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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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Study for System of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative 被引量:1
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作者 Yanping Zheng Wenxia Wang 《Applied Mathematics》 2013年第7期5-8,共4页
In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where Dα denotes standard Riemann-Liouville fractional derivative, 0 and A ?is a square matrix... In this work, we study existence theorem of the initial value problem for the system of fractional differential equations where Dα denotes standard Riemann-Liouville fractional derivative, 0 and A ?is a square matrix. At the same time, power-type estimate for them has been given. 展开更多
关键词 riemann-liouville fractional derivATIVE WEIGHTED Cauchy-Type Problem fractional Differential EQUATIONS
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An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid 被引量:2
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作者 Bo Yu Xiaoyun Jiang Haitao Qi 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2015年第2期153-161,共9页
In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit n... In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid. 展开更多
关键词 riemann-liouville fractional derivative Generalized second grade fluid Inverse problem Implicit numerical method fractional sensitivity equation
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Some New Delay Integral Inequalities Based on Modified Riemann-Liouville Fractional Derivative and Their Applications
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作者 Zhimin Zhao Run Xu 《Journal of Applied Mathematics and Physics》 2015年第5期465-477,共13页
By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the res... By using the properties of modified Riemann-Liouville fractional derivative, some new delay integral inequalities have been studied. First, we offered explicit bounds for the unknown functions, then we applied the results to the research concerning the boundness, uniqueness and continuous dependence on the initial for solutions to certain fractional differential equations. 展开更多
关键词 MODIFIED riemann-liouville fractional derivATIVE INTEGRAL INEQUALITIES DELAY fractional Differential Equation
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A note on the definition of fractional derivatives applied in rheology 被引量:1
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作者 Fan Yang Ke-Qin Zhu 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2011年第6期866-876,共11页
It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives-Riemann-Liouville (R-L) definition and Caputo definition. The multiple definitions of fracti... It is known that there exist obivious differences between the two most commonly used definitions of fractional derivatives-Riemann-Liouville (R-L) definition and Caputo definition. The multiple definitions of fractional derivatives in fractional calculus have hindered the application of fractional calculus in rheology. In this paper, we clarify that the R-L definition and Caputo definition are both rheologically imperfect with the help of mechanical analogues of the fractional element model (Scott-Blair model). We also clarify that to make them perfect rheologically, the lower terminals of both definitions should be put to ∞. We further prove that the R-L definition with lower terminal a →∞ and the Caputo definition with lower terminal a →∞ are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points. Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal a →∞ (or, equivalently, the Caputo derivatives with lower terminal a →∞) not only for steady-state processes, but also for transient processes. Based on the above definition, the problems of composition rules of fractional operators and the initial conditions for fractional differential equations are discussed, respectively. As an example we study a fractional oscillator with Scott-Blair model and give an exact solution of this equation under given initial conditions. 展开更多
关键词 fractional derivatives RHEOLOGY riemann-liouville definition Caputo definition
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ON A COUPLED INTEGRO-DIFFERENTIAL SYSTEM INVOLVING MIXED FRACTIONAL DERIVATIVES AND INTEGRALS OF DIFFERENT ORDERS
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作者 Bashir AHMAD Ravi P.AGARWAL +1 位作者 Abrar BROOM Ahmed ALSAEDI 《Acta Mathematica Scientia》 SCIE CSCD 2021年第4期1366-1384,共19页
By applying the standard fixed point theorems,we prove the existence and uniqueness results for a system of coupled differential equations involving both left Caputo and right Riemann-Liouville fractional derivatives ... By applying the standard fixed point theorems,we prove the existence and uniqueness results for a system of coupled differential equations involving both left Caputo and right Riemann-Liouville fractional derivatives and mixed fractional integrals,supplemented with nonlocal coupled fractional integral boundary conditions.An example is also constructed for the illustration of the obtained results. 展开更多
关键词 fractional differential equations Caputo and riemann-liouville fractional derivatives systems EXISTENCE fixed point theorems
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On the definition of fractional derivatives in rheology
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作者 Fan Yang~(1,2,a)) and Ke-Qin Zhu~(1,b)) 1)Department of Engineering Mechanics,Tsinghua University,Beijing 100084,China 2)Department of Physics,the Chinese University of Hong Kong,Hong Kong,China 《Theoretical & Applied Mechanics Letters》 CAS 2011年第1期62-65,共4页
During the last two decades fractional calculus has been increasingly applied to physics, especially to rheology.It is well known that there are obivious differences between Riemann-Liouville (R-L) definition and Capu... During the last two decades fractional calculus has been increasingly applied to physics, especially to rheology.It is well known that there are obivious differences between Riemann-Liouville (R-L) definition and Caputo definition,which are the two most commonly used definitions of fractional derivatives.The multiple definitions of fractional derivatives have hindered the application of fractional calculus in rheology.In this paper,we clarify that the R-L definition and Caputo definition are both Theologically unreasonable with the help of the mechanical analogues of the fractional element model.We also find that to make them more reasonable Theologically,the lower terminals of both definitions should be put to—∞.We further prove that the R-L definition with lower terminal—∞and the Caputo definition with lower terminal—∞are equivalent in the differentiation of functions that are smooth enough and functions that have finite number of singular points.Thus we can define the fractional derivatives in rheology as the R-L derivatives with lower terminal—∞(or,equivalently,the Caputo derivatives with lower terminal—∞) not only for steady-state processes,but also for transient processes. 展开更多
关键词 fractional derivative Caputo definition riemann-liouville definition Scott-Blair model
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Exact solutions for nonlinear partial fractional differential equations 被引量:22
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作者 Khaled A.Gepreel Saleh Omran 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第11期32-38,共7页
′In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved (G′/G)-expansion func... ′In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved (G′/G)-expansion function method to calculate the exact solutions to the time- and space-fractional derivative foam drainage equation and the time- and space-fractional derivative nonlinear KdV equation. This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations. 展开更多
关键词 fractional calculus complex transformation modified riemann-liouville derivative im- proved (G′/G)-expansion function method
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New analytical exact solutions of time fractional KdV KZK equation by Kudryashov methods 被引量:4
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作者 S Saha Ray 《Chinese Physics B》 SCIE EI CAS CSCD 2016年第4期30-36,共7页
In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For thi... In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equa- tion are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the non- linear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation. 展开更多
关键词 KdV-Khokhlov-Zabolotskaya-Kuznetsov equation Kudryashov method modified Kudryashovmethod fractional complex transform modified riemann-liouville derivative
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Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions 被引量:4
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作者 Boling GUO Qiang XU Zhe YIN 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2016年第3期403-416,共14页
An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation (FPE) with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for ... An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation (FPE) with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples. 展开更多
关键词 fractional percolation equation (FPE) riemann-liouville derivative frac-tional boundary condition finite difference method stability and convergence Toeplitzmatrix
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Fractional Birkhoffian Dynamics Based on Quasi-fractional Dynamics Models and Its Lie Symmetry 被引量:3
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作者 JIA Yundie ZHANG Yi 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI CSCD 2021年第1期84-95,共12页
In order to investigate the dynamic behavior of non-conservative systems,the Lie symmetries and conserved quantities of fractional Birkhoffian dynamics based on quasi-fractional dynamics model are proposed and studied... In order to investigate the dynamic behavior of non-conservative systems,the Lie symmetries and conserved quantities of fractional Birkhoffian dynamics based on quasi-fractional dynamics model are proposed and studied.The quasi-fractional dynamics model here refers to the variational problem based on the definition of RiemannLiouville fractional integral(RLFI),the variational problem based on the definition of extended exponentially fractional integral(EEFI),and the variational problem based on the definition of fractional integral extended by periodic laws(FIEPL).First,the fractional Pfaff-Birkhoff principles based on quasi-fractional dynamics models are established,and the corresponding Birkhoff’s equations and the determining equations of Lie symmetry are obtained.Second,for fractional Birkhoffian systems based on quasi-fractional models,the conditions and forms of conserved quantities are given,and Lie symmetry theorems are proved.The Pfaff-Birkhoff principles,Birkhoff’s equations and Lie symmetry theorems of quasi-fractional Birkhoffian systems and classical Birkhoffian systems are special cases of this article.Finally,some examples are given. 展开更多
关键词 quasi-fractional dynamics model Lie symmetry conserved quantity fractional Birkhoffian system riemann-liouville derivative
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Existence of positive solutions for integral boundary value problem of fractional differential equations 被引量:4
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作者 Xiping Liu Guiyun Wu 《上海师范大学学报(自然科学版)》 2014年第5期496-505,共10页
In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By u... In this paper,we concern ourselves with the existence of positive solutions for a type of integral boundary value problem of fractional differential equations with the fractional order linear derivative operator. By using the fixed point theorem in cone,the existence of positive solutions for the boundary value problem is obtained. Some examples are also presented to illustrate the application of our main results. 展开更多
关键词 fractional differential equations riemann-liouville fractional derivative fixed point theorem fractional order linear derivative operator
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Fractional Versions of the Fundamental Theorem of Calculus 被引量:2
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作者 Eliana Contharteze Grigoletto Edmundo Capelas de Oliveira 《Applied Mathematics》 2013年第7期23-33,共11页
The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we... The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions. 展开更多
关键词 fractional INTEGRAL fractional derivATIVE riemann-liouville derivATIVE LIOUVILLE derivATIVE Caputo derivATIVE WEYL derivATIVE and RIESZ derivATIVE
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Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis 被引量:1
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作者 Vidhya Saraswathy Krishnasamy Somayeh Mashayekhi Mohsen Razzaghi 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2017年第1期98-106,共9页
In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional int... In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique. 展开更多
关键词 Caputo derivative fractional differential equations(FEDs) fractional Taylor basis operational matrix riemann-liouville fractional integral operator
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Application of the Improved Kudryashov Method to Solve the Fractional Nonlinear Partial Differential Equations 被引量:2
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作者 Md. Abdus Salam Umme Habiba 《Journal of Applied Mathematics and Physics》 2019年第4期912-920,共9页
Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Bur... Our purpose of this paper is to apply the improved Kudryashov method for solving various types of nonlinear fractional partial differential equations. As an application, the time-space fractional Korteweg-de Vries-Burger (KdV-Burger) equation is solved using this method and we get some new travelling wave solutions. To acquire our purpose a complex transformation has been also used to reduce nonlinear fractional partial differential equations to nonlinear ordinary differential equations of integer order, in the sense of the Jumarie’s modified Riemann-Liouville derivative. Afterwards, the improved Kudryashov method is implemented and we get our required reliable solutions where the results are justified by mathematical software Maple-13. 展开更多
关键词 IMPROVED Kudryashov METHOD Time-Space fractional KdV-Burger Equation TRAVELLING Wave Solutions Jumarie’s Modified riemann-liouville derivative
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Numerical Methods for Solving Space Fractional Partial Differential Equations Using Hadamard Finite-Part Integral Approach 被引量:1
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作者 Yanyong Wang Yubin Yan Ye Hu 《Communications on Applied Mathematics and Computation》 2019年第4期505-523,共19页
We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case.The approximation of the space fractional Riemann-Liouville derivative is based on th... We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case.The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order O(h^3-a),where h is the space step size and α∈(1,2)is the order of Riemann-Liouville fractional derivative.Based on this scheme,we introduce a shifted finite difference method for solving space fractional partial differential equations.We obtained the error estimates with the convergence orders O(τ+h^3-a+h^β),where τ is the time step size and β>0 is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation.Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Griinwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the firstorder convergence,the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition.Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Griinwald-Letnikov formula or Lubich's higher order approximation schemes. 展开更多
关键词 riemann-liouville fractional derivative SPACE fractional partial differential equation Error ESTIMATES
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On the Approximate Solution of Fractional Logistic Differential Equation Using Operational Matrices of Bernstein Polynomials
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作者 R. F. Al-Bar 《Applied Mathematics》 2015年第12期2096-2103,共8页
In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouv... In this paper, operational matrices of Bernstein polynomials (BPs) are presented for solving the non-linear fractional Logistic differential equation (FLDE). The fractional derivative is described in the Riemann-Liouville sense. The operational matrices for the fractional integration in the Riemann-Liouville sense and the product are used to reduce FLDE to the solution of non-linear system of algebraic equations using Newton iteration method. Numerical results are introduced to satisfy the accuracy and the applicability of the proposed method. 展开更多
关键词 fractional LOGISTIC Equation riemann-liouville fractional derivatives riemann-liouville fractional Integral OPERATIONAL Matrix BERNSTEIN POLYNOMIALS
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Pseudo-Spectral Method for Space Fractional Diffusion Equation
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作者 Yiting Huang Minling Zheng 《Applied Mathematics》 2013年第11期1495-1502,共8页
This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogo... This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions. 展开更多
关键词 riemann-liouville derivATIVE Pseudo-Spectral METHOD COLLOCATION METHOD fractional DIFFUSION EQUATION
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The Exact Solution of the Space-Time Fractional Modified Kdv-Zakharov-Kuznetsov Equation
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作者 Qiuyan Jin Tiecheng Xia Jinbo Wang 《Journal of Applied Mathematics and Physics》 2017年第4期844-852,共9页
In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based... In this paper, we get many new analytical solutions of the space-time nonlinear fractional modified KDV-Zakharov Kuznetsov (mKDV-ZK) equation by means of a new approach namely method of undetermined coefficients based on a fractional complex transform. These solutions have physics meanings in natural sciences. This method can be used to other nonlinear fractional differential equations. 展开更多
关键词 Analytical Solutions the SPACE-TIME fractional MODIFIED KDV-ZK EQUATION Nonlinear fractional Differential EQUATION MODIFIED riemann-liouville derivative
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Approximate Solutions for a Class of Fractional-Order Model of HIV Infection via Linear Programming Problem
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作者 Samaneh Soradi Zeid Mostafa Yousefi Ali Vahidian Kamyad 《American Journal of Computational Mathematics》 2016年第2期141-152,共12页
In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of... In this paper, we provide a new approach to solve approximately a system of fractional differential equations (FDEs). We extend this approach for approximately solving a fractional-order differential equation model of HIV infection of CD4<sup>+</sup>T cells with therapy effect. The fractional derivative in our approach is in the sense of Riemann-Liouville. To solve the problem, we reduce the system of FDE to a discrete optimization problem. By obtaining the optimal solutions of new problem by minimization the total errors, we obtain the approximate solution of the original problem. The numerical solutions obtained from the proposed approach indicate that our approximation is easy to implement and accurate when it is applied to a systems of FDEs. 展开更多
关键词 riemann-liouville derivative fractional HIV Model Optimization Linear Programming Discritezation
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