I have the following system of ode's $$ \begin{cases} \frac{du_1}{dx}=u_1+u_2,\\ \frac{du_2}{dx}=u_1u_2 \end{cases} $$ BCS $$\frac{du_1}{dx}|_{x=0}=1,\frac{du_2}{dx}|_{x=1}=2 $$ $$x \in\Omega=[0,1]$$ I want to solve it with Bubnov-Galerkin method
So my approximation to functions is: $$u_1(x)=\alpha_1x+\alpha_2x^2$$ $$u_2(x)=\beta_1x+\beta_2x^2$$ And weight functions are: $$w_1(x)=\alpha_{11}x+\alpha_{22}x^2$$ $$w_2(x)=\beta_{11}x+\beta_{22}x^2$$ Now I write weighted residual form of equations:
$$\int_0^1w_1(x)(\frac{du_1}{dx}-u_1-u_2)dx +w_1(0)(\frac{du_1}{dx}-1)|_{x=0}=0$$ $$\int_0^1w_2(x)(\frac{du_2}{dx}-u_1u_2)dx +w_2(1)(\frac{du_2}{dx}-2)|_{x=1}=0$$ Now paste my approximation: $$\int_0^1(\alpha_{11}x+\alpha_{22}x^2)(\alpha_1+2\alpha_2x-\alpha_1x-\alpha_2x^2-\beta_{1}x-\beta_{2}x^2)dx = 0$$ $$\int_0^1(\beta_{11}x+\beta_{22}x^2)(\beta_1+2\beta_2x-u_1u_2)dx + (\beta_{11}+\beta_{22})(\beta_1+2\beta_2-2)=0$$ Since the parameters $\alpha_{11},\alpha_{22},\beta_{11},\beta_{22}$ are arbitrary we get the following four equations: $$ \begin{cases} \int_0^1x(\alpha_1+2\alpha_2x-\alpha_1x-\alpha_2x^2-\beta_{1}x-\beta_{2}x^2)dx = 0\\ \int_0^1x^2(\alpha_1+2\alpha_2x-\alpha_1x-\alpha_2x^2-\beta_{1}x-\beta_{2}x^2)dx = 0\\ \int_0^1x(\beta_1+2\beta_2x-u_1u_2)dx + (\beta_1+2\beta_2-2)=0\\ \int_0^1x^2(\beta_1+2\beta_2x-u_1u_2)dx + (\beta_1+2\beta_2-2)=0\\ \end{cases} $$ As you can see in the last two integrals there is a term $u_1u_2$, which. is nonlinear and gives us coeeficients like $\alpha_1\beta_1$. How can I get rid of them, and make system linear? Or maybe I should make another approximations at the beginning?
The question was already answered in another thread
here
with two different solutions:
Regards