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Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives
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作者 Nasser Sweilam Seham Al-Mekhlafi +2 位作者 Aya Ahmed Ahoud Alsheri Emad Abo-Eldahab 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第8期1619-1645,共27页
In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators... In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings. 展开更多
关键词 Cancer diseases hybrid variable-order fractional derivatives adams bashfourth fifth step generalized fifth order Runge-Kutta method
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Study of Fractional Order Dynamical System of Viral Infection Disease under Piecewise Derivative 被引量:2
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作者 Kamal Shah Hafsa Naz +1 位作者 Thabet Abdeljawad Bahaaeldin Abdalla 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第7期921-941,共21页
This research aims to understand the fractional order dynamics of the deadly Nipah virus(NiV)disease.We focus on using piecewise derivatives in the context of classical and singular kernels of power operators in the C... This research aims to understand the fractional order dynamics of the deadly Nipah virus(NiV)disease.We focus on using piecewise derivatives in the context of classical and singular kernels of power operators in the Caputo sense to investigate the crossover behavior of the considered dynamical system.We establish some qualitative results about the existence and uniqueness of the solution to the proposed problem.By utilizing the Newtonian polynomials interpolation technique,we recall a powerful algorithm to interpret the numerical findings for the aforesaid model.Here,we remark that the said viral infection is caused by an RNA type virus which can transmit from animals and also from an infected person to person.Fruits bats which are also known as flying foxes are one of the sources of transmission of NiV disease.Here in this work,we investigate its transmission mechanism through some new concepts of fractional calculus for further analysis and prediction.We present the approximate results for different compartments using different fractional orders.By using the piecewise derivative concept,we detect the crossover ormulti-steps behavior in the transmission dynamics of the mentioned disease.Therefore,the considered form of the derivative is used to deal with problems exhibiting crossover behaviors. 展开更多
关键词 NiV disease fractional calculus piecewise derivative qualitative results newton polynomial RNA virus
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Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives
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作者 Muath Awadalla Kinda Abuasbeh +1 位作者 Yves Yannick Yameni Noupoue Mohammed S.Abdo 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第6期2767-2785,共19页
This study focuses on the dynamics of drug concentration in the blood.In general,the concentration level of a drug in the blood is evaluated by themean of an ordinary and first-order differential equation.More precise... This study focuses on the dynamics of drug concentration in the blood.In general,the concentration level of a drug in the blood is evaluated by themean of an ordinary and first-order differential equation.More precisely,it is solved through an initial value problem.We proposed a newmodeling technique for studying drug concentration in blood dynamics.This technique is based on two fractional derivatives,namely,Caputo and Caputo-Fabrizio derivatives.We first provided comprehensive and detailed proof of the existence of at least one solution to the problem;we later proved the uniqueness of the existing solution.The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques.Stability via theUlam-Hyers(UH)technique was also investigated.The application of the proposedmodel on two real data sets revealed that the Caputo derivative wasmore suitable in this study.Indeed,for the first data set,the model based on the Caputo derivative yielded a Mean Squared Error(MSE)of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360.Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem,with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532.For the second experiment,Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066,whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299.Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach. 展开更多
关键词 PHARMACOKINETICS caputo fractional derivative stability study Caputo-Fabrizio fractional derivative
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Fractional derivative statistical damage model of unsaturated expansive soil based on unified hydraulic effect
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作者 ZHANG Hua WANG Peng 《Journal of Mountain Science》 SCIE CSCD 2023年第9期2769-2782,共14页
Unsaturated expansive soil is widely distributed in China and has complex engineering properties.This paper proposes the unified hydraulic effect shear strength theory of unsaturated expansive soil based on the effect... Unsaturated expansive soil is widely distributed in China and has complex engineering properties.This paper proposes the unified hydraulic effect shear strength theory of unsaturated expansive soil based on the effective stress principle,swelling force principle,and soil–water characteristics.Considering the viscoelasticity and structural damage of unsaturated expansive soil during loading,a fractional hardening–damage model of unsaturated expansive soil was established.The model parameters were established on the basis of the proposed calculation method of shear strength and the triaxial shear experiment on unsaturated expansive soil.The proposed model was verified by the experimental data and a traditional damage model.The proposed model can satisfactorily describe the entire process of the strain-hardening law of unsaturated expansive soil.Finally,by investigating the damage variables of the proposed model,it was found that:(a)when the values of confining pressure and matric suction are close,the coupling of confining pressure and matric suction contributes more to the shear strength;(b)there is a damage threshold for unsaturated expansive soil,and is mainly reflected by strength criterion of infinitesimal body;(c)the strain hardening law of unsaturated expansive soil is mainly reflected by fractional derivative operator. 展开更多
关键词 Unsaturated expansive soil Unified hydraulic effect Shear strength theory Hardening-damage model fractional derivative
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On the Fractional Derivatives with an Exponential Kernel
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作者 Enyu Fan Jingshu Wu Shaoying Zeng 《Communications on Applied Mathematics and Computation》 EI 2023年第4期1655-1673,共19页
The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional d... The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional derivatives are proposed,including the Fourier transform and the Laplace transform.Then,the L2 discretisation for the exponential Caputo derivative with a∈(1,2)is established.The estimation of the truncation error and the properties of the coefficients are discussed.In addition,a numerical example is given to verify the correctness of the derived L2 discrete formula. 展开更多
关键词 Exponential fractional derivative Integral transform L2 discretisation Truncation error
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Three Kinds of Discrete Formulae for the Caputo Fractional Derivative
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作者 Zhengnan Dong Enyu Fan +1 位作者 Ao Shen Yuhao Su 《Communications on Applied Mathematics and Computation》 EI 2023年第4期1446-1468,共23页
In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The trunc... In this paper,three kinds of discrete formulae for the Caputo fractional derivative are studied,including the modified L1 discretisation forα∈(O,1),and L2 discretisation and L2C discretisation forα∈(1,2).The truncation error estimates and the properties of the coeffcients of all these discretisations are analysed in more detail.Finally,the theoretical analyses areverifiedby thenumerical examples. 展开更多
关键词 Caputo fractional derivative Modified L1 discretisation L2 discretisation L2C discretisation Truncation error
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AN EXPLANATION ON FOUR NEW DEFINITIONS OF FRACTIONAL OPERATORS
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作者 Jiangen LIU Fazhan GENG 《Acta Mathematica Scientia》 SCIE CSCD 2024年第4期1271-1279,共9页
Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new f... Fractional calculus has drawn more attentions of mathematicians and engineers in recent years.A lot of new fractional operators were used to handle various practical problems.In this article,we mainly study four new fractional operators,namely the CaputoFabrizio operator,the Atangana-Baleanu operator,the Sun-Hao-Zhang-Baleanu operator and the generalized Caputo type operator under the frame of the k-Prabhakar fractional integral operator.Usually,the theory of the k-Prabhakar fractional integral is regarded as a much broader than classical fractional operator.Here,we firstly give a series expansion of the k-Prabhakar fractional integral by means of the k-Riemann-Liouville integral.Then,a connection between the k-Prabhakar fractional integral and the four new fractional operators of the above mentioned was shown,respectively.In terms of the above analysis,we can obtain this a basic fact that it only needs to consider the k-Prabhakar fractional integral to cover these results from the four new fractional operators. 展开更多
关键词 k-Prabhakar fractional operator Caputo-Fabrizio operator atangana-baleanu operator Sun-Hao-Zhang-Baleanu operator generalized Caputo type operator
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Analytical solutions fractional order partial differential equations arising in fluid dynamics
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作者 Sidheswar Behera Jasvinder Singh Pal Virdi 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2024年第3期458-468,共11页
This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectio... This article describes the solution procedure of the fractional Pade-Ⅱ equation and generalized Zakharov equation(GSEs)using the sine-cosine method.Pade-Ⅱ is an important nonlinear wave equation modeling unidirectional propagation of long-wave in dispersive media and GSEs are used to model the interaction between one-dimensional high,and low-frequency waves.Classes of trigonometric and hyperbolic function solutions in fractional calculus are discussed.Graphical simulations of the numerical solutions are flaunted by MATLAB. 展开更多
关键词 the sine-cosine method He's fractional derivative analytical solution fractional Pade-Ⅱequation fractional generalized Zakharov equation
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A New Scheme of the ARA Transform for Solving Fractional-Order Waves-Like Equations Involving Variable Coefficients
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作者 Yu-Ming Chu Sobia Sultana +2 位作者 Shazia Karim Saima Rashid Mohammed Shaaf Alharthi 《Computer Modeling in Engineering & Sciences》 SCIE EI 2024年第1期761-791,共31页
The goal of this research is to develop a new,simplified analytical method known as the ARA-residue power series method for obtaining exact-approximate solutions employing Caputo type fractional partial differential e... The goal of this research is to develop a new,simplified analytical method known as the ARA-residue power series method for obtaining exact-approximate solutions employing Caputo type fractional partial differential equations(PDEs)with variable coefficient.ARA-transform is a robust and highly flexible generalization that unifies several existing transforms.The key concept behind this method is to create approximate series outcomes by implementing the ARA-transform and Taylor’s expansion.The process of finding approximations for dynamical fractional-order PDEs is challenging,but the ARA-residual power series technique magnifies this challenge by articulating the solution in a series pattern and then determining the series coefficients by employing the residual component and the limit at infinity concepts.This approach is effective and useful for solving a massive class of fractional-order PDEs.Five appealing implementations are taken into consideration to demonstrate the effectiveness of the projected technique in creating solitary series findings for the governing equations with variable coefficients.Additionally,several visualizations are drawn for different fractional-order values.Besides that,the estimated findings by the proposed technique are in close agreement with the exact outcomes.Finally,statistical analyses further validate the efficacy,dependability and steady interconnectivity of the suggested ARA-residue power series approach. 展开更多
关键词 ARA-transform Caputo fractional derivative residue-power seriesmethod analytical solutions statistical analysis
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Multiple Solutions for a Class of Singular Boundary Value Problems of Hadamard Fractional Differential Systems with p-Laplacian Operator
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作者 Chen Wang Yansheng Liu 《Journal of Applied Mathematics and Physics》 2024年第9期3114-3134,共21页
This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the ... This paper discusses the existence and multiplicity of positive solutions for a class of singular boundary value problems of Hadamard fractional differential systems involving the p-Laplacian operator. First, for the sake of overcoming the singularity, sequences of approximate solutions to the boundary value problem are obtained by applying the fixed point index theory on the cone. Next, it is demonstrated that these sequences of approximate solutions are uniformly bounded and equicontinuous. The main results are then established through the Ascoli-Arzelà theorem. Ultimately, an instance is worked out to test and verify the validity of the main results. 展开更多
关键词 Multiple Solutions Fixed Point Index Theory Nonlinear fractional Differential Systems Hadamard fractional derivative
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A Hybrid ESA-CCD Method for Variable-Order Time-Fractional Diffusion Equations
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作者 Xiaoxue Lu Chunhua Zhang +1 位作者 Huiling Xue Bowen Zhong 《Journal of Applied Mathematics and Physics》 2024年第9期3053-3065,共13页
In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order a... In this paper, we study the solutions for variable-order time-fractional diffusion equations. A three-point combined compact difference (CCD) method is used to discretize the spatial variables to achieve sixth-order accuracy, while the exponential-sum-approximation (ESA) is used to approximate the variable-order Caputo fractional derivative in the temporal direction, and a novel spatial sixth-order hybrid ESA-CCD method is implemented successfully. Finally, the accuracy of the proposed method is verified by numerical experiments. 展开更多
关键词 Variable-Order Caputo fractional derivative Combined Compact Difference Method Exponential-Sum-Approximation
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Numerical Simulation and Parameter Estimation of Fractional-Order Dynamic Epidemic Model for COVID-19
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作者 Rong Kang Tianzeng Li 《Journal of Applied Mathematics and Physics》 2024年第10期3469-3495,共27页
The outbreak of COVID-19 in 2019 resulted in numerous infections and deaths. In order to better study the transmission of COVID-19, this article adopts an improved fractional-order SIR model. Firstly, the properties o... The outbreak of COVID-19 in 2019 resulted in numerous infections and deaths. In order to better study the transmission of COVID-19, this article adopts an improved fractional-order SIR model. Firstly, the properties of the model are studied, including the feasible domain and bounded solutions of the system. Secondly, the stability of the system is discussed, among other things. Then, the GMMP method is introduced to obtain numerical solutions for the COVID-19 system and combined with the improved MH-NMSS-PSO parameter estimation method to fit the real data of Delhi, India from April 1, 2020 to June 30, 2020. The results show that the fitting effect is quite ideal. Finally, long-term predictions were made on the number of infections. We accurately estimate that the peak number of infections in Delhi, India, can reach around 2.1 million. This paper also compares the fitting performance of the integer-order COVID-19 model and the fractional-order COVID-19 model using the real data from Delhi. The results indicate that the fractional-order model with different orders, as we proposed, performs the best. 展开更多
关键词 Parameter Estimation COVID-19 Infectious Disease Model fractional-Order derivative
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Establishment of a Fractional Order COVID-19 Model and Its Feasibility Analysis
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作者 Rong Kang Tianzeng Li Yu Zhao 《Journal of Computer and Communications》 2024年第10期62-77,共16页
This paper investigates an improved SIR model for COVID-19 based on the Caputo fractional derivative. Firstly, the properties of the model are studied, including the feasible domain and bounded solutions of the system... This paper investigates an improved SIR model for COVID-19 based on the Caputo fractional derivative. Firstly, the properties of the model are studied, including the feasible domain and bounded solutions of the system. Secondly, the stability of the system is discussed, among other things. Then, the GMMP method is introduced to obtain numerical solutions for the COVID-19 system. Numerical simulations were conducted using MATLAB, and the results indicate that our model is valuable for studying virus transmission. 展开更多
关键词 Grid Approximation Method COVID-19 Infectious Disease Model fractional-Order derivative
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Noether's theorems of a fractional Birkhoffian system within Riemann-Liouville derivatives 被引量:17
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作者 周燕 张毅 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第12期281-288,共8页
The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff's equations and the corresponding ... The Noether symmetry and the conserved quantity of a fractional Birkhoffian system are studied within the Riemann–Liouville fractional derivatives. Firstly, the fractional Birkhoff's equations and the corresponding transversality conditions are given. Secondly, from special to general forms, Noether's theorems of a standard Birhoffian system are given, which provide an approach and theoretical basis for the further research on the Noether symmetry of the fractional Birkhoffian system. Thirdly, the invariances of the fractional Pfaffian action under a special one-parameter group of infinitesimal transformations without transforming the time and a general one-parameter group of infinitesimal transformations with transforming the time are studied, respectively, and the corresponding Noether's theorems are established. Finally, an example is given to illustrate the application of the results. 展开更多
关键词 fractional Birkhoffian system Noether's theorem fractional conserved quantity Riemann–Liouville fractional derivative
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UNSTEADY FLOWS OF A GENERALIZED SECOND GRADE FLUID WITH THE FRACTIONAL DERIVATIVE MODEL BETWEEN TWO PARALLEL PLATES 被引量:19
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作者 谭文长 徐明喻 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2004年第5期471-476,共6页
The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced.Exact analytical solutions are obtained for a class of unsteady flows for the generalized secon... The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced.Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus.The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates.In addition,the solutions of the shear stresses at the plates are also determined. 展开更多
关键词 fractional derivative unsteady flows generalized second grade fluid parallel plates
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Fractional differential equations of motion in terms of combined Riemann-Liouville derivatives 被引量:15
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作者 张毅 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第8期302-306,共5页
In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defi... In this paper, we focus on studying the fractional variational principle and the differential equations of motion for a fractional mechanical system. A combined Riemann-Liouville fractional derivative operator is defined, and a fractional Hamilton principle under this definition is established. The fractional Lagrange equations and the fractional Hamilton canonical equations are derived from the fractional Hamilton principle. A number of special cases are given, showing the universality of our conclusions. At the end of the paper, an example is given to illustrate the application of the results. 展开更多
关键词 fractional Hamilton principle fractional Lagrange equation fractional Hamilton canon-ical equation combined Riemann-Liouville fractional derivative
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Hamilton formalism and Noether symmetry for mechanico electrical systems with fractional derivatives 被引量:7
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作者 张世华 陈本永 傅景礼 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第10期9-16,共8页
This paper presents extensions to the traditional calculus of variations for mechanico-electrical systems containing fractional derivatives. The Euler Lagrange equations and the Hamilton formalism of the mechanico-ele... This paper presents extensions to the traditional calculus of variations for mechanico-electrical systems containing fractional derivatives. The Euler Lagrange equations and the Hamilton formalism of the mechanico-electrical systems with fractional derivatives are established. The definition and the criteria for the fractional generalized Noether quasi- symmetry are presented. Furthermore, the fractional Noether theorem and conseved quantities of the systems are obtained by virtue of the invariance of the Hamiltonian action under the infinitesimal transformations. An example is presented to illustrate the application of the results. 展开更多
关键词 fractional derivative mechanico-electrical system Noether symmetry Hamiltonian formulation
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Lagrange equations of nonholonomic systems with fractional derivatives 被引量:7
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作者 周莎 傅景礼 刘咏松 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第12期25-29,共5页
This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, ba... This paper obtains Lagrange equations of nonholonomic systems with fractional derivatives. First, the exchanging relationships between the isochronous variation and the fractional derivatives are derived. Secondly, based on these exchanging relationships, the Hamilton's principle is presented for non-conservative systems with fractional derivatives. Thirdly, Lagrange equations of the systems are obtained. Furthermore, the d'Alembert-Lagrange principle with fractional derivatives is presented, and the Lagrange equations of nonholonomic systems with fractional derivatives are studied. An example is designed to illustrate these results. 展开更多
关键词 fractional derivative d'Alembert-Lagrange principle Lagrange equation nonholonomic system
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ON THE EXISTENCE AND STABILITY OF BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATION WITH HILFER-KATUGAMPOLA FRACTIONAL DERIVATIVE 被引量:4
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作者 E.M.ELSAYED S.HARIKRISHNAN K.KANAGARAJAN 《Acta Mathematica Scientia》 SCIE CSCD 2019年第6期1568-1578,共11页
In this paper, we discuss the existence, uniqueness and stability of boundary value problem for differential equation with Hilfer-Katugampola fractional derivative. The arguments are based upon Schaefer's fixed po... In this paper, we discuss the existence, uniqueness and stability of boundary value problem for differential equation with Hilfer-Katugampola fractional derivative. The arguments are based upon Schaefer's fixed point theorem, Banach contraction principle and Ulam type stability. 展开更多
关键词 Hilfer-Katugampola fractional derivative boundary condition EXISTENCE Ulam STABILITY
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Variational Calculus With Conformable Fractional Derivatives 被引量:4
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作者 Matheus J.Lazo Delfim F.M.Torres 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2017年第2期340-352,共13页
Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different ... Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives. 展开更多
关键词 Conformable fractional derivative fractional calculus of variations fractional optimal control invariant variational conditions Noether’s theorem
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