Journal Description
Fractal and Fractional
Fractal and Fractional
is an international, scientific, peer-reviewed, open access journal of fractals and fractional calculus and their applications in different fields of science and engineering published monthly online by MDPI.
- Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
- High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, and other databases.
- Journal Rank: JCR - Q1 (Mathematics, Interdisciplinary Applications) / CiteScore - Q1 (Analysis)
- Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 18.9 days after submission; acceptance to publication is undertaken in 3.5 days (median values for papers published in this journal in the second half of 2023).
- Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Impact Factor:
5.4 (2022);
5-Year Impact Factor:
4.7 (2022)
Latest Articles
A Fractional Heston-Type Model as a Singular Stochastic Equation Driven by Fractional Brownian Motion
Fractal Fract. 2024, 8(6), 330; https://doi.org/10.3390/fractalfract8060330 (registering DOI) - 30 May 2024
Abstract
This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven
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This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven by a fractional Brownian motion with a Hurst parameter . We establish the Malliavin differentiability of the fHt model and derive an expression for the expected payoff function, revealing potential discontinuities. Simulation experiments are conducted to illustrate the dynamics of the stock price process and option prices.
Full article
(This article belongs to the Special Issue Analysis of Fractional Stochastic Differential Equations and Their Applications)
Open AccessArticle
Existence of Weak Solutions for the Class of Singular Two-Phase Problems with a ψ-Hilfer Fractional Operator and Variable Exponents
by
Tahar Bouali, Rafik Guefaifia, Rashid Jan, Salah Boulaaras and Taha Radwan
Fractal Fract. 2024, 8(6), 329; https://doi.org/10.3390/fractalfract8060329 - 30 May 2024
Abstract
In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a -Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here,
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In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a -Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results.
Full article
Open AccessArticle
Matrix-Wigner Distribution
by
Long Wang, Manjun Cui, Ze Qin, Zhichao Zhang and Jianwei Zhang
Fractal Fract. 2024, 8(6), 328; https://doi.org/10.3390/fractalfract8060328 - 30 May 2024
Abstract
In order to achieve time–frequency superresolution in comparison to the conventional Wigner distribution (WD), this study generalizes the well-known -Wigner distribution ( -WD) with only one parameter to the multiple-parameter matrix-Wigner distribution (M-WD) with the parameter matrix . According
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In order to achieve time–frequency superresolution in comparison to the conventional Wigner distribution (WD), this study generalizes the well-known -Wigner distribution ( -WD) with only one parameter to the multiple-parameter matrix-Wigner distribution (M-WD) with the parameter matrix . According to operator theory, we construct Heisenberg’s inequalities on the uncertainty product in M-WD domains and formulate two kinds of attainable lower bounds dependent on . We solve the problem of lower bound minimization and obtain the optimality condition of , under which the M-WD achieves superior time–frequency resolution. It turns out that the M-WD breaks through the limitation of the -WD and gives birth to some novel distributions other than the WD that could generate the highest time–frequency resolution. As an example, the two-dimensional linear frequency-modulated signal is carried out to demonstrate the time–frequency concentration superiority of the M-WD over the short-time Fourier transform and wavelet transform.
Full article
Open AccessArticle
Abundant Closed-Form Soliton Solutions to the Fractional Stochastic Kraenkel–Manna–Merle System with Bifurcation, Chaotic, Sensitivity, and Modulation Instability Analysis
by
J. R. M. Borhan, M. Mamun Miah, Faisal Alsharif and Mohammad Kanan
Fractal Fract. 2024, 8(6), 327; https://doi.org/10.3390/fractalfract8060327 - 29 May 2024
Abstract
An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional
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An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination.
Full article
(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)
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Open AccessArticle
Darbo’s Fixed-Point Theorem: Establishing Existence and Uniqueness Results for Hybrid Caputo–Hadamard Fractional Sequential Differential Equations
by
Muhammad Yaseen, Sadia Mumtaz, Reny George, Azhar Hussain and Hossam A. Nabwey
Fractal Fract. 2024, 8(6), 326; https://doi.org/10.3390/fractalfract8060326 - 29 May 2024
Abstract
This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The
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This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo’s fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo’s theorem to analyze the solutions’ existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory.
Full article
(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)
Open AccessArticle
Dynamical Analysis of Two-Dimensional Fractional-Order-in-Time Biological Population Model Using Chebyshev Spectral Method
by
Ishtiaq Ali
Fractal Fract. 2024, 8(6), 325; https://doi.org/10.3390/fractalfract8060325 - 29 May 2024
Abstract
In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics,
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In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, which are problems that are frequently faced with integer-order models. We use the Chebyshev spectral approach for spatial derivatives, which is known for its faster convergence rate, in conjunction with the scheme for time-fractional derivatives because of its high accuracy and robustness in handling nonlocal effects. A detailed theoretical analysis, followed by a number of numerical experiments, is performed to confirmed the theoretical justification. Our simulation results show that our numerical technique significantly improves the convergence rates, effectively tackles computing difficulties, and provides a realistic simulation of biological population dynamics.
Full article
(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)
Open AccessArticle
Multivalued Variational Inequalities with Generalized Fractional Φ-Laplacians
by
Vy Khoi Le
Fractal Fract. 2024, 8(6), 324; https://doi.org/10.3390/fractalfract8060324 - 29 May 2024
Abstract
In this article, we examine variational inequalities of the form ,
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In this article, we examine variational inequalities of the form , where is a generalized fractional Φ-Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions.
Full article
(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)
Open AccessArticle
A Proportional-Integral-One Plus Double Derivative Controller-Based Fractional-Order Kepler Optimizer for Frequency Stability in Multi-Area Power Systems with Wind Integration
by
Mohammed H. Alqahtani, Sulaiman Z. Almutairi, Ali S. Aljumah, Abdullah M. Shaheen, Ghareeb Moustafa and Attia A. El-Fergany
Fractal Fract. 2024, 8(6), 323; https://doi.org/10.3390/fractalfract8060323 - 29 May 2024
Abstract
This study proposes an enhanced Kepler Optimization (EKO) algorithm, incorporating fractional-order components to develop a Proportional-Integral-First-Order Double Derivative (PI–(1+DD) controller for frequency stability control in multi-area power systems with wind power integration. The fractional-order element facilitates efficient information and past experience sharing among
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This study proposes an enhanced Kepler Optimization (EKO) algorithm, incorporating fractional-order components to develop a Proportional-Integral-First-Order Double Derivative (PI–(1+DD) controller for frequency stability control in multi-area power systems with wind power integration. The fractional-order element facilitates efficient information and past experience sharing among participants, hence increasing the search efficiency of the EKO algorithm. Furthermore, a local escaping approach is included to improve the search process for avoiding local optimization. Applications were performed through comparisons with the 2020 IEEE Congress on Evolutionary Computation (CEC 2020) benchmark tests and applications in a two-area system, including thermal and wind power. In this regard, comparisons were implemented considering three different controllers of PI, PID, and PI–(1+DD) designs. The simulations show that the EKO algorithm demonstrates superior performance in optimizing load frequency control (LFC), significantly improving the stability of power systems with renewable energy systems (RES) integration.
Full article
(This article belongs to the Special Issue Fractional Modelling, Analysis and Control for Power System)
Open AccessArticle
A New Fractional Discrete Memristive Map with Variable Order and Hidden Dynamics
by
Othman Abdullah Almatroud, Amina-Aicha Khennaoui, Adel Ouannas, Saleh Alshammari and Sahar Albosaily
Fractal Fract. 2024, 8(6), 322; https://doi.org/10.3390/fractalfract8060322 - 29 May 2024
Abstract
This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like
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This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like difference operator, is employed to establish the fractional form of the map with short memory. The numerical simulation results highlight its excellent dynamical behavior, revealing that the addition of the piecewise fractional order makes the memristive-based Duffing map even more chaotic. It is characterized by distinct features, including the absence of an equilibrium point and the presence of multiple hidden chaotic attractors.
Full article
(This article belongs to the Special Issue Bifurcation, Chaos, and Fractals in Fractional-Order Electrical and Electronic Systems)
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Analyzing a Dynamical System with Harmonic Mean Incidence Rate Using Volterra–Lyapunov Matrices and Fractal-Fractional Operators
by
Muhammad Riaz, Faez A. Alqarni, Khaled Aldwoah, Fathea M. Osman Birkea and Manel Hleili
Fractal Fract. 2024, 8(6), 321; https://doi.org/10.3390/fractalfract8060321 - 28 May 2024
Abstract
This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free
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This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics.
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(This article belongs to the Section Numerical and Computational Methods)
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Open AccessArticle
Modified MF-DFA Model Based on LSSVM Fitting
by
Minzhen Wang, Caiming Zhong, Keyu Yue, Yu Zheng, Wenjing Jiang and Jian Wang
Fractal Fract. 2024, 8(6), 320; https://doi.org/10.3390/fractalfract8060320 - 28 May 2024
Abstract
This paper proposes a multifractal least squares support vector machine detrended fluctuation analysis (MF-LSSVM-DFA) model. The system is an extension of the traditional MF-DFA model. To address potential overfitting or underfitting caused by the fixed-order polynomial fitting in MF-DFA, LSSVM is employed as
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This paper proposes a multifractal least squares support vector machine detrended fluctuation analysis (MF-LSSVM-DFA) model. The system is an extension of the traditional MF-DFA model. To address potential overfitting or underfitting caused by the fixed-order polynomial fitting in MF-DFA, LSSVM is employed as a superior alternative for fitting. This approach enhances model accuracy and adaptability, ensuring more reliable analysis results. We utilize the p model to construct a multiplicative cascade time series to evaluate the performance of MF-LSSVM-DFA, MF-DFA, and two other models that improve upon MF-DFA from recent studies. The results demonstrate that our proposed modified model yields generalized Hurst exponents and scaling exponents that align more closely with the analytical solutions, indicating superior correction effectiveness. In addition, we explore the sensitivity of MF-LSSVM-DFA to the overlapping window size s. We find that the sensitivity of our proposed model is less than that of MF-DFA. We find that when s exceeds the limited range of the traditional MF-DFA, and are closer than those obtained in MF-DFA when s is in a limited range. Meanwhile, we analyze the performances of the fitting of the two models and the results imply that MF-LSSVM-DFA achieves a better outstanding performance. In addition, we put the proposed MF-LSSVM-DFA into practice for applications in the medical field, and we found that MF-LSSVM-DFA improves the accuracy of ECG signal classification and the stability and robustness of the algorithm compared with MF-DFA. Finally, numerous image segmentation experiments are adopted to verify the effectiveness and robustness of our proposed method.
Full article
(This article belongs to the Special Issue Modern Methods for Fractal and Multifractal Analysis of Time Series: Theoretical Frameworks and Practical Applications)
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Open AccessArticle
Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function
by
Aljowhara H. Honain, Khaled M. Furati, Ibrahim O. Sarumi and Abdul Q. M. Khaliq
Fractal Fract. 2024, 8(6), 319; https://doi.org/10.3390/fractalfract8060319 - 27 May 2024
Abstract
The two-parameter Mittag–Leffler function is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that
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The two-parameter Mittag–Leffler function is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When , the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for , , with . This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.
Full article
(This article belongs to the Special Issue Mittag-Leffler Function: Generalizations and Applications)
Open AccessArticle
Fixed Point Results with Applications to Fractional Differential Equations of Anomalous Diffusion
by
Zhenhua Ma, Hanadi Zahed and Jamshaid Ahmad
Fractal Fract. 2024, 8(6), 318; https://doi.org/10.3390/fractalfract8060318 - 27 May 2024
Abstract
The main objective of this manuscript is to define the concepts of F-(⋏,h)-contraction and ( , -Reich type interpolative contraction in the framework of orthogonal -metric space and prove some fixed point results. Our primary result
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The main objective of this manuscript is to define the concepts of F-(⋏,h)-contraction and ( , -Reich type interpolative contraction in the framework of orthogonal -metric space and prove some fixed point results. Our primary result serves as a cornerstone, from which established findings in the literature emerge as natural consequences. To enhance the clarity of our novel contributions, we furnish a significant example that not only strengthens the innovative findings but also facilitates a deeper understanding of the established theory. The concluding section of our work is dedicated to the application of these results in establishing the existence and uniqueness of a solution for a fractional differential equation of anomalous diffusion.
Full article
(This article belongs to the Section General Mathematics, Analysis)
Open AccessArticle
Exploring the Depths: Soliton Solutions, Chaotic Analysis, and Sensitivity Analysis in Nonlinear Optical Fibers
by
Muhammad Shakeel, Xinge Liu and Fehaid Salem Alshammari
Fractal Fract. 2024, 8(6), 317; https://doi.org/10.3390/fractalfract8060317 - 27 May 2024
Abstract
This paper discusses the time-fractional nonlinear Schrödinger model with optical soliton solutions. We employ the -expansion method to attain the optical solution solutions. An important tool for explaining the particular explosion of brief pulses in optical
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This paper discusses the time-fractional nonlinear Schrödinger model with optical soliton solutions. We employ the -expansion method to attain the optical solution solutions. An important tool for explaining the particular explosion of brief pulses in optical fibers is the nonlinear Schrödinger model. It can also be utilized in a telecommunications system. The suggested method yields trigonometric solutions such as dark, bright, kink, and anti-kink-type optical soliton solutions. Mathematica 11 software creates 2D and 3D graphs for many physically important parameters. The computational method is effective and generally appropriate for solving analytical problems related to complicated nonlinear issues that have emerged in the recent history of nonlinear optics and mathematical physics. Furthermore, we venture into uncharted territory by subjecting our model to chaotic and sensitivity analysis, shedding light on its robustness and responsiveness to perturbations. The proposed technique is being applied to this model for the first time.
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(This article belongs to the Section Mathematical Physics)
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A Preconditioned Policy–Krylov Subspace Method for Fractional Partial Integro-Differential HJB Equations in Finance
by
Xu Chen, Xin-Xin Gong, Youfa Sun and Siu-Long Lei
Fractal Fract. 2024, 8(6), 316; https://doi.org/10.3390/fractalfract8060316 - 27 May 2024
Abstract
To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden
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To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy- -stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method.
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(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)
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Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem
by
Özge Arıbaş, İsmet Gölgeleyen and Mustafa Yıldız
Fractal Fract. 2024, 8(6), 315; https://doi.org/10.3390/fractalfract8060315 - 26 May 2024
Abstract
In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting
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In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting of the sum of two terms, one of which is linear and the other is nonlinear. The well-posedness of the direct problem is examined and the results are used to investigate the stability of an inverse problem of determining a function in the linear part of the source. The main tools in our study are the generalized eigenfunction expansions theory for nonlinear fractional diffusion equations, contraction mapping, Young’s convolution and generalized Grönwall’s inequalities. We present a stability estimate for the solution of the inverse source problem by means of observation data at a given point in the domain.
Full article
(This article belongs to the Special Issue Recent Advances in the Equation with Nonlinear Fractional Diffusion)
Open AccessArticle
Correlation between Agglomerates Hausdorff Dimension and Mechanical Properties of Denture Poly(methyl methacrylate)-Based Composites
by
Houda Taher Elhmali, Cristina Serpa, Vesna Radojevic, Aleksandar Stajcic, Milos Petrovic, Ivona Jankovic-Castvan and Ivana Stajcic
Fractal Fract. 2024, 8(6), 314; https://doi.org/10.3390/fractalfract8060314 - 26 May 2024
Abstract
The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with
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The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with SrTiO3, MnO2 and SrTiO3/MnO2 were used for fractal reconstructions of particle agglomerates in the polymer matrix. Fractal analysis represents a valuable mathematical tool for the characterization of the microstructure and finding correlation between microstructural features and mechanical properties. Utilizing the mathematical affine fractal regression model, the Fractal Real Finder software was employed to reconstruct agglomerate shapes and estimate the Hausdorff dimensions (HD). Controlled energy impact and tensile tests were used to evaluate the mechanical performance of PMMA-MnO2, PMMA-SrTiO3 and PMMA-SrTiO3/MnO2 composites. It was determined that PMMA-SrTiO3/MnO2 had the highest total absorbed energy value (Etot), corresponding to the lowest HD value of 1.03637 calculated for SrTiO3/MnO2 agglomerates. On the other hand, the highest HD value of 1.21521 was calculated for MnO2 agglomerates, while the PMMA-MnO2 showed the lowest Etot. The linear correlation between the total absorbed impact energy of composites and the HD of the corresponding agglomerates was determined, with an R2 value of 0.99486, showing the potential use of this approach in the optimization of composite materials’ microstructure–property relationship.
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(This article belongs to the Special Issue Advanced Research in Fractal Properties of Nanoparticle and Its Application)
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New Perturbation–Iteration Algorithm for Nonlinear Heat Transfer of Fractional Order
by
Mohammad Abdel Aal
Fractal Fract. 2024, 8(6), 313; https://doi.org/10.3390/fractalfract8060313 - 25 May 2024
Abstract
Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on
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Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on extending the nonlinear heat equations to a fractional order in a Caputo order. A new perturbation iteration algorithm (PIA) of the fractional order is applied to solve the nonlinear heat equations. Solving numerical problems that involve fractional differential equations can be challenging due to their inherent complexity and high computational cost. To overcome these challenges, there is a need to develop numerical schemes such as the PIA method. This method can provide approximate solutions to problems that involve classical fractional derivatives. The results obtained from this algorithm are compared with those obtained from the perturbation iteration method (PIM), the variational iteration method (VIM), and the Bezier curve method (BCM). All solutions are tested with numerical simulations. The study found that the new PIA algorithm performs better than the PIM, VIM, and BCM, achieving high accuracy and low computational cost. One significant advantage of this algorithm is that the solutions obtained have established that the fractional values of alpha, specifically , significantly influencing the accuracy of the outcome and the associated computational cost.
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(This article belongs to the Section Mathematical Physics)
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Multifractal Properties of Human Chromosome Sequences
by
J. P. Correia, R. Silva, D. H. A. L. Anselmo, M. S. Vasconcelos and L. R. da Silva
Fractal Fract. 2024, 8(6), 312; https://doi.org/10.3390/fractalfract8060312 - 24 May 2024
Abstract
The intricacy and fractal properties of human DNA sequences are examined in this work. The core of this study is to discern whether complete DNA sequences present distinct complexity and fractal attributes compared with sequences containing exclusively exon regions. In this regard, the
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The intricacy and fractal properties of human DNA sequences are examined in this work. The core of this study is to discern whether complete DNA sequences present distinct complexity and fractal attributes compared with sequences containing exclusively exon regions. In this regard, the entire base pair sequences of DNA are extracted from the NCBI (National Center for Biotechnology Information) database. In order to create a time series representation for the base pair sequence , we use the Chaos Game Representation (CGR) approach and a mapping rule f, which enables us to apply the metric known as the Complexity–Entropy Plane (CEP) and multifractal detrended fluctuation analysis (MF-DFA). To carry out our investigation, we divided human DNA into two groups: the first is composed of the 24 chromosomes, which comprises all the base pairs that form the DNA sequence, and another group that also includes the 24 chromosomes, but the DNA sequences rely only on the exons’ presence. The results show that both sets provide fractal patterns in their structure, as obtained by the CGR approach. Complete DNA sequences show a sharper visual fractal pattern than sequences composed only of exons. Moreover, the sequences occupy distinct areas of the complexity–entropy plane, and the complete DNA sequences lead to greater statistical complexity and lower entropy than the exon sequences. Also, we observed that different fractal parameters between chromosomes indicate diversity in genomic sequences. All these results occur in different scales for all chromosomes.
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(This article belongs to the Special Issue Modern Methods for Fractal and Multifractal Analysis of Time Series: Theoretical Frameworks and Practical Applications)
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The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions
by
Peiguang Wang, Bing Han and Junyan Bao
Fractal Fract. 2024, 8(6), 311; https://doi.org/10.3390/fractalfract8060311 - 24 May 2024
Abstract
In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the
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In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results.
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(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)
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