Research

19 pages, 401 KiB

Open AccessArticle

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

by Mustafa Zeki, Ramazan Tinaztepe, Salih Tatar, Suleyman Ulusoy and Rami Al-Hajj

Fractal Fract. 2023, 7(5), 371; https://doi.org/10.3390/fractalfract7050371 - 29 Apr 2023

Cited by 1 | Viewed by 1011

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has [...] Read more.

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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23 pages, 381 KiB

Open AccessArticle

Stabilization of a Non-linear Fractional Problem by a Non-Integer Frictional Damping and a Viscoelastic Term

by Banan Al-Homidan and Nasser-eddine Tatar

Fractal Fract. 2023, 7(5), 367; https://doi.org/10.3390/fractalfract7050367 - 28 Apr 2023

Cited by 1 | Viewed by 790

Abstract

In this paper, we consider a fractional equation of order between one and two which may be looked at as an interpolation between the heat and wave equations. The problem is non-linear as it involves a power-type non-linearity. We investigate the possibilities of [...] Read more.

In this paper, we consider a fractional equation of order between one and two which may be looked at as an interpolation between the heat and wave equations. The problem is non-linear as it involves a power-type non-linearity. We investigate the possibilities of stabilizing the system by a lower-order fractional term and/or a memory term involving the Laplacian. We prove a global Mittag–Leffler stability result in case a fractional frictional damping is active and a local Mittag–Leffler stability result when the material is viscoelastic in case of small relaxation functions. Unlike the integer-order problems, additional serious difficulties arise in the present case. These difficulties are highlighted clearly in the introduction. They are mainly due to the memory dependence of the fractional derivatives which is the cause of the invalidity of the product rule in particular. We utilize several properties in fractional calculus. Moreover, we introduce new Lyapunov-type functionals in the context of the multiplier technique. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

21 pages, 441 KiB

Open AccessArticle

Compact Difference Schemes with Temporal Uniform/Non-Uniform Meshes for Time-Fractional Black–Scholes Equation

by Jie Gu, Lijuan Nong, Qian Yi and An Chen

Fractal Fract. 2023, 7(4), 340; https://doi.org/10.3390/fractalfract7040340 - 19 Apr 2023

Cited by 2 | Viewed by 974

Abstract

In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in the equivalent form using the piecewise linear interpolation, while the [...] Read more.

In this paper, we are interested in the effective numerical schemes of the time-fractional Black–Scholes equation. We convert the original equation into an equivalent integral-differential equation and then discretize the time-integral term in the equivalent form using the piecewise linear interpolation, while the compact difference formula is applied in the spatial direction. Thus, we derive a fully discrete compact difference scheme with second-order accuracy in time and fourth-order accuracy in space. Rigorous proofs of the corresponding stability and convergence are given. Furthermore, in order to deal effectively with the non-smooth solution, we extend the obtained results to the case of temporal non-uniform meshes and obtain a temporal non-uniform mesh-based compact difference scheme as well as the numerical theory. Finally, extensive numerical examples are included to demonstrate the effectiveness of the proposed compact difference schemes. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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15 pages, 586 KiB

Open AccessArticle

Fast and Accurate Numerical Algorithm with Performance Assessment for Nonlinear Functional Volterra Equations

by Chinedu Nwaigwe and Sanda Micula

Fractal Fract. 2023, 7(4), 333; https://doi.org/10.3390/fractalfract7040333 - 17 Apr 2023

Cited by 3 | Viewed by 794

Abstract

An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit [...] Read more.

An efficient numerical algorithm is developed for solving nonlinear functional Volterra integral equations. The core idea is to define an appropriate operator, then combine the Krasnoselskij iterative scheme with collocation at discrete points and the Newton–Cotes quadrature rule. This results in an explicit scheme that does not require solving a nonlinear or linear algebraic system. For the convergence analysis, the discretization error is estimated and proved to converge via a recurrence relation. The discretization error is combined with the Krasnoselskij iteration error to estimate the total approximation error, hence establishing the convergence of the method. Then, numerical experiments are provided, first, to demonstrate the second order convergence of the proposed method, and secondly, to show the better performance of the scheme over the existing nonlinear-based approach. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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10 pages, 507 KiB

Open AccessArticle

Photothermal Response for the Thermoelastic Bending Effect Considering Dissipating Effects by Means of Fractional Dual-Phase-Lag Theory

by Aloisi Somer, Andressa Novatski, Marcelo Kaminski Lenzi, Luciano Rodrigues da Silva and Ervin Kaminski Lenzi

Fractal Fract. 2023, 7(3), 276; https://doi.org/10.3390/fractalfract7030276 - 22 Mar 2023

Cited by 3 | Viewed by 874

Abstract

We analyze an extension of the dual-phase lag model of thermal diffusion theory to accurately predict the contribution of thermoelastic bending (TE) to the Photoacoustic (PA) signal in a transmission configuration. To achieve this, we adopt the particular case of Jeffrey’s equation, an [...] Read more.

We analyze an extension of the dual-phase lag model of thermal diffusion theory to accurately predict the contribution of thermoelastic bending (TE) to the Photoacoustic (PA) signal in a transmission configuration. To achieve this, we adopt the particular case of Jeffrey’s equation, an extension of the Generalized Cattaneo Equations (GCEs). Obtaining the temperature distribution by incorporating the effects of fractional differential operators enables us to determine the TE effects in solid samples accurately. This study contributes to understanding the mechanisms that contribute to the PA signal and highlights the importance of considering fractional differential operators in the analysis of thermoelastic bending. As a result, we can determine the PA signal’s TE component. Our findings demonstrate that the fractional differential operators lead to a wide range of behaviors, including dissipative effects related to anomalous diffusion. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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15 pages, 3140 KiB

Open AccessArticle

Euler Wavelet Method as a Numerical Approach for the Solution of Nonlinear Systems of Fractional Differential Equations

by Sadiye Nergis Tural Polat and Arzu Turan Dincel

Fractal Fract. 2023, 7(3), 246; https://doi.org/10.3390/fractalfract7030246 - 08 Mar 2023

Cited by 1 | Viewed by 1224

Abstract

In this paper, a numerical approach for solving systems of nonlinear fractional differential equations (FDEs) is presented Using the Euler wavelets technique and associated operational matrices for fractional integration, we try to solve those systems of FDEs. The method’s major objective is to [...] Read more.

In this paper, a numerical approach for solving systems of nonlinear fractional differential equations (FDEs) is presented Using the Euler wavelets technique and associated operational matrices for fractional integration, we try to solve those systems of FDEs. The method’s major objective is to transform the nonlinear FDE into a nonlinear system of algebraic equations that is straightforward to solve with matrix techniques. The Euler wavelets are constructed using Euler polynomials, which have fewer terms than most other polynomials used to construct other types of wavelets, therefore, using Euler wavelets for the numerical approach provides sparse operational matrices. Thanks to the sparsity of those operational matrices, the proposed numerical approach requires less computation and takes less time to evaluate. The approach described here is also applicable to systems of fractional differential equations with variable orders. To illustrate the strength and performance of the method, four numerical examples are provided. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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18 pages, 350 KiB

Open AccessArticle

Backward and Non-Local Problems for the Rayleigh-Stokes Equation

by Ravshan Ashurov and Nafosat Vaisova

Fractal Fract. 2022, 6(10), 587; https://doi.org/10.3390/fractalfract6100587 - 13 Oct 2022

Cited by 4 | Viewed by 1040

Abstract

This paper presents the method of separation of variables to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Further, in the Rayleigh-Stokes problem, instead of the initial [...] Read more.

This paper presents the method of separation of variables to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Further, in the Rayleigh-Stokes problem, instead of the initial condition, the non-local condition is considered:

u (x, T) = β u (x, 0) + φ (x)

, where

β

is equal to zero or one. It is well known that if

β = 0

, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e., a small change in

u (x, T)

leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists, it is unique and stable. It will also be shown that if

β = 1

, then the corresponding non-local problem is well-posed and inequalities of coercive type are satisfied. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

19 pages, 661 KiB

Open AccessArticle

Non-Local Seismo-Dynamics: A Fractional Approach

by Vladimir Uchaikin and Elena Kozhemiakina

Fractal Fract. 2022, 6(9), 513; https://doi.org/10.3390/fractalfract6090513 - 13 Sep 2022

Cited by 1 | Viewed by 965

Abstract

This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville [...] Read more.

This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville differential equation of the first order in time, it is described by a fractional differential equation of a distributed equation in the interval (0, 1] order. The key characteristic of its motion is a spectral function, representing the order distribution over the interval. As a specific case of the process, a system with single-point spectrum is investigated. It follows the fractional Poisson process method evolution, obeying via a time-fractional differential equation with a unique order. The article ends with description of statistical estimation of parameters of seismic shocks imitated by Monte Carlo simulated fractional Poisson process. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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15 pages, 376 KiB

Open AccessArticle

Fast Compact Difference Scheme for Solving the Two-Dimensional Time-Fractional Cattaneo Equation

by Lijuan Nong, Qian Yi, Jianxiong Cao and An Chen

Fractal Fract. 2022, 6(8), 438; https://doi.org/10.3390/fractalfract6080438 - 11 Aug 2022

Cited by 4 | Viewed by 1292

Abstract

The time-fractional Cattaneo equation is an equation where the fractional order

α \in (1, 2)

has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation [...] Read more.

The time-fractional Cattaneo equation is an equation where the fractional order

α \in (1, 2)

has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space’s semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1

_{σ}

formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1

_{σ}

difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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15 pages, 341 KiB

Open AccessArticle

Strong Solution for Fractional Mean Field Games with Non-Separable Hamiltonians

by Hailong Ye, Wenzhong Zou and Qiang Liu

Fractal Fract. 2022, 6(7), 362; https://doi.org/10.3390/fractalfract6070362 - 29 Jun 2022

Viewed by 1192

Abstract

In this paper, we establish the existence and uniqueness of a strong solution to a fractional mean field games system with non-separable Hamiltonians, where the fractional exponent

σ \in (\frac{1}{2}, 1)

. Our result is new for fractional mean [...] Read more.

In this paper, we establish the existence and uniqueness of a strong solution to a fractional mean field games system with non-separable Hamiltonians, where the fractional exponent

σ \in (\frac{1}{2}, 1)

. Our result is new for fractional mean field games with non-separable Hamiltonians, which generalizes the work of D.M. Ambrose for the integral case. The important step is to choose the new appropriate fractional order function spaces and use the Banach fixed-point theorem under stronger assumptions for the Hamiltonians. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

19 pages, 982 KiB

Open AccessArticle

Application of the Explicit Euler Method for Numerical Analysis of a Nonlinear Fractional Oscillation Equation

by Valentine Aleksandrovich Kim and Roman Ivanovich Parovik

Fractal Fract. 2022, 6(5), 274; https://doi.org/10.3390/fractalfract6050274 - 19 May 2022

Cited by 3 | Viewed by 2130

Abstract

In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s [...] Read more.

In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform computational grid. For the first time, the issues of approximation, stability and convergence of the proposed explicit finite-difference scheme are considered. To compare the results, the Adams–Bashford–Moulton scheme was constructed as an experimental method. The theoretical results were confirmed using test examples, the computational accuracy of the method was evaluated, which is consistent with the theoretical one, and the simulation results were visualized. Using the example of a fractional Duffing oscillator, waveforms and phase trajectories, as well as its amplitude–frequency characteristics, were constructed using a finite-difference scheme. To identify chaotic regimes, the spectra of maximum Lyapunov exponents and Poincaré points were constructed. It is shown that an explicit finite-difference scheme can be acceptable under the condition of a step of the computational grid. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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19 pages, 370 KiB

Open AccessArticle

Average Process of Fractional Navier–Stokes Equations with Singularly Oscillating Force

by Chunjiao Han, Yi Cheng, Ranzhuo Ma and Zhenhua Zhao

Fractal Fract. 2022, 6(5), 241; https://doi.org/10.3390/fractalfract6050241 - 27 Apr 2022

Viewed by 1207

Abstract

The averaging process between two-dimensional fractional Navier–Stokes equations driven by a singularly oscillating external force and the averaged equations corresponding to the limiting case are investigated. The uniform boundedness of the global attractors for a fractional Navier–Stokes equation with a singularly external force [...] Read more.

The averaging process between two-dimensional fractional Navier–Stokes equations driven by a singularly oscillating external force and the averaged equations corresponding to the limiting case are investigated. The uniform boundedness of the global attractors for a fractional Navier–Stokes equation with a singularly external force is established. Furthermore, these global attractors converge uniformly to the attractor of the averaged equations under suitable assumptions on the singularly external force, and the explicit convergence rate of the global attractors is guaranteed. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

35 pages, 3431 KiB

Open AccessArticle

Application of the Fractional Riccati Equation for Mathematical Modeling of Dynamic Processes with Saturation and Memory Effect

by Dmitriy Tverdyi and Roman Parovik

Fractal Fract. 2022, 6(3), 163; https://doi.org/10.3390/fractalfract6030163 - 16 Mar 2022

Cited by 10 | Viewed by 1821

Abstract

In this study, the model Riccati equation with variable coefficients as functions, as well as a derivative of a fractional variable order (VO) of the Gerasimov-Caputo type, is used to approximate the data for some physical processes with saturation. In particular, the proposed [...] Read more.

In this study, the model Riccati equation with variable coefficients as functions, as well as a derivative of a fractional variable order (VO) of the Gerasimov-Caputo type, is used to approximate the data for some physical processes with saturation. In particular, the proposed model is applied to the description of solar activity (SA), namely the number of sunspots observed over the past 25 years. It is also used to describe data from Johns Hopkins University on coronavirus infection COVID-19, in particular data on the Russian Federation and the Republic of Uzbekistan. Finally, it is used to study issues related to seismic activity, in particular, the description of data on the volumetric activity of Radon (RVA). The Riccati equation used in the mathematical model was numerically solved by constructing an implicit finite difference scheme (IFDS) and its implementation by the modified Newton method (MNM). The calculated curves obtained in the study are compared with known experimental data. It is shown that if the model parameters are chosen appropriately, the model curves will give results that correlate well with real experimental data. Moreover, with other parameters of the model, it is possible to make some prediction about the possible course of the considered processes. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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