董立華,劉艷芹
德州學(xué)院 數(shù)學(xué)科學(xué)學(xué)院,山東 德州 253023
分數(shù)階非線性方程近似解析解的新解法
董立華,劉艷芹
德州學(xué)院 數(shù)學(xué)科學(xué)學(xué)院,山東 德州 253023
分數(shù)階非線性方程已廣泛地應(yīng)用到黏彈性流體、擴散過程、生物力學(xué)、固態(tài)物理等很多領(lǐng)域[1-4]。伴隨著非線性科學(xué)的發(fā)展,涌現(xiàn)出了很多求解分數(shù)階非線性方程的解析和數(shù)值方法[5-7]。而分數(shù)階偏微分方程的解析解難以推導(dǎo),而現(xiàn)有的解析近似方法又有其自身的缺點[8-9],需要計算復(fù)雜的Adomian多項式和Lagrange乘子等,近年來很多學(xué)者對這些方法做了改進[10-11]。Wu[12-13]將Laplace變換和變分迭代法相結(jié)合,克服了分數(shù)階Lagrange乘子難以計算的困難。在前人研究的基礎(chǔ)上本文提出一種新的修正方法,將變分迭代法、同倫擾動法和Laplace變換相結(jié)合,并將該方法應(yīng)用于分數(shù)階非線性發(fā)展方程的求解,其中利用Laplace變換推導(dǎo)分數(shù)階的Lagrange乘子,而He的多項式則用來處理方程中出現(xiàn)的非線性項,該方法簡單有效。
考慮如下的時間分數(shù)階方程:
首先根據(jù)修正的變分迭代法[12-13],方程(1)和(2)兩邊作Laplace變換并得到方程的迭代格式為:考慮L[R[un(x,t)]+N[un(x,t)]]為變分項,可得到Lagrange乘子為 λ(s)=-1/sα,方程(3)兩邊再作逆Laplace變換 L-1得到:
u0(x,t)為方程(1)的初始迭代值,包含了初始值和源匯項的信息:
齊次方程非線性項的處理。根據(jù)同倫擾動法假設(shè)方程的解可以表示為p的冪級數(shù):
考察如下的求解時間分數(shù)階耦合的MKdV方程[15]。
γ是任意常數(shù),當α=1時精確解為:
這里 t>0,0<α≤1,初始條件為:方程(5)~(6)兩邊作Laplace變換得到如下迭代格式:
u0,v0是式(5)和(6)的初始迭代值,取 u0=u(x,0),v0= v(x,0),應(yīng)用上述修正的新方法得到:
圖1 方程(5)~(6)的精確解 u(x,t)當 λ=1,γ=0.1
圖3 方程(5)~(6)的二階近似解 u0+u1當 α=λ=1,γ=0.1
圖4 方程(5)~(6)的二階近似解 v0+v1當 α=λ=1,γ=0.1
圖 1和圖 2分別表示方程(5)~(6)精確解 u(x,t),v(x,t)的圖像。圖3和圖4分別表示方程(5)~(6)二級近似解的圖像當α=1。從圖像可以看出這種新方法具有較高的精確度,而且不需要復(fù)雜的計算。
這種新的方法結(jié)合了變分迭代法、同倫分析法和Laplace變換三種處理方法的優(yōu)點,便于簡單計算Lagrange乘子和方便處理非線性項。實例證明該方法可以用于其他分數(shù)階的非線性方程。
[1]Podlubny I.Fractional differential equations:an introduction to fractional derivatives,fractional differential equations,to methods of their solution and some of their applications[M].New York:Academic Press,1999:69-265.
[2]Ma J H,Liu Y Q.Exact solution for a generalized nonlinear fractional Fokker-Planck equation[J].Nonlinear Analysis:Real World Applications,2010,11(1):515-521.
[3]Metzler R,Klafter J.The random walks guide to anomalous diffusion:a fractional dynamics approach[J].Physics Reports,2000,339(1):1-77.
[4]Yildirim A.Application of the homotopy perturbation method forthe Fokker-Planck equation[J].Communications in Numerical Methods in Engineering,2010,26(9):1144-1154.
[5]He J H.Homotopy perturbation method:a new nonlinear analytical technique[J].Applied Mathematics and Computation,2003,135(1):73-79.
[6]Gupta P K,Yildirim A,Rai K N.Application of He’s homotopy perturbation method for multi-dimensional fractional Helmholtz equation[J].Int J Num Meth Heat& Fl Flow,2012,22(4):424-435.
[7]Liu Y Q,Ma J H.Exact solutions of a generalized multifractional nonlinear diffusion equation in radial symmetry[J]. Communications in Theoretical Physics,2009,52(5):857-861.
[8]Liu Y Q.Approximate solutions of fractional nonlinear equations using homotopy perturbation transformation method[J].Abstract and Applied Analysis,2012.
[9]Liu Y Q.Study on space-time fractional nonlinear biologicalequation in radialsymmetry[J].MathematicalProblems in Engineering,2013.
[10]Liu Y Q.Variational homotopy perturbation method for solving fractional initial boundary value problems[J]. Abstract and Applied Analysis,2012.
[11]Guo S M,Mei L Q,Li Y.Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation[J].Applied Mathematics and Computation,2013,219(11):5909-5917.
[12]Wu G C,Baleanu D.Vatiational iteration method for fractional calculus-a universal approach by Laplace transform[J].Advances in Difference Equations,2013,18:1-9.
[13]Wu G C.Laplace transform overcoming principle drawbacks in application of the variation iteration method to fractional heat equations[J].Thermal Science,2012,16(4):1257-1261.
[14]Ghorbani A.Beyond Adomian polynomials:He polynomials[J].Chaos Solitons&Fractals,2009,39(3):1486-1492.
[15]Liu J C,Hong L.Approximate analytic solutions of timefractionalHirota-Satsumacoupled KdV equation and coupled MKdV equation[J].Abstract and Applied Analysis,2013.
DONG Lihua,LIU Yanqin
School of Mathematical Sciences,Dezhou University,Dezhou,Shandong 253023,China
A novel method which is based on variational iteration method,Laplace transform and homotopy perturbation method is proposed,and this new method is applied to obtain the approximate solutions of the fractional coupled MKdV equation.The fractional Lagrange multiplier is accurately determined by the Laplace transform and the nonlinear term can be easily handled by He’s polynomials.The results demonstrate accuracy and fast convergence of this new algorithm.
variational iteration method;Laplace transform;homotopy perturbation method;fractional equation;nonlinear equation
將變分迭代法、同倫擾動法和Laplace變換相結(jié)合應(yīng)用于分數(shù)階非線性發(fā)展方程近似解的求解,其中Laplace變換可準確方便地求得分數(shù)階的Lagrange乘子,而He的多項式可簡單地處理方程中出現(xiàn)的非線性項,將新的處理方法應(yīng)用到分數(shù)階耦合的MKdV方程,結(jié)果表明該方法具有較高的精度和收斂性。
變分迭代法;Laplace變換;同倫擾動法;分數(shù)階方程;非線性方程
A
O175.29
10.3778/j.issn.1002-8331.1404-0192
DONG Lihua,LIU Yanqin.New approximate solutions of fractional nonlinear equations.Computer Engineering and Applications,2014,50(23):1-3.
山東省優(yōu)秀中青年科學(xué)家科研獎勵基金(No.BS2013HZ026);山東省自然科學(xué)基金(No.ZR2013AQ005)。
董立華(1965—),女,教授,研究方向為函數(shù)論;劉艷芹(1981—),女,博士,副教授。E-mail:yanqinliu@dzu.edu.cn
2014-04-14
2014-05-27
1002-8331(2014)23-0001-03
CNKI網(wǎng)絡(luò)優(yōu)先出版:2014-06-26,http://www.cnki.net/kcms/doi/10.3778/j.issn.1002-8331.1404-0192.html