3,842 results on '"Caputo fractional derivative"'
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2. Delayed analogue of three‐parameter Mittag‐Leffler functions and their applications to Caputo‐type fractional time delay differential equations.
- Author
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Huseynov, Ismail T. and Mahmudov, Nazim I.
- Subjects
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FRACTIONAL differential equations , *CAPUTO fractional derivatives , *TIME delay systems , *MATRIX functions , *INTEGRAL transforms - Abstract
In this paper, we consider a Cauchy problem for a Caputo‐type time delay linear system of fractional differential equations with permutable matrices. First, we provide a new representation of solutions to linear homogeneous fractional differential equations using the Laplace integral transform and variation of constants formula via a newly defined delayed Mittag‐Leffler type matrix function introduced through a three‐parameter Mittag‐Leffler function. Second, with the help of a delayed perturbation of a Mittag‐Leffler type matrix function, we attain an explicit formula for solutions to a linear nonhomogeneous time delay fractional order system using the superposition principle. Furthermore, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations using the contraction mapping principle. Finally, we present an example to illustrate the applicability of our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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3. Instantaneous blow‐up for a fractional‐in‐time evolution equation arising in plasma theory.
- Author
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Jleli, Mohamed
- Subjects
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CAPUTO fractional derivatives , *PLASMA waves , *EVOLUTION equations , *TEST methods - Abstract
We consider a fractional‐in‐time evolution equation arising in the theory of ion‐sound waves in plasma, where the spatial variable varies on the half‐line. We first provide sufficient conditions for which there exist no global‐in‐time weak solutions and obtain an upper bound of the lifespan. Next, we find a class of initial data for which local‐in‐time weak solutions do not exist, that is, an instantaneous blow‐up of weak solutions occurs. The proofs of our results are based on some integral inequalities and the test function method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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4. A generalized electro-osmotic MHD flow of hybrid ferrofluid through Fourier and Fick's law in inclined microchannel.
- Author
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Khan, Dolat, Ali, Gohar, Kumam, Poom, and Suttiarporn, Panawan
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BOUNDARY layer (Aerodynamics) , *ELECTRO-osmosis , *BOUNDARY layer control , *CAPUTO fractional derivatives , *HEAT transfer - Abstract
There are several applications for electro-osmotic MHD flow of hybrid Ferrofluid in the present era, notably in the biochemical as drug delivery systems, microfluidic devices, biomedical diagnostics, microscale systems and medical industries. The electro-osmotic MHD flow of a hybrid Ferrofluid containing Cobalt Ferrite, magnetite nanoparticles via a vertically inclined microchannel is investigated in this study. In furthermore, the perpendicular magnetic field is considered. Investigations are also carried into the effects of mass and heat transfer in this moving fluid. Partial differential equations provide as a representation for the aforementioned physical phenomenon, using suitable dimensionless nondimensional variables. Also, the classical system is fractionalized using the generalized Fourier and Fick's law. Generalizations are made based on the account of the Caputo derivative. The solution for the velocity, concentration, and temperature outlines is developed by using the Fourier and Laplace techniques. Moreover, the parametric impact of many physical factors as the Brinkman parameter, the temperature, velocity, concentration and stress parameters (Schmidt, Grashof, and Prandtl numbers). Graphs and discussions of concentration distributions is also discussed. The Sherwood number, rate of heat transmission, and skin friction are calculated and summarized. Since the fractional models are more accurate, they also provide a broader variety of possible solutions. Considering the relevant data, these solutions could be the best. Additionally, the heat transfer rate is higher as compare to nanofluid and regular fluid. The Hybrid Ferrofluid having capability to control velocity boundary layer rapidly as compare to nanofluid and regular fluid. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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5. Utilizing fractional derivatives and sensitivity analysis in a random framework: a model-based approach to the investigation of random dynamics of malware spread.
- Author
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Bekiryazici, Zafer
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FRACTIONAL differential equations , *CAPUTO fractional derivatives , *BASIC reproduction number , *EULER method , *RANDOM variables - Abstract
In this study, an ordinary-deterministic equation system modeling the spread dynamics of malware under mutation is analyzed with fractional derivatives and random variables. The original model is transformed into a system of fractional-random differential equations (FRDEs) using Caputo fractional derivatives. Normally distributed random variables are defined for the parameters of the original system that are related to the mutations and infections of the nodes in the network. The resulting system of FRDEs is simulated using the predictor-corrector method based fde12 algorithm and the forward fractional Euler method (ffEm) for various values of the model components such as the standard deviations, orders of derivation, and repetition numbers. Additionally, the sensitivity analysis of the original model is investigated in relation to the random nature of the components and the basic reproduction number ( R 0 ) to underline the correspondence of random dynamics and sensitivity indices. Both the normalized forward sensitivity indices (NFSI) and the standard deviation of R 0 with random components give matching results for analyzing the changes in the spread rate. Theoretical results are backed by the simulation outputs on the numerical characteristics of the fractional-random model for the expected number of infections and mutations, expected timing of the removal of mutations from the network, and measurement of the variability in the results such as the coefficients of variation. Comparison of the results from the original model and the fractional-random model shows that the fractional-random analysis provides a more generalized perspective while facilitating a versatile investigation with ease and can be used on different models as well. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. Dynamics of the time-fractional reaction–diffusion coupled equations in biological and chemical processes.
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Ghafoor, Abdul, Fiaz, Muhammad, Hussain, Manzoor, Ullah, Asad, Ismail, Emad A. A., and Awwad, Fuad A.
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REACTION-diffusion equations , *CHEMICAL equations , *CHEMICAL processes , *FINITE differences , *NONLINEAR equations , *SIMULTANEOUS equations - Abstract
This paper aims to demonstrate a numerical strategy via finite difference formulations for time fractional reaction–diffusion models which are ubiquitous in chemical and biological phenomena. The time-fractional derivative is considered in the Caputo sense for both linear and nonlinear problems. First, the Caputo derivative is replaced with a quadrature formula, then an implicit method is used for the remaining part. In the linear case, the proposed strategy reduces the time fractional models into linear simultaneous equations. In nonlinear cases, Quasilinearization is utilized to tackle the nonlinear parts. With this strategy, solutions of the fractional system transform into linear algebraic systems which are easy to solve. Next, the Von Neumann method is implemented to examine the stability of the scheme which discloses that the scheme is unconditionally stable. Further, the applicability of the presented scheme is tested with different linear and nonlinear models which include the one dimensional Schnakenberg and Gray–Scott models, and one and two dimensional Brusselator models. To analyze the accuracy of the present technique two norms namely, L ∞ and L 2 , and relative error are addressed. Moreover, the obtained outcomes are shown tabulated and graphically which identifies that the scheme properly works for the time fractional reaction–diffusion systems. [ABSTRACT FROM AUTHOR]
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- 2024
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7. A fast <italic>L</italic>1 formula on tanh meshes for time fractional Burgers equations.
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Xing, Zhiyong, Sun, Wenbing, and Zhu, Xiaogang
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CAPUTO fractional derivatives , *COMPUTATIONAL complexity , *COMPUTER simulation , *BURGERS' equation - Abstract
In this paper, a fast
L 1 formula on tanh meshes is proposed for time fractional Burgers equations with Caputo fractional derivative. The solvability, boundness and convergence of the numerical scheme are rigorously established. Several numerical experiments are provided to support the theoretical results. The results of the experiments showed that the proposed numerical method cannot only effectively deal with the weak singularity of the problem near t = 0, but also significantly reduce the computational complexity of numerical simulation. [ABSTRACT FROM AUTHOR]- Published
- 2024
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8. Stability analysis of linear fractional neutral delay differential equations.
- Author
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Zhao, Jingjun, Wang, Xingchi, and Xu, Yang
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DELAY differential equations , *CAPUTO fractional derivatives - Abstract
This paper investigates the analytical stability region and the asymptotic stability of linear fractional neutral delay differential equations. Employing boundary locus techniques, the stability region of this problem is analyzed. Furthermore, we derive the fundamental solution of linear fractional neutral delay differential equations, and prove the exponential boundedness, the asymptotic stability and the algebraic decay rate. Finally, numerical tests are conducted to verify the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. Extreme solution for fractional differential equation with nonlinear boundary condition.
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FUFAN LUO, PIAO LIU, and WEIBING WANG
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BOUNDARY value problems , *FRACTIONAL differential equations , *NONLINEAR equations , *CAPUTO fractional derivatives , *LINEAR equations - Abstract
In this paper, we investigate a class of fractional equations with nonlinear boundary condition. We establish a new comparison principle related to linear fractional equation and show the existence of extreme solution by using monotone iterative method and lower and upper solutions method. [ABSTRACT FROM AUTHOR]
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- 2024
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10. Study and mathematical analysis of the novel fractional bone mineralization model.
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Agarwal, Ritu and Midha, Chhaya
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MATHEMATICAL analysis , *TAPHONOMY , *CAPUTO fractional derivatives , *BIOLOGICAL models , *LEAD , *BONE diseases - Abstract
Different biological models can be evaluated using mathematical models in both qualitative and quantitative ways. A fractional bone mineralization model involving Caputo’s fractional derivative is presented in this work. The fractional mathematical model is beneficial because of its memory carrying property. An appropriate fractional order of the derivative can be chosen that is more closely related to experimental or actual data. The dynamical system of equations for the process of bone mineralization is examined qualitatively and quantitatively in this article. A numerical simulation has been performed for the model. The model’s parameters have undergone sensitivity analysis and their effects on the model variables have been explored. By studying the mineralization patterns in bone, different diseases can be cured, and it can also be examined how the deviations from healthy mineral distributions lead to specific bone diseases. [ABSTRACT FROM AUTHOR]
- Published
- 2024
11. Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment.
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Kengne, Jacques Ndé and Tadmon, Calvin
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BEHAVIOR modification , *NONLINEAR analysis , *FINITE differences , *FRACTIONAL differential equations ,TREATMENT of Ebola virus diseases - Abstract
This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bistability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018e2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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12. Numerical inspection of (3 + 1)- perturbed Zakharov–Kuznetsov equation via fractional variational iteration method with Caputo fractional derivative.
- Author
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Nawaz, Rashid, Fewster-Young, Nicholas, Katbar, Nek Muhammad, Ali, Nasir, Zada, Laiq, Ibrahim, Rabha W., Jamshed, Wasim, and Alqahtani, Haifa
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CAPUTO fractional derivatives , *DIFFERENTIAL forms , *FRACTIONAL differential equations , *EQUATIONS - Abstract
In this paper, the implementation of the Fractional Variational iteration method (FVIM) is used to approximate the generalized fractional perturbed Zakharove-Kuznetsov equation. The obtained results show that the suggested approach is very efficient for dealing with various forms of differential equations of fractional order. The numerical comparison for the attained resolution is made through the q-homotopy analysis transmute manner (q-HATM) which confirms the convergence of the devised method. The applicability of the method is observed by applying the proposed technique to two numerical examples. Furthermore, the solutions obtained by the FVIM are completely well-matched with the solutions presented in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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13. Lie symmetry analysis of time fractional nonlinear partial differential equations in Hilfer sense.
- Author
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Thomas, Reetha and Bakkyaraj, T.
- Subjects
SIMILARITY transformations ,NONLINEAR differential equations ,ORDINARY differential equations ,FRACTIONAL differential equations ,CAPUTO fractional derivatives - Abstract
We derive the prolongation formula of the one-parameter Lie point transformations to the Hilfer fractional derivative and show that the existing prolongation formula for the Riemann Liouville and Caputo fractional derivatives are special cases of the proposed formula, corresponding to the type parameter γ = 0 and γ = 1 , respectively. The applicability of the proposed formula is demonstrated by deriving the Lie point symmetries of the time-fractional heat equation, the fractional Burgers equation, and the fractional KdV equation in Hilfer's sense. We use the obtained Lie point symmetries to find the similarity variables and transformations. Using the similarity transformations, we show that each is converted into a nonlinear fractional ordinary differential equation with a new independent variable. The fractional derivative in the reduced equation can be either the Hilfer-type modification of the Erdélyi Kober fractional derivative or the Hilfer fractional derivative itself. We demonstrate that the exact solution of the time-fractional differential equation in the Hilfer sense can be reduced to the exact solutions of the corresponding time-fractional differential equations in the Riemann–Liouville and Caputo senses by setting the type parameter to γ = 0 and γ = 1 , respectively. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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14. A fractional control model to study Monkeypox transport network-related transmission.
- Author
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Zhang, Nan, Emmanuel, Addai, Mezue, Mary Nwaife, Rashid, Saima, Akinnubi, Abiola, Abdul-Hamid, Zalia, and Asamoah, Joshua Kiddy K.
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CAPUTO fractional derivatives , *MONKEYPOX , *PUBLIC transit , *NUMERICAL analysis , *PUBLIC health - Abstract
Effective disease control measures to manage the spread of Monkeypox (Mpox) virus are crucial, especially given the serious public health risks posed by the ongoing global epidemic in regions where the virus is both prevalent and not. This study introduces a precise model, based on the Caputo fractional derivative, which takes into account both human and non-human populations as well as public transportation, to delve into the transmission characteristics of Mpox outbreaks. By employing the fixed point theorem, we have precisely determined the solutions regarding existence and uniqueness. We have analyzed the stability of various equilibrium states within the model to assess Mpox’s transmission capabilities. Additionally, through detailed numerical simulations, we have gauged the impact of critical model parameters that contribute to enhancing Mpox prevention and management strategies. The insights gained from our research significantly enrich epidemiological understanding and lay the foundation for improved disease containment approaches. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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15. Package Guidance Problem for a Fractional-Order System.
- Author
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Surkov, P. G.
- Abstract
The problem of guaranteed closed-loop guidance to a given set at a given time is studied for a linear dynamic control system described by differential equations with a fractional derivative of the Caputo type. The initial state is a priori unknown, but belongs to a given finite set. The information on the position of the system is received online in the form of an observation signal. The solvability of the guidance problem for the control system is analyzed using the method of Osipov–Kryazhimskii program packages. The paper provides a brief overview of the results that develop the program package method and use it in guidance problems for various classes of systems. This method allows us to connect the solvability condition of the guaranteed closed-loop guidance problem for an original system with the solvability condition of the open-loop guidance problem for a special extended system. Following the technique of the program package method, a criterion for the solvability of the considered guidance problem is derived for a fractional-order system. In the case where the problem is solvable, a special procedure for constructing a guiding program package is given. The developed technique for analyzing the guaranteed closed-loop guidance problem and constructing a guiding control for an unknown initial state is illustrated by the example of a specific linear mechanical control system with a Caputo fractional derivative. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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16. A hybrid yang transform adomian decomposition method for solving time-fractional nonlinear partial differential equation.
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Bekela, Alemu Senbeta and Deresse, Alemayehu Tamirie
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PARTIAL differential equations , *NONLINEAR differential equations , *DECOMPOSITION method , *CAPUTO fractional derivatives , *STOCHASTIC systems - Abstract
Nonlinear time-fractional partial differential equations (NTFPDEs) play a great role in the mathematical modeling of real-world phenomena like traffic models, the design of earthquakes, fractional stochastic systems, diffusion processes, and control processing. Solving such problems is reasonably challenging, and the nonlinear part and fractional operator make them more problematic. Thus, developing suitable numerical methods is an active area of research. In this paper, we develop a new numerical method called Yang transform Adomian decomposition method (YTADM) by mixing the Yang transform and the Adomian decomposition method for solving NTFPDEs. The derivative of the problem is considered in sense of Caputo fractional order. The stability and convergence of the developed method are discussed in the Banach space sense. The effectiveness, validity, and practicability of the method are demonstrated by solving four examples of NTFPEs. The findings suggest that the proposed method gives a better solution than other compared numerical methods. Additionally, the proposed scheme achieves an accurate solution with a few numbers of iteration, and thus the method is suitable for handling a wide class of NTFPDEs arising in the application of nonlinear phenomena. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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17. Heat source determining inverse problem for nonlocal in time equation.
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Serikbaev, Daurenbek, Ruzhansky, Michael, and Tokmagambetov, Niyaz
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CAPUTO fractional derivatives , *INVERSE problems , *POSITIVE operators , *OPERATOR equations , *HEAT equation - Abstract
In this paper, we consider the inverse problem of determining the time‐dependent source term in the general setting of Hilbert spaces and for general additional data. We prove the well‐posedness of this inverse problem by reducing the problem to an operator equation for the source function. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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18. Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law.
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Alimbekova, Nurlana, Berdyshev, Abdumauvlen, Madiyarov, Muratkan, and Yergaliyev, Yerlan
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DIFFERENTIAL forms , *FINITE difference method , *FINITE element method , *CAPUTO fractional derivatives , *NEWTON-Raphson method - Abstract
In this article, a numerical method is proposed and investigated for an initial boundary value problem governed by a fractional differential generalization of the nonlinear transient filtration law which describes fluid motion in a porous medium. This type of equation is widely used to describe complex filtration processes such as fluid movement in horizontal wells in fractured geological formations. To construct the numerical method, a high-order approximation formula for the fractional derivative in the sense of Caputo is applied, and a combination of the finite difference method with the finite element method is used. The article proves the uniqueness and continuous dependence of the solution on the input data in differential form, as well as the stability and convergence of the proposed numerical scheme. The linearization of nonlinear terms is carried out by the Newton method, which allows for achieving high accuracy in solving complex problems. The research results are confirmed by a series of numerical tests that demonstrate the applicability of the developed method in real engineering problems. The practical significance of the presented approach lies in its ability to accurately and effectively model filtration processes in shale formations, which allows engineers and geologists to make more informed decisions when designing and operating oil fields. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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19. New results on fractional advection–dispersion equations.
- Author
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Qiao, Yan, Chen, Fangqi, An, Yukun, and Lu, Tao
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CRITICAL point theory , *CAPUTO fractional derivatives , *EQUATIONS , *BASIC needs - Abstract
In this paper, a class of fractional Sturm–Liouville advection–dispersion equations with instantaneous and noninstantaneous impulses is considered, in particular, the nonlinearities discussed here include Caputo fractional derivatives. Since the nonlinear terms contain fractional derivatives, this problem does not directly have variational structure, we need to combine critical point theory and an iterative method to deal with such problems. Finally, the existence of at least one nontrivial solution is proved by the mountain pass theorem and the iterative method. At the same time, an example is given to illustrate the main result. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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20. On the analyzing of bifurcation properties of the one‐dimensional Mackey–Glass model by using a generalized approach.
- Author
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Zhang, Shuai, Wang, Yaya, Geng, Hongyin, Gao, Wei, Ilhan, Esin, and Mehmet Baskonus, Haci
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CAPUTO fractional derivatives , *DIFFERENTIAL equations , *POWER spectra , *FORECASTING - Abstract
The goal of this work is to look at how a nonlinear model describes hematopoiesis and its complexities utilizing commonly used techniques with historical and material links. Based on time delay, the Mackey–Glass model is explored in two instances. To offer a range, the relevance of the parameter impacting stability (bifurcation) is recorded. The power spectrum of the considered model is collected in order to analyze the periodic behavior of a solution in a differential equation. The complex nature of the system is relayed on a parameter which is illustrated in the bifurcation plot. Due to the fact that the considered model is associated with blood‐related diseases, the effect coefficients are effectively captured. The corresponding parameters‐based consequences of the generalized model in different order are deduced. The parametric charts for both examples reveal intriguing results. The current work enables investigations into complex real‐world problems as well as forecasts of essential techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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21. Navigating climate complexity and its control via hyperchaotic dynamics in a 4D Caputo fractional model.
- Author
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Naik, Manisha Krishna, Baishya, Chandrali, Premakumari, R. N., and Samei, Mohammad Esmael
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CAPUTO fractional derivatives , *CHAOS theory , *FRACTIONAL differential equations , *SEA ice , *GLOBAL warming - Abstract
This interdisciplinary study critically analyzes current research, establishing a profound connection between sea water, sea ice, sea temperature, and surface temperature through a 4D hyperchaotic Caputo fractional differential equation. Emphasizing the collective impact on climate, focusing on challenges from anthropogenic global warming, the study scrutinizes theoretical aspects, including existence and uniqueness. Two sliding mode controllers manage chaos in this 4D fractional system, assessed amid uncertainties and disruptions. The global stability of these controlled systems is also confirmed, considering both commensurate and non-commensurate 4D fractional order. To demonstrate the intricate chaotic motion within the system, we employ the Lyapunov exponent and Poincare sections. Numerical simulations are conducted by using the predictor-corrector method. The effects of surface temperature on chaotic dynamics are discussed. The crucial role of sea ice reflection in climate stability is highlighted in two scenarios. Correlation graphs, comparing model and observational data using the predictor-corrector method, enhance the proposed 4D hyperchaotic model's credibility. Subsequently, numerical simulations validate theoretical assertions about the controllers' influence. These controllers indicate which variable significantly contributes to controlling the chaos. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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22. On viscosity solutions of path-dependent Hamilton–Jacobi–Bellman–Isaacs equations for fractional-order systems.
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Gomoyunov, M.I.
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VISCOSITY solutions , *DIFFERENTIABLE dynamical systems , *FRACTIONAL differential equations , *CAUCHY problem , *HAMILTON-Jacobi-Bellman equation , *ZERO sum games - Abstract
This paper deals with a two-person zero-sum differential game for a dynamical system described by a Caputo fractional differential equation of order α ∈ (0 , 1) and a Bolza cost functional. The differential game is associated to the Cauchy problem for the path-dependent Hamilton–Jacobi–Bellman–Isaacs equation with so-called fractional coinvariant derivatives of order α and the corresponding right-end boundary condition. A notion of a viscosity solution of the Cauchy problem is introduced, and the value functional of the differential game is characterized as a unique viscosity solution of this problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Well-posedness and stability of a fractional heat-conductor with fading memory.
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Kerbal, Sebti, Tatar, Nasser-eddine, and Al-Salti, Nasser
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MEMORY , *HEAT flux , *CAPUTO fractional derivatives - Abstract
We consider a problem which describes the heat diffusion in a complex media with fading memory. The model involves a fractional time derivative of order between zero and one instead of the classical first order derivative. The model takes into account also the effect of a neutral delay. We discuss the existence and uniqueness of a mild solution as well as a classical solution. Then, we prove a Mittag-Leffler stability result. Unlike the integer-order case, we run into considerable difficulties when estimating some problematic terms. It is found that even without the memory term in the heat flux expression, the stability is still of Mittag-Leffler type. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. On the convergence of the Galerkin method for random fractional differential equations.
- Author
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Jornet, Marc
- Subjects
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FRACTIONAL differential equations , *GALERKIN methods , *CAPUTO fractional derivatives , *PARTIAL differential equations , *INDEPENDENT sets - Abstract
In the context of forward uncertainty quantification, we investigate the convergence of the Galerkin projections for random fractional differential equations. The governing system is formed by a finite set of independent input random parameters (a germ) and by a fractional derivative in the Caputo sense. Input uncertainty arises from biased measurements, and a fractional derivative, defined by a convolution, takes past history into account. While numerical experiments on the gPC-based Galerkin method are already available in the literature for random ordinary, partial and fractional differential equations, a theoretical analysis of mean-square convergence is still lacking for the fractional case. The aim of this contribution is to fill this gap, by establishing new inequalities and results and by raising new open problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Application of subordination principle to coefficient inverse problem for multi-term time-fractional wave equation.
- Author
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Bazhlekova, Emilia
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INVERSE problems , *WAVE equation , *INTEGRAL transforms , *CAPUTO fractional derivatives - Abstract
An initial-boundary value problem for the multi-term time-fractional wave equation on a bounded domain is considered. For the largest and smallest orders of the involved Caputo fractional time-derivatives, α and α m , it is assumed 1 < α < 2 and α - α m ≤ 1 . Subordination principle with respect to the corresponding single-term time-fractional wave equation of order α is deduced. Injectivity of the integral transform, defined by the subordination relation, is established. The subordination identity is used to prove uniqueness for a coefficient inverse problem for the multi-term equation, based on an analogous property for the related single-term one. In addition, the subordination relation is applied for deriving a regularity estimate. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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26. A tempered subdiffusive Black–Scholes model.
- Author
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Krzyżanowski, Grzegorz and Magdziarz, Marcin
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BLACK-Scholes model , *FINITE difference method , *FRACTIONAL differential equations , *CAPUTO fractional derivatives , *SCATTERING (Mathematics) - Abstract
In this paper, we focus on the tempered subdiffusive Black–Scholes model. The main part of our work consists of the finite difference method as a numerical approach to option pricing in the considered model. We derive the governing fractional differential equation and the related weighted numerical scheme. The proposed method has an accuracy order 2 - α with respect to time, where α ∈ (0 , 1) is the subdiffusion parameter and 2 with respect to space. Furthermore, we provide stability and convergence analysis. Finally, we present some numerical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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27. Existence Result for a Class of Time-Fractional Nonstationary Incompressible Navier–Stokes–Voigt Equations.
- Author
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Xu, Keji and Zeng, Biao
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CAPUTO fractional derivatives , *OPERATOR equations , *WORKING class , *SURJECTIONS , *EQUATIONS - Abstract
We are devoted in this work to dealing with a class of time-fractional nonstationary incompressible Navier–Stokes–Voigt equation involving the Caputo fractional derivative. By exploiting the properties of the operators in the equation, we use the Rothe method to show the existence of weak solutions to the equation by verifying all the conditions of the surjectivity theorem for nonlinear weakly continuous operators. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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28. Polynomial decay of a linear system of PDEs via Caputo fractional‐time derivative.
- Author
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Arfaoui, Hassen
- Subjects
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CAPUTO fractional derivatives , *LINEAR systems , *FOURIER analysis , *FINITE difference method , *POLYNOMIALS , *STABILITY of linear systems - Abstract
An in‐depth study and analysis of the stability of one‐dimensional Linear System of PDEs via Caputo time fractional derivative (LSCFD) was presented. We proved some stability results for the LSCFD in different Hilbert spaces. Indeed, by using Fourier analysis method and the properties of the Mittag–Leffler Function (MLF), some polynomial stability results for LSCFD have been established. Finally, as an application, we used finite difference methods well suited to integer and fractional order derivatives, and performed some numerical experiments to confirm the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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29. Advanced neural network approaches for coupled equations with fractional derivatives.
- Author
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Alfalqi, Suleman, Boukhari, Boumediene, Bchatnia, Ahmed, and Beniani, Abderrahmane
- Abstract
We investigate numerical solutions and compare them with Fractional Physics-Informed Neural Network (FPINN) solutions for a coupled wave equation involving fractional partial derivatives. The problem explores the evolution of functions u and v over time t and space x. We employ two numerical approximation schemes based on the finite element method to discretize the system of equations. The effectiveness of these schemes is validated by comparing numerical results with exact solutions. Additionally, we introduce the FPINN method to tackle the coupled equation with fractional derivative orders and compare its performance against traditional numerical methods. Key findings reveal that both numerical approaches provide accurate solutions, with the FPINN method demonstrating competitive performance in terms of accuracy and computational efficiency. Our study highlights the significance of employing FPINNs in solving fractional differential equations and underscores their potential as alternatives to conventional numerical methods. The novelty of this work lies in its comparative analysis of traditional numerical techniques and FPINNs for solving coupled wave equations with fractional derivatives, offering insights into advancing computational methods for complex physical systems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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30. Efficient Numerical Implementation of the Time-Fractional Stochastic Stokes–Darcy Model.
- Author
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Baishemirov, Zharasbek, Berdyshev, Abdumauvlen, Baigereyev, Dossan, and Boranbek, Kulzhamila
- Subjects
- *
CAPUTO fractional derivatives , *COLLOCATION methods , *FLUID flow , *POROUS materials , *STOCHASTIC models - Abstract
This paper presents an efficient numerical method for the fractional-order generalization of the stochastic Stokes–Darcy model, which finds application in various engineering, biomedical and environmental problems involving interaction between free fluid flow and flows in porous media. Unlike the classical model, this model allows taking into account the hereditary properties of the process under uncertainty conditions. The proposed numerical method is based on the combined use of the sparse grid stochastic collocation method, finite element/finite difference discretization, a fast numerical algorithm for computing the Caputo fractional derivative, and a cost-effective ensemble strategy. The hydraulic conductivity tensor is assumed to be uncertain in this problem, which is modeled by the reduced Karhunen–Loève expansion. The stability and convergence of the deterministic numerical method have been rigorously proved and validated by numerical tests. Utilizing the ensemble strategy allowed us to solve the deterministic problem once for all samples of the hydraulic conductivity tensor, rather than solving it separately for each sample. The use of the algorithm for computing the fractional derivatives significantly reduced both computational cost and memory usage. This study also analyzes the influence of fractional derivatives on the fluid flow process within the fractional-order Stokes–Darcy model under uncertainty conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Analysis of Caputo Sequential Fractional Differential Equations with Generalized Riemann–Liouville Boundary Conditions.
- Author
-
Gunasekaran, Nallappan, Manigandan, Murugesan, Vinoth, Seralan, and Vadivel, Rajarathinam
- Subjects
- *
NONLINEAR boundary value problems , *CAPUTO fractional derivatives , *FRACTIONAL integrals , *SEQUENTIAL analysis - Abstract
This paper delves into a novel category of nonlocal boundary value problems concerning nonlinear sequential fractional differential equations, coupled with a unique form of generalized Riemann–Liouville fractional differential integral boundary conditions. For single-valued maps, we employ a transformation technique to convert the provided system into an equivalent fixed-point problem, which we then address using standard fixed-point theorems. Following this, we evaluate the stability of these solutions utilizing the Ulam–Hyres stability method. To elucidate the derived findings, we present constructed examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
32. Chaos and stability of a fractional model of the cyber ecosystem.
- Author
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Gómez-Aguilar, José F., Naik, Manisha Krishna, George, Reny, Baishya, Chandrali, Avcı, İbrahim, and Pérez-Careta, Eduardo
- Subjects
CAPUTO fractional derivatives ,COMPUTERS ,COMPUTER software ,NUMERICAL analysis ,COMPUTER users ,PREDATION - Abstract
The widespread use of computer hardware and software in society has led to the emergence of a type of criminal conduct known as cybercrime, which has become a major worldwide concern in the 21st century spanning multiple domains. As a result, in the present setting, academics and practitioners are showing a great deal of interest in conducting research on cybercrime. In this work, a fractional-order model was replaced by involving three sorts of human populations: online computer users, hackers, and cyber security professionals, in order to examine the online computer user-hacker system. The existence, uniqueness and boundedness were studied. To support our theoretical conclusions, a numerical analysis of the influence of the various logical parameters was conducted and we derived the necessary conditions for the different equilibrium points to be locally stable. We examined the effects of the fear level and refuge factor on the equilibrium densities of prey and predators in order to explore and understand the dynamics of the system in a better way. Using some special circumstances, the model was examined. Our theoretical findings and logical parameters were validated through a numerical analysis utilizing the generalized Adams-Bashforth-Moulton technique. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. Application of Improved SPH Method in Solving Time Fractional Schrödinger Equation.
- Author
-
Ma, Luyang and Imin, Rahmatjan
- Subjects
SCHRODINGER equation ,FINITE difference method ,NONLINEAR Schrodinger equation ,ANALYTICAL solutions ,NUMERICAL calculations ,CAPUTO fractional derivatives - Abstract
In this paper, a pure meshless method for solving the time fractional Schrödinger equation (TFSE) based on KDF-SPH method is presented. The method is used for the first time to numerically solve the TFSE. The method utilizes the finite difference method (FDM) to approximate the time fractional-order derivative defined in the Caputo sense. The spatial derivatives are discretized by the KDF-SPH meshless method. Expressions for the kernel approximation and the particle approximation are provided. To ensure the validity and flexibility of the numerical calculations, we conducted numerical simulations of one- and two-dimensional linear/nonlinear time Schrödinger equations (1D/2D TFLSE/TF-NLSE) in both bounded and unbounded regions. We also examined nonlinear time fractional Schrödinger equations that lack analytical solutions and compared our method with other meshless methods. Numerical results show that the proposed method can approximate to the second-order precision in space, which verifies the effectiveness and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. An optimal solution for tumor growth model using generalized Bessel polynomials.
- Author
-
Saeidi, Hojat, Dahaghin, M. Sh., Mehrabi, Samrad, and Hassani, Hossein
- Subjects
- *
CAPUTO fractional derivatives , *PARTIAL differential equations , *ONCOLYTIC virotherapy , *LAGRANGE multiplier , *MATHEMATICAL models - Abstract
In this paper, a mathematical model is given that depicts the interactions between cancer cells and viruses in the setting of oncolytic virotherapy. The model is separated into three classes, namely, concentrations of uninfected tumor cells in the population “ U$$ \mathcal{U} $$”, free virus “ V$$ \mathcal{V} $$”, and cancerous cells infected “ I$$ \mathcal{I} $$”. Applying Caputo fractional derivative, the model is fractionalized, and using generalized Bessel polynomials, an optimal problem is solved utilizing Lagrange multipliers method. The results show that the presented method has high accuracy and is suitable for solving the nonlinear systems based on partial differential equations especially tumors models. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. An efficient approximate analytical technique for the fractional model describing the solid tumor invasion.
- Author
-
Chethan, H. B., Saadeh, Rania, Prakasha, D. G., Qazza, Ahmad, Malagi, Naveen S., Nagaraja, M., Sarwe, Deepak Umrao, and Özdemir, Necati
- Subjects
SOLID solutions ,CAPUTO fractional derivatives - Abstract
In this manuscript, we derive and examine the analytical solution for the solid tumor invasion model of fractional order. The main aim of this work is to formulate a solid tumor invasion model using the Caputo fractional operator. Here, the model involves a system of four equations, which are solved using an approximate analytical method. We used the fixed-point theorem to describe the uniqueness and existence of the model's system of solutions and graphs to explain the results we achieved using this approach. The technique used in this manuscript is more efficient for studying the behavior of this model, and the results are accurate and converge swiftly. The current study reveals that the investigated model is time-dependent, which can be explored using the fractional-order calculus concept. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Study of the results of Hilbert transformation for some fractional derivatives.
- Author
-
Nazeer, Nazakat, Akgül, Ali, and Ali, Faeem
- Subjects
- *
CAPUTO fractional derivatives , *FRACTIONAL integrals , *FRACTIONAL calculus - Abstract
Caputo Fractional Derivative is a base for the Riemann-Liouville Derivative, Katugampola fractional derivative, and other fractional derivatives. In this paper, we have illustrated several critical features of the Caputo Derivative via Hilbert Transformation that have not yet been discussed. We have also investigated the behavior of the Riemann-Liouville integral and studied the results of the
k -Riemann-Liouville fractional integral with the applications of Hilbert transformation. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
37. Qualitative Analysis for the Solutions of Fractional Stochastic Differential Equations.
- Author
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Mohammed Djaouti, Abdelhamid and Imran Liaqat, Muhammad
- Subjects
- *
STOCHASTIC differential equations , *CAPUTO fractional derivatives , *FRACTIONAL differential equations , *DIFFERENTIAL equations , *PANTOGRAPH - Abstract
Fractional pantograph stochastic differential equations (FPSDEs) combine elements of fractional calculus, pantograph equations, and stochastic processes to model complex systems with memory effects, time delays, and random fluctuations. Ensuring the well-posedness of these equations is crucial as it guarantees meaningful, reliable, and applicable solutions across various disciplines. In differential equations, regularity refers to the smoothness of solution behavior. The averaging principle offers an approximation that balances complexity and simplicity. Our research contributes to establishing the well-posedness, regularity, and averaging principle of FPSDE solutions in L p spaces with p ≥ 2 under Caputo derivatives. The main ingredients in the proof include the use of Hölder, Burkholder–Davis–Gundy, Jensen, and Grönwall–Bellman inequalities, along with the interval translation approach. To understand the theoretical results, we provide numerical examples at the end. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. Fractional order prey–predator model incorporating immigration on prey: Complexity analysis and its control.
- Author
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Uddin, Md. Jasim and Podder, Chandra Nath
- Subjects
- *
CAPUTO fractional derivatives , *CHAOS theory , *FRACTIONAL calculus , *FRACTIONAL differential equations , *BIFURCATION theory , *EMIGRATION & immigration , *HYBRID systems , *LOTKA-Volterra equations - Abstract
In this paper, the Caputo fractional derivative is assumed to be the prey–predator model. In order to create Caputo fractional differential equations for the prey–predator model, a discretization process is first used. The fixed points of the model are categorized topologically. We identify requirements for the fixed points of the suggested prey–predator model's local asymptotic stability. We demonstrate analytically that, under specific parametric conditions, a fractional order prey–predator model supports both a Neimark–Sacker (NS) bifurcation and a Flip bifurcation. We present evidence for NS and Flip bifurcations using central manifold and bifurcation theory. The parameter values and the initial conditions have been found to have a profound impact on the dynamical behavior of the fractional order prey–predator model. As the bifurcation parameter is increased, the system displays chaotic behavior. Numerical simulations are shown to demonstrate chaotic behaviors like bifurcations, phase portraits, invariant closed cycles, and attractive chaotic sets in addition to validating analytical conclusions. The suggested prey–predator dynamical system's chaotic behavior will be controlled by the OGY and hybrid control methodology, which will also visualize the chaotic state for various biological parameters. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions.
- Author
-
Cen, Zhongdi, Huang, Jian, and Xu, Aimin
- Subjects
- *
FRACTIONAL differential equations , *BOUNDARY value problems , *FRACTIONAL integrals , *INTEGRAL transforms , *INTEGRALS , *INTEGRAL equations - Abstract
In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Comparison of unsteady MHD flow of second grade fluid by two fractional derivatives.
- Author
-
Nazar, Mudassar, Abbas, Shajar, Asghar, Sumbal, Saleem, Salman, Abutuqayqah, Hajar, AL Garalleh, Hakim, and Jastaneyah, Zuhair
- Subjects
- *
UNSTEADY flow , *DARCY'S law , *FREE convection , *CONVECTIVE flow , *CAPUTO fractional derivatives , *FLUIDS - Abstract
AbstractA study is conducted on the unsteady motion of a free convective flow of second grade fluid, energy transfer, and Darcy’s law over an oscillatory smooth vertical plate. The study compares two different approaches in developing a fractional model: the Caputo-Fabrizio operator with nonsingular kernel and the constant proportional Caputo fractional operator with Fourier’s and Fick’s laws. By applying the Laplace method and transforming the provided set of equations into nondimensional form, we obtained semi-analytical results and presented these results through graphical analysis. The study examines how different flow parameters, including the fractional parameters, affect the velocity, mass, and heat profiles of the physical system. The results suggest that the physical model using constant proportional Caputo derivative leads to higher temperature, stronger concentration, and increased velocity compared to the Caputo-Fabrizio model. This highlights the importance of selecting an appropriate fractional model when studying complex physical systems. It is also depicted from the whole analysis that field variables with novel hybrid fractional derivative constant proportional Caputo exhibits a more effective and declining tendency than Caputo-Fabrizio. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Analysis of COVID-19 epidemic with intervention impacts by a fractional operator.
- Author
-
Bhatter, Sanjay, Kumawat, Sangeeta, Bhatia, Bhamini, and Purohit, Sunil Dutt
- Subjects
- *
BASIC reproduction number , *CAPUTO fractional derivatives , *INFECTIOUS disease transmission , *COVID-19 pandemic , *NORMALIZED measures - Abstract
This study introduces an innovative fractional methodology for analyzing the dynamics of COVID-19 outbreak, examining the impact of intervention strategies like lockdown, quarantine, and isolation on disease transmission. The analysis incorporates the Caputo fractional derivative to grasp long-term memory effects and non-local behavior in the advancement of the infection. Emphasis is placed on assessing the boundedness and non-negativity of the solutions. Additionally, the Lipschitz and Banach contraction theorem are utilized to validate the existence and uniqueness of the solution. We determine the basic reproduction number associated with the model utilizing the next generation matrix technique. Subsequently, by employing the normalized sensitivity index, we perform a sensitivity analysis of the basic reproduction number to effectively identify the controlling parameters of the model. To validate our theoretical findings, numerical simulations are conducted for various fractional order values, utilizing a two-step Lagrange interpolation technique. Furthermore, the numerical algorithms of the model are represented graphically to illustrate the effectiveness of the proposed methodology and to analyze the effect of arbitrary order derivatives on disease dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. Existence and uniqueness study for partial neutral functional fractional differential equation under Caputo derivative.
- Author
-
Sene, Ndolane and Ndiaye, Ameth
- Subjects
- *
FRACTIONAL differential equations , *CAPUTO fractional derivatives , *RESOLVENTS (Mathematics) , *FUNCTIONAL differential equations - Abstract
The partial neutral functional fractional differential equation described by the fractional operator is considered in the present investigation. The used fractional operator is the Caputo derivative. In the present paper, the fractional resolvent operators have been defined and used to prove the existence of the unique solution of the fractional neutral differential equations. The fixed point theorem has been used in existence investigations. For an illustration of our results in this paper, an example has been provided as well. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. On the multi-cluster flocking of the fractional Cucker–Smale model.
- Author
-
Ahn, Hyunjin
- Subjects
CAPUTO fractional derivatives ,COMPUTER simulation ,CLUSTER analysis (Statistics) ,MATHEMATICAL models ,ALGEBRAIC equations - Abstract
This paper demonstrates several sufficient frameworks for the multi-cluster flocking behavior of the fractional Cucker–Smale (CS) model. For this, we first employ the Caputo fractional derivative instead of the usual derivative to propose the fractional CS model with the memory effect. Then, using mathematical tools based on fractional calculus, we present suitable sufficient conditions in terms of properly separated initial data close to the multi-cluster, and well-prepared system parameters for the multi-cluster flocking of the fractional system to emerge. Finally, we offer several numerical simulations and compare them with the analytical results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. A HYBRID CHELYSHKOV WAVELET-FINITE DIFFERENCES METHOD FOR TIME-FRACTIONAL BLACK-SCHOLES EQUATION.
- Author
-
HASHEMI, S. A. SAMAREH, SAEEDI, H., and BASTANI, A. FOROUSH
- Subjects
WAVELETS (Mathematics) ,BLACK-Scholes model ,DISCRETIZATION methods ,ERROR analysis in mathematics ,CAPUTO fractional derivatives - Abstract
In this paper, a hybrid method for solving time-fractional Black-Scholes equation is introduced for option pricing. The presented method is based on time and space discretization. A second order finite difference formula is used to time discretization and space discretization is done by a spectral method based on Chelyshkov wavelets and an op- erational process by defining Chelyshkov wavelets operational matrices. Convergence and error analysis for Chelyshkov wavelets approximation and also for the proposed method are discussed. The method is validated and its accuracy, convergency and efficiency are demonstrated through some cases with given accurate solutions. The method is also utilize for pricing various European options conducted by a time-fractional Black- Scholes model. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. On Fractional Semilinear Nonlocal Initial Value Problem with State Dependent Delay.
- Author
-
Alam, Md Mansur and Dubey, Shruti
- Abstract
In this paper, we consider a class of fractional order semilinear abstract Cauchy problem with state dependent delay subject to nonlocal initial conditions, and enlarge the existence theory with two different sets of assumptions. Under the first set of assumptions, we establish the existence of Hölder classical solution. Since the Hölder exponent appears as an exponent on the metric function in contraction inequality, it is not suitable to use Banach contraction mapping principle. Krasnoselskii's fixed point theorem becomes effective to overcome this situation. Under the second set of assumptions, we obtain only the existence of mild solution using Schauder's fixed point theorem. Few examples have been provided to illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Upper and lower solutions for fractional integro-differential equation of higher-order and with nonlinear boundary conditions.
- Author
-
El Allaoui, Abdelati, Allaoui, Youssef, Melliani, Said, and El Khalfi, Hamid
- Subjects
FRACTIONAL integrals ,INTEGRO-differential equations ,NONLINEAR boundary value problems ,CAPUTO fractional derivatives ,MATHEMATICAL formulas - Abstract
This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Existence and Hyers–Ulam Stability of Jerk-Type Caputo and Hadamard Mixed Fractional Differential Equations.
- Author
-
Ma, Yanli, Maryam, Maryam, Riaz, Usman, Popa, Ioan-Lucian, Ragoub, Lakhdar, and Zada, Akbar
- Abstract
This article is concerned with existence of mild solutions for jerk-type fractional differential equations in the sense of Hadamard and Caputo fractional derivatives with separated boundary conditions. For the uniqueness of mild solutions in both cases, Banach contraction principle are followed. Moreover, at least one mild solution of jerk-type Caputo–Hadamard and Hadamard–Caputo fractional differential equations can be analyzed using Krasnoselskii’s and Leray–Schauder fixed point theorems. Hyers–Ulam stability and its generalized case for both type of mentioned jerk-type problems can be find out with the help of some conditions and definitions. For the illustration of main results, an example is provided. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Bazykin’s Predator–Prey Model Includes a Dynamical Analysis of a Caputo Fractional Order Delay Fear and the Effect of the Population-Based Mortality Rate on the Growth of Predators.
- Author
-
Kumar, G. Ranjith, Ramesh, K., Khan, Aziz, Lakshminarayan, K., and Abdeljawad, Thabet
- Abstract
In this paper, we investigate a system of two differential equations of fractional order for the fear effect in prey-predator interactions, in which the density of predators controls the mortality pace of the prey population. The non-integer order differential equation is interpreted in terms of the Caputo derivative, and the development of the non-integer order scheme is described in terms of the influence of memory on population increase. The primary goal of existing research is to explore how the changing aspects of the current scheme are impacted by various types of parameters, including time delay, fear effect, and fractional order. The solutions’ positivity, existence-uniqueness, and boundedness are established with precise mathematical conclusions. The requirements necessary for the local asymptotic stability of different equilibrium points and the global stability of coexistence equilibrium are established. Hopf bifurcation occurs in the system at various delay times. The model’s fractional-order derivatives enhance the model behaviours and provide stability findings for the solutions. We have observed that fractional order plays an important role in population dynamics. Also, Hopf bifurcation for the proposed system have been observed for certain values of order of derivatives. Thus, the stability conditions of the equilibrium points may be changed by changing the order of the derivatives without changing other parametric values. Finally, a numerical simulation is run to verify our conclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator
- Author
-
Wei Fan and Kangqun Zhang
- Subjects
fractional diffusion equation ,erdélyi-kober fractional derivative ,caputo fractional derivative ,nonlinearity ,well-posedness ,Mathematics ,QA1-939 - Abstract
In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity.
- Published
- 2024
- Full Text
- View/download PDF
50. Utilizing fractional derivatives and sensitivity analysis in a random framework: a model-based approach to the investigation of random dynamics of malware spread
- Author
-
Zafer Bekiryazici
- Subjects
Fractional differential equation ,Random parameters ,Normal distribution ,Fractional Euler method ,Caputo fractional derivative ,Sensitivity ,Analysis ,QA299.6-433 - Abstract
Abstract In this study, an ordinary-deterministic equation system modeling the spread dynamics of malware under mutation is analyzed with fractional derivatives and random variables. The original model is transformed into a system of fractional-random differential equations (FRDEs) using Caputo fractional derivatives. Normally distributed random variables are defined for the parameters of the original system that are related to the mutations and infections of the nodes in the network. The resulting system of FRDEs is simulated using the predictor-corrector method based fde12 algorithm and the forward fractional Euler method (ffEm) for various values of the model components such as the standard deviations, orders of derivation, and repetition numbers. Additionally, the sensitivity analysis of the original model is investigated in relation to the random nature of the components and the basic reproduction number ( R 0 $R_{0}$ ) to underline the correspondence of random dynamics and sensitivity indices. Both the normalized forward sensitivity indices (NFSI) and the standard deviation of R 0 $R_{0}$ with random components give matching results for analyzing the changes in the spread rate. Theoretical results are backed by the simulation outputs on the numerical characteristics of the fractional-random model for the expected number of infections and mutations, expected timing of the removal of mutations from the network, and measurement of the variability in the results such as the coefficients of variation. Comparison of the results from the original model and the fractional-random model shows that the fractional-random analysis provides a more generalized perspective while facilitating a versatile investigation with ease and can be used on different models as well.
- Published
- 2024
- Full Text
- View/download PDF
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