Fractal and Fractional

Research

22 pages, 1032 KiB

Open AccessArticle

Properties and Applications of Complex Fractal–Fractional Operators in the Open Unit Disk

by Adel A. Attiya, Soheil Salahshour, Rabha W. Ibrahim and Mansour F. Yassen

Fractal Fract. 2024, 8(10), 584; https://doi.org/10.3390/fractalfract8100584 - 3 Oct 2024

Viewed by 524

Abstract

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function:

Γ_{ℓ} (.)

). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows [...] Read more.

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function:

Γ_{ℓ} (.)

). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete

ℓ -

fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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14 pages, 1503 KiB

Open AccessArticle

Modelling Yeast Prion Dynamics: A Fractional Order Approach with Predictor–Corrector Algorithm

by Daasara Keshavamurthy Archana, Doddabhadrappla Gowda Prakasha and Nasser Bin Turki

Fractal Fract. 2024, 8(9), 542; https://doi.org/10.3390/fractalfract8090542 - 19 Sep 2024

Viewed by 484

Abstract

This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Predictor–Corrector approach. An analysis has [...] Read more.

This work aims to comprehend the dynamics of neurodegenerative disease using a mathematical model of fractional-order yeast prions. In the context of the Caputo fractional derivative, we here study and examine the solution of this model using the Predictor–Corrector approach. An analysis has been conducted on the existence and uniqueness of the selected model. Also, we examined the model’s stability and the existence of equilibrium points. With the purpose of analyzing the dynamics of the Sup35 monomer and Sup35 prion population, we displayed the graphs to show the obtained solutions over time. Graphical simulations show that the behaviour of the populations can change based on fractional orders and threshold parameter values. This work may present a good example of how biological theories and data can be better understood via mathematical modelling. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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12 pages, 325 KiB

Open AccessArticle

L^p(L^q)-Maximal Regularity for Damped Equations in a Cylindrical Domain

by Edgardo Alvarez, Stiven Díaz and Carlos Lizama

Fractal Fract. 2024, 8(9), 516; https://doi.org/10.3390/fractalfract8090516 - 30 Aug 2024

Viewed by 728

Abstract

We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are [...] Read more.

We show maximal regularity estimates for the damped hyperbolic and strongly damped wave equations with periodic initial conditions in a cylindrical domain. We prove that this property strongly depends on a critical combination on the parameters of the equation. Noteworthy, our results are still valid for fractional powers of the negative Laplacian operator. We base our methods on the theory of operator-valued Fourier multipliers on vector-valued Lebesgue spaces of periodic functions. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

26 pages, 657 KiB

Open AccessArticle

Spectral Galerkin Methods for Riesz Space-Fractional Convection–Diffusion Equations

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

Fractal Fract. 2024, 8(7), 431; https://doi.org/10.3390/fractalfract8070431 - 22 Jul 2024

Cited by 1 | Viewed by 758

Abstract

This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using [...] Read more.

This paper applies the spectral Galerkin method to numerically solve Riesz space-fractional convection–diffusion equations. Firstly, spectral Galerkin algorithms were developed for one-dimensional Riesz space-fractional convection–diffusion equations. The equations were solved by discretizing in space using the Galerkin–Legendre spectral approaches and in time using the Crank–Nicolson Leap-Frog (CNLF) scheme. In addition, the stability and convergence of semi-discrete and fully discrete schemes were analyzed. Secondly, we established a fully discrete form for the two-dimensional case with an additional complementary term on the left and then obtained the stability and convergence results for it. Finally, numerical simulations were performed, and the results demonstrate the effectiveness of our numerical methods. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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25 pages, 3439 KiB

Open AccessArticle

Split-Step Galerkin FE Method for Two-Dimensional Space-Fractional CNLS

by Xiaogang Zhu, Yaping Zhang and Yufeng Nie

Fractal Fract. 2024, 8(7), 402; https://doi.org/10.3390/fractalfract8070402 - 5 Jul 2024

Viewed by 928

Abstract

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in [...] Read more.

In this paper, we study a split-step Galerkin finite element (FE) method for the two-dimensional Riesz space-fractional coupled nonlinear Schrödinger equations (CNLSs). The proposed method adopts a second-order split-step technique to handle the nonlinearity and FE approximation to discretize the fractional derivatives in space, which avoids iteration at each time layer. The analysis of mass conservative and convergent properties for this split-step FE scheme is performed. To test its capability, some numerical tests and the simulation of the double solitons intersection and plane wave are carried out. The results and comparisons with the algorithm combined with Newton’s iteration illustrate its effectiveness and advantages in computational efficiency. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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10 pages, 290 KiB

Open AccessArticle

Time-Stepping Error Estimates of Linearized Grünwald–Letnikov Difference Schemes for Strongly Nonlinear Time-Fractional Parabolic Problems

by Hongyu Qin, Lili Li, Yuanyuan Li and Xiaoli Chen

Fractal Fract. 2024, 8(7), 390; https://doi.org/10.3390/fractalfract8070390 - 29 Jun 2024

Cited by 2 | Viewed by 733

Abstract

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through [...] Read more.

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several 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, 2nd Edition)

10 pages, 972 KiB

Open AccessArticle

Ultrafast Diffusion Modeling via the Riemann–Liouville Nonlocal Structural Derivative and Its Application in Porous Media

by Wei Xu, Hui Liu, Lijuan Chen and Yongtao Zhou

Fractal Fract. 2024, 8(2), 110; https://doi.org/10.3390/fractalfract8020110 - 12 Feb 2024

Viewed by 1289

Abstract

Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of [...] Read more.

Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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13 pages, 460 KiB

Open AccessArticle

Error Analysis of the Nonuniform Alikhanov Scheme for the Fourth-Order Fractional Diffusion-Wave Equation

by Zihao An and Chaobao Huang

Fractal Fract. 2024, 8(2), 106; https://doi.org/10.3390/fractalfract8020106 - 10 Feb 2024

Viewed by 1250

Abstract

This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a [...] Read more.

This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a weak singularity at the initial time. Using the order reduction method, the fourth-order fractional diffusion-wave equation can be rewritten as a coupled system of low order, which is approximated by the nonuniform Alikhanov scheme in time and the finite difference method in space. Furthermore, the

H^{2}

-norm stability result is obtained. With the help of this result and a priori bounds of the solution, an

α

-robust error estimate with optimal convergence order is derived. In order to further verify the accuracy of our theoretical analysis, some numerical results are provided. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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17 pages, 417 KiB

Open AccessArticle

Collocation-Based Approximation for a Time-Fractional Sub-Diffusion Model

by Kaido Lätt, Arvet Pedas, Hanna Britt Soots and Mikk Vikerpuur

Fractal Fract. 2023, 7(9), 657; https://doi.org/10.3390/fractalfract7090657 - 31 Aug 2023

Viewed by 933

Abstract

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary [...] Read more.

We consider the numerical solution of a one-dimensional time-fractional diffusion problem, where the order of the Caputo time derivative belongs to (0, 1). Using the technique of the method of lines, we first develop from the original problem a system of fractional ordinary differential equations. Using an integral equation reformulation of this system, we study the regularity properties of the exact solution of the system of fractional differential equations and construct a piecewise polynomial collocation method to solve it numerically. We also investigate the convergence and the convergence order of the proposed method. To conclude, we present the results of some numerical experiments. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

10 pages, 296 KiB

Open AccessArticle

Existence and Stability Results for Piecewise Caputo–Fabrizio Fractional Differential Equations with Mixed Delays

by Doha A. Kattan and Hasanen A. Hammad

Fractal Fract. 2023, 7(9), 644; https://doi.org/10.3390/fractalfract7090644 - 24 Aug 2023

Cited by 5 | Viewed by 1316

Abstract

In this article, by using the differential Caputo–Fabrizio operator, we suggest a novel family of piecewise differential equations (DEs). The issue under study contains a mixed delay period under the criteria of anti-periodic boundaries. It is possible to utilize the piecewise derivative to [...] Read more.

In this article, by using the differential Caputo–Fabrizio operator, we suggest a novel family of piecewise differential equations (DEs). The issue under study contains a mixed delay period under the criteria of anti-periodic boundaries. It is possible to utilize the piecewise derivative to describe a variety of complex, multi-step, real-world situations that arise from nature. Using fixed point (FP) techniques, like Banach’s FP theorem, Schauder’s FP theorem, and Arzelá Ascoli’s FP theorem, the Hyer–Ulam (HU) stability and the existence theorem conclusions are investigated for the considered problem. Eventually, a supportive example is given to demonstrate the applicability and efficacy of the applied concept. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications, 2nd Edition)

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