An analytic approach to solve multiple solutions of a strongly nonlinear problem
Introduction
The Gelfand equation [1], [2]is well-known for the existence and multiplicity of its nontrivial solutions, where parameter λ denotes the reaction term, x is a spatial variable, Ω and ∂Ω denote the spatial domain and its boundary, respectively. This equation contains the exponent term exp(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 fixed temperature. The Gelfand equation represents the steady state of diffusion and transfer of heat conduction [2], [3]. There exists steady-state heat transfer for small λ. As λ is greater than a critical value, the reaction will lead to explosion so that Eq. (1) has no solutions. We state the critical λ as λ*. For details about Gelfand’s equation, please refer to [4], [5], [6], [7], [8].
It is hard to solve the Gelfand equation in a general domain Ω. The classical Gelfand problem is with radially symmetric domain, i.e. u = u(r). In this case, one haswhere 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 (λ, u(0)) plane. When N = 1, Eq. (2) is equivalent to the equationwhere λ is a physical parameter, and the prime denotes the differentiation with respect to x. Obviously, u has the maximum value at x = 1/2, denoted by μ. In 1853, Liouville [10] found an analytic expression between μ and λ for N = 1. When 0 < λ < λ*, where λ* ≃ 3.51383, there exist two values of μ, 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 existence 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 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 δ-expansion 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–Fermi’s 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 example.
Section snippets
Homotopy analysis solution
Under the transformation
Eq. (3) becomessubject to the boundary conditions
From (6), it is obvious that w(x) can be expressed by the power serieswhere an is a coefficient. So, we can further make the transformationwhere γ is unknown and is dependent on λ. Here, γ is introduced to search for the multiple solutions, as described later. Using the above transformation, Eq. (5) becomes
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 example. The approach is convenient and efficient. Its validity is verified by comparing the approximation series with the known exact solution. Note that the Gelfand equation contains an exponent term exp(u) and thus has very strong nonlinearity. And different from perturbation
Acknowledgement
This work is supported by the “National Science Fund for Distinguished Young Scholars” (Approval No. 50125923) of Natural Science Foundation of China for the financial support.
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