论文研究 变系数时间分数阶子扩散方程的数值解.pdf

对于变系数的时间分数阶子扩散方程，提出了一种数值方法，该方法在时间方向使用由Lagrange插值函数所得的递推公式，在空间方向，利用二次样条插值函数做为基函数，构成了最优紧二次样条配置法。理论分析和数值例子证明了该方法在配置点处具有超收敛性。罗卫华,吴国成:变系数时间分数阶子扩散方程的数值解2017,53(4)为如下形式的线性方程组01A+o, B)u=6(15)IIT-<[1+>o2B I llo AlrKo2Br'l<其中1+2(o2 B)'IlloyAllhe1-|02B)li1A‖+124k(tn+2=-k(nn+1可知对于足够小的z,h,H如是有界的。=(u2,n3…,aM+1),b=(b1,b2,…,b)其次,在每个时间层j=2,3,…,N+1),用式且M阶矩阵A,B分别为(12)减去式(13)可得o1△7,)+2D2△n(7,)=Oh4+x22)利用H的有界性,可得这里结合引理1中式(6)的后两个等式,引理2中 Caputo导数算子的误差式(8),使用三角不等式,立即可得式=04)-24)8()-(4+n2(n)(16),定理证毕。b=038(7-1)g(n+1),i=2,3,…,M-14数值例子b+1=a8(m2-1)-24Dg1x+)-(4+n12-)在本章,给出数值例子,以验证上·章所提出数值而ag=1h方法的精度。为了分别验证在时间方向和空间方向的24(tn+1)精度,在下列所给表格中,表1列出了固定时间步长下,对于配置方程(13)和相应的线性方程组(15),可证各空间步长的精度,为了尽量避免时间方向带来的影明其解的存在唯一性以及所导致的误差响,这里取分数阶=0.01。而在表2,3中,列出了同定定理1设解函数a(x,)∈C4(a,b1×10,TD,则对步长下,各时间步长的精度,在这两个表中,为了调查分于足够小的正数bz,配置方程(13)存在唯一解数阶对精度的影响,分别取=0.L..0.9。在所有l、,),而由其引起的误差为:表格当中,都以无穷范数L。来计算误差,表格中E代,t)-a,川=O(h4+r2")t=1,2,…,M+2表在所有配置点=1,2,…,M+2处的误差,而收敛率1=12…N+1,lu-wl|=Oh2+x8(16)Rate的计算公式为:RateIn(E(N/2)E(N)/In(2)证明:记△a(m,4)=l2(m,4)-、m,4),以及I=a1A+2表1空间方向的误差(z=1100008=0.01)首先,对于配置方程(13)和线性方程组(15),当例1算例2kt)>0时,显然系数矩阵Ⅰ为个严格对角占优矩阵ERateRate从而是非奇异的,这意味着近似解a2(x,t唯一存在。5.8725E-65.2759E-6此外,由文献20可知矩阵B是可逆的,从而对于163.7759-73.95913.3931E-73.9587有界的正函数k)有2.3949E.97882.1529E-83.97831.5136E-93.98391.3561E-93.9887(a2B)1A+1)(a2B)而当步长τ,h足够小时,显然有算例1t(x,)=(cosx+e)k)=sint,而f(x,)=a1A<1,o2B)l<1T(4)t(cos.+e)-t(e+cos risin t所以可以利用矩阵的 Neumann级数,可得IT=I-2(1(02)]o2 B)算例2a(x,)=ax,k)=cost,而f(x,)=12.71t.r cost从而有表2算例1中时间方向的误差(h=1/64)B=0.1B=0.5B=0.9ERateEE8.8403E-74.2244E-59.9981E-4l0002.3827F-71.89151.5022F-5149174.6672F-409910006.4003E-81.89645.3341E-61.49382.1782E-41.099440001.7209E-81.895018935E-61.4942l.0168E1.0991782017,53(4)Computer Engineering and4 pplications计算机工程与应用表3算例2中时间方向的误差(=LB=0.1=0.5B=0.9ERateEE1477E-72.3010E-63.2517E-51.9192H-71.89708.1355F-71.5001.53l2E-5l.0)655.1113E-82.8820E-71.49727.4119E-61.046740001.3627E-81.90721.0253E-7149103.7737E-60.9739从表1,2,3的这些实验结果中,可以看到,该数值权显式有限差分方法[四川师范大学学报:自然科学版,方法在x,)∈Cx2(ab×0,T]的条件下,在时间方向2016,39(1):76-8[9 Gao G H, Sun 7. 7.A compact finite difference scheme和空间方向分别达到了数值精度O(h和O(x),特别for the fractional sub-diffusion equations.J Comput Ph地,在空间方向达到了超收敛性。这和上节的理论分2011,230:586-595析是相吻合的[10] JiC C, Sun ZZ A high-order compact finite differencescheme for the fractional sub-diffusion equation[J]. J Sci5纬束语Comput,201564(3):959-985本文基于二次样条插值函数,对变系数的时间分数「1 Jin B, Lazarov r, Zhou Z Error estimates for a semidis阶子扩散方程提出了一种高效的数值方法,理论分析和crete finite element method for fractional order parabolic数值实验都证明了该方法在一定的光滑条件下,在配置equations[J]. SIAM J Numer Anal, 2013, 51(1): 445-466点处可以达到数值精度Oh4+x2-)。此外相比于纯1212ngF,LC,LiuF, et al.The use of finite difference/粹的有限差分方法,本文所提的方法还可以计算空问方element approaches for solving the time- fractional subdiffusion equation[J].SIAM J Sci Comput, 2013, 3.5(6)向的任何一点处的数值解。2976-3000.13 Zeng F, Li C, Liu F et al. Numerical algorithms for time参势文献fractional subdiflusion equation with second-order accu[l Metzler R, Klafter J. The random walk's guide to anomaracy[J. SIAM J Sci Comput, 2015, 37: 55-78lous diflusion: A fractional dynamics approach[C]Phys [14] Azizi H, Loghmani G B Solution of time fractional dif-Rep,2000,339:1-77fusion equations using a semi-discrete scheme and col-[2] Ingo C, Magin R, Parrish T New insights into the fractionallocation method based on Chebyshev polynomials[J]order diffusion equation using entropy and kurtosis[J]Iran J Sci Tech, 2013, 37(1): 23-28Entropy,2014,l6:5838-5852L15 Chen F, Xu Q, Hesthaven J SA multi-domain spectral3 Wu G C, Baleanu D, Zeng S D, et al. Discrete fractionalmethod for time-fractional differential equations[J].J CoImdiffusion equation[J] Nonlinear Dynamics, 2015, 80(172)put Phys,2013,293:157-172281-28616]梂世敏,许传炬分数阶微分方程的理论和数值方法研[4] Das S, Kumar R Fractional diffusion equations in the pre究[计算数学,2016,38(1):1-24ence of reaction terms[J]. Comput J Complex Appl, 2015, [171 Christara CC Quadratic spline collocation methods for1(1):15-21elliptic partial differential equations[J]. BIT, 1994, 34: 33-615] Zheng M, Liu F, Turner I, et al. A novel high order space- [18] Houstis E N, Christard CC, Rice J R. Quadralic-splinetime spectral method for the time fractional fokkercollocation methods for two-point boundary value probplanck equation[J].SIAM J Sci Comput, 2015,37: 701-724lems[J]. J Numer Methods Eng, 1988, 26: 935-952[6] Liu F, Zhuang P, Turner I, et al. A new fractional finite [191 Gao G H, Sun Z Z, Zhang H WA new fractional numericalvolume method for solving the fractional diffusion eyuadifferentiation formula to approximate the Caputo frac-tion[j]. Appl Math Model, 2014, 38: 3871-3878tional derivative and its applications[J].J Comput Phys[7 Yuste S B, Acedo L, Lindenberg KReaction front in an2014,259(2):33-50A+B-C reaction-subdiffusion process[J]. Phys Rev E, [20] Luo W H, Huang T Z,Wu G C, et al. Quadratic spline2004,69:1-10collocation method for the time fractional subdiffusion[8]马亮亮,刘冬兵.两边空间分数阶对流扩散方程的一种加equation[J]. Appl Math Comput, 2016, 276: 252-265