Research

16 pages, 328 KiB

Open AccessArticle

Investigating Asymptotic Stability for Hybrid Cubic Integral Inclusion with Fractal Feedback Control on the Real Half-Axis

by Hind H. G. Hashem, Ahmed M. A. El-Sayed and Shorouk M. Al-Issa

Fractal Fract. 2023, 7(6), 449; https://doi.org/10.3390/fractalfract7060449 - 31 May 2023

Cited by 3 | Viewed by 666

Abstract

In this paper, we discuss the existence of solutions for a hybrid cubic delayed integral inclusion with fractal feedback control. We are seeking solutions for these hybrid cubic delayed integral inclusions that are defined, continuous, and bounded on the semi-infinite interval. Our proof [...] Read more.

In this paper, we discuss the existence of solutions for a hybrid cubic delayed integral inclusion with fractal feedback control. We are seeking solutions for these hybrid cubic delayed integral inclusions that are defined, continuous, and bounded on the semi-infinite interval. Our proof is based on the technique associated with measures of noncompactness by a given modulus of continuity in the space in

B C (R_{+}) .

In addition, some sufficient conditions are investigated to demonstrate the asymptotic stability of the solutions of that integral inclusion. Finally, some cases analyzed are in the presence and absence of the control variable, and two examples are provided in order to indicate the validity of the assumptions. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

27 pages, 417 KiB

Open AccessArticle

Total Controllability for a Class of Fractional Hybrid Neutral Evolution Equations with Non-Instantaneous Impulses

by Ahmed Salem and Kholoud N. Alharbi

Fractal Fract. 2023, 7(6), 425; https://doi.org/10.3390/fractalfract7060425 - 24 May 2023

Cited by 1 | Viewed by 4375

Abstract

This study demonstrates the total control of a class of hybrid neutral fractional evolution equations with non-instantaneous impulses and non-local conditions. The boundary value problem with non-local conditions is created using the Caputo fractional derivative of order

1 < α \leq 2

. [...] Read more.

This study demonstrates the total control of a class of hybrid neutral fractional evolution equations with non-instantaneous impulses and non-local conditions. The boundary value problem with non-local conditions is created using the Caputo fractional derivative of order

1 < α \leq 2

. In order to create novel, strongly continuous associated operators, the infinitesimal generator of the sine and cosine families is examined. Additionally, two approaches are used to discuss the solution’s total controllability. A compact strategy based on the non-linear Leray–Schauder alternative theorem is one of them. In contrast, a measure of a non-compactness technique is implemented using the Sadovskii fixed point theorem with the Kuratowski measure of non-compactness. These conclusions are applied using simulation findings for the non-homogeneous fractional wave equation. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

16 pages, 367 KiB

Open AccessArticle

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

by Zhongshu Wu, Xinxia Zhang, Jihan Wang and Xiaoyan Zeng

Fractal Fract. 2023, 7(5), 374; https://doi.org/10.3390/fractalfract7050374 - 30 Apr 2023

Cited by 2 | Viewed by 1007

Abstract

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, [...] Read more.

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, the fractional differential equations were transformed into linear systems, which are easier to solve. Two types of fractional differential equations were used for the numerical simulations, and the numerical results demonstrated the fast convergence and high accuracy of the proposed methods. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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23 pages, 380 KiB

Open AccessArticle

The Rates of Convergence for Functional Limit Theorems with Stable Subordinators and for CTRW Approximations to Fractional Evolutions

by Vassili N. Kolokoltsov

Fractal Fract. 2023, 7(4), 335; https://doi.org/10.3390/fractalfract7040335 - 17 Apr 2023

Cited by 1 | Viewed by 1041

Abstract

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the [...] Read more.

From the initial development of probability theory to the present days, the convergence of various discrete processes to simpler continuous distributions remains at the heart of stochastic analysis. Many efforts have been devoted to functional central limit theorems (also referred to as the invariance principle), dealing with the convergence of random walks to Brownian motion. Though quite a lot of work has been conducted on the rates of convergence of the weighted sums of independent and identically distributed random variables to stable laws, the present paper is the first to supply the rates of convergence in the functional limit theorem for stable subordinators. On the other hand, there is a lot of activity on the convergence of CTRWs (continuous time random walks) to processes with memory (subordinated Markov process) described by fractional PDEs. Our second main result is the first one yielding rates of convergence in such a setting. Since CTRW approximations may be used for numeric solutions of fractional equations, we obtain, as a direct consequence of our results, the estimates for error terms in such numeric schemes. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

16 pages, 873 KiB

Open AccessArticle

Laplace-Residual Power Series Method for Solving Time-Fractional Reaction–Diffusion Model

by Moa’ath N. Oqielat, Tareq Eriqat, Osama Ogilat, Ahmad El-Ajou, Sharifah E. Alhazmi and Shrideh Al-Omari

Fractal Fract. 2023, 7(4), 309; https://doi.org/10.3390/fractalfract7040309 - 02 Apr 2023

Cited by 7 | Viewed by 1322

Abstract

Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the nonlinear time-fractional reaction–diffusion model. [...] Read more.

Despite the fact the Laplace transform has an appreciable efficiency in solving many equations, it cannot be employed to nonlinear equations of any type. This paper presents a modern technique for employing the Laplace transform LT in solving the nonlinear time-fractional reaction–diffusion model. The new approach is called the Laplace-residual power series method (L-RPSM), which imitates the residual power series method in determining the coefficients of the series solution. The proposed method is also adapted to find an approximate series solution that converges to the exact solution of the nonlinear time-fractional reaction–diffusion equations. In addition, the method has been applied to many examples, and the findings are found to be impressive. Further, the results indicate that the L-RPSM is effective, fast, and easy to reach the exact solution of the equations. Furthermore, several actual and approximate solutions are graphically represented to demonstrate the efficiency and accuracy of the proposed method. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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12 pages, 1448 KiB

Open AccessArticle

A Novel Scheme of the ARA Transform for Solving Systems of Partial Fractional Differential Equations

by Aliaa Burqan

Fractal Fract. 2023, 7(4), 306; https://doi.org/10.3390/fractalfract7040306 - 31 Mar 2023

Cited by 1 | Viewed by 784

Abstract

In this article, a new analytical scheme of the ARA transform is introduced to solve systems of fractional partial differential equations. The principle of the proposed technique is based on combining the ARA transform with the residual power series method to create an [...] Read more.

In this article, a new analytical scheme of the ARA transform is introduced to solve systems of fractional partial differential equations. The principle of the proposed technique is based on combining the ARA transform with the residual power series method to create an approximate series solution for a system of partial differential equations of fractional order on the form of a rapid convergent series. To illustrate the effectiveness, accuracy, and validity of the suggested technique, an Attractive physical system, the fractional neutron diffusion equation with one delayed neutrons group, is discussed and solved. Two different neutron flux initial conditions are presented numerically to clarify various cases in order to ensure the theoretical results. The necessary Mathematica codes are run using vital nuclear reactor cross-section data, and the results for various values of time are tabulated and graphically represented. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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22 pages, 436 KiB

Open AccessArticle

Existence of Sobolev-Type Hilfer Fractional Neutral Stochastic Evolution Hemivariational Inequalities and Optimal Controls

by Sivajiganesan Sivasankar, Ramalingam Udhayakumar, Venkatesan Muthukumaran, Saradha Madhrubootham, Ghada AlNemer and Ahmed M. Elshenhab

Fractal Fract. 2023, 7(4), 303; https://doi.org/10.3390/fractalfract7040303 - 30 Mar 2023

Cited by 3 | Viewed by 1387

Abstract

This article concentrates on a control system with a nonlocal condition that is driven by neutral stochastic evolution hemivariational inequalities (HVIs) of Sobolev-type Hilfer fractional (HF). In order to illustrate the necessary requirements for the existence of mild solutions to the required control [...] Read more.

This article concentrates on a control system with a nonlocal condition that is driven by neutral stochastic evolution hemivariational inequalities (HVIs) of Sobolev-type Hilfer fractional (HF). In order to illustrate the necessary requirements for the existence of mild solutions to the required control system, we first use the characteristics of the modified Clarke sub-differential and a fixed point approach for multivalued functions. Then, we show that there are optimal state-control sets that are driven by Sobolev-type HF neutral stochastic evolution HVIs utilizing constrained Lagrange optimal systems. The optimal control (OC) results are created without taking the uniqueness of the control system solutions into account. Finally, the main finding is shown by an example. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

26 pages, 977 KiB

Open AccessArticle

Efficient Spectral Collocation Method for Tempered Fractional Differential Equations

by Tinggang Zhao

Fractal Fract. 2023, 7(3), 277; https://doi.org/10.3390/fractalfract7030277 - 22 Mar 2023

Viewed by 2016

Abstract

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. In this paper, we present a spectral collocation method with tempered fractional Jacobi functions (TFJFs) as basis functions and obtain an efficient algorithm to solve tempered-type fractional differential equations. We set [...] Read more.

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. In this paper, we present a spectral collocation method with tempered fractional Jacobi functions (TFJFs) as basis functions and obtain an efficient algorithm to solve tempered-type fractional differential equations. We set up the approximation error as

O (N^{μ - ν})

for projection and interpolation by the TFJFs, which shows “spectral accuracy” for a certain class of functions. We derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e., the fractional Helmholtz equation, and the fractional Burgers equation. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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18 pages, 1454 KiB

Open AccessArticle

Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended Tanh-Function Method

by Mohammed Bakheet Almatrafi

Fractal Fract. 2023, 7(3), 252; https://doi.org/10.3390/fractalfract7030252 - 10 Mar 2023

Cited by 14 | Viewed by 1158

Abstract

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in [...] Read more.

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in understanding the long-term dynamics of these events. This article primarily focuses on using the improved modified extended tanh-function algorithm to determine certain traveling wave solutions to the space-time fractional symmetric regularized long wave (SRLW) equation, which is used to discuss space-charge waves, shallow water waves, etc. The Jumarie’s modified Riemann-Liouville derivative is successfully used to deal with the fractional derivatives, which appear in the SRLW problem. We find many traveling wave solutions on the form of trigonometric, hyperbolic, complex, and rational functions. Furthermore, the performance of the employed technique is investigated in comparison to other techniques such as the Oncoming

e x p (- Θ (q))

-expansion method and the extended Jacobi elliptic function expansion strategy. Some obtained results are graphically displayed to show their physical features. The findings of this article demonstrate that the used approach enables us to handle more NLFPDEs that emerge in mathematical physics. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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17 pages, 1775 KiB

Open AccessArticle

Analytical Computational Scheme for Multivariate Nonlinear Time-Fractional Generalized Biological Population Model

by Mohammad Alaroud, Abedel-Karrem Alomari, Nedal Tahat and Anuar Ishak

Fractal Fract. 2023, 7(2), 176; https://doi.org/10.3390/fractalfract7020176 - 10 Feb 2023

Cited by 3 | Viewed by 1127

Abstract

This work provides exact and analytical approximate solutions for a non-linear time-fractional generalized biology population model (FGBPM) with suitable initial data under the time-Caputo fractional derivative, in view of a novel effective and applicable scheme, based upon elegant amalgamation between the Laplace transform [...] Read more.

This work provides exact and analytical approximate solutions for a non-linear time-fractional generalized biology population model (FGBPM) with suitable initial data under the time-Caputo fractional derivative, in view of a novel effective and applicable scheme, based upon elegant amalgamation between the Laplace transform operator and the generalized power series method. The solution form obtained by the proposed algorithm of considered FGBPM is an infinite multivariable convergent series toward the exact solutions for the integer fractional order. Some applications of the posed model are tested to confirm the theoretical aspects and highlight the superiority of the proposed scheme in predicting the analytical approximate solutions in closed forms compared to other existing analytical methods. Associated figure representations and the results are displayed in different dimensional graphs. Numerical analyses are performed, and discussions regarding the errors and the convergence of the scheme are presented. The simulations and results report that the proposed modern scheme is, indeed, direct, applicable, and effective to deal with a wide range of non-linear time multivariable fractional models. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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11 pages, 899 KiB

Open AccessArticle

A Family of Transformed Difference Schemes for Nonlinear Time-Fractional Equations

by Hongyu Qin, Xiaoli Chen and Boya Zhou

Fractal Fract. 2023, 7(1), 96; https://doi.org/10.3390/fractalfract7010096 - 14 Jan 2023

Cited by 2 | Viewed by 1229

Abstract

In this paper, we present a class of finite difference methods for numerically solving fractional differential equations. Such numerical schemes are developed based on the change in variable and piecewise interpolations. Error analysis of the numerical schemes is obtained by using a Grönwall-type [...] Read more.

In this paper, we present a class of finite difference methods for numerically solving fractional differential equations. Such numerical schemes are developed based on the change in variable and piecewise interpolations. Error analysis of the numerical schemes is obtained by using a Grönwall-type inequality. Numerical examples are given to confirm the theoretical results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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16 pages, 2941 KiB

Open AccessArticle

Adaptive Stochastic Gradient Descent Method for Convex and Non-Convex Optimization

by Ruijuan Chen, Xiaoquan Tang and Xiuting Li

Fractal Fract. 2022, 6(12), 709; https://doi.org/10.3390/fractalfract6120709 - 29 Nov 2022

Cited by 3 | Viewed by 2009

Abstract

Stochastic gradient descent is the method of choice for solving large-scale optimization problems in machine learning. However, the question of how to effectively select the step-sizes in stochastic gradient descent methods is challenging, and can greatly influence the performance of stochastic gradient descent [...] Read more.

Stochastic gradient descent is the method of choice for solving large-scale optimization problems in machine learning. However, the question of how to effectively select the step-sizes in stochastic gradient descent methods is challenging, and can greatly influence the performance of stochastic gradient descent algorithms. In this paper, we propose a class of faster adaptive gradient descent methods, named AdaSGD, for solving both the convex and non-convex optimization problems. The novelty of this method is that it uses a new adaptive step size that depends on the expectation of the past stochastic gradient and its second moment, which makes it efficient and scalable for big data and high parameter dimensions. We show theoretically that the proposed AdaSGD algorithm has a convergence rate of

O (1 / T)

in both convex and non-convex settings, where T is the maximum number of iterations. In addition, we extend the proposed AdaSGD to the case of momentum and obtain the same convergence rate for AdaSGD with momentum. To illustrate our theoretical results, several numerical experiments for solving problems arising in machine learning are made to verify the promise of the proposed method. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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10 pages, 3047 KiB

Open AccessArticle

The Effect of a Nonlocal Thermoelastic Model on a Thermoelastic Material under Fractional Time Derivatives

by Aatef Hobiny and Ibrahim Abbas

Fractal Fract. 2022, 6(11), 639; https://doi.org/10.3390/fractalfract6110639 - 02 Nov 2022

Cited by 4 | Viewed by 1132

Abstract

This article develops a novel nonlocal theory of generalized thermoelastic material based on fractional time derivatives and Eringen’s nonlocal thermoelasticity. An ultra-short pulse laser heats the surface of the medium’s surrounding plane. Using the Laplace transform method, the basic equations and their accompanying [...] Read more.

This article develops a novel nonlocal theory of generalized thermoelastic material based on fractional time derivatives and Eringen’s nonlocal thermoelasticity. An ultra-short pulse laser heats the surface of the medium’s surrounding plane. Using the Laplace transform method, the basic equations and their accompanying boundary conditions were numerically solved. The distribution of thermal stress, temperature and displacement are physical variables for which the eigenvalues approach was employed to generate the analytical solution. Visual representations were used to examine the influence of the nonlocal parameters and fractional time derivative parameters on the wave propagation distributions of the physical fields for materials. The consideration of the nonlocal thermoelasticity theory (nonlocal elasticity and heat conduction) with fractional time derivatives may lead us to conclude that the variations in physical quantities are considerably impacted. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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17 pages, 344 KiB

Open AccessArticle

Controllability and Hyers–Ulam Stability of Fractional Systems with Pure Delay

by Barakah Almarri, Xingtao Wang and Ahmed M. Elshenhab

Fractal Fract. 2022, 6(10), 611; https://doi.org/10.3390/fractalfract6100611 - 20 Oct 2022

Cited by 4 | Viewed by 1193

Abstract

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for [...] Read more.

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii’s fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

14 pages, 8404 KiB

Open AccessArticle

A Stochastic Bayesian Regularization Approach for the Fractional Food Chain Supply System with Allee Effects

by Basma Souayeh, Zulqurnain Sabir, Najib Hdhiri, Wael Al-Kouz, Mir Waqas Alam and Tarfa Alsheddi

Fractal Fract. 2022, 6(10), 553; https://doi.org/10.3390/fractalfract6100553 - 29 Sep 2022

Cited by 4 | Viewed by 1200

Abstract

This motive of current research is to provide a stochastic platform based on the artificial neural networks (ANNs) along with the Bayesian regularization approach for the fractional food chain supply system (FFSCS) with Allee effects. The investigations based on the fractional derivatives are [...] Read more.

This motive of current research is to provide a stochastic platform based on the artificial neural networks (ANNs) along with the Bayesian regularization approach for the fractional food chain supply system (FFSCS) with Allee effects. The investigations based on the fractional derivatives are applied to achieve the accurate and precise results of FFSCS. The dynamical FFSCS is divided into special predator category

P (η)

, top-predator class

Q (η)

, and prey population dynamics

R (η)

. The computing numerical performances for three different variations of the dynamical FFSCS are provided by using the ANNs along with the Bayesian regularization approach. The data selection for the dynamical FFSCS is selected for train as 78% and 11% for both test and endorsement. The accuracy of the proposed ANNs along with the Bayesian regularization method is approved using the comparison performances. For the rationality, ability, reliability, and exactness are authenticated by using the ANNs procedure enhanced by the Bayesian regularization method through the regression measures, correlation values, error histograms, and transition of state performances. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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21 pages, 423 KiB

Open AccessArticle

A Fast High-Order Predictor–Corrector Method on Graded Meshes for Solving Fractional Differential Equations

by Xinxin Su and Yongtao Zhou

Fractal Fract. 2022, 6(9), 516; https://doi.org/10.3390/fractalfract6090516 - 13 Sep 2022

Cited by 2 | Viewed by 1220

Abstract

In this paper, we focus on the computation of Caputo-type fractional differential equations. A high-order predictor–corrector method is derived by applying the quadratic interpolation polynomial approximation for the integral function. In order to deal with the weak singularity of the solution near the [...] Read more.

In this paper, we focus on the computation of Caputo-type fractional differential equations. A high-order predictor–corrector method is derived by applying the quadratic interpolation polynomial approximation for the integral function. In order to deal with the weak singularity of the solution near the initial time of the fractional differential equations caused by the fractional derivative, graded meshes were used for time discretization. The error analysis of the predictor–corrector method is carefully investigated under suitable conditions on the data. Moreover, an efficient sum-of-exponentials (SOE) approximation to the kernel function was designed to reduce the computational cost. Lastly, several numerical examples are presented to support our theoretical analysis. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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17 pages, 499 KiB

Open AccessArticle

Error Bounds of a Finite Difference/Spectral Method for the Generalized Time Fractional Cable Equation

by Ying Ma and Lizhen Chen

Fractal Fract. 2022, 6(8), 439; https://doi.org/10.3390/fractalfract6080439 - 11 Aug 2022

Cited by 2 | Viewed by 1416

Abstract

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is [...] Read more.

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have

\min {2 - α, 2 - β}

-order convergence in time and spectral accuracy in space for smooth solutions, where

α, β

are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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