In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principl...In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.展开更多
In this paper,we prove the transportation cost-information inequalities on the space of continuous paths with respect to the L~2-metric and the uniform metric for the law of the mild solution to the stochastic heat eq...In this paper,we prove the transportation cost-information inequalities on the space of continuous paths with respect to the L~2-metric and the uniform metric for the law of the mild solution to the stochastic heat equation defined on[0,T]×[0,1]driven by double-parameter fractional noise.展开更多
Fractional differential equations(FDEs)provide a powerful tool for modeling systems with memory and non-local effects,but understanding their underlying structure remains a significant challenge.While numerous numeric...Fractional differential equations(FDEs)provide a powerful tool for modeling systems with memory and non-local effects,but understanding their underlying structure remains a significant challenge.While numerous numerical and semi-analytical methods exist to find solutions,new approaches are needed to analyze the intrinsic properties of the FDEs themselves.This paper introduces a novel computational framework for the structural analysis of FDEs involving iterated Caputo derivatives.The methodology is based on a transformation that recasts the original FDE into an equivalent higher-order form,represented as the sum of a closed-form,integer-order component G(y)and a residual fractional power seriesΨ(x).This transformed FDE is subsequently reduced to a first-order ordinary differential equation(ODE).The primary novelty of the proposed methodology lies in treating the structure of the integer-order component G(y)not as fixed,but as a parameterizable polynomial whose coefficients can be determined via global optimization.Using particle swarm optimization,the framework identifies an optimal ODE architecture by minimizing a dual objective that balances solution accuracy against a high-fidelity reference and the magnitude of the truncated residual series.The effectiveness of the approach is demonstrated on both a linear FDE and a nonlinear fractional Riccati equation.Results demonstrate that the framework successfully identifies an optimal,low-degree polynomial ODE architecture that is not necessarily identical to the forcing function of the original FDE.This work provides a new tool for analyzing the underlying structure of FDEs and gaining deeper insights into the interplay between local and non-local dynamics in fractional systems.展开更多
In this paper,we prove the existence of martingale solutions of a class of stochastic equations with a monotone drift of polynomial growth of arbitrary order and a continuous diffusion term with superlinear growth.Bot...In this paper,we prove the existence of martingale solutions of a class of stochastic equations with a monotone drift of polynomial growth of arbitrary order and a continuous diffusion term with superlinear growth.Both the nonlinear drift and diffusion terms are not required to be locally Lipschitz continuous.We then apply the abstract result to establish the existence of martingale solutions of the fractional stochastic reaction-diffusion equation with polynomial drift driven by a superlinear noise.The pseudo-monotonicity techniques and the Skorokhod-Jakubowski representation theorem in a topological space are used to pass to the limit of a sequence of approximate solutions defined by the Galerkin method.展开更多
In this paper,we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations{(-△)^(s)u+λu=|u|^(p-2)u-|u|^(q-2)u,x∈R^(N),∫_(R^(N))|u|^(2)dx=c>0,wher...In this paper,we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations{(-△)^(s)u+λu=|u|^(p-2)u-|u|^(q-2)u,x∈R^(N),∫_(R^(N))|u|^(2)dx=c>0,where N≥2,s∈(0,1),2+4s/N<p<q≤2_(s)^(*)=2N/N-2s,(-△)^(s)represents the fractional Laplacian operator of order s,and the frequencyλ∈R is unknown and appears as a Lagrange multiplier.Specifically,we show that there exists a c>0 such that if c>c,then the problem(P)has at least two normalized solutions,including a normalized ground state solution and a mountain pass type solution.We mainly extend the results in[Commun Pure Appl Anal,2022,21:4113–4145],which dealt with the problem(P)for the case 2<p<q<2+4s/N.展开更多
In this paper we study the Freidlin-Wentzell's large deviation principle for the following nonlinear fractional stochastic heat equation driven by Gaussian noise∂/∂tu^(ε)=D_(δ)^(α)(t,x)+√εσ(u^(ε)(t,x))W(t,x...In this paper we study the Freidlin-Wentzell's large deviation principle for the following nonlinear fractional stochastic heat equation driven by Gaussian noise∂/∂tu^(ε)=D_(δ)^(α)(t,x)+√εσ(u^(ε)(t,x))W(t,x),(t,x)∈[0,T]×R,where D_(δ)^(α)is a nonlocal fractional differential operator and W is the Gaussian noise which is white in time and behaves as a fractional Brownian motion with Hurst index H satisfying 3-α/4<H<1/2,in the space variable.The weak convergence approach plays an important role.展开更多
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This me...In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.展开更多
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
Fractional differential equations have garnered significant attention within the mathematical and physical sciences due to the diverse range of fractional operators available.Fractional calculus has demonstrated its u...Fractional differential equations have garnered significant attention within the mathematical and physical sciences due to the diverse range of fractional operators available.Fractional calculus has demonstrated its utility across various disciplines,including biological modeling[1–5],applications in physics[6,7],most notably in the formulation of fractional diffusion equations,in robotics,and emerging areas such as intelligent artificial systems,among others.Numerous types of fractional operators exist,including those characterized by singular kernels,such as the Caputo and Riemann-Liouville derivatives[8,9].It is important to highlight that the Riemann-Liouville derivative exhibits certain limitations;most notably,the derivative of a constant is not zero,which poses a significant inconvenience.To circumvent this issue,the Caputo derivative was introduced.Additionally,there are fractional derivatives with non-singular kernels,such as the Caputo-Fabrizio derivative[10]and the Atangana-Baleanu fractional derivative[11],each providing unique advantages for modeling purposes.Given the growing interest in utilizing fractional operators for various modeling scenarios,it is imperative to propose robust methodologies for obtaining both approximate and exact solutions.Consequently,this special issue emphasizes the exploration of diverse numerical schemes aimed at deriving approximate solutions for the models under consideration.Furthermore,analytical methods have also been discussed,providing additional avenues for obtaining exact solutions.展开更多
Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integ...Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.展开更多
In this paper,the Lie symmetry analysis method is applied to the(2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation.We obtain all the Lie symmetries admitted by the governing equation and re...In this paper,the Lie symmetry analysis method is applied to the(2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation.We obtain all the Lie symmetries admitted by the governing equation and reduce the corresponding(2+1)-dimensional fractional partial differential equations with the Riemann–Liouville fractional derivative to(1+1)-dimensional counterparts with the Erdélyi–Kober fractional derivative.Then,we obtain the power series solutions of the reduced equations,prove their convergence and analyze their dynamic behavior graphically.In addition,the conservation laws for all the obtained Lie symmetries are constructed using the new conservation theorem and the generalization of Noether operators.展开更多
The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solu...The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solution to the Boussinesq equations provided the real parameter α satisfies α≥1/2 ＋n/4.展开更多
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
In this paper,we investigate the integrable fractional coupled Gerdjikov-Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions.Based on the Riemann-Hilbert metho...In this paper,we investigate the integrable fractional coupled Gerdjikov-Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions.Based on the Riemann-Hilbert method,we construct the fractional N-soliton solutions.We find that as the powerεof the Riesz fractional derivative increases,the amplitudes of the fractional soliton solutions remain invariant,while their widths decrease and the absolute values of the wave velocity,group velocity,and phase velocity increase.Additionally,we examine the long-time asymptotic behavior of the fractional N-soliton solution.The results show that as t→±∞,the solution can be approximated by the sum of N fractional one-soliton solutions,with each soliton's amplitude and velocity remaining constant,whereas both position and phase shifts are observed.展开更多
We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorou...We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.展开更多
In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified...In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney （mBBM） equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions： amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense.展开更多
We study the Cauchy problem of the fractional Kramers-Fokker-Planck equation with moderate soft potential and show that the solution to the Cauchy problem enjoys an analytic Gelfand-Shilov regularizing effect for posi...We study the Cauchy problem of the fractional Kramers-Fokker-Planck equation with moderate soft potential and show that the solution to the Cauchy problem enjoys an analytic Gelfand-Shilov regularizing effect for positive time.展开更多
In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is prop...In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie＇s modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.展开更多
The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is nece...The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s equations.The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics.Here,the fractional Pfaff-Birkhoff variational problem is presented and studied.The definitions of fractional derivatives,the formulae for integration by parts and some other preliminaries are firstly given.Secondly,the fractional Pfaff-Birkhoff principle and the fractional Birkhoff′s equations in terms of RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.展开更多
This paper focuses on applying the barycentric Lagrange interpolation collocation method(BLICM)for solving 2D time-fractional diffusion-wave equation(TFDWE).In order to obtain the discrete format of the equation,we co...This paper focuses on applying the barycentric Lagrange interpolation collocation method(BLICM)for solving 2D time-fractional diffusion-wave equation(TFDWE).In order to obtain the discrete format of the equation,we construct the multivariate barycentric Lagrange interpolation approximation function and process the integral terms by using the Gauss-Legendre quadrature formula.We provide a detailed error analysis of the discrete format on the second kind of Chebyshev nodes.The efficacy of the proposed method is substantiated by some numerical experiments.The results of these experiments demonstrate that our method can obtain high-precision numerical solutions for fractional partial differential equations.Additionally,the method's capability to achieve high precision with a reduced number of nodes is confirmed.展开更多
文摘In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.
基金Partially supported by Postgraduate Research and Practice Innovation Program of Jiangsu Province(Nos.KYCX22-2211,KYCX22-2205)。
文摘In this paper,we prove the transportation cost-information inequalities on the space of continuous paths with respect to the L~2-metric and the uniform metric for the law of the mild solution to the stochastic heat equation defined on[0,T]×[0,1]driven by double-parameter fractional noise.
基金Research Council of Lithuania(LMTLT),agreement No.S-PD-24-120Research Council of Lithuania(LMTLT),agreement No.S-PD-24-120funded by the Research Council of Lithuania.
文摘Fractional differential equations(FDEs)provide a powerful tool for modeling systems with memory and non-local effects,but understanding their underlying structure remains a significant challenge.While numerous numerical and semi-analytical methods exist to find solutions,new approaches are needed to analyze the intrinsic properties of the FDEs themselves.This paper introduces a novel computational framework for the structural analysis of FDEs involving iterated Caputo derivatives.The methodology is based on a transformation that recasts the original FDE into an equivalent higher-order form,represented as the sum of a closed-form,integer-order component G(y)and a residual fractional power seriesΨ(x).This transformed FDE is subsequently reduced to a first-order ordinary differential equation(ODE).The primary novelty of the proposed methodology lies in treating the structure of the integer-order component G(y)not as fixed,but as a parameterizable polynomial whose coefficients can be determined via global optimization.Using particle swarm optimization,the framework identifies an optimal ODE architecture by minimizing a dual objective that balances solution accuracy against a high-fidelity reference and the magnitude of the truncated residual series.The effectiveness of the approach is demonstrated on both a linear FDE and a nonlinear fractional Riccati equation.Results demonstrate that the framework successfully identifies an optimal,low-degree polynomial ODE architecture that is not necessarily identical to the forcing function of the original FDE.This work provides a new tool for analyzing the underlying structure of FDEs and gaining deeper insights into the interplay between local and non-local dynamics in fractional systems.
文摘In this paper,we prove the existence of martingale solutions of a class of stochastic equations with a monotone drift of polynomial growth of arbitrary order and a continuous diffusion term with superlinear growth.Both the nonlinear drift and diffusion terms are not required to be locally Lipschitz continuous.We then apply the abstract result to establish the existence of martingale solutions of the fractional stochastic reaction-diffusion equation with polynomial drift driven by a superlinear noise.The pseudo-monotonicity techniques and the Skorokhod-Jakubowski representation theorem in a topological space are used to pass to the limit of a sequence of approximate solutions defined by the Galerkin method.
基金supported by the NNSF of China(12471103)the Natural Science Foundation of Guangdong Province(2024A1515012370)the Guangzhou Basic and Applied Basic Research(2023A04J1316)。
文摘In this paper,we investigate the existence and multiplicity of normalized solutions for the following fractional Schrödinger equations{(-△)^(s)u+λu=|u|^(p-2)u-|u|^(q-2)u,x∈R^(N),∫_(R^(N))|u|^(2)dx=c>0,where N≥2,s∈(0,1),2+4s/N<p<q≤2_(s)^(*)=2N/N-2s,(-△)^(s)represents the fractional Laplacian operator of order s,and the frequencyλ∈R is unknown and appears as a Lagrange multiplier.Specifically,we show that there exists a c>0 such that if c>c,then the problem(P)has at least two normalized solutions,including a normalized ground state solution and a mountain pass type solution.We mainly extend the results in[Commun Pure Appl Anal,2022,21:4113–4145],which dealt with the problem(P)for the case 2<p<q<2+4s/N.
基金Partially supported by NSFC(No.11701304)the K.C.Wong Education Foundation。
文摘In this paper we study the Freidlin-Wentzell's large deviation principle for the following nonlinear fractional stochastic heat equation driven by Gaussian noise∂/∂tu^(ε)=D_(δ)^(α)(t,x)+√εσ(u^(ε)(t,x))W(t,x),(t,x)∈[0,T]×R,where D_(δ)^(α)is a nonlocal fractional differential operator and W is the Gaussian noise which is white in time and behaves as a fractional Brownian motion with Hurst index H satisfying 3-α/4<H<1/2,in the space variable.The weak convergence approach plays an important role.
基金Supported by Natural Science Foundation of Shandong Province of China under Grant No.ZR2013AQ009National Training Programs of Innovation and Entrepreneurship for Undergraduates under Grant No.201310433031Doctoral initializing Foundation of Shandong University of Technology of China under Grant No.4041-413030
文摘In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann–Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method,we apply this method to solve the space-time fractional Whitham–Broer–Kaup(WBK) equations and the nonlinear fractional Sharma–Tasso–Olever(STO) equation, and as a result, some new exact solutions for them are obtained.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10972151)
文摘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.
文摘Fractional differential equations have garnered significant attention within the mathematical and physical sciences due to the diverse range of fractional operators available.Fractional calculus has demonstrated its utility across various disciplines,including biological modeling[1–5],applications in physics[6,7],most notably in the formulation of fractional diffusion equations,in robotics,and emerging areas such as intelligent artificial systems,among others.Numerous types of fractional operators exist,including those characterized by singular kernels,such as the Caputo and Riemann-Liouville derivatives[8,9].It is important to highlight that the Riemann-Liouville derivative exhibits certain limitations;most notably,the derivative of a constant is not zero,which poses a significant inconvenience.To circumvent this issue,the Caputo derivative was introduced.Additionally,there are fractional derivatives with non-singular kernels,such as the Caputo-Fabrizio derivative[10]and the Atangana-Baleanu fractional derivative[11],each providing unique advantages for modeling purposes.Given the growing interest in utilizing fractional operators for various modeling scenarios,it is imperative to propose robust methodologies for obtaining both approximate and exact solutions.Consequently,this special issue emphasizes the exploration of diverse numerical schemes aimed at deriving approximate solutions for the models under consideration.Furthermore,analytical methods have also been discussed,providing additional avenues for obtaining exact solutions.
文摘Exact solutions to conformable time fractional (3+1)-dimensional equations are derived by using the modified form of the Kudryashov method. The compatible wave transformation reduces the equations to an ODE with integer orders. The predicted solution of the finite series of a rational exponential function is substituted into this ODE.The resultant polynomial equation is solved by using algebraic operations. The method works for the Jimbo–Miwa, the Zakharov–Kuznetsov, and the modified Zakharov–Kuznetsov equations in conformable time fractional forms. All the solutions are expressed in explicit forms.
基金supported by the State Key Program of the National Natural Science Foundation of China(72031009).
文摘In this paper,the Lie symmetry analysis method is applied to the(2+1)-dimensional time-fractional Heisenberg ferromagnetic spin chain equation.We obtain all the Lie symmetries admitted by the governing equation and reduce the corresponding(2+1)-dimensional fractional partial differential equations with the Riemann–Liouville fractional derivative to(1+1)-dimensional counterparts with the Erdélyi–Kober fractional derivative.Then,we obtain the power series solutions of the reduced equations,prove their convergence and analyze their dynamic behavior graphically.In addition,the conservation laws for all the obtained Lie symmetries are constructed using the new conservation theorem and the generalization of Noether operators.
基金partially supported by NSFC(1117102611371059)+1 种基金BNSF(2112023)the Fundamental Research Funds for the Central Universities of China
文摘The goal of this paper is to consider the global well-posedness to n-dimensional （n 〉 3） Boussinesq equations with fractional dissipation. More precisely, it is proved that there exists a unique global regular solution to the Boussinesq equations provided the real parameter α satisfies α≥1/2 ＋n/4.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11072218 and 10672143)
文摘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.
基金funded by the National Natural Science Foundation of China(Grant Nos.12305003,12101246,12471237)。
文摘In this paper,we investigate the integrable fractional coupled Gerdjikov-Ivanov equation and derive its explicit form by employing the completeness relation of squared eigenfunctions.Based on the Riemann-Hilbert method,we construct the fractional N-soliton solutions.We find that as the powerεof the Riesz fractional derivative increases,the amplitudes of the fractional soliton solutions remain invariant,while their widths decrease and the absolute values of the wave velocity,group velocity,and phase velocity increase.Additionally,we examine the long-time asymptotic behavior of the fractional N-soliton solution.The results show that as t→±∞,the solution can be approximated by the sum of N fractional one-soliton solutions,with each soliton's amplitude and velocity remaining constant,whereas both position and phase shifts are observed.
基金supported by NSFC Project(11301446,11271145)China Postdoctoral Science Foundation Grant(2013M531789)+3 种基金Specialized Research Fund for the Doctoral Program of Higher Education(2011440711009)Program for Changjiang Scholars and Innovative Research Team in University(IRT1179)Project of Scientific Research Fund of Hunan Provincial Science and Technology Department(2013RS4057)the Research Foundation of Hunan Provincial Education Department(13B116)
文摘We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.
文摘In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney （mBBM） equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions： amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense.
基金Supported by the National Natural Science Foundation of China (Grant No.12031006)the Fundamental Research Funds for the Central Universities of China。
文摘We study the Cauchy problem of the fractional Kramers-Fokker-Planck equation with moderate soft potential and show that the solution to the Cauchy problem enjoys an analytic Gelfand-Shilov regularizing effect for positive time.
文摘In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie＇s modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.
基金Supported by the National Natural Science Foundation of China(10972151,11272227)the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province(CXZZ11_0949)the Innovation Program for Postgraduate of Suzhou University of Science and Technology(SKCX11S_050)
文摘The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s equations.The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics.Here,the fractional Pfaff-Birkhoff variational problem is presented and studied.The definitions of fractional derivatives,the formulae for integration by parts and some other preliminaries are firstly given.Secondly,the fractional Pfaff-Birkhoff principle and the fractional Birkhoff′s equations in terms of RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.
基金Supported by the Scientific Research Foundation for Talents Introduced of Guizhou University of Finance and Economics(Grant No.2023YJ16)the Institute of Complexity Science,Henan University of Technology(Grant No.CSKFJJ-2025-33)the International Science and Technology Cooperation Project of Henan Province(Grant No.252102520007).
文摘This paper focuses on applying the barycentric Lagrange interpolation collocation method(BLICM)for solving 2D time-fractional diffusion-wave equation(TFDWE).In order to obtain the discrete format of the equation,we construct the multivariate barycentric Lagrange interpolation approximation function and process the integral terms by using the Gauss-Legendre quadrature formula.We provide a detailed error analysis of the discrete format on the second kind of Chebyshev nodes.The efficacy of the proposed method is substantiated by some numerical experiments.The results of these experiments demonstrate that our method can obtain high-precision numerical solutions for fractional partial differential equations.Additionally,the method's capability to achieve high precision with a reduced number of nodes is confirmed.
