An analytic approach to solve multiple solutions of a strongly nonlinear problem

Transcription

1 Applied Mathematics and Computation 169 (2005) An analytic approach to solve multiple solutions of a strongly nonlinear problem Shuicai Li, Shi-Jun Liao * School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai , China Abstract Based on a new kind of analytic method, namely the homotopy analysis method, an analytic approach to solve multiple solutions of strongly nonlinear problems is described by using Gelfand equation as an eample. Its validity is verified by comparing the approimation series with the known eact solution. And different from perturbation techniques, this approach is independent upon any small/large perturbation quantities. So, the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Gelfand problem; Multiple solutions; Strong nonlinearity; Bifurcation; Heat transfer; Homotopy analysis method * Corresponding author. address: sjliao@sjtu.edu.cn (S.-J. Liao) /$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi: /j.amc

2 1. Introduction S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) The Gelfand equation [1,2] Du þ ke u ¼ 0; 2 X; ð1þ u ¼ 0; 2 ox is well-known for the eistence and multiplicity of its nontrivial solutions, where parameter k denotes the reaction term, is a spatial variable, X and ox denote the spatial domain and its boundary, respectively. This equation contains the eponent term ep(u) and thus has very strong nonlinearity. It comes from the theory of combustion, and is used as a model for the thermal reaction process such as that when a combustible medium is placed in a vessel whose walls are maintained at a fied temperature. The Gelfand equation represents the steady state of diffusion and transfer of heat conduction [2,3]. There eists steady-state heat transfer for small k. Ask is greater than a critical value, the reaction will lead to eplosion so that Eq. (1) has no solutions. We state the critical k as k*. For details about GelfandÕs equation, please refer to [4 8]. It is hard to solve the Gelfand equation in a general domain X. The classical Gelfand problem is with radially symmetric domain, i.e. u = u(r). In this case, one has u 00 þ N 1 u 0 þ ke u ¼ 0; r 2ð0; 1Š; N ¼ 1; 2; 3;...; r ð2þ u 0 ð0þ ¼uð1Þ ¼0; where N = 1, 2, and 3 correspond to the infinite slab, infinite circular cylinder, and sphere, respectively. In 1973, Joseph and Lundgren [9] obtained the numerical solutions for all N =1,2,3,... for the domain of a unit ball. He made an interesting conclusion that all solutions of Eq. (2) lie on a unique curve in (k,u(0)) plane. When N = 1, Eq. (2) is equivalent to the equation u 00 þ ke u ¼ 0; 2ð0; 1Þ; ð3þ uð0þ ¼uð1Þ ¼0; where k is a physical parameter, and the prime denotes the differentiation with respect to. Obviously, u has the maimum value at = 1/2, denoted by l. In 1853, Liouville [10] found an analytic epression between l and k for N =1. When 0 < k < k*, where k* , there eist two values of l, as shown in Fig. 4. Thus, Eq. (3) has multiple solutions. Generally speaking, it is difficult to get multiple solutions of a nonlinear problem, especially by means of analytic methods. Perturbation techniques are dependent upon the eistence of small or large parameters. This greatly restricts their applications. Currently, a new analytic technique, namely the homotopy analysis method [11,12], is developed. Different from perturbation techniques, the homotopy analysis method does not depend upon any small

3 856 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) or large parameters and thus is valid for more problems in science and engineering. Besides, it logically contains other nonperturbation techniques such as Lyapunov small parameter method [13], the d-epansion method [14], and Adomian decomposition method [15]. The homotopy analysis method has been successfully applied to many nonlinear problems such as nonlinear vibration [16], nonlinear water waves [17], viscous flows of nonnewtonian fluids [18], Thomas FermiÕs equation [19], nonlinear heat transfer [20], a third grade fluid past a porous plate [21], the flow of an Oldroyd 6-constant fluid [22], and so on. In this paper, based on the homotopy analysis method, a new approach is proposed to solve multiple solutions of strongly nonlinear problems by using Gelfand equation (3) as an eample. 2. Homotopy analysis solution Under the transformation wðþ ¼e u : Eq. (3) becomes ð4þ wðþw 00 ðþ ½w 0 ðþš 2 kwðþ ¼0; 2ð0; 1Þ; ð5þ subject to the boundary conditions wð0þ ¼wð1Þ ¼1: From (6), it is obvious that w() can be epressed by the power series ð6þ wðþ ¼1 þ Xþ1 a n n ; where a n is a coefficient. So, we can further make the transformation wðþ ¼1 þ csðþ; ð7þ where c is unknown and is dependent on k. Here, c is introduced to search for the multiple solutions, as described later. Using the above transformation, Eq. (5) becomes c½1 þ csðþšs 00 ðþ c 2 ½s 0 ðþš 2 k½1 þ csðþš ¼ 0; 2ð0; 1Þ; ð8þ subject to the boundary conditions sð0þ ¼sð1Þ ¼0: From (9), it is obvious that s() can be epressed by the power series sðþ ¼ Xþ1 b n n ; ð9þ ð10þ

4 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) where b n is a coefficient. This provides us with the so-called Rule of Solution Epression. According to the Rule of Solution Epression denoted by (10) and the boundary conditions (9), it is straightforward to choose the initial solution s 0 ðþ ¼4ð1 Þ; ð11þ which satisfies s 0 (1/2) = 1. According to the Rule of Solution Epression denoted by (10) and from Eq. (8), it is obvious to choose the auiliary linear operator L/ ¼ d2 / d : 2 According to Eq. (8), we define the nonlinear operator ð12þ N½/ð; qþ; CðqÞŠ ¼ CðqÞ½1 þ CðqÞ/ð; qþš o2 /ð; qþ o 2 2 C 2 o/ð; qþ ðqþ k½1 þ CðqÞ/ð; qþš; ð13þ o where q 2 [0, 1] is an embedding parameter. Let H() 5 0 denote an auiliary function and h 5 0 an auiliary parameter. We construct the zero-order deformation equation ð1 qþl½/ð; qþ s 0 ðþš ¼ hhðþn½/ð; qþ; CðqÞŠ; 2ð0; 1Þ; ð14þ subject to the boundary conditions /ð0; qþ ¼/ð1; qþ ¼0: ð15þ When q = 0 and 1, the above equation has the solution /ð; 0Þ ¼s 0 ðþ ð16þ and /ð; 1Þ ¼sðÞ; Cð1Þ ¼c; ð17þ respectively. Assume that the auiliary function H() and the auiliary parameter h are so properly chosen that /(;q) andc(q) can be epressed by the Taylor series /ð; qþ ¼/ð; 0Þþ Xþ1 s n ðþq n ; CðqÞ ¼c 0 þ Xþ1 c n q n ; ð18þ ð19þ

5 858 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) where s n ðþ ¼ 1 o n /ð; qþ n! oq n ; c n ¼ 1 o n CðqÞ q¼0 n! oq n ; ð20þ q¼0 and besides that the above two series are convergent at q = 1. Then, using (16) and (17), we have sðþ ¼s 0 ðþþ Xþ1 s n ðþ; c ¼ c 0 þ Xþ1 c n : For the sake of simplicity, define the vectors ~s m ðþ ¼fs 0 ðþ; s 1 ðþ; s 2 ðþ;...; s m ðþg and ð21þ ð22þ ~c m ¼ fc 0 ; c 1 ; c 2 ; ; c m g: Differentiating the zero-order deformation Eqs. (14) and (15) n times with respect to the embedding parameter q, then setting q = 0, and finally dividing by n!, we have the nth-order deformation equation L½s n ðþ v n s n 1 ðþš ¼ hhðþr n ½~s n 1 ðþ;~c n 1 Š; ð23þ subject to the boundary conditions s n ð0þ ¼s n ð1þ ¼0; ð24þ where R n ½~s n 1 ðþ;~c n 1 Š ¼ Xn 1 1 v iþ1 þ A i Dn 1 i Xn 1 B i B n 1 i i¼0 i¼0 under the definitions kð1 v n þ A n 1 Þ ð25þ A m ¼ Xm i¼0 B m ¼ Xm i¼0 D m ¼ Xm i¼0 c m i s i ðþ; c m i s 0 i ðþ; c m i s 00 i ðþ; ð26þ ð27þ ð28þ

6 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) and 0; k 6 1; v k ¼ 1; k > 1: Obviously, the solution of Eq. (23) is s n ðþ ¼v n s n 1 ðþþh Z Z g 0 0 ð29þ HðnÞR n ½~s n 1 ðnþ;~c n 1 Šdndg þ C 1 þ C 2 ; ð30þ where the integral constants C 1 and C 2 are determined by the boundary condition (24). Due to the Rule of Solution Epression denoted by (10), H() can be in the form HðÞ ¼ j ; j P 0: For simplicity, we choose here HðÞ ¼1: ð31þ Note that, s 1 () contains the unknown c 0. In general, s n () contains the unknown c n 1 for n P 1. Obviously, /(;q) is dependent on C(q). Thus, the convergence of the series (21) is dependent on C(q) and also the corresponding series (22). Note that we have great freedom to choose c 0 = C(0). Therefore, although C(1) = c is determined by k, the function C(q) can be quite different. Let b 0 þ Xþ1 b n ð32þ denote a known convergent series, where b 0 ¼ s 0 ð1=2þ ¼1 ð33þ and b n (n P 1) is a given constant. We can choose C(q) in such a way that the series s 0 ðþþ Xþ1 s n ðþ is convergent at = 1/2, and besides s n ð1=2þ ¼b n ; n P 1: ð34þ Solving above algebraic equation, we obtain the unknown c n 1. An infinite number of convergent series can be employed as the series (32). We provide here two approaches. First, we can choose such a convergent series b n ¼ 0; n P 1; ð35þ

7 860 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) which gives the algebraic equation s n ð1=2þ ¼0; n P 1 ð36þ for c n 1. In this case, it holds wð1=2þ ¼1 þ c: When n = 1, we have 7 c þ 1 þ 5k c 48 0 þ k 8 ¼ 0; ð37þ which has two branches of solutions pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5k þ 48 25k 2 864k þ 2304 c 0 ¼ ð38þ 112 and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5k þ 48 þ 25k 2 864k þ 2304 c 0 ¼ ; ð39þ 112 when k < 48 pffiffiffiffiffi :91204; 25 as shown in Fig. 1. Each of them corresponds to a different solution. When n P 2, Eq. (35) is a linear algebraic equation of c n 1. Note that, the epressions (38) and (39) give comple number when k > 48 pffiffiffiffiffi :91204: 25 For k > , we employ the second approach and simply set c 0 a proper value. In this case, s 1 () is determined by the chosen value of c 0, and b 1 = s 1 (1/ 2) (n P 2) might be nonzero. The unknown c n is determined by s n ð1=2þ ¼0; and it holds wð1=2þ ¼1 þ cð1 þ b 1 Þ: The key of the above mentioned method is to ensure the convergence of the series (21) and (32). Fortunately, we have great freedom to choose the auiliary parameter h. As mentioned by Liao [11], itish that provides us with a simple way to control the convergence of the series given by the homotopy analysis method. A convenient method is provided by Liao [11] to choose the value of h. Simply speaking, one can plot the so-called h-curve of c via h at high enough order of approimations and then choose a proper value of h in a region

8 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) γ λ Fig. 1. The two branches of c 0. Solid line: c 0 given by (38); dashed line: c 0 given by (39). corresponding to a horizontal line segment of c via h. For details, please refer to Liao [11]. Using above-mentioned method, we successfully obtain two different solutions. When 0 < k < 2.9, the first approach is used. For k P 2.9, we employ the second approach. There eist two branches of solutions with two different values l = u(1/2) at = 1/2. Our results agree well with eact solutions, as shown for eample in Tables 1 and 2 for k = 2. In general, Table 1 Comparison of the HAM approimation of l = u(1/2) with the eact solution l = when k =2,h = 3, and c 0 = / Order of approimation l Relative error (%) 1th th th th th th th th th th th E 05 20th E 08

9 862 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) Table 2 Comparison of the HAM approimation of u 0 (0) with the eact solution u 0 (0) = when k =2,h = 3, and c 0 = / Order of approimation u 0 (0) Relative error (%) 1th-order th-order th-order th-order th-order th-order th-order th-order th-order th-order th-order th-order E th-order appro th-order appro th-order a ppro th-order a ppro Fig. 2. Comparisons of the HAM approimation with the eact solution for smaller l when k =3. Symbol: eact solution; solid line: the HAM approimation given by c 0 ¼ 9 and h = the series with smaller value of l converges more quickly than the series with larger value of l, as shown in Figs. 2 and 3 for k = 3. The whole curve of l via k agrees well with the eact ones, as shwon in Fig. 4. And it is found that the series diverge when k > k*, indicating that there eist no solutions for k > k*. All of these verify that our analytic approach is valid

10 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) th-orderappro th-order appro th-order appro th-order appro Fig. 3. Comparisons of the HAM approimation with the eact solution for the larger l when k = 3. Symbol: the eact solution; solid line: the HAM approimation given by c 0 ¼ 4 and 5 h = µ λ Fig. 4. The comparison between the eact value of l with the HAM approimations. Symbol: the HAM approimation; solid line: the eact result.

11 864 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) for searching for multiple solutions of strongly nonlinear problems such as Gelfand problem. 3. Conclusion Based on a new kind of analytic method, namely the homotopy analysis method [11], an analytic approach of searching for multiple solutions of strongly nonlinear problems is described by using Gelfand equation as an eample. The approach is convenient and efficient. Its validity is verified by comparing the approimation series with the known eact solution. Note that the Gelfand equation contains an eponent term ep(u) and thus has very strong nonlinearity. And different from perturbation techniques, this approach is independent upon any small/large perturbation quantities. So, the basic ideas of this approach can be employed to search for multiple solutions of strongly nonlinear problems in science and engineering. Acknowledgement This work is supported by the National Science Fund for Distinguished Young Scholars (Approval No ) of Natural Science Foundation of China for the financial support. References [1] I.M. Gelfand, Some problems in the theory of quasi-linear equations, Am. Math. Soc. Transl. Ser. 2 (29) (1963) [2] J. Jacobsen, The Liouville Bratu Gelfand problem for radial operators, J. Diff. Equat. 184 (2002) [3] D.A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Kinetics, Princeton University Press, Princeton,NJ, [4] P. Korman, An accurate computation of the global solution curve for the Gelfand problem through a two point approimation, Appl. Math. Computat. 139 (2003) [5] J.S. McGough, Numerical continuation and the gelfand problem, Appl. Math. Computat. 89 (1998) [6] M. Plum, C. Wieners, New solution of the Gelfand problem, J. Math. Anal. Appl. 269 (2002) [7] M. Plum, Computer-assisted enclosure methods for elliptic differential equations, Linear Algebra Appl. 324 (2001) [8] E. Balakrishnan, A. Swift, G.C. Wake, Critical values for some non-class a geometries in thermal ignition theory, Math. Comput. Model. 24 (8) (1996) [9] D.D. Joseph, T.S. Lundgren, Quasilinear dirichlet problem driven by positive source, Arch. Rat. Mech. Anal. 49 (1973)

12 S. Li, S.-J. Liao / Appl. Math. Comput. 169 (2005) [10] J. Liouville, Sur iõéquation au différence partielles d2 log k du dv k2a2 ¼ 0, J. Math Pures Appl. 18 (1853) [11] S.J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/ CRC Press, Boca Raton, [12] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Computat. 147 (2004) [13] A.M. Lyapunov, (1892). General Problem on Stability of Motion, London, Taylor & Francis, 1992 (English translation). [14] A.V. Karmishin, A.T. Zhukov, V.G. Kolosov, Methods of Dynamics Calculation and Testing for Thin-walled Structures, Mashinostroyenie, Moscow, 1990 (in Russian). [15] G. Adomian, Nonlinear stochastic differential equations, J. Math. Anal. Appl. 55 (1976) [16] S.J. Liao, An analytic approimate approach for free oscillations of self-ecited systems, Int. J. Non-Linear Mech. 39 (2) (2004) [17] S.J. Liao, K.F. Cheung, Homotopy analysis of nonlinear progressive waves in deep water, J. Eng. Math. 45 (2) (2003) [18] S.J. Liao, On the analytic solution of magnetohydrodynamic flows of non-newtonian fluids over a stretching sheet, J. Fluid Mech. 488 (2003) [19] S.J. Liao, An eplicit analytic solution to the Thomas Fermi equation, Appl. Math. Computat. 144 (2003) [20] C. Wang, et al., On the eplicit analytic solution of Cheng Chang equation, Int. J. Heat Mass Transfer 46 (10) (2003) [21] M. Ayub, A. Rasheed, T. Hayat, Eact flow of a third grade fluid past a porous plate using homotopy analysis method, Int. J. Eng. Sci. 41 (2003) [22] T. Hayat, M. Khan, M. Ayub, On the eplicit analytic solutions of an Oldroyd 6-constant fluid, Int. J. Eng. Sci. 42 (2004)