## 4 The Optimal Auxiliary Functions Method  
Let us consider a general form of nonlinear differential equations and boundary conditions:  
\[L[T(t)]+N[T(t)]=0,\;\;\;t\in D,\;B\bigg{(}T(t),\frac{dT(t)}{dt}\bigg{)}=0 \tag{70}\]  
where in the last two equations, \(L\), \(N\), \(t\), \(T(t)\), \(D\), and \(B\) are the linear operator, nonlinear operator, time, the unknown function of time, the domain of interest, and the boundary operator, respectively. According to OAFM, an approximate solution of Equations (69) and (70), \(\overline{T}(t)\), has only two components:  
\[\overline{T}(t)=T_{0}(t)+T_{1}(t,C_{1},C_{2},\ldots C_{n}) \tag{71}\]  
where \(n\) is an arbitrary integer number, and \(C_{1}\), \(C_{2}\), \(\ldots\), \(C_{n}\) are unknown parameters.  
The initial approximation \(T_{0}(t)\) can be obtained from the linear equation:  
\[L[T_{0}(t)]=0,\;B\bigg{(}T_{0}(t),\frac{dT_{0}(t)}{dt}\bigg{)}=0 \tag{72}\]  
In general, the solution of this equation has the form:  
\[T_{0}(t)=\sum_{i=1}^{m}K_{i}f_{i}(t) \tag{73}\]  
where the coefficients \(K_{i}\), the functions \(f_{i}\), and the integer number m are known by elementary calculations. The nonlinear operator \(N\) from Equation (69) calculated for the value of \(T_{0}(t)\) and given by Equation (73) may be easily written in the form:  
\[N[T_{0}(t)]=\sum_{i=1}^{m_{1}}p_{i}g_{j}(x) \tag{74}\]  
where the coefficients \(p_{i}\), the function \(g_{i}\), and the integer number m\({}_{1}\) are known.  
With the help of the method of influence function, operator method, and Cauchy method, we considered the unknown first approximation \(T_{1}(t,C_{1},C_{2},\ldots C_{n})\) as the solution of the linear differential equation:  
\[L[T_{1}(t,C_{1},C_{2},\ldots C_{n})]+\sum_{k=1}^{n}C_{k}F_{k}(f_{i},g_{j})=0, \;B\bigg{(}T_{1},\frac{dT_{1}}{dt}\bigg{)}=0 \tag{75}\]  
in which \(C_{k}\) represents \(n\) unknown parameters, and \(F_{k}\) represents the \"auxiliary functions\" depending on only some functions, \(f_{i}\) and \(g_{j}\), involved in Equations (73) and (74). More precisely, the auxiliary functions \(F_{k}\) are combinations of known functions \(f_{i}\) and \(g_{j}\). We noticed that we had the freedom to choose the values of the integer \(n\) and of the auxiliary functions \(F_{k}\)[31].  
It is clear from Equation (71) that the boundary conditions were fulfilled, and Equation (69) responded to all initial/boundary conditions. Following our technique, the parameters \(C_{k}\), \(k=\overline{1,n}\) were optimally identified by some known procedures, such as the Ritz method, the Galerkin method, the least squares method, the Kantorovich method, and the collocation method, or by minimizing the square residual error.  
In this way, the auxiliary functions \(F_{k}\), the parameters \(C_{k}\), and analytical solution \(\overline{T}(t)\) were known. It was remarkable that if the number of the convergence-control parameters increased, then the accuracy of the results obtained by our procedure grew. Finally, we can emphasize that any nonlinear differential equation can be reduced to only two linear differential equations, and OAFM does not suppose the presence of small parameters in the governing equations or in the initial/boundary conditions.