Adomian’s decomposition method for eigenvalue problems Yee-Mou Kao 1 and T. F. Jiang 2 1 Institute of Applied Mathematics, National Chiao-Tung University, Hsinchu 30010, Taiwan 2 Institute of Physics, National Chiao-Tung University, Hsinchu 30010, Taiwan Received 10 September 2004; published 8 March 2005 We extend the Adomian’s decomposition method to work for the general eigenvalue problems, in addition to the existing applications of the method to boundary and initial value problems with nonlinearity. We develop the Hamiltonian inverse iteration method which will provide the ground state eigenvalue and the explicit form eigenfunction within a few iterations. The method for finding the excited states is also proposed. We present a space partition method for the case that the usual way of series expansion failed to converge. DOI: 10.1103/PhysRevE.71.036702 PACS numbers: 02.70.Wz, 03.65.Ge I. INTRODUCTION Adomian’s method solves nonlinear differential equations with decompositions. Neither linearization nor perturbation is applied to the nonlinear part. The method has been widely applied to various domains in science and engineering, but is less popular in physics. Actually, in Chap. 14 of Adomian’s comprehensive book 1, he treated many physical topics, namely, the Navier-Stokes equations, onset of turbu- lence, Burger’s equation, nonlinear transport, advection- diffusion equation, Korteweg–de Vries equation, nonlinear Schrödinger equation NLSE, and classical N-body dynam- ics, etc. It shows that Adomian’s decomposition method ADM is extremely versatile in nonlinear physical prob- lems. For some other examples, Adomian and co-workers also formulated the solutions for Thomas-Fermi equation 2 and the Ginzburg-Landau equation 3. Wazwaz employed ADM to give the soliton and periodic solutions of the Bouss- inesq equation 4. Abbaoui et al. discussed the convergence of the ADM 5. Guellal et al. gave the ADM explicit solu- tion of the Lorenz system 6. The ADM is generally applicable to nonlinear differential equations for either initial value problems or boundary prob- lems. The basic theory is clearly described in Adomian’s book 1. On the other hand, we are not able to find out systematic treatment for the eigenvalue problems by ADM. Since the eigenvalue problem is fundamentally important for the structure of a system, the pursuit of ADM for the eigen- value problem is a worthwhile work. It is, nevertheless, not straightforward. Also, the ADM gives an explicit form of solutions that the numerical grids method cannot do. Thus the ADM treatment of the eigenvalue problem is valuable to computational physics. In this paper, we develop the method for solving the eigenvalue problems by ADM. We will briefly describe the ADM first, and then present our method for the eigenvalue problem. Some paradigmatic examples of both linear and nonlinear eigenvalue equations are given. The paper is organized as follows. In Sec. II, we introduce the Hamiltomian inverse iteration scheme for the ADM of eigenvalue problems. In Sec. III, we apply the method to the problem of a particle in a box. In Sec. IV, the method is applied to the simple harmonic oscillator. Section V is a treatment of the anharmonic oscillator, and in Sec. VI, we try to solve the nonlinear Gross-Pitaevskii equation that de- scribes the Bose-Einstein condensate by the new scheme. We find that the straightforward way of ADM failed to converge. We introduce in Sec. VII the space partition method to over- come the trouble of divergence encountered in the previous section. Section VII is devoted to concluding remarks. II. HAMILTONIAN INVERSE ITERATION Consider the general eigenvalue problem Hux = ux , 1 where H = L + V„ux… . 2 L is usually a differential operator such as -1  2 d 2 / dx 2  and V(ux) is the potential function, either dependent or independent of ux. The former case is a linear problem while the latter case is nonlinear and is called a nonlinear Schrödinger equation. We describe the difficulty in the ADM for eigenvalue problems first. Adomian wrote the solution as the sum of decompositions ux ;  =  n=0   n u n x . 3 Expand the potential V(ux ; ), V„ux ; … =  n=0   n A n ; 4 here we introduce the parameter  to collect the coefficients of same order in both sides to find out the decompositions A n for a general V. The introduced  is set equal to 1 at last. Some examples of A n can be found in Ref. 1 and will not be repeated here. Let L -1 be the inverse operator of L. Op- erating the L -1 to Eq. 1, we have L -1 Lu = L -1 u - L -1 V . 5 The solution of decomposition orders are obtained system- atically 1. That is, PHYSICAL REVIEW E 71, 036702 2005 1539-3755/2005/713/0367027/$23.00 ©2005 The American Physical Society 036702-1