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Search Results (1,235)

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Keywords = Caputo’s derivatives

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19 pages, 1767 KiB  
Article
Dynamics of a Fractional-Order Within-Host Virus Model with Adaptive Immune Responses and Two Routes of Infection
by Taofeek O. Alade, Furaha M. Chuma, Muhammad Javed, Samson Olaniyi, Adekunle O. Sangotola and Gideon K. Gogovi
Math. Comput. Appl. 2025, 30(4), 80; https://doi.org/10.3390/mca30040080 (registering DOI) - 2 Aug 2025
Abstract
This paper introduces a novel fractional-order model using the Caputo derivative operator to investigate the virus dynamics of adaptive immune responses. Two infection routes, namely cell-to-cell and virus-to-cell transmissions, are incorporated into the dynamics. Our research establishes the existence and uniqueness of positive [...] Read more.
This paper introduces a novel fractional-order model using the Caputo derivative operator to investigate the virus dynamics of adaptive immune responses. Two infection routes, namely cell-to-cell and virus-to-cell transmissions, are incorporated into the dynamics. Our research establishes the existence and uniqueness of positive and bounded solutions through the application of the generalized mean-value theorem and Banach fixed-point theory methods. The fractional-order model is shown to be Ulam–Hyers stable, ensuring the model’s resilience to small errors. By employing the normalized forward sensitivity method, we identify critical parameters that profoundly influence the transmission dynamics of the fractional-order virus model. Additionally, the framework of optimal control theory is used to explore the characterization of optimal adaptive immune responses, encompassing antibodies and cytotoxic T lymphocytes (CTL). To assess the influence of memory effects, we utilize the generalized forward–backward sweep technique to simulate the fractional-order virus dynamics. This study contributes to the existing body of knowledge by providing insights into how the interaction between virus-to-cell and cell-to-cell dynamics within the host is affected by memory effects in the presence of optimal control, reinforcing the invaluable synergy between fractional calculus and optimal control theory in modeling within-host virus dynamics, and paving the way for potential control strategies rooted in adaptive immunity and fractional-order modeling. Full article
18 pages, 1025 KiB  
Article
The Analysis of Three-Dimensional Time-Fractional Helmholtz Model Using a New İterative Method
by Yasin Şahin, Mehmet Merdan and Pınar Açıkgöz
Symmetry 2025, 17(8), 1219; https://doi.org/10.3390/sym17081219 (registering DOI) - 1 Aug 2025
Abstract
This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes, [...] Read more.
This paper proposes a novel analytical method to address the Helmholtz fractional differential equation by combining the Aboodh transform with the Adomian Decomposition Method, resulting in the Aboodh–Adomian Decomposition Method (A-ADM). Fractional differential equations offer a comprehensive framework for describing intricate physical processes, including memory effects and anomalous diffusion. This work employs the Caputo–Fabrizio fractional derivative, defined by a non-singular exponential kernel, to more precisely capture these non-local effects. The classical Helmholtz equation, pivotal in acoustics, electromagnetics, and quantum physics, is extended to the fractional domain. Following the exposition of fundamental concepts and characteristics of fractional calculus and the Aboodh transform, the suggested A-ADM is employed to derive the analytical solution of the fractional Helmholtz equation. The method’s validity and efficiency are evidenced by comparisons of analytical and approximation solutions. The findings validate that A-ADM is a proficient and methodical approach for addressing fractional differential equations that incorporate Caputo–Fabrizio derivatives. Full article
30 pages, 9514 KiB  
Article
FPGA Implementation of Secure Image Transmission System Using 4D and 5D Fractional-Order Memristive Chaotic Oscillators
by Jose-Cruz Nuñez-Perez, Opeyemi-Micheal Afolabi, Vincent-Ademola Adeyemi, Yuma Sandoval-Ibarra and Esteban Tlelo-Cuautle
Fractal Fract. 2025, 9(8), 506; https://doi.org/10.3390/fractalfract9080506 (registering DOI) - 31 Jul 2025
Viewed by 87
Abstract
With the rapid proliferation of real-time digital communication, particularly in multimedia applications, securing transmitted image data has become a vital concern. While chaotic systems have shown strong potential for cryptographic use, most existing approaches rely on low-dimensional, integer-order architectures, limiting their complexity and [...] Read more.
With the rapid proliferation of real-time digital communication, particularly in multimedia applications, securing transmitted image data has become a vital concern. While chaotic systems have shown strong potential for cryptographic use, most existing approaches rely on low-dimensional, integer-order architectures, limiting their complexity and resistance to attacks. Advances in fractional calculus and memristive technologies offer new avenues for enhancing security through more complex and tunable dynamics. However, the practical deployment of high-dimensional fractional-order memristive chaotic systems in hardware remains underexplored. This study addresses this gap by presenting a secure image transmission system implemented on a field-programmable gate array (FPGA) using a universal high-dimensional memristive chaotic topology with arbitrary-order dynamics. The design leverages four- and five-dimensional hyperchaotic oscillators, analyzed through bifurcation diagrams and Lyapunov exponents. To enable efficient hardware realization, the chaotic dynamics are approximated using the explicit fractional-order Runge–Kutta (EFORK) method with the Caputo fractional derivative, implemented in VHDL. Deployed on the Xilinx Artix-7 AC701 platform, synchronized master–slave chaotic generators drive a multi-stage stream cipher. This encryption process supports both RGB and grayscale images. Evaluation shows strong cryptographic properties: correlation of 6.1081×105, entropy of 7.9991, NPCR of 99.9776%, UACI of 33.4154%, and a key space of 21344, confirming high security and robustness. Full article
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13 pages, 345 KiB  
Article
An Application of Liouville–Caputo-Type Fractional Derivatives on Certain Subclasses of Bi-Univalent Functions
by Ibtisam Aldawish, Hari M. Srivastava, Sheza M. El-Deeb, Gangadharan Murugusundaramoorthy and Kaliappan Vijaya
Fractal Fract. 2025, 9(8), 505; https://doi.org/10.3390/fractalfract9080505 (registering DOI) - 31 Jul 2025
Viewed by 79
Abstract
In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients |c2|, |c3| for functions in these subclasses of [...] Read more.
In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients |c2|, |c3| for functions in these subclasses of bi-univalent functions. Additionally, by using the values of a2,a3 we determine the Fekete–Szegö inequality results. Moreover, a few new subclasses are deduced that have not been studied in relation to Liouville–Caputo fractional derivatives so far. The implications of the results are also emphasized. Our results are concrete examples of several earlier discoveries that are not only improved but also expanded upon. Full article
19 pages, 1791 KiB  
Article
A Novel Approach to Solving Generalised Nonlinear Dynamical Systems Within the Caputo Operator
by Mashael M. AlBaidani and Rabab Alzahrani
Fractal Fract. 2025, 9(8), 503; https://doi.org/10.3390/fractalfract9080503 (registering DOI) - 31 Jul 2025
Viewed by 74
Abstract
In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and [...] Read more.
In this study, we focus on solving the nonlinear time-fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) and modified Korteweg–de Vries (MKdV) equations, using the Yang transform iterative method (YTIM). This method combines the Yang transform with a new iterative scheme to construct reliable and efficient solutions. Readers can understand the procedures clearly, since the implementation of Yang transform directly transforms fractional derivative sections into algebraic terms in the given problems. The new iterative scheme is applied to generate series solutions for the provided problems. The fractional derivatives are considered in the Caputo sense. To validate the proposed approach, two numerical examples are analysed and compared with exact solutions, as well as with the results obtained from the fractional reduced differential transform method (FRDTM) and the q-homotopy analysis transform method (q-HATM). The comparisons, presented through both tables and graphical illustrations, confirm the enhanced accuracy and reliability of the proposed method. Moreover, the effect of varying the fractional order is explored, demonstrating convergence of the solution as the order approaches an integer value. Importantly, the time-fractional Hirota–Satsuma coupled KdV and modified Korteweg–de Vries (MKdV) equations investigated in this work are not only of theoretical and computational interest but also possess significant implications for achieving global sustainability goals. Specifically, these equations contribute to the Sustainable Development Goal (SDG) “Life Below Water” by offering advanced modelling capabilities for understanding wave propagation and ocean dynamics, thus supporting marine ecosystem research and management. It is also relevant to SDG “Climate Action” as it aids in the simulation of environmental phenomena crucial to climate change analysis and mitigation. Additionally, the development and application of innovative mathematical modelling techniques align with “Industry, Innovation, and Infrastructure” promoting advanced computational tools for use in ocean engineering, environmental monitoring, and other infrastructure-related domains. Therefore, the proposed method not only advances mathematical and numerical analysis but also fosters interdisciplinary contributions toward sustainable development. Full article
(This article belongs to the Special Issue Recent Trends in Computational Physics with Fractional Applications)
12 pages, 441 KiB  
Article
G-Subdiffusion Equation as an Anomalous Diffusion Equation Determined by the Time Evolution of the Mean Square Displacement of a Diffusing Molecule
by Tadeusz Kosztołowicz, Aldona Dutkiewicz and Katarzyna D. Lewandowska
Entropy 2025, 27(8), 816; https://doi.org/10.3390/e27080816 (registering DOI) - 31 Jul 2025
Viewed by 71
Abstract
Normal and anomalous diffusion processes are characterized by the time evolution of the mean square displacement of a diffusing molecule σ2(t). When σ2(t) is a power function of time, the process is described by [...] Read more.
Normal and anomalous diffusion processes are characterized by the time evolution of the mean square displacement of a diffusing molecule σ2(t). When σ2(t) is a power function of time, the process is described by a fractional subdiffusion, fractional superdiffusion or normal diffusion equation. However, for other forms of σ2(t), diffusion equations are often not defined. We show that to describe diffusion characterized by σ2(t), the g-subdiffusion equation with the fractional Caputo derivative with respect to a function g can be used. Choosing an appropriate function g, we obtain Green’s function for this equation, which generates the assumed σ2(t). A method for solving such an equation, based on the Laplace transform with respect to the function g, is also described. Full article
(This article belongs to the Section Complexity)
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17 pages, 333 KiB  
Article
Hille–Yosida-Type Theorem for Fractional Differential Equations with Dzhrbashyan–Nersesyan Derivative
by Vladimir E. Fedorov, Wei-Shih Du, Marko Kostić, Marina V. Plekhanova and Darya V. Melekhina
Fractal Fract. 2025, 9(8), 499; https://doi.org/10.3390/fractalfract9080499 - 30 Jul 2025
Viewed by 197
Abstract
It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type [...] Read more.
It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem. Full article
(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
19 pages, 5262 KiB  
Article
A Conservative Four-Dimensional Hyperchaotic Model with a Center Manifold and Infinitely Many Equilibria
by Surma H. Ibrahim, Ali A. Shukur and Rizgar H. Salih
Modelling 2025, 6(3), 74; https://doi.org/10.3390/modelling6030074 (registering DOI) - 29 Jul 2025
Viewed by 214
Abstract
This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis [...] Read more.
This paper presents a novel four-dimensional autonomous conservative model characterized by an infinite set of equilibrium points and an unusual algebraic structure in which all eigenvalues of the Jacobian matrix are zero. The linearization of the proposed model implies that classical stability analysis is inadequate, as only the center manifolds are obtained. Consequently, the stability of the system is investigated through both analytical and numerical methods using Lyapunov functions and numerical simulations. The proposed model exhibits rich dynamics, including hyperchaotic behavior, which is characterized using the Lyapunov exponents, bifurcation diagrams, sensitivity analysis, attractor projections, and Poincaré map. Moreover, in this paper, we explore the model with fractional-order derivatives, demonstrating that the fractional dynamics fundamentally change the geometrical structure of the attractors and significantly change the system stability. The Grünwald–Letnikov formulation is used for modeling, while numerical integration is performed using the Caputo operator to capture the memory effects inherent in fractional models. Finally, an analog electronic circuit realization is provided to experimentally validate the theoretical and numerical findings. Full article
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17 pages, 420 KiB  
Article
Assessing the Efficiency of the Homotopy Analysis Transform Method for Solving a Fractional Telegraph Equation with a Bessel Operator
by Said Mesloub and Hassan Eltayeb Gadain
Fractal Fract. 2025, 9(8), 493; https://doi.org/10.3390/fractalfract9080493 - 28 Jul 2025
Viewed by 144
Abstract
In this study, we apply the Laplace Transform Homotopy Analysis Method (LTHAM) to numerically solve a fractional-order telegraph equation with a Bessel operator. The iterative scheme developed is tested on multiple examples to evaluate its efficiency. Our observations indicate that the method generates [...] Read more.
In this study, we apply the Laplace Transform Homotopy Analysis Method (LTHAM) to numerically solve a fractional-order telegraph equation with a Bessel operator. The iterative scheme developed is tested on multiple examples to evaluate its efficiency. Our observations indicate that the method generates an approximate solution in series form, which converges rapidly to the analytic solution in each instance. The convergence of these series solutions is assessed both geometrically and numerically. Our results demonstrate that LTHAM is a reliable, powerful, and straightforward approach to solving fractional telegraph equations, and it can be effectively extended to solve similar types of equations. Full article
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17 pages, 2333 KiB  
Article
Numerical Investigation of the Time-Fractional Black–Scholes Problem Using the Caputo Fractional Derivative in the Financial Industry
by Muhammad Nadeem, Bitao Cheng and Loredana Florentina Iambor
Fractal Fract. 2025, 9(8), 490; https://doi.org/10.3390/fractalfract9080490 - 25 Jul 2025
Viewed by 237
Abstract
The present study addresses the European option pricing problem based on the Black–Scholes (B-S) model using a hybrid analytical approach known as the Sawi homotopy perturbation transform scheme (SHPTS). We formulate fractional derivatives in the Caputo sense to effectively capture the memory effects [...] Read more.
The present study addresses the European option pricing problem based on the Black–Scholes (B-S) model using a hybrid analytical approach known as the Sawi homotopy perturbation transform scheme (SHPTS). We formulate fractional derivatives in the Caputo sense to effectively capture the memory effects inherent in financial models. The competency and reliability of the SHPTS are demonstrated through two illustrative examples. This method produces a closed-form series solution that converges to the precise solution. We perform convergence and visual analyses to demonstrate the competency and reliability of the proposed scheme. The numerical findings further reveal that the strategy is straightforward to apply and very successful in resolving the fractional form of the B-S problem. Full article
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24 pages, 1197 KiB  
Article
Fractional Gradient-Based Model Reference Adaptive Control Applied on an Inverted Pendulum-Cart System
by Maibeth Sánchez-Rivero, Manuel A. Duarte-Mermoud, Lisbel Bárzaga-Martell, Marcos E. Orchard and Gustavo Ceballos-Benavides
Fractal Fract. 2025, 9(8), 485; https://doi.org/10.3390/fractalfract9080485 - 24 Jul 2025
Viewed by 255
Abstract
This study introduces a novel model reference adaptive control (MRAC) framework that incorporates fractional-order gradients (FGs) to regulate the displacement of an inverted pendulum-cart system. Fractional-order gradients have been shown to significantly improve convergence rates in domains such as machine learning and neural [...] Read more.
This study introduces a novel model reference adaptive control (MRAC) framework that incorporates fractional-order gradients (FGs) to regulate the displacement of an inverted pendulum-cart system. Fractional-order gradients have been shown to significantly improve convergence rates in domains such as machine learning and neural network optimization. Nevertheless, their integration with fractional-order error models within adaptive control paradigms remains unexplored and represents a promising avenue for research. The proposed control scheme extends the classical MRAC architecture by embedding Caputo fractional derivatives into the adaptive law governing parameter updates, thereby improving both convergence dynamics and control flexibility. To ensure optimal performance across multiple criteria, the controller parameters are systematically tuned using a multi-objective Particle Swarm Optimization (PSO) algorithm. Two fractional-order error models (FOEMs) incorporating fractional gradients (FOEM2-FG, FOEM3-FG) are investigated, with their stability formally analyzed via Lyapunov-based methods under conditions of sufficient excitation. Validation is conducted through both simulation and real-time experimentation on a physical pendulum-cart setup. The results demonstrate that the proposed fractional-order MRAC (FOMRAC) outperforms conventional MRAC, proportional-integral-derivative (PID), and fractional-order PID (FOPID) controllers. Specifically, FOMRAC-FG achieved superior tracking performance, attaining the lowest Integral of Squared Error (ISE) of 2.32×105 and the lowest Integral of Squared Input (ISI) of 6.40 in simulation studies. In real-time experiments, FOMRAC-FG maintained the lowest ISE (5.11×106). Under real-time experiments with disturbances, it still achieved the lowest ISE (1.06×105), highlighting its practical effectiveness. Full article
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20 pages, 506 KiB  
Article
Efficient Numerical Methods for Time-Fractional Diffusion Equations with Caputo-Type Erdélyi–Kober Operators
by Ruilian Du and Jianhua Tang
Fractal Fract. 2025, 9(8), 486; https://doi.org/10.3390/fractalfract9080486 - 24 Jul 2025
Viewed by 184
Abstract
This study proposes an L1 discretization scheme (an accurate second-order finite difference method) for time-fractional diffusion equations involving the Caputo-type Erdélyi–Kober operator, which models anomalous diffusion. Our key contributions include the following: (i) reformulation of the original problem into an equivalent fractional integral [...] Read more.
This study proposes an L1 discretization scheme (an accurate second-order finite difference method) for time-fractional diffusion equations involving the Caputo-type Erdélyi–Kober operator, which models anomalous diffusion. Our key contributions include the following: (i) reformulation of the original problem into an equivalent fractional integral equation to facilitate analysis; (ii) development of a novel L1 scheme for temporal discretization, which is rigorously proven to realize second-order accuracy in time; (iii) derivation of positive definiteness properties for discrete kernel coefficients; (iv) discretization of the spatial derivative using the classical second-order centered difference scheme, for which its second-order spatial convergence is rigorously verified through numerical experiments (this results in a fully discrete scheme, enabling second-order accuracy in both temporal and spatial dimensions); (v) a fast algorithm leveraging sum-of-exponential approximation, reducing the computational complexity from O(N2) to O(NlogN) and memory requirements from O(N) to O(logN), where N is the number of grid points on a time scale. Our numerical experiments demonstrate the stability of the scheme across diverse parameter regimes and quantify significant gains in computational efficiency. Compared to the direct method, the fast algorithm substantially reduces both memory requirements and CPU time for large-scale simulations. Although a rigorous stability analysis is deferred to subsequent research, the proven properties of the coefficients and numerical validation confirm the scheme’s reliability. Full article
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15 pages, 436 KiB  
Article
Optimal Control of the Inverse Problem of the Fractional Burgers Equation
by Jiale Qin, Jun Zhao, Jing Xu and Shichao Yi
Fractal Fract. 2025, 9(8), 484; https://doi.org/10.3390/fractalfract9080484 - 24 Jul 2025
Viewed by 202
Abstract
This paper investigates the well-posedness of the inverse problem for the time-fractional Burgers equation, which aims to reconstruct initial conditions from terminal observations. Such equations are crucial for the modeling of hydrodynamic phenomena with memory effects. The inverse problem involves inferring initial conditions [...] Read more.
This paper investigates the well-posedness of the inverse problem for the time-fractional Burgers equation, which aims to reconstruct initial conditions from terminal observations. Such equations are crucial for the modeling of hydrodynamic phenomena with memory effects. The inverse problem involves inferring initial conditions from terminal observation data, and such problems are typically ill-posed. A framework based on optimal control theory is proposed, addressing the ill-posedness via H1 regularization. Three substantial results are achieved: (1) a rigorous mathematical framework transforming the ill-posed inverse problem into a well-posed optimization problem with proven existence of solutions; (2) theoretical guarantee of solution uniqueness when the regularization parameter is α>0 and the stability is of order O(δ) with respect to observation noise (δ); and (3) the discovery of a “super-stability” phenomenon in numerical experiments, where the actual stability index (0.046) significantly outperforms theoretical expectations (1.0). Finally, the theoretical framework is validated through comprehensive numerical experiments, demonstrating the accuracy and practical effectiveness of the proposed optimal control approach for the reconstruction of hydrodynamic initial conditions. Full article
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22 pages, 10576 KiB  
Article
Numerical Simulation of Double-Layer Nanoplates Based on Fractional Model and Shifted Legendre Algorithm
by Qianqian Fan, Qiumei Liu, Yiming Chen, Yuhuan Cui, Jingguo Qu and Lei Wang
Fractal Fract. 2025, 9(7), 477; https://doi.org/10.3390/fractalfract9070477 - 21 Jul 2025
Viewed by 292
Abstract
This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the [...] Read more.
This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the Fractional Kelvin–Voigt (FKV) model with the Caputo fractional derivative and the theory of nonlocal elasticity. Then, the shifted Legendre polynomial is used to approximate the displacement function, and the governing equations are transformed into algebraic equations to facilitate the numerical solution in the time domain. Moreover, the systematic convergence analysis is carried out to verify the convergence of the ternary displacement function and its fractional derivatives in the equation, ensuring the rigor of the mathematical model. Finally, a dimensionless numerical example is given to verify the feasibility of the proposed algorithm, and the effects of material parameters on plate displacement are analyzed for double-layer plates with different materials. Full article
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22 pages, 320 KiB  
Article
A New Caputo Fractional Differential Equation with Infinite-Point Boundary Conditions: Positive Solutions
by Jing Ren, Zijuan Du and Chengbo Zhai
Fractal Fract. 2025, 9(7), 466; https://doi.org/10.3390/fractalfract9070466 - 18 Jul 2025
Viewed by 227
Abstract
This paper mainly studies a different infinite-point Caputo fractional differential equation, whose nonlinear term may be singular. Under some conditions, we first use spectral analysis and fixed-point index theorem to explore the existence of positive solutions of the equation, and then use Banach [...] Read more.
This paper mainly studies a different infinite-point Caputo fractional differential equation, whose nonlinear term may be singular. Under some conditions, we first use spectral analysis and fixed-point index theorem to explore the existence of positive solutions of the equation, and then use Banach fixed-point theorem to prove the uniqueness of positive solutions. Finally, an interesting example is used to explain the main result. Full article
(This article belongs to the Section General Mathematics, Analysis)
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