﻿ Higher-Order Numeric Solutions for Nonlinear Systems Based on the Modified Decomposition Method Journal of Applied Mathematics and Physics, 2014, 2, 1-7 Published Online January 2014 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2014.21001 OPEN ACCESS JAMP Higher-Order Numeric Solutions for Nonlinear Systems Based on the Modified Decomposition Method Junsheng Duan School of Sciences, Shanghai Institute of Technology, Shanghai, China Email: dua njssdu@sina.com, duanjs@sit.edu.cn Received July 2013 ABSTRACT Higher-order numeric solutions for nonlinear differential equations based on the Rach-Adomian-Meyers mod- ified decomposition method are designed in this work. The presented one-step numeric algorithm has a high effi- ciency due to the new, efficient algorithms of the Adomian polynomials, and it enables us to easily generate a higher-order numeric scheme such as a 10th-order scheme, while for the Runge-Kutta method, there is no gen- eral procedure to generate higher-order numeric solutions. Finally, the method is demonstrated by using the Duffing equation and the pendulum equation. KEYWORDS Adomian Polynomials; Modified Decomposition Method; Adomian-Rach Theorem; Nonlinear Differential Equations; Numeric Solution 1. Introduction The Adomian decomposition method (ADM) [1-4] is a practical technology for solving linear or nonlinear or-dinary differential equations, partial differential equations, integral equations, etc. The ADM provides an effi-cient analytic approximate solution of nonlinear equations, which model real-world applications in engineering and the applied sciences. The Adomian decomposition series has been shown to be equivalent to a Banach-space analog of the Taylor series expansion about the initial solution component function, instead of the classic Taylor series expansion about a constant that is the initial point . The ADM decomposes the pre-existent, unique, analytic solution into a series 0,nnuu∞==∑ (1) and decomposes the nonlinearity ()Nuf u= into the series of the Adomian polynomials 0() ,nnfu A∞==∑ (2) where the Adomian polynomials ,nA depend on the solution component functions 01,,,,nuu u and are de-fined by the formula  001() ,0.!nknknkdAfu nndλλλ∞=== ≥∑ (3) For convenient reference, we list the first five Adomian polynomials 00( ).A fu= 1 01'() .A fuu= 212 02 0'( )''( ).2!uAfuufu= + J. S. DUAN OPEN ACCESS JAMP 2 313030 120'( )''( )'''( ).3!uAfuuf uuufu=++ 2 24(4)212 140 401300'( )''( )()'''( )( ).2!2! 4!uuu uAfu ufuuufufu= ++++ Some algorithms for symbolic programming have since been devised to efficiently generate the Adomian po-lynomials quickly to high orders, such as in [5-10]. Rach’s Rule for the Adomian polynomials reads () 01( ),1.nkknnkAfuC n== ≥∑ (4) where the coefficients knC are the sums of all possible products of k components from 12 1,,, ,nkuu u−+ whose subscripts sum to n, divided by the factorial of the number of repeated subscripts . New, more efficient algorithms and subroutines in MATHEMATICA for fast generation of the one-variable and multi-variable Adomian polynomials to high orders have been provided in [8-10]. Here we list Corollary 3 algorithm  for the one-variable Adomian polynomials. Corollary 3 algorithm : For 1n≥, 1.nnCu= (5) For 2kn≤≤, 11101( 1).nkkknj njjCj uCn−−+ −−== +∑ (6) Then the Adomian polynomials are given by the formula () 01() .nkknnkAf uC==∑ The recurrence procedure for knC does not involve the differentiation operator, only requires the operations of addition and multiplication, which is eminently convenient for computer algebra systems. In 1992, Rach, Adomian and Meyers  proposed a modified decomposition method based on the nonlinear transformation of series by the Adomian-Rach theorem [12,13]: If 00()( ),nnnuxa xx∞== −∑ then 00(())( ),nnnfuxA xx∞== −∑ (7) where 01(,,,)nn nA Aaaa= are the Adomian polynomials in terms of the solution coefficients. The Rach-Adomian-Meyers modified decomposition method  combines the power series solution and the Ado mia n -Rach theorem [12,13], and has been efficiently applied to solve various nonlinear models [2,14-16]. In this work, higher-order numeric one-step methods are designed for solving nonlinear differential equations based on the Rach-Adomian-Meyers modified decomposition method  and the previous research . In the next section, we develop the numeric solution based on the modified decomposition method for nonli-near second-order differential equations, and demonstrate its application. 2. Higher-Order Numeric Solutions Based on the Modified Decomposition Method We consider the IVP for the second-order ODE 22()() ()()( )(),d uduttuttf ugtdtdtαβ γ++ += (8) 000 1(),'() ,utCutC= = (9) where 0,t tT≤≤ (),(), (),()ttt gtαβγ are specified bounded, analytic functions, and f is an analytic nonli- J. S. DUAN OPEN ACCESS JAMP 3 near operator. The modified decomposition method supposes an analytic solution 00()().mmmuta t t∞== −∑ (10) Then the functions (),(), (),()tttgtαβγare decomposed into the Taylor expansions 00()(),mmmt ttαα∞== −∑ (11) 00()() ,mmmt ttββ∞== −∑ (12) 00()() ,mmmt ttγγ∞== −∑ (13) 00()() ,mmmgtgt t∞== −∑ (14) and the analytic nonlinearity ()fuis decomposed into the Taylor series 00()( ),mmmfuA tt∞== −∑ (15) where the coefficients 01(,,,)nn nA Aaaa= are the Adomian polynomials in terms of the solution coefficients ka due to the Adomian-Rach theorem [12,13]. Substituting Equation s (10)-(15) in Equations (8), regrouping terms, equating the coefficients of like powers of 0()tt−, and using the initial condition we obtain the recurrence scheme for the solution coefficients 001 1,,aCa C= = 2101[((1 ))],( 1)(2)mmmlmllmllmllagm laaAmmα βγ++− − −==− +−+ +++ ∑ (16) where 0m≥ and the 01(,,,)nn nA Aaaa= are the Adomian polynomials in terms of the coefficients ka for the nonlinear function ()fu. In particular, if ()tαα=,()tββ= and ()tγ= γ are constants, then the recurrence formula becomes001 1,,aCa C= = 211[( 1)],0.( 1)(2)mmm mmagmaaAmmmα βγ++=−+− −≥++ (17) Further if ()gt g= is also a constant, then the recurrence formula becomes 001 1,,aCa C= = 2 1001[ ],2aga aAαβγ=−− − 211[( 1)],1.( 1)(2)mm mmamaaAmmmα βγ++−=+ ++≥++ (18) We denote the (n + 1)-term approximation of the solution as 1 00100(,,,)() .nmnmmtt C Cattφ+== −∑ (19) We regard 001,,tCC as three parameters, and generate the numeric solutions by using the (n + 1)-term ap-proximation 1nφ+. Partition the interval 0[,]Ntt into01 Ntt t<< <. Here we consider an equal step-size partition with1kkht t−= −. The numeric solution generated by 1nφ+ is of order n. We denote the nth-order numeric solution by nku<>, k = 0, 1, …, N. The one-step recurrence scheme is as follows: 000 1,,nuCuC<>= = J. S. DUAN OPEN ACCESS JAMP 4 111 1( 1)112(, ,,),nnkn kkkknn kmkk mmutt uuuuha hφ<> <>+−− −<> −−−===++∑ 111 1( 1)112(, ,,),knknkk kttnkmkmmdutt uudtuma hφ<>+−− −=−−−===+ ∑ k = 1, 2, ···, N, where (0)ma are the ma in (16), and for k = 2, ···, N, ( 1)kma−, 2,3, ,mn=, are determined by a recursion similar to (16) with ( 1)01knkau− <>−= and ( 1)11kkau−−=. Example 1. Consider the IVP for the Duffing equation 232d2sinsin 2,duuut tt−+ =− (0)1, '(0)0.uu= = The IVP has the exact solution *( )cos.ut t= The Adomian polynomials for the nonlinearity 3()ft u= are 300,Aa= 21 013,A aa= 22201023 3,A aa aa= + 3231012036 3,A aaaaaa=++ 22 241202013043 3 63,Aaa aaaaa aa=++ + . The 8th-order numeric solutions on the interval [0,45] are plotted in Figure 1 with the step-size h = 0.5. The numeric solution is suitable for a larger domain as the order increases. Example 2. Consider the IVP for the pendulum equation 2225sin0, (0)0,'(0)9.du uu udt +=== The exact solution can be expressed in terms of a Jacobi elliptic function as *9 81()2arcsin(sn(5 ,)).10 100ut t= The Adomian polynomials in terms of the decomposition coefficients ka for the sinusoidal nonlinearity sinu are Figure 1. The exact solution (solid line) and the 8th-order numeric solution on [0,45] with h = 0.5 (dots). J. S. DUAN OPEN ACCESS JAMP 5 00sin ,Aa= 11 0cos ,Aa a= 2122 00cossin ,2aAa aa= − 3130 12030cossincos ,6aAaaaaaa=−− − . Using the initial conditions, the coefficients of solution series are calculated to be 0123450,9,0,75 /2,0,795 / 4,.aaaaaa==== −= = The 5-term, 10-term and 20-term approximations 5( ,0,0,9),tφ 10( ,0,0,9),tφ 20 (,0,0,9)tφ are plotted in Figure 2. It is shown that the decomposition solution has a radius of convergence of more than 0.2. The 5-term approximation 5 001(, ,,)tt C Cφ under the general initial conditions are calculated to be 235001 01000100420 00111(,,,)()()sin()()cos()261()sin()(cos()).24ttCCCCttttCCttCttCC Cφ γγγγ=+ −−−−−+− + The 4th-order and 5th-order numeric solutions on the interval [0,6] with h = 0.1 are plotted in Figures 3 and 4, respectively. The 9th-order numeric solutions on the interval [0,10] with 0.2h= are plotted in F igure 5. We observe that the higher-order numeric solutions permit a larger step-size, and enlarge the effective region. Figure 2. The exact solution ( )*ut (solid line), the 5-term approximation ,,,5(t009)φ (dot line), the 10-term appro- ximation ,,,10(t009)φ (dash line) and the 20-term approximation ,,,20(t009)φ (dot-dash line). Figure 3. The exact solution (solid line) and the 4th-order numeric solution on [0,6] with h = 0.1 (dots). J. S. DUAN OPEN ACCESS JAMP 6 Figure 4. The exact solution (solid line) and the 5th-order numeric solution on [0,6] with h = 0.1 (dots). Figure 5. The exact solution (solid line) and the 9th-order numeric solution on [0,10] with h = 0.2 (dots). 3. Conclusions We have developed higher-order numeric solutions for nonlinear differential equations based on the Rach- Ado mia n -Meyers modified decomposition method. Due to the new, efficient algorithms of the Adomian poly-nomials, the one-step numeric algorithm has a high efficiency, and permits us to easily generate a higher-order numeric scheme such as a 10th-order scheme, while for the Runge-Kutta method, there is no general procedure to generate higher-order numeric solutions. We demonstrated the presented numeric method by two nonlinear physical models. Acknowledgements This work was supported by the NNSF of China (11201308) and the Innovation Program of Shanghai Municipal Education Commission (14ZZ161). REFERENCES  G. Adomian, “Nonlinear Stochastic Operator Equations,” Academic, Orlando, 1986.  G. 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