Telescoping Decomposition Method for Solving First Order Nonlinear Differential Equations

Transcription

1 Telescoping Decomposition Method or Solving First Order Nonlinear Dierential Equations 1 Mohammed Al-Reai 2 Maysem Abu-Dalu 3 Ahmed Al-Rawashdeh Abstract The Telescoping Decomposition Method TDM is a new iterative method to obtain numerical and analytical solutions or irst order nonlinear dierential equations The method is a modiied orm o the well-known Adomian Decomposition Method ADM where the Adomian polynomials have not to be calculating The TDM is easier to apply and oers better accuracy than the ADM Also, it can be applied to other systems where the ADM does not work The TDM is proved to be convergent to the exact solution while it is not the case in the ADM The idea o the TDM can be developed to deal with various types o unctional equations, as well Keywords: IVP s; ADM MSC: 34A12; 49M27 1 Introduction Dierential equations appear in various applications in the physical sciences and engineering They also, have been used to study solutions o partial dierential equations, since most techniques reduce the partial dierential equation into a dierential or a system o dierential equations Most o dierential equations coming rom real lie applications are nonlinear and the seek o analytical solutions is a diicult task Thereore, numerical methods have been introduced The Runge-Kutta methods, linear multistep methods and Galerkin method can be used to integrate dierential equations numerically and obtain accurate solutions [7, 8] Recently, certain methods that produce accurate and analytical solutions have been introduced Such as the tanh method [11], the variational iteration method [9] and the Adomian decomposition method ADM The ADM method was irst introduced by Adomian [1, 2], and it has been used to integrate various systems o unctional equations [3, 4] A main part o the ADM is inding the Adomian polynomials Several Authors have discussed this issue and obtained dierent approaches or calculating the Adomian polynomials [6, 12] However, the most popular one is 1 Department o Mathematical Sciences, United Arab Emirates University, PO Box 17551, Al Ain, UAE m alreai@uaeuacae 2+3 Department o Mathematics and Statistics, Jordan University o Science and Technology, PO Box 33 - Irbid Jordan the ormula obtained in [1, 3] A n = 1 d n dλ n u i λ i λ=, where A n denotes the Adomian polynomial o degree n, u = u i is the exact solution o the problem and u is the nonlinear term in the equation It is worth noting that calculating the Admian polynomials is diicult or large n and the above ormula can not be applied i is a unction o more than one variable, such as = u, u Also, the ADM is shown to be divergent or certain problems [1] In this paper we introduce the Telescoping decomposition method TDM or solving irst order nonlinear initial value problems We will use the idea o the Adomain method but avoid calculating the Adomian polynomials In section 2, we present the expansion procedure o the TDM and then show the convergence o the TDM in Section 3 We present some numerical results and comparison with the ADM in Section 4, and inally we write some concluding remarks in Sections 5 2 The Expansion Procedure We consider the initial value problem u t = t, u, u t, t Ω 1 u = u, 2 where Ω = [, T ] is compact subset o R By integrating the above equation we have ut = u + u u τ τdτ 3 We consider a solution o the orm ut = n= un t, where u n t has to be determined sequentially upon the ollowing algorithm: u = u, 4 u 1 = u 2 = u u τ τ d 1 1 u k u k τ τ dτ k= k= ISBN: IMECS 28

2 k= u 3 = u 4 = u n = u u τ τ d 2 2 u k u k τ τ dτ k= 1 u k k= 3 u k k= 2 u k k= k= k= 1 u k τ τ d k= 3 u k τ τ dτ k= 2 u k τ τ d k= u k u k τ τ dτ k= k= u k u k τ τ dτ 5 k= k= k= 6 We remind here that the choice o u in 4 is not unique, we can chose it to be any unction o t and this depends on the problem as we will see in Section 4 Also, i t, u, u t = u the simple linear case then the ADM and TDM will coincide and give the exact solution o the problem 3 Convergence analysis In this section we prove that k= uk converges uniormly to the exact solution o 1-2 We have the ollowing lemmas beore writing the main result Lemma 1 Consider the sequence o unctions u n t, n, as deined in 4-5 I is dierentiable on Ω and is continuous on Ω, then the ininite series un converges uniormly on Ω Proo 1 As and are continuous on Ω, then there exist positive real numbers M and L such that u, u τ M and L First we shall use mathematical induction to prove the inequality u n L Mt n, n 1 7 The result is true or n = 1, since u 1 = t, u, u τ dτ Mt Suppose that or k 2, u k L Mt k k!, we shall show that the inequality holds or k + 1 Recall t k k u k+1 = [ u i u i τ τ dτ u i u i τ τ ]dτ By applying the Mean Value Theorem on the second variable o, there exists ζ k such that k u i k u i τ τ u i u i τ τ = ζ k u k where ζ k lies on the line segment 1 λ k ui τ + λ ui or λ, 1 Then we have Adding equations between 4 and 5 we have u k+1 n ζ k u k d t u k t = u + u k u k τ τ d n 1 thereore by the induction hypothesis, we get u k+1 L M k! Lk Mt k+1 k + 1!, and the inequality is proved ζ k t k dτ The supremum norm o u n on Ω satisies u n Ω = sup{ u n t ; t Ω} L MT Now as n= L is convergent series o positive real numbers, then by using the Weierstrass M-test [5],p317, we have n= un is uniormly convergent on Ω, hence the lemma has been checked Lemma 2 Consider the sequence o unctions u n t, n, as deined in 4-5 I is dierentiable on Ω and is continuous on Ω, then the ininite series un t converges on Ω Proo 2 As and are continuous on Ω, then there exist positive real numbers M and L such that u, u τ M and L Thereore u1 τ = t, u, M Now recall that or all n 2, u n τ = u i u i τ τ u i u i τ τ ISBN: IMECS 28

3 By applying the Mean value theorem on the second variable o, there exists ζ such that u i u i τ τ u i u i τ τ = ζ u where ζ lies on the line segment 1 λ ui t + λ ui t, or some λ [, 1] Then by using inequality 7 we have u n τ t = ζ u t L Mt L n 1! = L Mt n 1! Hence we proved the validity o the argument or all n 1 Thereore, u n t Ω = sup{ u n t t ; t Ω} L Mβ n 1!, or some positive real number β As n= L is convergent series o positive real numbers, then by the Weierstrass M-test, the lemma has been proved Theorem 1 Consider the initial value problem 1-2, and the sequence o unctions u n t, as deined in 4-5 I is dierentiable on Ω and is continuous on Ω, then un converges uniormly to the exact solution o the initial value problem Proo 3 Recall that or every positive integer n, Let n u k = u + k= s n = u k, k= u k, u k τ dτ k= k= s n = k= and deine the sequence o unctions g n by g n := s n, s n By Lemma 1, we have that s n is uniormly convergent on Ω to some unction V, and by Lemma 2, we get s n converges uniormly on Ω to V [5],p317 Moreover, as is uniormly continuous on Ω, then the sequence g n converges uniormly to V, V Thereore, u k = u + lim g n dτ n k= u k τ = u + lim g ndτ n = u + = u + and hence the result is obtained 4 Numerical Results lim s n, lim s ndτ n n V V τdτ In the ollowing we present various applications to illustrate the validity and eectiveness o the TDM We denote u T = n k= uk and u A = n k= u k the approximate solutions obtained by the TDM and ADM, respectively Example 1 Consider the initial value problem u = 2 + 2t + 2t 2 + 2t 3 + t t 2 u 2 u = 1 The problem has been discussed in [1] and the ininite series solution obtained by the ADM is proved to be divergent Since t, u is dierentiable and = 21 + t2 u is continuous, then the ininite series solution obtained by the TDM converges to the exact solution ut = 1 + t Integrating the equation with respect to t we have ut = 1 + 2t + t t t t5 1 + τ 2 u 2 τdτ We start with u = 1 + 2t + t t t t5 and the approximate solution u T = n k= uk is obtained by applying the TDM algorithm Figures 1 and 2 show the exact and approximate solutions o the problem obtained by the TDM or n=3,4,5 and 6 with t [, 5] Figure 1 show that the approximate solutions u T are close enough to the exact solution u or t [ 25] While they are little bit ar rom the exact solution in [25 5] However, or n = 4 and 5 the approximate solutions are very close to the exact one in [ 5] as shown in Figure 2 That is, we have to increase the number o terms n in order to achieve accurate approximate solution as the time increases Example 2 Consider the initial value problem u = 1 + u 2 u = The Taylor series expansion o the exact solution ut = tant = t + t t t t ISBN: IMECS 28

4 We have ut = u+t+ u2 τd and since u =, we start with u = t Applying the TDM we have u 1 = 1 3 t3, u 2 = 2 15 t t7, u 3 = u 4 = Hence, 4 15 t t t t t15, t t t t t t31 u + u 1 + u 2 + u 3 + u 4 + = t + t t t t Ot 11, which coincides with the Taylor series o the exact solution That is, the exact solution o the problem is obtained by the TDM Example 3 Consider the initial value problem u = 1 + 2u u 2 u = The exact solution o the problem ut = 1 + 2t 2 tanh + 2 log Figure 3 and 4 present the exact and approximate solutions u A and u T o the problem obtained by the ADM and TDM, respectively, or n = 2, 3, 4 and t [ 15] One can see that the ADM and the TDM produce solutions which converge to the exact solution o the problem but the TDM converges aster To illustrate this conclusion, we compute the errors e A and e T between the exact and approximate solutions obtained by the ADM and TDM, respectively We deine T e A = u u A 2 = u u A 2 dt, and in a similar manner we deine e T Tables 1-5 present the errors e A and e T or dierent values o n and T One can see that both errors increase as T increases For T = 15 the error e A is not necessarily decreasing with n, which indicates the ADM produces unstable solution While, the error e T is decreasing with n in all cases o T Also, we have e T < e A except or the case when T = 5 and n = 2 as indicated in Table 1 The above discussion demonstrates that the TDM is more eicient and converges aster than the ADM and we expect it to have the same eature or other problems, as well Table 1: The errors between the exact and approximate solutions obtained by the ADM and TDM or n=2 e A e T t t t Table 2: The errors between the exact and approximate solutions obtained by the ADM and TDM or n=3 e A e T t t t Table 3: The errors between the exact and approximate solutions obtained by the ADM and TDM or n=4 e A e T t t t Concluding Remarks We have presented and analyzed the Telescoping Decomposition Method TDM or solving nonlinear irst order initial value problem o the orm u t = t, u, u t, u = u, t Ω, where Ω is a compact subset o R We proved that the TDM converges uniormly to the exact solution o the problem provided that is dierentiable and is continuous on Ω We have applied the TDM or a variety o applications and obtained an accurate solutions as well as, exact solutions or certain problems The TDM has several advantages over the Adomian Decomposition Method ADM: no need to calculate the Adomian polynomials and replace them by a simple ormula, the TDM always converges to the exact solution while the ADM is not, and it converges aster to the exact solution Also, there is no general ormula or calculating the Adomian polynomials with nonlinear term t, u, u t The idea o the TDM can easily be modiied to obtain solutions o higher order dierential equations but proving the convergence o the method is not an easy task Also, it can be applied or various systems o partial and integral equations, as well However, we leave these issues or a uture work ISBN: IMECS 28

5 Figure 1: The exact solution u and the approximate solution u T obtained by the TDM or n = 3 let and n = 4 right and t [, 5] Figure 2: The exact solution u and the approximate solution u T obtained by the TDM or n = 5 let and n = 6 right and t [, 5] Figure 3: The exact solution u and the approximate solutions u A and u T t [, 15] or n = 2 let and n = 3 right and Figure 4: The exact solution u and the approximate solutions u A and u T or n = 4 and t [, 15] ISBN: IMECS 28

6 Table 4: The errors between the exact and approximate solutions obtained by the ADM and TDM or n=5 ADM TDM t t t [11] W Malliet, Solitary Wave Solutions o Noblinear Wave Equations, Am J Phys, 61992, p [12] A Wazwaz and S El-Sayed, A New Modiication o the Adomian Decomposition Method or Linear and Nonlinear Operators, Apl Math Compu 12821, p Table 5: The errors between the exact and approximate solutions obtained by the ADM and TDM or n=6 ADM TDM t t t Reerences [1] G Adomian, Nonlinear Stochastic Operator Equations, Academic Press, 1986 [2] G Adomian, A Review o the Decomposition Method in Applied Mathematics, JMathAnalAppl ,p [3] G Adomian, A Review o the Decomposition Method and Some Recent Results or Nonlinear Equations, MathComput Model ,p17-43 [4] M Al-Quran and M Al-Reai, Exact Solutions o Partial Dierential Equations with Continuous Delay by the Adomian Decomposition Method, Submitted to Canadian Mathematical Bulletin, February 27 [5] R Bartle, The Elements o Real Analysis, John Wiley and Sons, New York1975 [6] J Biazar, M Ilie and A Khoshkenar, An Improvement to an Alternate Algorithm or Computing Adomian Polynomials in Special Cases, Appl Math Comp, 13726, p [7] J Butcher, The Numerical Analysis o Ordinary Dierential Equations:Runge-Kutta and Generalized Linear Methods, John Wiley and Sons, Great Britain1987 [8] C Fletcher, Computational Galerkin Methods, Springer Verlag, New York1984 [9] Ji-Huan He, Variational Iteration Method or Autonomous Ordinary Dierential Equations, Appl Math Comput, 1142, p [1] M Hosseini and H Nasabzadeh, On the Convergence o Adomian Decomposition Method, Appl Math Comp, Issue 1, November26, p ISBN: IMECS 28