求解分数阶延迟微分方程的卷积Runge-Kutta方法
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摘要
本文利用强A-稳定Runge-Kutta方法求解一类非线性分数阶延迟微分方程初值问题,并给出了算法的稳定性和误差分析.数值算例验证算法的有效性及其相关理论结果.
In this paper, a strongly A-stable Runge-Kutta method is constructed to solve a class of nonlinear fractional differential equation with delay and Caputo fractional derivative. Stability and error analysis of the numerical algorithm are given. Numerical experiments demonstrate the validity of the proposed numerical algorithm and related theoretical results.
In this paper, a strongly A-stable Runge-Kutta method is constructed to solve a class of nonlinear fractional differential equation with delay and Caputo fractional derivative. Stability and error analysis of the numerical algorithm are given. Numerical experiments demonstrate the validity of the proposed numerical algorithm and related theoretical results.
引文
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[2]DENG J,QU H.New uniqueness results of solutions for fractional differential equations with infinite delay[J].Comput.Math.Appl.,2010,60(8):2253-2259.
[3]MOGHADDAM B P,MOSTAGHIM Z S.A novel matrix approach to fractional finite difference for solving models based on nonlinear fractional delay differential equations[J].Ain.Shams Eng.J.,2014,5(2):585-594.
[4]MORGADO M L,FORD N J,LIMA P M.Analysis and numerical methods for fractional differential equations with delay[J].J.Comput.App.Math.,2013,252(252):159-168.
[5]BHALEKAR S,DAFTARDAR-GEJJI V.A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order[J].J.Fract.Calc.Appl.,2011,1(5):1-9.
[6]ZHEN Wang.A numerical method for delayed fractional-order differential equations[J].J.Appl.Math.,2013,2013(2):707-724.
[7]DAFTARDARGEJJI V,SUKALE Y,BHALEKAR S.Solving fractional delay differential equations:a new approach[J].Fract.Calc.Appl.Anal.,2015,18(2):400-418.
[8]LUBICH C.Discretized fractional calculus[J].SIAM J.Math.Anal.,1986,17(3):704-719.
[9]LUBICH C,OSTERMANN A.Runge-Kutta methods for parabolic equations and convolution quadrature[J].Math.Comput.,1993,60(201):105-131.
[10]BANJAI L,LUBICH C.An error analysis of Runge-Kutta convolution quadrature[J].BIT.Numer.Math.,2011,51(3):483-496.
[11]DEHGHAN M,YOUSEFI S A,LOTFI A.The use of He’s variational iteration method for solving the telegraph and fractional telegraph equations[J].Int.J.Numer.Meth.Biomed.,2011,27(2):219-231.
[12]曹学年,王海燕.非线性分数阶微分方程的Radau IIA方法[J].系统仿真学报,2011,23(10):2075-2078.
[13]LUBICH C.Convolution quadrature and discretized operational calculus.I[J].Numer.Math.,1988,52(2):129-145.
[14]LUBICH C.Convolution quadrature and discretized operational calculus.II[J].Numer.Math.,1988,52(4):413-425.
[15]PODLUBNY I.Fractional Differential Equations[M].New York:Academic Press,1999.
[16]杨水平,肖爱国.分数阶延迟微分方程的样条配置方法[J].数学的实践与认识,2014,44(6):247-254.
[17]潘新元.两类分数延迟微分方程及其数值方法的渐近稳定性[D].湖南:湘潭大学硕士学位论文,2009.
[18]邱宁.时间分数阶延迟微分方程在流体力学中的应用[J].沈阳大学学报(自然科学版),2016,28(2):170-172.
[19]李艳.分数阶非线性时滞系统的迭代学习控制及稳定性[D].安徽:安徽大学硕士学位论文,2015.
[20]TORVIK P J,BAGLEY R L.On the appearance of the fractional derivative in the behavior of real materials[J].J.Appl.Mech.,1984,51(2):294-298.
[21]SAEEDI H,MOGHADAM M M,MOLLAHASANI N,CHUEV G N.A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order[J].Commun.Nonlinear.Sci.Numer.Simul.,2011,16(3):1154-1163.
[22]XU D.Numerical solution of evolutionary integral equations with completely monotonic kernel by Runge-Kutta convolution quadrature[J].Numer.Meth.Part.Differ.Equ.,2015,31(1):105-142.