li

Fractal Fract. 2022, 6(8), 417; https://doi.org/10.3390/fractalfract6080417 - 29 Jul 2022

Viewed by 1023

Abstract

An exponential-type function was discovered to transform known difference formulas by involving a shifted parameter

θ

to approximate fractional calculus operators. In contrast to the known

θ

methods obtained by polynomial-type transformations, our exponential-type

θ

methods take the advantage of the fact that [...] Read more.

An exponential-type function was discovered to transform known difference formulas by involving a shifted parameter

θ

to approximate fractional calculus operators. In contrast to the known

θ

methods obtained by polynomial-type transformations, our exponential-type

θ

methods take the advantage of the fact that they have no restrictions in theory on the range of

θ

such that the resultant scheme is asymptotically stable. As an application to investigate the subdiffusion problem, the second-order fractional backward difference formula is transformed, and correction terms are designed to maintain the optimal second-order accuracy in time. The obtained exponential-type scheme is robust in that it is accurate even for very small

α

and can naturally resolve the initial singularity provided

θ = - \frac{1}{2}

, both of which are demonstrated rigorously. All theoretical results are confirmed by extensive numerical tests. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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22 pages, 406 KiB

Open AccessArticle

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

by Zhen Wang, Luhan Sun and Jianxiong Cao

Fractal Fract. 2022, 6(7), 349; https://doi.org/10.3390/fractalfract6070349 - 22 Jun 2022

Cited by 1 | Viewed by 1194

Abstract

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order

α \in (0, 1)

. Considering the weak singularity of the solution

u (x, t)

[...] Read more.

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order

α \in (0, 1)

. Considering the weak singularity of the solution

u (x, t)

at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as

t \to 0^{+}

albeit the function itself being right continuous at

t = 0

, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-

1_{σ}

formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

15 pages, 1578 KiB

Open AccessArticle

Supervised Neural Network Procedures for the Novel Fractional Food Supply Model

by Basma Souayeh, Zulqurnain Sabir, Muhammad Umar and Mir Waqas Alam

Fractal Fract. 2022, 6(6), 333; https://doi.org/10.3390/fractalfract6060333 - 16 Jun 2022

Cited by 11 | Viewed by 1502

Abstract

This work presents the numerical performances of the fractional kind of food supply (FKFS) model. The fractional kinds of the derivatives have been used to acquire the accurate and realistic solutions of the FKFS model. The FKFSM system contains three types, special kind [...] Read more.

This work presents the numerical performances of the fractional kind of food supply (FKFS) model. The fractional kinds of the derivatives have been used to acquire the accurate and realistic solutions of the FKFS model. The FKFSM system contains three types, special kind of the predator L(x), top-predator M(x) and prey populations N(x). The numerical solutions of three different cases of the FKFS model are provided through the stochastic procedures of the scaled conjugate gradient neural networks (SCGNNs). The data selection for the FKFS model is chosen as 82%, for training and 9% for both testing and authorization. The precision of the designed SCGNNs is provided through the achieved and Adam solutions. To rationality, competence, constancy, and correctness is approved by using the stochastic SCGNNs along with the simulations of the regression actions, mean square error, correlation performances, error histograms values and state transition measures. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations)

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