The First Alternative of the Optimal Homotopy Asymptotic Method

Abstract

In this chapter, we consider m = 2 into Eq. 2.29 such that we obtain the second-order approximate solution in the form

$$ \overline{u}\left(x,{C}_i\right)={u}_0(x)+{u}_1\left(x,{C}_i\right)+{u}_2\left(x,{C}_i\right) $$

where the terms u₀, u₁ and u₂ are given by the linear differential equations 2.19, 2.20, 2.23, 2.25 and 2.26 respectively or, more precisely, by the following equations:

$$ \begin{array}{cc}\hfill L\left({u}_0(x)\right)+g(x)=0,\hfill & \hfill B\left({u}_0,\frac{d{u}_0}{dx}\right)\hfill \end{array}=0 $$

$$ \begin{array}{cc}\hfill L\left({u}_1\left(x,{C}_i\right)\right)={H}_1\left(x,{C}_i\right){N}_0\left({u}_0(x)\right),\hfill & \hfill B\left({u}_1,\frac{d{u}_1}{dx}\right)=0\hfill \end{array} $$

$$ \begin{array}{c}\hfill \begin{array}{l}L\left({u}_2\left(x,{C}_i\right)\right)-L\left({u}_1\left(x,{C}_i\right)\right)=\\ {}={H}_1\left(x,{C}_i\right)\left[L\left({u}_1\left(x,{C}_i\right)\right)+{N}_1\left({u}_0(x),{u}_1\left(x,{C}_i\right)\right)\right]+{H}_2^{*}\left(x,{C}_i\right){N}_0\left({u}_0(x)\right),\end{array}\hfill \\ {}\hfill B\left({u}_2,\frac{d{u}_2}{dx}\right)=0\hfill \end{array} $$