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34 pages, 592 KiB

Open AccessArticle

A Fractional Adams Method for Caputo Fractional Differential Equations with Modified Graded Meshes

by Yuhui Yang and Yubin Yan

Mathematics 2025, 13(5), 891; https://doi.org/10.3390/math13050891 - 6 Mar 2025

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In this paper, we introduce an Adams-type predictor–corrector method based on a modified graded mesh for solving Caputo fractional differential equations. This method not only effectively handles the weak singularity near the initial point but also reduces errors associated with large intervals in [...] Read more.

In this paper, we introduce an Adams-type predictor–corrector method based on a modified graded mesh for solving Caputo fractional differential equations. This method not only effectively handles the weak singularity near the initial point but also reduces errors associated with large intervals in traditional graded meshes. We prove the error estimates in detail for both

0 < α < 1

and

1 < α < 2

cases, where

α

is the order of the Caputo fractional derivative. Numerical experiments confirm the convergence of the proposed method and compare its performance with the traditional graded mesh approach. Full article

(This article belongs to the Section E: Applied Mathematics)

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16 pages, 942 KiB

Open AccessFeature PaperArticle

Deformed Boson Algebras and W_α,β,ν-Coherent States: A New Quantum Framework

by Riccardo Droghei

Mathematics 2025, 13(5), 759; https://doi.org/10.3390/math13050759 - 25 Feb 2025

Viewed by 298

Abstract

We introduce a novel class of coherent states, termed

W_{α, β, ν}

-coherent states, constructed using a deformed boson algebra based on the generalised factorial

{[n]}_{α, β, ν}!

. This algebra extends conventional factorials, [...] Read more.

We introduce a novel class of coherent states, termed

W_{α, β, ν}

-coherent states, constructed using a deformed boson algebra based on the generalised factorial

{[n]}_{α, β, ν}!

. This algebra extends conventional factorials, incorporating advanced special functions such as the Mittag-Leffler and Wright functions, enabling the exploration of a broader class of quantum states. The mathematical properties of these states, including their continuity, completeness, and quantum fluctuations, are analysed. A key aspect of this work is the resolution of the Stieltjes moment problem associated with these states, achieved through the inverse Mellin transformation method. The framework provides insights into the interplay between the classical and quantum regimes, with potential applications in quantum optics and fractional quantum mechanics. By extending the theoretical landscape of coherent states, this study opens avenues for further exploration in mathematical physics and quantum technologies. Full article

(This article belongs to the Section E4: Mathematical Physics)

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13 pages, 427 KiB

Open AccessArticle

Existence of Solutions to the Variable Order Caputo Fractional Thermistor Problem

by John R. Graef, Kadda Maazouz, Sandra Pinelas, Zineb Bellabes and Naima Boussekkine

Fractal Fract. 2025, 9(3), 139; https://doi.org/10.3390/fractalfract9030139 - 22 Feb 2025

Viewed by 332

Abstract

The thermistor model captures the complex interaction between heat dissipation, electrical current conduction, and Joule heat generation. Our research examines the diverse properties and implications of employing fractional calculus in the analysis with a focus on fixed-point principles. This paper addresses the existence [...] Read more.

The thermistor model captures the complex interaction between heat dissipation, electrical current conduction, and Joule heat generation. Our research examines the diverse properties and implications of employing fractional calculus in the analysis with a focus on fixed-point principles. This paper addresses the existence and uniqueness of solutions to a variable order Caputo fractional thermistor problem by applying Schauder’s fixed-point theorem. Full article

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29 pages, 975 KiB

Open AccessArticle

Theoretical Results on the pth Moment of ϕ-Hilfer Stochastic Fractional Differential Equations with a Pantograph Term

by Abdelhamid Mohammed Djaouti and Muhammad Imran Liaqat

Fractal Fract. 2025, 9(3), 134; https://doi.org/10.3390/fractalfract9030134 - 20 Feb 2025

Viewed by 328

Abstract

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the

ϕ

-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle [...] Read more.

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the

ϕ

-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for

ϕ (t)

=

t

,

a = 1

. Our findings match the Riemann–Liouville fractional operator for

ϕ (t) = t

,

a = 0

. They agree with the Hadamard and Caputo–Hadamard fractional operators when

ϕ (t) = ln (t)

,

a = 0

and

ϕ (t) = ln (t)

,

a = 1

, respectively. Second, regarding the space, we are make generalizations for the case

p = 2

. Full article

(This article belongs to the Special Issue Fractional Differential Equations: Computation and Modelling with Applications)

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15 pages, 422 KiB

Open AccessArticle

New Results on the Stability and Existence of Langevin Fractional Differential Equations with Boundary Conditions

by Rahman Ullah Khan, Maria Samreen, Gohar Ali and Ioan-Lucian Popa

Fractal Fract. 2025, 9(2), 127; https://doi.org/10.3390/fractalfract9020127 - 18 Feb 2025

Viewed by 228

Abstract

This manuscript aims to establish the existence, uniqueness, and stability of solutions for Langevin fractional differential equations involving the generalized Liouville-Caputo derivative. Using a novel approach, we derive existence and uniqueness results through fixed-point theorems, extending and generalizing several existing findings in the [...] Read more.

This manuscript aims to establish the existence, uniqueness, and stability of solutions for Langevin fractional differential equations involving the generalized Liouville-Caputo derivative. Using a novel approach, we derive existence and uniqueness results through fixed-point theorems, extending and generalizing several existing findings in the literature. To demonstrate the applicability of our results, we provide a practical example that validates the theoretical framework. Full article

(This article belongs to the Section General Mathematics, Analysis)

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12 pages, 501 KiB

Open AccessArticle

Concentrated Couple in a Plane and in a Half-Plane in the Framework of Fractional Nonlocal Elasticity

by Yuriy Povstenko, Tamara Kyrylych, Bożena Woźna-Szcześniak, Ireneusz Szcześniak and Andrzej Yatsko

Appl. Sci. 2025, 15(4), 2048; https://doi.org/10.3390/app15042048 - 15 Feb 2025

Viewed by 360

Abstract

In nonlocal elasticity, the constitutive equation for the stress tensor is written in an integral form, with the weight function, referred to as the nonlocality kernel, often being the Green’s function for the partial differential equation. In this paper, we obtain solutions to [...] Read more.

In nonlocal elasticity, the constitutive equation for the stress tensor is written in an integral form, with the weight function, referred to as the nonlocality kernel, often being the Green’s function for the partial differential equation. In this paper, we obtain solutions to elasticity problems for a concentrated couple in a plane and on the boundary of a half-plane within framework of a new theory of nonlocal elasticity, where the nonlocal kernel is the Green’s function of the Cauchy problem for the fractional diffusion equation. The obtained solutions are free from nonphysical singularities that appear in the classical local elasticity solutions. Full article

(This article belongs to the Special Issue Deformation and Fracture Behaviors of Materials)

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21 pages, 1488 KiB

Open AccessArticle

Exploring Fractional Damped Burgers’ Equation: A Comparative Analysis of Analytical Methods

by Azzh Saad Alshehry and Rasool Shah

Fractal Fract. 2025, 9(2), 107; https://doi.org/10.3390/fractalfract9020107 - 10 Feb 2025

Viewed by 516

Abstract

This investigation focuses on the study of the fractional damped Burgers’ equation by using the natural residual power series method coupled with the new iteration transform method in the context of the Caputo operator. The equation of Burgers under the damped context is [...] Read more.

This investigation focuses on the study of the fractional damped Burgers’ equation by using the natural residual power series method coupled with the new iteration transform method in the context of the Caputo operator. The equation of Burgers under the damped context is useful when studying one-dimensional nonlinear waves involving damping effect, and is used in fluid dynamics, among other applications. Two new mathematical methods that can be used to obtain an approximate solution to this complex non-linear problem are the natural residual power series method and the new iteration transform method. Therefore, it can be deduced that the Caputo operator aids in modeling of the fractional derivatives, as it provides a better description of the physical realities. Thus, the objective of the present work is to advance the knowledge accumulated on the behavior of solutions to the damped Burgers’ equation, as well as to check the applicability of the proposed approaches to other nonlinear fractional partial differential equations. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

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24 pages, 619 KiB

Open AccessArticle

Nonlinear Fractional Evolution Control Modeling via Power Non-Local Kernels: A Generalization of Caputo–Fabrizio, Atangana–Baleanu, and Hattaf Derivatives

by F. Gassem, Mohammed Almalahi, Osman Osman, Blgys Muflh, Khaled Aldwoah, Alwaleed Kamel and Nidal Eljaneid

Fractal Fract. 2025, 9(2), 104; https://doi.org/10.3390/fractalfract9020104 - 8 Feb 2025

Viewed by 445

Abstract

This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. [...] Read more.

This paper presents a novel framework for modeling nonlinear fractional evolution control systems. This framework utilizes a power non-local fractional derivative (PFD), which is a generalized fractional derivative that unifies several well-known derivatives, including Caputo–Fabrizio, Atangana–Baleanu, and generalized Hattaf derivatives, as special cases. It uniquely features a tunable power parameter “p”, providing enhanced control over the representation of memory effects compared to traditional derivatives with fixed kernels. Utilizing the fixed-point theory, we rigorously establish the existence and uniqueness of solutions for these systems under appropriate conditions. Furthermore, we prove the Hyers–Ulam stability of the system, demonstrating its robustness against small perturbations. We complement this framework with a practical numerical scheme based on Lagrange interpolation polynomials, enabling efficient computation of solutions. Examples illustrating the model’s applicability, including symmetric cases, are supported by graphical representations to highlight the approach’s versatility. These findings address a significant gap in the literature and pave the way for further research in fractional calculus and its diverse applications. Full article

(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function, 2nd Edition)

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23 pages, 707 KiB

Open AccessArticle

Novel Fractional Boole’s-Type Integral Inequalities via Caputo Fractional Operator and Their Implications in Numerical Analysis

by Wali Haider, Abdul Mateen, Hüseyin Budak, Asia Shehzadi and Loredana Ciurdariu

Mathematics 2025, 13(4), 551; https://doi.org/10.3390/math13040551 - 7 Feb 2025

Viewed by 401

Abstract

The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) [...] Read more.

The advancement of fractional calculus, particularly through the Caputo fractional derivative, has enabled more accurate modeling of processes with memory and hereditary effects, driving significant interest in this field. Fractional calculus also extends the concept of classical derivatives and integrals to noninteger (fractional) orders. This generalization allows for more flexible and accurate modeling of complex phenomena that cannot be adequately described using integer-order derivatives. Motivated by its applications in various scientific disciplines, this paper establishes novel n-times fractional Boole’s-type inequalities using the Caputo fractional derivative. For this, a fractional integral identity is first established. Using the newly derived identity, several novel Boole’s-type inequalities are subsequently obtained. The proposed inequalities generalize the classical Boole’s formula to the fractional domain. Further extensions are presented for bounded functions, Lipschitzian functions, and functions of bounded variation, providing sharper bounds compared to their classical counterparts. To demonstrate the precision and applicability of the obtained results, graphical illustrations and numerical examples are provided. These contributions offer valuable insights for applications in numerical analysis, optimization, and the theory of fractional integral equations. Full article

(This article belongs to the Section E: Applied Mathematics)

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35 pages, 2025 KiB

Open AccessArticle

Fractional Calculus for Type 2 Interval-Valued Functions

by Mostafijur Rahaman, Dimplekumar Chalishajar, Kamal Hossain Gazi, Shariful Alam, Soheil Salahshour and Sankar Prasad Mondal

Fractal Fract. 2025, 9(2), 102; https://doi.org/10.3390/fractalfract9020102 - 5 Feb 2025

Viewed by 444

Abstract

This paper presents a contemporary introduction of fractional calculus for Type 2 interval-valued functions. Type 2 interval uncertainty involves interval uncertainty with the goal of more assembled perception with reference to impreciseness. In this paper, a Riemann–Liouville fractional-order integral is constructed in Type [...] Read more.

This paper presents a contemporary introduction of fractional calculus for Type 2 interval-valued functions. Type 2 interval uncertainty involves interval uncertainty with the goal of more assembled perception with reference to impreciseness. In this paper, a Riemann–Liouville fractional-order integral is constructed in Type 2 interval delineated vague encompassment. The exploration of fractional calculus is continued with the manifestation of Riemann–Liouville and Caputo fractional derivatives in the cited phenomenon. In addition, Type 2 interval Laplace transformation is proposed in this text. Conclusively, a mathematical model regarding economic lot maintenance is analyzed as a conceivable implementation of this theoretical advancement. Full article

(This article belongs to the Special Issue Mathematical and Physical Analysis of Fractional Dynamical Systems)

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31 pages, 817 KiB

Open AccessArticle

Qualitative Analysis of Generalized Power Nonlocal Fractional System with p-Laplacian Operator, Including Symmetric Cases: Application to a Hepatitis B Virus Model

by Mohamed S. Algolam, Mohammed A. Almalahi, Muntasir Suhail, Blgys Muflh, Khaled Aldwoah, Mohammed Hassan and Saeed Islam

Fractal Fract. 2025, 9(2), 92; https://doi.org/10.3390/fractalfract9020092 - 1 Feb 2025

Viewed by 471

Abstract

This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a [...] Read more.

This paper introduces a novel framework for modeling nonlocal fractional system with a p-Laplacian operator under power nonlocal fractional derivatives (PFDs), a generalization encompassing established derivatives like Caputo–Fabrizio, Atangana–Baleanu, weighted Atangana–Baleanu, and weighted Hattaf. The core methodology involves employing a PFD with a tunable power parameter within a non-singular kernel, enabling a nuanced representation of memory effects not achievable with traditional fixed-kernel derivatives. This flexible framework is analyzed using fixed-point theory, rigorously establishing the existence and uniqueness of solutions for four symmetric cases under specific conditions. Furthermore, we demonstrate the Hyers–Ulam stability, confirming the robustness of these solutions against small perturbations. The versatility and generalizability of this framework is underscored by its application to an epidemiological model of transmission of Hepatitis B Virus (HBV) and numerical simulations for all four symmetric cases. This study presents findings in both theoretical and applied aspects of fractional calculus, introducing an alternative framework for modeling complex systems with memory processes, offering opportunities for more sophisticated and accurate models and new avenues for research in fractional calculus and its applications. Full article

(This article belongs to the Special Issue Advanced Numerical Methods for Fractional Functional Models)

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19 pages, 389 KiB

Open AccessFeature PaperArticle

On the Existence and Uniqueness of Solutions for Neutral-Type Caputo Fractional Differential Equations with Iterated Delays: Hyers–Ulam–Mittag–Leffler Stability

by Ekaterina Madamlieva and Mihail Konstantinov

Mathematics 2025, 13(3), 484; https://doi.org/10.3390/math13030484 - 31 Jan 2025

Viewed by 524

Abstract

This study investigates nonlinear Caputo-type fractional differential equations with iterated delays, focusing on the neutral type. Initially formulated by D. Bainov and the second author of the current paper between 1972 and 1978, these superneutral equations have been extensively studied in scholarly inquiry. [...] Read more.

This study investigates nonlinear Caputo-type fractional differential equations with iterated delays, focusing on the neutral type. Initially formulated by D. Bainov and the second author of the current paper between 1972 and 1978, these superneutral equations have been extensively studied in scholarly inquiry. The present research seeks to reinvigorate interest in such delays within sophisticated frameworks of differential equations, particularly those involving fractional calculus. The primary objectives are to thoroughly examine neutral-type fractional differential equations with iterated delays and provide novel insights into their existence and uniqueness by applying Bielecki’s and Chebyshev’s norms for solution constraints analysis. Additionally, this work establishes Hyers–Ulam–Mittag–Leffler stability for these equations. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 3rd Edition)

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Plot of <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mo>Γ</mo> <mo>(</mo> <mi>z</mi> <mo>)</mo> <mo>Γ</mo> <mo>(</mo> <mn>2</mn> <mo>−</mo> <mi>z</mi> <mo>)</mo> </mrow> </mfrac> </mstyle> <mo><</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>z</mi> <mo>∈</mo> <mrow> <mo>[</mo> <mo>−</mo> <mn>15</mn> <mo>,</mo> <mn>16</mn> <mo>]</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> Full article ">Figure 2
Plot of <math display="inline"><semantics> <mrow> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mn>1</mn> <mrow> <mo>Γ</mo> <mo>(</mo> <mi>α</mi> <mo>)</mo> <mo>Γ</mo> <mo>(</mo> <mn>2</mn> <mo>−</mo> <mi>α</mi> <mo>)</mo> </mrow> </mfrac> </mstyle> <mo><</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt"/> <mi>α</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> Full article ">Figure 3
Plot of the solutions <math display="inline"><semantics> <mrow> <msup> <mi>x</mi> <mo>*</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="4pt"/> <msub> <mi>x</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> of Example 2. Full article ">

18 pages, 497 KiB

Open AccessArticle

Strict Stability of Fractional Differential Equations with a Caputo Fractional Derivative with Respect to Another Function

by Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Mathematics 2025, 13(3), 452; https://doi.org/10.3390/math13030452 - 29 Jan 2025

Viewed by 571

Abstract

In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider [...] Read more.

In this paper, we study nonlinear systems of fractional differential equations with a Caputo fractional derivative with respect to another function (CFDF) and we define the strict stability of the zero solution of the considered nonlinear system. As an auxiliary system, we consider a system of two scalar fractional equations with CFDF and define a strict stability in the couple. We illustrate both definitions with several examples and, in these examples, we show that the applied function in the fractional derivative has a huge influence on the stability properties of the solutions. In addition, we use Lyapunov functions and their CFDF to obtain several sufficient conditions for strict stability. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

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19 pages, 894 KiB

Open AccessArticle

Fixed/Preassigned Time Synchronization of Impulsive Fractional-Order Reaction–Diffusion Bidirectional Associative Memory (BAM) Neural Networks

by Rouzimaimaiti Mahemuti, Abdujelil Abdurahman and Ahmadjan Muhammadhaji

Fractal Fract. 2025, 9(2), 88; https://doi.org/10.3390/fractalfract9020088 - 28 Jan 2025

Viewed by 489

Abstract

This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, [...] Read more.

This study delves into the synchronization issues of the impulsive fractional-order, mainly the Caputo derivative of the order between 0 and 1, bidirectional associative memory (BAM) neural networks incorporating the diffusion term at a fixed time (FXT) and a predefined time (PDT). Initially, this study presents certain characteristics of fractional-order calculus and several lemmas pertaining to the stability of general impulsive nonlinear systems, specifically focusing on FXT and PDT stability. Subsequently, we utilize a novel controller and Lyapunov functions to establish new sufficient criteria for achieving FXT and PDT synchronizations. Finally, a numerical simulation is presented to ascertain the theoretical dependency. Full article

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12 pages, 3493 KiB

Open AccessArticle

On a Preloaded Compliance System of Fractional Order: Numerical Integration

by Marius-F. Danca

Fractal Fract. 2025, 9(2), 84; https://doi.org/10.3390/fractalfract9020084 - 26 Jan 2025

Viewed by 446

Abstract

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, [...] Read more.

In this paper, the use of a class of fractional-order dynamical systems with discontinuous right-hand side defined with Caputo’s derivative is considered. The existence of the solutions is analyzed. For this purpose, differential inclusions theory is used to transform, via the Filippov regularization, the discontinuous right-hand side into a set-valued function. Next, via Cellina’s Theorem, the obtained set-valued differential inclusion of fractional order can be restarted as a single-valued continuous differential equation of fractional order, to which the existing numerical schemes for fractional differential equations can be applied. In this way, the delicate problem of integrating discontinuous problems of fractional order, as well as integer order, is solved by transforming the discontinuous problem into a continuous one. Also, it is noted that even the numerical methods for fractional-order differential equations can be applied abruptly to the discontinuous problem, without considering the underlying discontinuity, so the results could be incorrect. The technical example of a single-degree-of-freedom preloaded compliance system of fractional order is presented. Full article

(This article belongs to the Special Issue Numerical Solution and Applications of Fractional Differential Equations, 2nd Edition)

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