Caputo Derivative Research Articles

L1‐type finite element method for time‐fractional diffusion‐wave equations on nonuniform grids

AbstractAs is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments.

Multidimensional Fractional Calculus: Theory and Applications

In this paper, we introduce several new types of partial fractional derivatives in the continuous setting and the discrete setting. We analyze some classes of the abstract fractional differential equations and the abstract fractional difference equations depending on several variables, providing a great number of structural results, useful remarks and illustrative examples. Concerning some specific applications, we would like to mention here our investigation of the fractional partial differential inclusions with Riemann–Liouville and Caputo derivatives. We also establish the complex characterization theorem for the multidimensional vector-valued Laplace transform and provide certain applications.

A numerical method for Ψ-fractional integro-differential equations by Bell polynomials

In this work, we focus on a class of Ψ− fractional integro-differential equations (Ψ-FIDEs) involving Ψ-Caputo derivative. The objective of this paper is to derive the numerical solution of Ψ-FIDEs in the truncated Bell series. Firstly, Ψ-FIDEs by using the definition of Ψ− Caputo derivative is converted into a singular integral equation. Then, a computational procedure based on the Bell polynomials, Gauss-Legendre quadrature rule, and collocation method is developed to effectively solve the singular integral equation. The convergence of the approximation obtained in the presented strategy is investigated. Finally, the effectiveness and superiority of our method are revealed by numerical samples. The results of the suggested approach are compared with the results obtained by extended Chebyshev cardinal wavelets method (EChCWM).

Fractional relaxation model with general memory effects and stability analysis

Voigt and Maxwell models are popularly used to model viscoelastic materials’ property. They are often presented in form of fractional relaxation equations. In order to describe rich viscoelasticity, a general Caputo derivative is introduced in fractional modeling. Then this work studies attractivity and asymptotic stability of the Caputo fractional relaxation equation with general memory effects. Firstly, the considered problem is transformed into an integral equation. A mapping and an attractive set are constructed. Furthermore, the existence of fixed points on the attractive set are investigated by using fixed point theorems. Finally, the effectiveness and convenience of the stability theory are verified through two numerical examples.

Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method

المعادلة التفاضلية الكسرية توفر إطارًا رياضيًا قويًا لنمذجة وفهم مجموعة واسعة من الظواهر المعقدة وغير المحلية والمعتمدة على الذاكرة في مختلف التطبيقات العلمية والهندسية والعالمية. تعد الاعتلالات الكبدية أورامًا سرطانية ثانوية تتطور في الكبد نتيجة انتشار خلايا السرطان منورام سرطاني أصلي ينشأ في جزء آخر من جسم الإنسان. التركيز الأساسي لهذه الدراسة هو فهم نمو الأورام في الكبد البشري، سواء بوجود أو بدون علاج دوائي. ولتحقيق ذلك، يتم استخدام معادلة تفاضلية جزئية معرفة بالترتيب الزمني والكسري، ويتم تحليلها باستخدام أساليب عددية. يتم استخدام المشتقة كابوتو لاستكشاف تأثير العلاج الدوائي على نمو الورم. لحل النموذج الرياضي عدديًا، تم تطوير طريقة الفروق المحدودة كرانك-نيكولسون. يتم اختيار هذه الطريقة بسبب خصائصها المميزة، بما في ذلك الاستقرار الغير مشروط والدقة من الدرجة الثانية في الأبعاد الزمنية والمكانية. الاستقرار والدقة التي توفرهما هاتان الخصائص أمران حاسمان لضمان موثوقية النتائج المحصلة في هذه البحث. يتم تقديم النتائج والتحققات المستمدة من هذه الدراسة بفعالية من خلال تمثيلات بصرية متنوعة. تعتبر هذه الوسائل البصرية أدوات لا غنى عنها لفهم التأثير العميق للعلاج الدوائي على نمو الأورام. من خلال هذا التمثيل البصري، يمكن للشخص الحصول على فهم أوضح وأكثر تفصيلًا للديناميات المعقدة التي تحدث. تم الوصول إلى الحل العددي لهذه المشكلة المعقدة من خلال تنفيذ خوارزميات تم تطويرها بدقة باستخدام لغة البرمجة بايثون المتعددة الاستخدامات والقوية. مرونة بايثون والمكتبات الشاملة وإمكانيات الحساب العددي القوية تجعلها الخيار المثالي للتعامل مع تعقيدات هذه الدراسة.

NEUTRON POINT KINETICS MODEL WITH A DISTRIBUTED-ORDER FRACTIONAL DERIVATIVE

In this paper, the solutions of an extended form of the Fractional-order Neutron Point Kinetics (FNPK) equation in terms of Caputo-time derivatives of the same order are investigated. Instead of using a Caputo derivative, a distributed-order fractional derivative in the Caputo sense was employed in the term of the FNPK equation which is multiplied by the reactivity. This term plays an important role in the description of neutron kinetics during the start-up, shutdown, and steady-state processes in nuclear reactors. The extended (DFNPK) model was solved using the beta, normal, bimodal and Dirac delta distributions to investigate their effect on the transient state solutions of the neutron density. Regardless of the distribution used, the most significant finding is that a destabilizing effect on the neutron density is induced when the mode (or the instant of application of the Dirac delta) of the distribution tends to one while maintaining the orders of the Caputo-time derivatives constant. What defines the destabilizing effect are large magnitude oscillations, a rapid decay, and an oscillation-free steady state with a monotonic increase that is parallel to but somewhat above the trend determined by the FNPK equation. The extended model is anticipated to be effective for modeling neutron density dispersion in a highly heterogeneous medium that may be described using distributed derivatives.

The transmission dynamics of an infectious disease model in fractional derivative with vaccination under real data

The resurgence of monkeypox causes considerable healthcare risks needing efficient immunization programs. This work investigates the monkeypox disease dynamics in the UK, focusing on the impact of vaccination under real data. The key difficulty is to correctly predict the spread of the disease and evaluate the success of immunization efforts. We construct a mathematical model for monkeypox infection and extend it to the fractional case considering the Caputo derivative. The analysis ensures the positivity, boundedness, and uniqueness of the solution for the non-integer system. We conduct a local asymptotical stability analysis (LAS) at the disease-free equilibrium (DFE) D0, showing the result for R0<1. Additionally, we demonstrate the existence of multiple endemic equilibria and provide conditions for backward bifurcation, which are illustrated graphically. Using real case data from the UK, we estimate model parameters via the nonlinear least square method. Our results show that, without vaccination, R2≈0.8, whereas vaccination reduces it to R2v=0.48. We perform sensitivity analysis to identify key parameters influencing disease elimination, presenting the outcomes through graphs. To solve numerically the fractional model, we outline a numerical scheme and provide detailed results under various parameter assumptions. Our findings suggest that high vaccine efficacy, a low waning rate of the vaccines, and increased vaccination of the infected people can significantly reduce the future cases of monkeypox in the UK. The present study offers a comprehensive framework for monkeypox dynamics and informs public health strategies for effective disease control and prevention.

On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces

As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation dψβ,μdtβdψα,μdtα+λx(t)=f(t,x(t)),t∈[a,b],λ∈R, for f∈C[a,b]×R and some critical orders β,α∈(0,1), combined with appropriate initial or boundary conditions, and with general classes of ψ-tempered Hilfer problems with ψ-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied.

Simulating time delays and space–time memory interactions: An analytical approach

This study introduces a novel analytical framework to explore the effects of Caputo spatial and temporal memory indices combined with a proportional time delay on (non)linear (1+2)-dimensional evolutionary models. The solution is expressed as a Cauchy product of an absolutely convergent series that effectively captures the dynamics of these parameters. By extending the differential transform method into higher-dimensional fractional space, we reformulate the evolution equation as a (non)linear higher-order recurrence relation, which enables the precise determination of fractional series coefficients. Our findings show that Caputo derivatives and time delay significantly impact the system’s behavior, with graphical analysis revealing a continuous transition from a stationary to an integer state solution. The study also identifies a quantitative analogy between the Caputo-time fractional derivative and proportional time delay that highlights the role of Caputo derivatives as memory indices. This method has proven highly effective in deriving solutions for fractional higher-dimensional extensions of evolutionary equations.

A Robust Higher-Order Scheme for Fractional Delay Differential Equations Involving Caputo Derivative

A Robust Higher-Order Scheme for Fractional Delay Differential Equations Involving Caputo Derivative

Fractional diffusion equations interpolate between damping and waves

Abstract The behaviour of the solutions of the time-fractional diffusion equation,&amp;#xD;based on the Caputo derivative, is studied and its dependence on the fractional&amp;#xD;exponent is analysed. The time-fractional convection-diffusion equation is also solved&amp;#xD;and an application to Pennes bioheat model is presented. Generically, a wave-like&amp;#xD;transport at short times passes over to a diffusion-like behaviour at later times.&amp;#xD;

High-Order Numerical Approximation for 2D Time-Fractional Advection–Diffusion Equation under Caputo Derivative

High-Order Numerical Approximation for 2D Time-Fractional Advection–Diffusion Equation under Caputo Derivative

Analysis of some dynamical systems by combination of two different methods

In this study, we introduce a novel iterative method combined with the Elzaki transformation to address a system of partial differential equations involving the Caputo derivative. The Elzaki transformation, known for its effectiveness in solving differential equations, is incorporated into the proposed iterative approach to enhance its efficiency. The system of partial differential equations under consideration is characterized by the presence of Caputo derivatives, which capture fractional order dynamics. The developed method aims to provide accurate and efficient solutions to this complex mathematical system, contributing to the broader understanding of fractional calculus applications in the context of partial differential equations. Through numerical experiments and comparisons, we demonstrate the efficacy of the proposed Elzaki-transform-based iterative method in handling the intricate dynamics inherent in the given system. The study not only showcases the versatility of the Elzaki transformation but also highlights the potential of the developed iterative technique for addressing similar problems in various scientific and engineering domains.

Wavelet Collocation Method for HIV-1/HTLV-I Co-Infection Model Using Hermite Polynomial.

In this study, the dynamic behavior of fractional order co-infection model with human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is analyzed using operational matrix of Hermite wavelet collocation method. Also, the uniqueness and existence of solutions are calculated based on the fixed point hypothesis. For the fractional order co-infection model, its positivity and boundedness are demonstrated. Furthermore, different types of Ulam-Hyres stability are also discussed. The numerical solution of the model are obtained by using the operational matrix of the Hermite wavelet approach. This scheme is used to solve the system of nonlinear equationsthat are very fruitful and easy to implement. Additionally, the stability analysis of the numerical scheme is explained. The mathematical model taken in this work incorporates the biological characteristics of both HIV-1 and HTLV-I. After that all the equilibrium points of the fractional order co-infection model are found and their existence conditions are explored with the help of the Caputo derivative. The global stability of all equilibrium points of this model are determined with the help of Lyapunov functions and the LaSalle invariance principle. Convergence analysis is also discussed. Hermite wavelet operational matrix methods are more accurate and convergent than other numerical methods. Lastly, variations in model dynamics are found when examining different fractional order values. These findings will be valuable to biologists in the treatment of HIV-1/HTLV-I.

An accurate numerical technique for solving fractional advection–diffusion equation with generalized Caputo derivative

An accurate numerical technique for solving fractional advection–diffusion equation with generalized Caputo derivative

Laplace Adomian Decomposition Method for Fractional Order SIS Epidemic Model

This research manuscript addresses finding a solution to the SIS Epidemic model using the Caputo derivative. To achieve the required results, Laplace transform together with the Adomian Decomposition Method (LADM) were utilized. This method is a powerful tool for dealing with various linear and nonlinear problems of fractional-order differential equations (FODEs). Additionally, some results related to the qualitative theory of the model of interest were studied. The computational results showed the verification of the performed analysis. The investigation of the approximate solution computed in the form of an infinite series was carried out. The results were graphically displayed to analyze the adopted procedure for solving nonlinear FODEs using the Caputo fractional derivative.

Three-dimensional fractional plastic models for saturated sand using Caputo derivative and R-L derivative: A comparative study

The strength and dilatancy of sand are mainly influenced by the void ratio, confining pressure and stress path. In engineering, the stress-strain relationship of sand in three-dimensional stress state is complicated, where the state-dependent properties of sand are difficult to be described by traditional constitutive models. Thus, fractional plastic models based on different fractional derivatives were developed to capture such state-dependent behavior of sand. This paper attempts to make a comparative study of the fractional models based on two typical fractional derivatives, i.e., the R-L derivative and Caputo derivative. The results show that the different types of fractional derivatives are mainly reflected in the different order of differentiation and integration, which will have a significant impact on the calculation of plastic flow direction, stress-dilatancy ratio and fractional order parameter βin the constitutive model. By determining the constitutive parameters of the two models, the constitutive behaviors predicted by different models were compared with the corresponding test results of saturated sands under different initial mean pressures and different stress paths. The models can reasonably simulate the test results of saturated sands, where the dilatancy characteristic can be captured. Compared with R-L model, Caputo model can predict more volumetric strain in the dilatancy process, and has better prediction effect. Overall, the predicted effects of the two models are close, with a maximum difference of about 7 %.

A Fast Second-Order ADI Finite Difference Scheme for the Two-Dimensional Time-Fractional Cattaneo Equation with Spatially Variable Coefficients

This paper presents an efficient finite difference method for solving the time-fractional Cattaneo equation with spatially variable coefficients in two spatial dimensions. The main idea is that the original equation is first transformed into a lower system, and then the graded mesh-based fast L2-1σ formula and second-order spatial difference operator for the Caputo derivative and the spatial differential operator are applied, respectively, to derive the fully discrete finite difference scheme. By adding suitable perturbation terms, we construct an efficient fast second-order ADI finite difference scheme, which significantly improves computational efficiency for solving high-dimensional problems. The corresponding stability and error estimate are proved rigorously. Extensive numerical examples are shown to substantiate the accuracy and efficiency of the proposed numerical scheme.

A new approximation to the first order fractional derivative in the Caputo-Fabrizio sense using Haar Wavelet integration formula

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. Due to the singularity of the kernel, the definition of Caputo-Fabrizio appeared, which has a non-singular kernel and mathematical properties similar to the derivative of the integer order. This last definition attracted many mathematicians and researchers to use it in modeling phenomena and obtaining historical information about the development of the studied phenomena, but usually the analytical solution does not exist, which necessitated numerical methods to find an approximate solution. These approximate methods depend on finding an approximate formula for the fractional derivative, and then the problem is transformed into a system of algebraic equations that is easy to solve. In fact, all the numerical methods that have been used have a polynomial rate of convergence, which calls for thinking about a new method that is more effective and has a better rate of convergence. For this reason, in this paper, we propose an efficient numerical method to approximate the first order fractional derivative in the Caputo-Fabrizio sense. This method develops a new quadratic formula using Haar wavelet integration method. Error analysis of our proposed method gives an exponential convergence rate of . To check the effectiveness of the proposed method, we examine some examples with different fractional orders. The quantative results demonstrated the stability and efficiency of the proposed method.

A Qualitative Analysis of Leukemia Fractional Order SICW Model

Using a series of fundamental differential equations, including the Caputo derivative, which makes it easier to specify the initial conditions of the differential equations, we present a fractional order concept of leukemia in this study. The universality, positivity, and boundedness of solutions are first established. The local stability properties of the equilibrium are studied using the fractional Routh-Hurwitz stability criteria. The differential equation system has been solved using unconventional finite difference techniques. The Leukemia Fractional Order SICW model introduces several innovative elements compared to traditional epidemiological and disease models. This stands out due to its integration of fractional-order differential equations, inclusion of leukemic cells and immune cells compartments, simulation of treatment strategies, consideration of waning immunity, and its application to leukemia-specific scenarios. These elements collectively make it a valuable tool for studying leukemia dynamics, exploring treatment options, and improving our understanding of how the immune system interacts with cancer cells in leukemia patients. Numerical simulations of the model are shown at the conclusion to interpret our theoretical outcomes in support of various fractional orders of derivative options. From there, we can observe how the evolution of the system components is impacted by the fractional derivative .