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yuanjian1987木蟲 (正式寫手)
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[求助]
分?jǐn)?shù)階微分方程Predictor-corrector PECE，程序怎么運行呢？
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附件是老外編寫的程序，但是我不會運行，誰能舉個例子運行一下呢？ 比如 D^q1=x^2+xy; D^q2=-x^2-2y; fdefun需要單獨編一個m文件嗎？ 里面有好多function，需要將它們單獨編一個m文件嗎？ Description of FDE12 FDE12 solves an initial value problem for a nonlinear differential equation of fractional order (FDE). The code implements the predictor-corrector PECE method of Adams-Bashforth-Moulton type described in [1]. [T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,h) integrates the initial value problem for the FDE, or the system of FDEs, of order ALPHA > 0 D^ALPHA Y(t) = FDEFUN(T,Y(T)) Y^(k)(T0) = Y0(:,k+1), k=0,...,m-1 where m is the smallest integer greater than ALPHA and D^ALPHA is the fractional derivative according to the Caputo's definition. FDEFUN is a function handle corresponding to the vector field of the FDE and for a scalar T and a vector Y, FDEFUN(T,Y) must return a column vector. The set of initial conditions Y0 is a matrix with a number of rows equal to the size of the problem (hence equal to the number of rows of the output of FDEFUN) and a number of columns depending on ALPHA and given by m. The step-size H>0 is assumed constant throughout the integration. [T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,H,PARAM) solves as above with the additional set of parameters for the FDEFUN as FDEFUN(T,Y,PARAM). [T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,H,PARAM,MU) solves the FDE with the selected number MU of multiple corrector iterations. The following values for MU are admissible: MU = 0 : the corrector is not evaluated and the solution is provided just by the predictor method (the first order rectangular rule); MU > 0 : the corrector is evaluated by the selected number MU of times; the classical PECE method is obtained for MU=1; MU = Inf : the corrector is evaluated for a certain number of times until convergence of the iterations is reached (for convergence the difference between two consecutive iterates is tested). The defalut value for MU is 1 [T,Y] = FDE12(ALPHA,FDEFUN,T0,TFINAL,Y0,H,PARAM,MU,MU_TOL) allows to specify the tolerance for testing convergence when MU = Inf. If not specified, the default value MU_TOL = 1.E-6 is used. FDE12 is an implementation of the predictor-corrector method of Adams-Bashforth -Moulton studied in [1]. Convergence and accuracy of the method are studied in [2]. The implementation with multiple corrector iterations has been proposed and discussed for multiterm FDEs in [3]. In this implementation the discrete convolutions are evaluated by means of the FFT algorithm described in [4] allowing to keep the computational cost proportional to N*log(N)^2 instead of N^2 as in the classical implementation; N is the number of time-point in which the solution is evaluated, i.e. N = (TFINAL-T)/H. The stability properties of the method implemented by FDE12 have been studied in [5]. [1] K. Diethelm, A.D. Freed, The Frac PECE subroutine for the numerical solution of differential equations of fractional order, in: S. Heinzel, T. Plesser (Eds.), Forschung und Wissenschaftliches Rechnen 1998, Gessellschaft fur Wissenschaftliche Datenverarbeitung, Gottingen, 1999, pp. 57-71. [2] K. Diethelm, N.J. Ford, A.D. Freed, Detailed error analysis for a fractional Adams method, Numer. Algorithms 36 (1) (2004) 31-52. [3] K. Diethelm, Efficient solution of multi-term fractional differential equations using P(EC)mE methods, Computing 71 (2003), pp. 305-319. [4] E. Hairer, C. Lubich, M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations, SIAM J. Sci. Statist. Comput. 6 (3) (1985) 532-541. [5] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations, Internat. J. Comput. Math. 87 (10) (2010) 2281-2290. Copyright (c) 2011-2012, Roberto Garrappa, University of Bari, Italy garrappa at dm dot uniba dot it Revision: 1.2 - Date: July, 6 2012 |
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2013-11-24 20:58:59, 9.11 K
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3樓2013-11-25 05:58:36
feixiaolin
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alpha：FDE的階次，必須為正. fdefun 是標(biāo)量T和向量Y定義的指向矢量場FDE的函數(shù)函數(shù)句柄，F(xiàn)DEFUN(T,Y) 必須返回一個列向量。. t0,tfinal: 參數(shù)t的初始值、終值. y0：一個矩陣，其行等于該問題的大�。ㄒ驳扔谛蠪DEFUN的輸出數(shù)） param：參數(shù)個數(shù). mu：校正迭代參數(shù)選擇參數(shù). MU = 0 : the corrector is not evaluated and the solution is provided just by the predictor method (the first order rectangular rule); MU > 0 : the corrector is evaluated by the selected number MU of times; the classical PECE method is obtained for MU=1; MU = Inf : the corrector is evaluated for a certain number of times until convergence of the iterations is reached (for convergence the difference between two consecutive iterates is tested). The defalut value for MU is 1 mu_tol：控制誤差. h:從N = ceil((tfinal-t0)/h) 看，h是時間跨距，N是評估解的時間點的數(shù)目。 FDE12詳見 http://www.mathworks.cn/matlabce ... ferential-equations |
2樓2013-11-24 21:44:53
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4樓2013-11-25 06:06:27
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5樓2013-12-04 10:26:46
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