# The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method
## 2 Galerkin Method for Solutions of Nonlinear Differential Equations  
To implement the [PERSON] method to solve nonlinear differential equations, the fundamental concepts and procedures are introduced to provide an essential understanding. The rigorous mathematical procedure and associated assumptions and proofs are not included in this paper, but they can be found in many mathematical textbooks and monographs [10; 32]. A general nonlinear differential second-order equation is  
\[N\left(x,y,y^{k},y^{\prime},y^{\prime\prime},p\right)=0,\ x\in[a,b], \tag{1}\]  
where \(y,y^{\prime}(y^{\prime\prime}),p,x,a\), and \(b\) are the function, derivatives with respect to the spatial variable, nonlinear term with the power of \(k\), parameter, spatial variable, lower bound, and upper bound, respectively. With prescribed boundary and initial conditions, the approximate solution, also generally asymptotic, can be assumed as  
\[y=\sum_{n=0}^{M}A_{n}Y_{n}(x), \tag{2}\]  
where the chosen function \(Y_{n}(x)\) satisfies the boundary conditions, and \(A_{n}\) are coefficients to be determined with the limitations by an integer \(M\). As a general rule of thumb, the trial solutions \(Y_{n}(x)\) can be conveniently chosen from the solutions of a simplified linear equation, so calculations will be simple and efficient. Furthermore, it is important to expect that these solutions are also orthogonal. There is abundant literature on the optimal selections of trial functions, and one must deal with some typical equation types and appropriate boundary conditions. Applying the [PERSON] method, the problem in Equation (1) is transformed to  
\[\int_{a}^{b}N\left(x,y,y^{k},y^{\prime},y^{\prime\prime},p\right)\sum_{n=0}^{M} \delta A_{n}Y_{n}(x)\;\mathrm{d}x=0. \tag{3}\]  
Because the [PERSON] method has proven its capability to provide accurate solutions to linear differential equations as a powerful and robust approximate procedure, it is also thought the nonlinear equations can be solved in a similar manner with equal strength. With this hypothesis in mind, the arbitrary variation in independent coefficients in Equation (3) results in a series of nonlinear algebraic equations of unknown amplitudes after integration as  
\[\overline{N}_{n}\left(A_{m},\ A_{m}^{q},p\right)=0,\ n,m=1,2,\cdots,M, \tag{4}\]  
using the equation and trial functions with parameters \(p\) and \(q\). By solving the system of nonlinear algebraic equations for amplitude \(A_{m}\), the approximate solution can be obtained. It is better to have orthogonal functions in the domain of integration for a simple form and least couplings of unknowns in Equation (4). After obtaining solutions of the coefficients, the solution is in Equation (2) for manipulations in relation to the problem. If a simplified procedure is sought, with a careful selection of trial functions satisfying the boundary conditions, the procedure starts with  
\[y=A_{0}Y_{0}(x), \tag{5}\]  
Then, the first approximate equation is  
\[\int_{a}^{b}N\left(x,A_{0}Y_{0}(x),A_{0}^{k}Y_{0}^{k}(x),A_{0}Y_{0}^{\prime}(x ),A_{0}Y_{0}^{\prime\prime}(x),p\right)Y_{0}(x)\mathrm{d}x=0. \tag{6}\]  
The coefficient \(A_{0}\) is to be solved from the above equation. To improve the solution with an additional function, it is further assumed that the updated solution is  
\[y=A_{0}Y_{0}(x)+A_{1}Y_{1}(x),\ \ \delta y=\delta A_{1}Y_{1}(x). \tag{7}\]  
Then, the following Equation is for the unknown amplitude \(A_{1}\)  
\[\int_{a}^{b}N\left(x,y,y^{k},y^{\prime},y^{\prime\prime},p\right)Y_{1}(x) \mathrm{d}x=0, \tag{8}\]  
by using known \(A_{0}\) from the procedure above in Equation (6). By repeating this procedure with the successive addition of higher-order solutions, other coefficients of higher-order approximations can be successively obtained for an improved solution. With the two steps for the approximate solution in the above demonstration, the results have to be compared before a conclusion can be drawn on the advantage of the Galerkin method.