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22 pages, 3829 KB

Open AccessArticle

The Crank-Nicolson Mixed Finite Element Scheme and Its Reduced-Order Extrapolation Model for the Fourth-Order Nonlinear Diffusion Equations with Temporal Fractional Derivative

by Jiahua Wang, Hong Li, Xuehui Ren and Xiaohui Chang

Fractal Fract. 2025, 9(12), 789; https://doi.org/10.3390/fractalfract9120789 - 1 Dec 2025

Viewed by 317

Abstract

This paper presents a Crank–Nicolson mixed finite element method along with its reduced-order extrapolation model for a fourth-order nonlinear diffusion equation with Caputo temporal fractional derivative. By introducing the auxiliary variable

v = - ε^{2} Δ u + f (u)

[...] Read more.

This paper presents a Crank–Nicolson mixed finite element method along with its reduced-order extrapolation model for a fourth-order nonlinear diffusion equation with Caputo temporal fractional derivative. By introducing the auxiliary variable

v = - ε^{2} Δ u + f (u)

, the equation is reformulated as a second-order coupled system. A Crank–Nicolson mixed finite element scheme is established, and its stability is proven using a discrete fractional Gronwall inequality. Error estimates for the variables u and v are derived. Furthermore, a reduced-order extrapolation model is constructed by applying proper orthogonal decomposition to the coefficient vectors of the first several finite element solutions. This scheme is also proven to be stable, and its error estimates are provided. Theoretical analysis shows that the reduced-order extrapolation Crank–Nicolson mixed finite approach reduces the degrees of freedom from tens of thousands to just a few, significantly cutting computational time and storage requirements. Numerical experiments demonstrate that both schemes achieve spatial second-order convergence accuracy. Under identical conditions, the CPU time required by the reduced-order extrapolation Crank–Nicolson mixed finite model is only

1 / 60

of that required by the Crank–Nicolson mixed finite scheme. These results validate the theoretical analysis and highlight the effectiveness of the methods. Full article

(This article belongs to the Section Numerical and Computational Methods)

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27 pages, 643 KB

Open AccessArticle

Fractional Modeling and Stability Analysis of Tomato Yellow Leaf Curl Virus Disease: Insights for Sustainable Crop Protection

by Mansoor Alsulami, Ali Raza, Marek Lampart, Umar Shafique and Eman Ghareeb Rezk

Fractal Fract. 2025, 9(12), 754; https://doi.org/10.3390/fractalfract9120754 - 21 Nov 2025

Viewed by 414

Abstract

Tomato Yellow Leaf Curl Virus (TYLCV) has recently caused severe economic losses in global tomato production. According to the International Plant Protection Convention (IPPC), yield reductions of 50–60% have been reported in several regions, including the Caribbean, Central America, and South Asia, with [...] Read more.

Tomato Yellow Leaf Curl Virus (TYLCV) has recently caused severe economic losses in global tomato production. According to the International Plant Protection Convention (IPPC), yield reductions of 50–60% have been reported in several regions, including the Caribbean, Central America, and South Asia, with losses in sensitive cultivars reaching up to 90–100%. In developing countries, TYLCV and mixed infections affect more than seven million hectares of tomato-growing land annually. In this study, we construct and analyze a nonlinear dynamic model describing the transmission of TYLCV, incorporating the Caputo fractional-order derivative operator. The existence and uniqueness of solutions to the proposed model are rigorously established. Equilibrium points are identified, and the Jacobian determinant approach is applied to compute the basic reproduction number,

R_{0}

. Suitable Lyapunov functions are formulated to analyze the global asymptotic stability of both the disease-free and endemic equilibria. The model is numerically solved using the Grünwald–Letnikov-based nonstandard finite difference method, and simulations assess how the memory index and preventive strategies influence disease propagation. The results reveal critical factors governing TYLCV transmission and suggest effective intervention measures to guide sustainable crop protection policies. Full article

(This article belongs to the Special Issue Applications of Fractional Calculus in Modern Mathematical Modeling)

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24 pages, 774 KB

Open AccessArticle

Electrical Analogy Approach to Fractional Heat Conduction Models

by Slobodanka Galovic, Marica N. Popovic and Dalibor Chevizovich

Fractal Fract. 2025, 9(10), 653; https://doi.org/10.3390/fractalfract9100653 - 9 Oct 2025

Viewed by 765

Abstract

Fractional heat conduction models extend classical formulations by incorporating fractional differential operators that capture multiscale relaxation effects. In this work, we introduce an electrical analogy that represents the action of these operators via generalized longitudinal impedance and admittance elements, thereby clarifying their physical [...] Read more.

Fractional heat conduction models extend classical formulations by incorporating fractional differential operators that capture multiscale relaxation effects. In this work, we introduce an electrical analogy that represents the action of these operators via generalized longitudinal impedance and admittance elements, thereby clarifying their physical role in energy transfer: fractional derivatives account for the redistribution of heat accumulation and dissipation within micro-scale heterogeneous structures. This analogy unifies different classes of fractional models—diffusive, wave-like, and mixed—as well as distinct fractional operator types, including the Caputo and Atangana–Baleanu forms. It also provides a general computational methodology for solving heat conduction problems through the concept of thermal impedance, defined as the ratio of surface temperature variations (relative to ambient equilibrium) to the applied heat flux. The approach is illustrated for a semi-infinite sample, where different models and operators are shown to generate characteristic spectral patterns in thermal impedance. By linking these spectral signatures of microstructural relaxation to experimentally measurable quantities, the framework not only establishes a unified theoretical foundation but also offers a practical computational tool for identifying relaxation mechanisms through impedance analysis in microscale thermal transport. Full article

(This article belongs to the Special Issue Analysis and Applications of Fractional Calculus in Computational Physics)

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22 pages, 968 KB

Open AccessArticle

Fractal–Fractional Coupled Systems with Constant and State- Dependent Delays: Existence Theory and Ecological Applications

by Faten H. Damag, Ashraf A. Qurtam, Arshad Ali, Abdelaziz Elsayed, Alawia Adam, Khaled Aldwoah and Salahedden Omer Ali

Fractal Fract. 2025, 9(10), 652; https://doi.org/10.3390/fractalfract9100652 - 9 Oct 2025

Cited by 1 | Viewed by 749

Abstract

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that [...] Read more.

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that vary with the system’s state. Such a formulation provides a closer representation of biological and physical processes in which delays are not fixed but evolve dynamically. Sufficient conditions for the existence and uniqueness of solutions are established using fixed-point theory, while the stability of the solution is investigated via the Hyers–Ulam (HU) stability approach. To demonstrate applicability, the approach is applied to two illustrative examples, including a predator–prey interaction model. The findings advance the theory of fractional-order systems with mixed delays and offer a rigorous foundation for developing realistic, application-driven dynamical models. Full article

(This article belongs to the Special Issue Fractional Calculus Applied in Environmental Biosystems)

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32 pages, 3446 KB

Open AccessArticle

Finite Element Method for Time-Fractional Navier–Stokes Equations with Nonlinear Damping

by Shahid Hussain, Xinlong Feng, Arafat Hussain and Ahmed Bakhet

Fractal Fract. 2025, 9(7), 445; https://doi.org/10.3390/fractalfract9070445 - 4 Jul 2025

Cited by 2 | Viewed by 1352

Abstract

We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (

0 < α < 1

) with mixed finite element methods (P1b–P1 and [...] Read more.

We propose a hybrid numerical framework for solving time-fractional Navier–Stokes equations with nonlinear damping. The method combines the finite difference L1 scheme for time discretization of the Caputo derivative (

0 < α < 1

) with mixed finite element methods (P1b–P1 and

P_{2}

–

P_{1}

) for spatial discretization of velocity and pressure. This approach addresses the key challenges of fractional models, including nonlocality and memory effects, while maintaining stability in the presence of the nonlinear damping term

{γ | u |}^{r - 2} u

, for

r \geq 2

. We prove unconditional stability for both semi-discrete and fully discrete schemes and derive optimal error estimates for the velocity and pressure components. Numerical experiments validate the theoretical results. Convergence tests using exact solutions, along with benchmark problems such as backward-facing channel flow and lid-driven cavity flow, confirm the accuracy and reliability of the method. The computed velocity contours and streamlines show close agreement with analytical expectations. This scheme is particularly effective for capturing anomalous diffusion in Newtonian and turbulent flows, and it offers a strong foundation for future extensions to viscoelastic and biological fluid models. Full article

(This article belongs to the Special Issue Analysis and Numerical Computations of Nonlinear Fractional and Classical Differential Equations)

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13 pages, 330 KB

Open AccessArticle

Existence of Solutions to Fractional Differential Equations with Mixed Caputo–Riemann Derivative

by Mahir Almatarneh, Sonuc Zorlu and Nazim I. Mahmudov

Fractal Fract. 2025, 9(6), 374; https://doi.org/10.3390/fractalfract9060374 - 12 Jun 2025

Cited by 1 | Viewed by 1329

Abstract

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to [...] Read more.

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo–Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem’s parameters. The results are derived by constructing the Green’s function for the initial value problem. Schauder’s fixed-point theorem is used to prove existence, while Banach’s contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results. Full article

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31 pages, 476 KB

Open AccessArticle

Strong Convergence of a Modified Euler—Maruyama Method for Mixed Stochastic Fractional Integro—Differential Equations with Local Lipschitz Coefficients

by Zhaoqiang Yang and Chenglong Xu

Fractal Fract. 2025, 9(5), 296; https://doi.org/10.3390/fractalfract9050296 - 1 May 2025

Viewed by 1257

Abstract

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using [...] Read more.

This paper presents a modified Euler—Maruyama (EM) method for mixed stochastic fractional integro—differential equations (mSFIEs) with Caputo—type fractional derivatives whose coefficients satisfy local Lipschitz and linear growth conditions. First, we transform the mSFIEs into an equivalent mixed stochastic Volterra integral equations (mSVIEs) using a fractional calculus technique. Then, we establish the well—posedness of the analytical solutions of the mSVIEs. After that, a modified EM scheme is formulated to approximate the numerical solutions of the mSVIEs, and its strong convergence is proven based on local Lipschitz and linear growth conditions. Furthermore, we derive the modified EM scheme under the same conditions in the

L^{2}

sense, which is consistent with the strong convergence result of the corresponding EM scheme. Notably, the strong convergence order under local Lipschitz conditions is inherently lower than the corresponding order under global Lipschitz conditions. Finally, numerical experiments are presented to demonstrate that our approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the

L^{2}

sense. Full article

(This article belongs to the Section Numerical and Computational Methods)

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22 pages, 343 KB

Open AccessArticle

Hyers–Ulam and Hyers–Ulam–Rassias Stability for a Class of Fractional Evolution Differential Equations with Neutral Time Delay

by Kholoud N. Alharbi

Symmetry 2025, 17(1), 83; https://doi.org/10.3390/sym17010083 - 7 Jan 2025

Cited by 2 | Viewed by 1098

Abstract

In this paper, we demonstrate that neutral fractional evolution equations with finite delay possess a stable mild solution. Our model incorporates a mixed fractional derivative that combines the Riemann–Liouville and Caputo fractional derivatives with orders

0 < α < 1

and [...] Read more.

In this paper, we demonstrate that neutral fractional evolution equations with finite delay possess a stable mild solution. Our model incorporates a mixed fractional derivative that combines the Riemann–Liouville and Caputo fractional derivatives with orders

0 < α < 1

and

1 < β < 2

. We identify the infinitesimal generator of the cosine family and analyze the stability of the mild solution using both Hyers–Ulam–Rassias and Hyers–Ulam stability methodologies, ensuring robust and reliable results for fractional dynamic systems with delay. In order to guarantee that the features of invariance under transformations, such as rotations or reflections, result in the presence of fixed points that remain unchanging and represent the consistency and balance of the underlying system, fixed-point theorems employ the symmetry idea. Lastly, the results obtained are applied to a fractional order nonlinear wave equation with finite delay with respect to time. Full article

(This article belongs to the Special Issue Nonlinear Differential and Integral Equations and Their Infinite Systems)

30 pages, 2587 KB

Open AccessArticle

A Local Radial Basis Function Method for Numerical Approximation of Multidimensional Multi-Term Time-Fractional Mixed Wave-Diffusion and Subdiffusion Equation Arising in Fluid Mechanics

by Kamran, Ujala Gul, Zareen A. Khan, Salma Haque and Nabil Mlaiki

Fractal Fract. 2024, 8(11), 639; https://doi.org/10.3390/fractalfract8110639 - 29 Oct 2024

Cited by 3 | Viewed by 1832

Abstract

This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation features a unique time–space coupled derivative while simultaneously [...] Read more.

This article develops a simple hybrid localized mesh-free method (LMM) for the numerical modeling of new mixed subdiffusion and wave-diffusion equation with multi-term time-fractional derivatives. Unlike conventional multi-term fractional wave-diffusion or subdiffusion equations, this equation features a unique time–space coupled derivative while simultaneously incorporating both wave-diffusion and subdiffusion terms. Our proposed method follows three basic steps: (i) The given equation is transformed into a time-independent form using the Laplace transform (LT); (ii) the LMM is then used to solve the transformed equation in the LT domain; (iii) finally, the time domain solution is obtained by inverting the LT. We use the improved Talbot method and the Stehfest method to invert the LT. The LMM is used to circumvent the shape parameter sensitivity and ill-conditioning of interpolation matrices that commonly arise in global mesh-free methods. Traditional time-stepping methods achieve accuracy only with very small time steps, significantly increasing the computational time. To overcome these shortcomings, the LT is used to provide a more powerful alternative by removing the need for fine temporal discretization. Additionally, the Ulam–Hyers stability of the considered model is analyzed. Four numerical examples are presented to illustrate the effectiveness and practical applicability of the method. Full article

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

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19 pages, 325 KB

Open AccessArticle

A Qualitative Analysis of a Non-Linear Coupled System under Two Types of Fractional Derivatives along with Mixed Boundary Conditions

by Abdelkader Amara, Mohammed El-Hadi Mezabia, Brahim Tellab, Khaled Zennir, Keltoum Bouhali and Loay Alkhalifa

Fractal Fract. 2024, 8(7), 366; https://doi.org/10.3390/fractalfract8070366 - 22 Jun 2024

Cited by 1 | Viewed by 1311

Abstract

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. [...] Read more.

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model’s solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method’s efficacy and potential for further applications. Full article

(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications)

24 pages, 469 KB

Open AccessArticle

Time-Varying Function Matrix Projection Synchronization of Caputo Fractional-Order Uncertain Memristive Neural Networks with Multiple Delays via Mixed Open Loop Feedback Control and Impulsive Control

by Hongguang Fan, Yue Rao, Kaibo Shi and Hui Wen

Fractal Fract. 2024, 8(5), 301; https://doi.org/10.3390/fractalfract8050301 - 20 May 2024

Cited by 20 | Viewed by 1673

Abstract

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization [...] Read more.

This paper shows solicitude for the generalized projective synchronization of Caputo fractional-order uncertain memristive neural networks (FOUMNNs) with multiple delays. By extending the constant scale factor to the time-varying function matrix, we establish an extraordinary synchronization mode called time-varying function matrix projection synchronization (TFMPS), which is a generalized version of traditional matrix projection synchronization, modified projection synchronization, complete synchronization, and anti-synchronization. To achieve the goal of TFMPS, we design a novel mixed controller including the open loop feedback control and impulsive control, which employs the state information in the time-delayed interval and the sampling information at the impulse instants. It has a prominent advantage that impulse intervals are not restricted by time delays. To establish the connection between the error system and the auxiliary system, a generalized fractional-order comparison theorem with time-varying coefficients and impulses is established. Applying the stability theory, the comparison theorem, and the Laplace transform, new synchronization criteria of FOUMNNs are acquired under the mixed impulsive control schemes, and the derived synchronization theorem and corollary can effectively expand the correlative synchronization achievements of fractional-order systems. Full article

(This article belongs to the Special Issue Modeling, Optimization, and Control of Fractional-Order Neural Networks and Nonlinear Systems)

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20 pages, 707 KB

Open AccessArticle

Existence and Uniqueness Result for Fuzzy Fractional Order Goursat Partial Differential Equations

by Muhammad Sarwar, Noor Jamal, Kamaleldin Abodayeh, Chanon Promsakon and Thanin Sitthiwirattham

Fractal Fract. 2024, 8(5), 250; https://doi.org/10.3390/fractalfract8050250 - 25 Apr 2024

Cited by 4 | Viewed by 1931

Abstract

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s

g H

-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s

g H

-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert [...] Read more.

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s

g H

-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s

g H

-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s

g H

-differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)

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17 pages, 922 KB

Open AccessArticle

A New Mixed Fractional Derivative with Applications in Computational Biology

by Khalid Hattaf

Computation 2024, 12(1), 7; https://doi.org/10.3390/computation12010007 - 4 Jan 2024

Cited by 67 | Viewed by 3850

Abstract

This study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, [...] Read more.

This study develops a new definition of a fractional derivative that mixes the definitions of fractional derivatives with singular and non-singular kernels. This developed definition encompasses many types of fractional derivatives, such as the Riemann–Liouville and Caputo fractional derivatives for singular kernel types, as well as the Caputo–Fabrizio, the Atangana–Baleanu, and the generalized Hattaf fractional derivatives for non-singular kernel types. The associate fractional integral of the new mixed fractional derivative is rigorously introduced. Furthermore, a novel numerical scheme is developed to approximate the solutions of a class of fractional differential equations (FDEs) involving the mixed fractional derivative. Finally, an application in computational biology is presented. Full article

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10 pages, 296 KB

Open AccessArticle

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

by Doha A. Kattan and Hasanen A. Hammad

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

Cited by 5 | Viewed by 1943

Abstract

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

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

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

10 pages, 1255 KB

Open AccessArticle

Sturm-Liouville Problem with Mixed Boundary Conditions for a Differential Equation with a Fractional Derivative and Its Application in Viscoelasticity Models

by Ludmila Kiryanova and Tatiana Matseevich

Axioms 2023, 12(8), 779; https://doi.org/10.3390/axioms12080779 - 11 Aug 2023

Cited by 1 | Viewed by 2015

Abstract

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system [...] Read more.

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system of eigenfunctions, the biorthogonal system, and the distribution of eigenvalues on the real axis were presented. The numerical behavior of eigenvalues was studied depending on the order of the fractional derivative for both definitions of the fractional derivative. Full article

(This article belongs to the Special Issue Mathematical Analysis in Application to Solving Mathematical and Technical Problems)

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