Galerkin method for nonlinear ode I'm trying to solve the following differential equation:
$$\frac{d^2u}{dx^2}=\frac{du}{dx}u+u^2+x$$
$$x \in \Omega=[0,1]$$
$$BCS:u|_{x=0}=1;\frac{du}{dx}|_{x=1}=1$$
You can see that the right side contains $u^2$. So when I paste it in the weighted residual form, I get nonlinear term.
For example, if I have approximation:
$$ u=1+\sum_{i=1}^n\alpha_i x^i$$
There will be nonlinear integral in weighted residuals
$$\int (1+\sum_{i=1}^n\alpha_i x^i)^2dx$$
That's why the system will be nonlinear. What am I missing?
I tried to switch from $u$ to $u^2$ in equation because $u\frac{du}{dx}=\frac{1}{2}\frac{du^2}{dx}$, but can't make it for $\frac{d^2u}{dx^2}$
Edit, according to the answer:
I won't write BCS integrals, because they don't make real sense in the question. I'll write only the integral in the main domain. So I have
$$\int_0^1w(\frac{d^2u}{dx^2}-\frac{du}{dx}u-u^2-x)dx=0$$
$w-$weight function. Paste approximation of $u$. Let's take $n = 2$
$$\int_0^1w(2\alpha_2-(\alpha_1 + 2\alpha_2 x)(1+\alpha_1 x +\alpha_2 x^2)-(1+\alpha_1 x +\alpha_2 x^2)^2-x)dx=0$$
Take in account Bubnov-Galerkin approximation of weight function:
$$ w=\beta_1x+\beta_2x^2$$
$$\int_0^1\beta_1x(2\alpha_2-(\alpha_1 + 2\alpha_2 x)(1+\alpha_1 x +\alpha_2 x^2)-(1+\alpha_1 x +\alpha_2 x^2)^2-x)dx +\int_0^1\beta_2x^2(2\alpha_2-(\alpha_1 + 2\alpha_2 x)(1+\alpha_1 x +\alpha_2 x^2)-(1+\alpha_1 x +\alpha_2 x^2)^2-x)dx=0$$
From here since $\beta_i $ arbitrary we have system
$$\begin{cases}
 \int_0^1x(2\alpha_2-(\alpha_1 + 2\alpha_2 x)(1+\alpha_1 x +\alpha_2 x^2)-(1+\alpha_1 x +\alpha_2 x^2)^2-x)dx =0\\
\int_0^1x^2(2\alpha_2-(\alpha_1 + 2\alpha_2 x)(1+\alpha_1 x +\alpha_2 x^2)-(1+\alpha_1 x +\alpha_2 x^2)^2-x)dx=0
\end{cases}
$$
Here we exactly have unknowns only $\alpha_i;i=1,2$.But if we extend polynomial to $2n=4$ we will have new  $\alpha_i;i=1..4$ with 2 equations only
Edit 2:
Actually I need two terms approximation, so I don't think that switching to 2n terms and then solving 2n equations is the key point. I suppose we should simplify ode, or choose another interpolation functions rather then $x^i$
 A: You missed nothing. The product is non-linear. However why don't you extend your polynomial expansion with
$$\int (1+\sum_{i=1}^n\alpha_i x^i)^2dx\equiv\int  (1 + \sum_{i=1}^{2n}\tilde{\alpha_i} x^i) dx.$$
The product of $u\cdot u$ is still a polynomial, however with a higher polynomial degree of at least $2n$.
Then you will get the Galerkin solution if you integrate $$\int (1 + \sum_{i=1}^{2n}\tilde{\alpha_i} x^i) dx.$$ The Galerkin solution are the first $n$ coefficients of $\tilde{\alpha_i}$. Simply spoken:

*

*You only consider the first $n$ coefficients

*Coefficients greater $n$ are neglected

The truncation of the additional $n$ modes can be  intepreted as projection in a $2n$ dimensional space onto a $n$ dimensional space where the solution is orthogonal to the chosen subspaces.
This is the key property of the Galerkin approach.
Regards
A: Considering instead the ODE
$$
u''+u'u+u^2-x=0\ \ \ \ \ \ \ (1)
$$
with better behavior regarding the polynomial approximation, the Galerkin procedure can be handled as follows.
1 - Choosing a convenient orthogonal basis into the interval as for instance the shifted Tchebicheff polynomials $\theta_k$ in $[0,1]$  we make an approximating sequence as
$$
u_n(x) = \sum_{k=0}^n a_k \theta_k(x)\ \ \ \ \ \ \ (2)
$$
2 - Calculate the residual $r_n(x,a_k)$ from $(1)$ after substitution of $(2)$
3 - Calculate the relationships
$$
g_i(a_k) = \int_0^1 r_n(x, a_k)\theta_i(x) dx, \ \ i = 1,\cdots, n
$$
4 - Calculate the boundary conditions
$$
\cases{b_1(a_k) = u_n(0)-1\\
       b_2(a_k) = u'_n(0)-1}
$$
5 - Solve the minimization problem
$$
\min_{a_k}\sum_{i=0}^n g_i^2(a_k)\ \ \ \text{s. t.}\ \ \{b_1(a_k) = 0,  b_2(a_k) = 0\}
$$
Follows a MATHEMATICA script to illustrate that
t[x, 0] = 1;
t[x, 1] = x;
t[x_, k_] := t[x, k] = 2 x t[x, k - 1] - t[x, k - 2]
n = 4;
theta = Table[t[x, k], {k, 0, n}];
thetas = theta /. {x -> 2 y - 1};
u[x_] := Sum[Subscript[a, k]  thetas[[k]], {k, 1, n}]
A = Table[Subscript[a, k], {k, 1, n}]
d[u_, x_] := D[u, x, x] + D[u, x] u + u^2 - x
equs = Table[Integrate[d[u[y], y] thetas[[k]], {y, 0, 1}], {k, 1, n}];
bc1 = (u[y] /. {y -> 0}) - 1
bc2 = (D[u[y], y] /. {y -> 0}) - 1
sol = NMinimize[{equs.equs, bc1 == bc2 == 0}, A]
u0 = u[x] /. sol[[2]];
solux = NDSolve[{d[v[x], x] == 0, v[0] == v'[0] == 1}, v, {x, 0, 1}][[1]];
plot1 = Plot[Evaluate[v[x] /. solux], {x, 0, 1}, PlotStyle -> Red];
plot2 = Plot[u0, {y, 0, 1}];
Show[plot1, plot2]

Attached a plot showing in red the solution for $(1)$ and in blue for $n = 4$ the approximation.

