HOMOTOPY PERTURBATION METHOD FOR THE SOLUTION OF THE ELECTROSTATIC POTENTIAL DIFFERENTIAL EQUATION

This paper obtains an explicit analytical solution for nonlinear Poisson-Boltzmann equation by the homotopy perturbation method, which does not require a small parameter in the equation under study, so it can be applied to both the weakly and strongly nonlinear problems. The obtained results show the evidence of the usefulness of the homotopy perturbation method for obtaining approximate analytical solutions for nonlinear equations.


Introduction
Recently Oyanader and Arce [11] suggested a more effective, accurate, and mathematically friendly solution for prediction of the electrostatic potential, commonly used on electrokinetic research and its related applications.In order to obtain an explicit mathematical expression for the electrostatic potential, we have to solve the following nonlinear Poisson-Boltzmann equation: where u is the dimensionless electrical potential, λ the dimensionless inverse Debye length.The nonlinear term on the right-hand side of (1.1) is related to the free charge density.A very common simplification invokes the Debye-Hückel approximation usually written as As a result, the nonlinear Poisson-Boltzmann equation reduces to the following linear equation: 2 HPM for Poisson-Boltzmann equation Such approximation is valid for the case when −1 ≤ u ≤ +1.In this paper, we will suggest an alternative approach to the search for an explicit analytical solution for (1.1) by homotopy perturbation method [2,3].

A brief introduction to homotopy perturbation method
Homotopy perturbation method [2,3] is a novel and effective method, and can solve various nonlinear equations.Recently, some rather extraordinary virtues of the homotopy perturbation method have been exploited, and there has been a considerable deal of research in applying homotopy technique for solving various strongly nonlinear equations [1, 4-7, 9, 12, 13], a complete review is available in [10], to illustrate its basic ideas of the new method, we consider the mathematical pendulum, the equation can be expressed as with initial conditions u(0) = A and u (0) = 0. Equation (2.1) can be approximated as follows: According to the homotopy perturbation method, we construct a homotopy in the form where p ∈ [0,1] is an embedding parameter, ω is the frequency of the mathematical pendulum which is to be determined further.
It is obvious that when p = 0, (2.3) becomes a linearized equation, When p = 1, it becomes the original nonlinear one.So the changing process of p from zero to unity is just that of (2.4) to (2.2).The embedding parameter p monotonically increases from zero to unity as the trivial problem (2.4) is continuously deformed to the studied problem (2.2).This is the basic idea of homotopy method which is to continuously deform a simple problem easy to solve into the difficult problem under study.
The homotopy perturbation method [2,3,9] uses the imbedding parameter p as a small parameter.The basic assumption is that the solution of (2.3) can be written as a power series in p: The convergence of the above series is discussed in [9], the asymptotic character of the series is illustrated in [8,10].
L.-N.Zhang and J.-H.He 3 Substituting (2.5) into (2.3),collecting terms of the same power of p, gives ) Equation (2.6) can be solved easily, giving u 0 = Acos ωt.If u 0 is substituted into (2.7), and the resulting equation is simplified, we obtain In order to avoid secular terms in u 1 , ω is chosen to eliminate the coefficient of cosωt.This condition determines ω to be (2.9) The approximate period for mathematical pendulum reads (2.10) When A = π/2, the value obtained from (2.10) is T = 1.20T 0 , while the exact one is T ex = 1.16T 0 , where T 0 = 2π/ω, the 3.4% accuracy is remarkably good in view of the crude approximation, sin x ≈ x − (1/6)x 3 .If we approximate the sine function to the order x 5 , sinx ≈ x − (1/6)x 3 + (1/120)x 5 , by the same manipulation as illustrated above, we obtain The accuracy, for example, when A = π/2, reaches 0.86%.

Explicit analytical solution for Poisson-Boltzmann equation
We approximate the nonlinear Poisson-Boltzmann equation in the form Suppose that the linearized equation of (1.1) can be expressed as Hereby, ω is a modified inverse Debye length.Equation (3.2) is different from the Debye-Hüchel approximation (1.3) in that the former is valid for the whole solution domain, while the latter for −1 ≤ u ≤ +1.

HPM for Poisson-Boltzmann equation
Now we construct a homotopy in the form As an illustrative example, we consider the following boundary conditions: Supposing that the solution of (3.3) can be expressed in the form of (2.5), by the same operation as illustrated in the above section, we obtain the differential equations for u 0 and u 1 : ) The solution of (3.5) is Substituting u 0 into (3.6)results in Collecting the cosh ωx term, and eliminating the cosh ωx term from (3.8) altogether, we have or With this requirement, (3.8) becomes The solution of (3.11), incorporating with the boundary conditions, u 1 (−1)=0, u 1 (+1) = 0, and u 1 (0) = 0, is u 1 = A 3 192ω 2 cosh 3 ω cosh 3ωx − cosh 3ω cosh ω cosh ωx . (3.12) So we obtain the first-order approximation, which reads as u = u 0 + u 1 = Acosh ωx cosh ω + A 3 192ω 2 cosh 3 ω cosh 3ωx − cosh 3ω cosh ω cosh ωx , (3.13) where ω is solved from (3.10).In practice, we always use its zero-order approximate solution for simplicity: .14) .11)       L.-N.Zhang and J.-H.He 5