## 2 Improved Homotopy Analysis Method
### Improved Homotopy Analysis Method
#### 2.3.3 Convergence Theorem  
**Theorem 1** (Convergence Theorem).: _If the series_  
\[w_{0}(\xi)+\sum_{n=1}^{+\infty}\overline{w}_{n}(\xi) \tag{36}\]  
_converges, where \(\overline{w}_{n}(\xi)\) satisfies the high-order deformation Equations (31) and (32), and the definitions in Equations (33) and (34) are true, it is the solution to Equations (3) and (4)._  
The above assumption is then proved as below.  
Letting  
\[\overline{S}(\xi)=w_{0}(\xi)+\sum_{n=1}^{+\infty}\overline{w}_{n}(\xi) \tag{37}\]  
being the convergent series, a necessary condition for the series to converge is:  
\[\underset{n\rightarrow+\infty}{lim}\overline{w}_{n}(\xi)=0 \tag{38}\]  
From Equations (31), (34) and (38), we have  
\[\begin{array}{ll}&\h\sum\limits_{n=1}^{+\infty}\overline{R}_{n}(\overline{ w}_{n-1},\xi)\\ &=\sum\limits_{n=1}^{+\infty}\mathcal{L}[\overline{w}_{n}(\xi)-\chi_{n} \overline{w}_{n-1}(\xi)]\\ &=\mathcal{L}\bigg{\{}\sum\limits_{n=1}^{+\infty}[\overline{w}_{n}(\xi)-\chi _{n}\overline{w}_{n-1}(\xi)]\bigg{\}}\\ &=\mathcal{L}[\overline{w}_{1}+(\overline{w}_{2}-\overline{w}_{1})+( \overline{w}_{3}-\overline{w}_{2})+\ldots+(\overline{w}_{n}-\overline{w}_{n-1 })+\ldots]\\ &=\mathcal{L}\bigg{[}\underset{n\rightarrow+\infty}{lim}\overline{w}_{n}( \xi)\bigg{]}=0\end{array}\]  
Because \(\h\
eq 0\), so  
\[\sum\limits_{n=1}^{+\infty}\overline{R}_{n}(\overline{w}_{n-1},\xi)=0 \tag{39}\]From Equation (33)  
\[\sum_{n=1}^{+\infty}\overline{R}_{n}(\overline{w}_{n-1},\xi)=\sum_{n=1}^{+\infty} \frac{1}{(n-1)!}\frac{\partial^{n-1}\overline{N}[\overline{\Psi}(\xi,q)]}{ \partial q^{n-1}}\Bigg{|}_{q-0}=0 \tag{40}\]  
Generally, \(\overline{\Psi}(\xi,q)\) does not satisfy the original nonlinear Equation (3). Letting  
\[\overline{\delta}(\xi,q)=\overline{N}\big{[}\overline{\Psi}(\xi,q)\big{]} \tag{41}\]  
represents the residual error of Equation (3), then  
\[\overline{\delta}(\xi,q)=0\]  
corresponds to the exact solution of Equation (3). According to the above definition, the McLaughlin series of residual error \(\overline{\delta}(\xi;q)\) with respect to the embedded variable \(q\) is:  
\[\sum_{n=0}^{+\infty}\frac{q^{n}}{n!}\frac{\partial^{n}\overline{\delta}(\xi,q) }{\partial q^{n}}\Bigg{|}_{q=0}=\sum_{n=0}^{+\infty}\frac{q^{n}}{n!}\frac{ \partial^{n}\overline{N}[\overline{\Psi}(\xi,q)]}{\partial q^{n}}\[PERSON]{|}_{q=0} \tag{42}\]  
According to Equation (40), when \(q=1\), Equation (42) becomes:  
\[\overline{\delta}(\xi,1)=\sum_{n=0}^{+\infty}\frac{1}{n!}\frac{\partial^{n} \overline{\delta}(\xi,q)}{\partial q^{n}}\[PERSON]{|}_{q-0}=0 \tag{43}\]  
According to the definition of \(\overline{\delta}(\xi;q)\), when \(q=1\), Equation (43) is the exact solution to the original Equation (3). Therefore, as long as the below series:  
\[\mid w_{0}(\xi)+\sum_{n=1}^{+\infty}\overline{w}_{n}(\xi)\]  
converges, there exists a solution to the original Equation (3).  
**Theorem 2**.: _If the series in Equation (36) converges, where \(\overline{w}_{n}(\xi)\) satisfies the high-order deformation Equations (31) and (32), and the definitions in Equations (33) and (34) are true, there is:_  
\[\sum_{n=1}^{+\infty}\overline{R}_{n}(\overline{w}_{n-1},\xi)=0 \tag{44}\]  
This theorem is self-explanatory with reference to Equation (40).  
According to Theorems 1 and 2, only the initial guessing solution \(w_{0}(\xi)\), auxiliary parameter \(\mathbb{h}\), auxiliary linear operator \(\mathcal{L}\), and auxiliary nonlinear operator \(\mathcal{N}\) to ensure the convergence of series in Equation (36) should be derived.