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1、Runge-kutta2010527Several Kinds of Runge-Kutta Methods for FractionalDierential EquationsCandidateSupervisorCollegeProgramSpecialityDegreeUniversityDateYunfei LiProfessor Xuenian CaoMathematics and Computational ScienceComputational MathematicsNumerical Solution of Fractional Dierential EquationsMas
2、ter of ScienceXiangtan UniversityApril 18th, 2010,L-Runge-KuttaRiemann-Liouville, RadauIA,L-.Runge-Kutta.Runge-KuttaIL-Runge-KuttaLobattaIIICAbstractBased on a high order approximation of L-stable Runge-Kutta methodsfor Riemann-Liouville fractional dirivatives,a class of high-order L-stable Runge-Ku
3、tta methods for solving the nonlinear fractional dierential equations is con-structed in this paper .Consistency, convergence, and stability analysis of thesemethods are given. In numerical experiments,fractional Radau IA methods andLobatto IIIC methods and singly-diagonal implicit Runge-Kutta metho
4、ds com-bining the short memory principle are chosen.These methods are ecient forsolving nonlinear fractional dierential equations.Keywords: Fractional dierential equation; L-stable Runge-Kutta method;consistency; convergence; stability;short memory principleII.1L-Runge-Kutta. 22.12.22.32.4. .4. 4. .
5、6. 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6、. 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25III.,1Abel(GorenoVessela, 1991)2,(RossikhinShitikova, 1997)3,(Benson, WheatcraftMeerschert,2000a, b)4,.(Wyss, 2000)5,.,.,.K.Diethelm,N.J.FordA.D. Freed6-.,7,910.Lubich8Runge-KuttaAb
7、el-Volterra.Lubich,1,11RungeKutta,13.RadauIIA.,12,.,.14Riemann-Liouville., FordSimpson15.,L-L-Runge-KuttaRunge-Kutta.,(,L-L-RK),Runge-Kutta,LobattoIIICRadauIA,Runge-Kutta.1Runge-Kutta:C yi(0) = y0i,f(t, y(t), 0 l 1 0.m ,9Runge-Kuttamj=1aijckj 1 =ckikq, k = 1, , q.2.2.19K(s),q,pL-Runge-Kutta,h h0,(2.
8、1.1)Riemann-Liouvillen, h0Tv=0,WnvYv 0Dty(t) = 1, t = (n + 1)h T,.,2.2.2 (Bellman), 0, h 0, 0, 1 , Nm + hm1j=0j, m = 0, 1, , N,m emh, m = 0, 1, , N.40Dt y(t)1 b A h h T 1 b A hk1 = O(hp) + O(hq+1+| log h|), 2 = O(hq+1), Un| arg(sc) | , c R, | K(s) | M|s|,i = 1, 2, , m,1b A hWn2.2.392.2.1,h h0(h0),Wn
9、 C(nh)1enh, n 1,n =0n =1,C, h0,.L-2.2.49 K(s)Runge-Kutta,m 2,(2.0.4)q,p q +1Fni| Fni F (tn + cih, y(tn + cih) |= 2, i = 1, 2, , m.2.2.1(2.0.3),q,pL-Runge-Kuttatn+1p q+1+Yvj ,Yvj = y(tv + cjh),(i).tv + cmh = tv+1,(2.1.3)Tnv=0WnvYv = f(tn+1, y(tn+1) + 3 +l1k=0tkn+1y(k)(0)(k + 1),3,2.2.1,(2.0.3)l1k=0tk
10、n+1y(k)(0)(k + 1),3 = 1.(ii).tv + cmh = tv+1,(2.1.3)Tn1v=0Tm1j=1qjw0,jy(tn + cjh) + qmhw0,myn+1= f(tn+1, yn+1) +l1k=0tkn+1y(k)(0)(k + 1), w0,j (j = 1, 2, , m), qj(j = 1, 2, , m)mbT A1W0 = h1(A1) mm . 2.2.1m+ 1 + qmhw0,m(yn+1 y(tn+1) = f(tn+1, yn+1) +l1k=0tkn+1y(k)(0)(k + 1),(2.0.3)qmhw0,m(yn+1 y(tn+
11、1) = f(tn+1, yn+1) f(tn+1, y(tn+1) 1.5(2.2.1)O(h ) + O(h| logh |).1b A h0Dt y(t) + 1 = f(tn+1, y(tn+1) + 3 +1b A h1WnvYv + b A h0Dtn+1y(t)qmhw0,m = hw0,m,h(2.2.1),w0,mw0,m.| yn+1 y(tn+1) |hL| w0,m | yn+1 y(tn+1) | +h| w0,m | 1 |,h0 ,0|w0,m| 1,0 h h0,| yn+1 y(tn+1) | = O(hp) + O(hq+1+ | log h |).2.3.
12、2.3.1 (2.0.3), q,L- Runge-Kuttapt = (n + 1)h, evj = y(tv + cjh) Yvj, j = 1, , m, Y (tv) =(y(tv + c1h), , y(tv + cmh)T , Ev = (ev1, ev2, , evm)T , v = 0, 1, , n,nhv=0WnvYv = F (Tn, Yn),(2.3.1)Tnv=0WnvYv = f(tn+1, yn+1) +l1k=0tkn+1y(k)(0)(k + 1),(2.3.2)2.2.4nhv=0WnvY (tv) = F (Tn, Y (tn) + Un,(2.3.3)2
13、.2.1,Tnv=0WnvY (tv) = f(tn+1, y(tn+1) +l1k=0tkn+1y(k)(0)(k + 1)+ 1,(2.3.4)(2.3.3)(2.3.1),nhv=0WnvEv = F (Tn, Y (tn) F (Tn, Yn) + Un,(2.3.5)6hLh0 | 1 | w0,m | Lh0p q +1(2.1.2) (2.1.3)O(h ) + O(hq+1+ | log h |) + O(hq+1).1b A h1b A h(2.3.4)(2.3.2),nbT A1hv=0WnvEv = f(tn+1, y(tn+1) f(tn+1, yn+1) + 1.(2
14、.3.6)(2.3.5)hW0En = F (Tn, Y (tn) F (Tn, Yn) + Un hn1v=0WnvEv,(2.3.7)W0 = h1(A1),(2.3.7)hAEn = hA(F (Tn, Y (tn) F (Tn, Yn) +1 v=0(2.3.8)F (Tn, Y (tn) F (Tn, Yn)=1im L 1max | y(tni) Yni |= L En ,(2.3.9)(2.3.8),(2.3.9),En+1n1v=0WnvEv+hAUn ,(2.3.10)h0,h0 LA 1,0 h h0EnhM(h A )1LhM0n1v=0n1v=0EvEv+ (h+ (h
15、1A )1L1A )1LUnUn ,M = max( Wv ), v = 1, 2, , n).C1 = (h A1 )1L,M1 = (h A M)1L,En C1Un+hM1n1v=0Ev ,2.2.2En eM1T C17Un ,(2.3.11)n1+h A Un hAWnvEv,max | f(tni, y(tni) f(tni, Yni) |iEn h LA+hA(h A )1L00 0f(tn+1, y(tn+1) f(tn+1, yn+1) = B(y(tn+1) yn+1),B,(2.3.6)nB(y(tn+1) yn+1) = bT A1hv=0WnvEv 1,y(tn+1)
16、 yn+1 = B1bT A1hnv=0WnvEv 1B1,(2.3.5), (2.3.9), (2.3.11)(C1LeM1T Un += O(hp) + O(hq+1+ | log h |) + O(hq+1).2.42.4.1(2.0.3),Runge-Kutta(2.1.2)-(2.1.3).y(k)(0), Yn = (Yn1, , Ynm)T ,Zn = (Zn1, , Znm)Tnz(k)(0)hv=0WnvYv = F (Tn, Yn),(2.4.1)Tnv=0WnvYv = f(tn+1, yn+1) +l1k=0tkn+1y(k)(0)(k + 1),(2.4.2)nhv=
17、0WnvZv = F (Tn, Zn),(2.4.3)Tnv=0WnvZv = f(tn+1, zn+1) +l1k=0tkn+1z(k)(0)(k + 1).(2.4.4)vj = Yvj Zvj, j = 1, , m, v = (v1, , vm)T , v = 0, , n, (k)(0) =y(k)(0) z(k)(0), k = 0, 1, , l 1,(2.4.1) (2.4.3) ,hnv=0Wnvv = F (Tn, Yn) F (Tn, Zn) +l1k=0Tnk(k)(0)(k + 1),(2.4.5)8| y(tn+1) yn+1 | B1bT A1Un )+ | B11 |yn+1zn+11b A h1b A h,F (Tn, Yn) = (f(tni, Yni) =1, F (Tn, Zn) = (f(tni, Zni)i=1,hW0n = F (Tn, Yn) F (Tn, Zn)n1 l1v=0 k=0,(2.4.6)F (Tn, Yn) F (Tn, Zn)= max1im L 1max | Yni Zni |= L n ,(2.4.7)k = 0, 1, , l 1,Tnk M2|1(k+1)()(l+1) M1,|(2.4.8)2.3,(2.4.6)hA,(2.4.7),(2.4.8)