How to use the Ritz method with the weak form to approximate solution of differential equation

Question

I am having trouble understanding how to approximate the solution to this problem using the Ritz method and the weak form:

$$\frac{d^2u}{dx^2} - u=0; \ \ x \in [0,1]$$ $$u(x=0)=0; \ \ \frac{du}{dx} \bigg|_{x=1} = 20$$

We multiply the strong form of the equation by a weight function $w$ and integrate over the domain by parts to get

$$ \int_0^1w\left[\frac{d^2u}{dx^2} - u \right] dx = \int_0^1 w \frac{d^2 u}{dx^2} dx - \int_0^1wu \ dx$$

$$\int_0^1 \frac{du}{dx} \frac{dw}{dx} dx - \int_0^1 wu \ dx - 20 w(1) = 0$$

which is the weak form. Now, I am confused because when I try to change this into a matrix equation, I practically try to solve

$$\int_0^1 \frac{du}{dx} \frac{dw}{dx} dx = \int_0^1 wu \ dx + 20 w(1) $$

If I try to use, say $$u\approx\hat{u}=a_1 \phi_1 + a_2 \phi_2$$

I should be able to form a linear problem $$\mathbf{K} \vec{a} = \vec{b}$$

where $\mathbf{K}$ is a $2 \times 2$ matrix obtained from the left hand side of the equation where $w \to \phi_i$ and $u \to \phi_j$ to get the entries $K_{ij}$, but this substitution would not work to obtain the components of $\vec{b}$ since I this substitution gives me terms of $\phi_i$ and $\phi_j$ and this makes no sense since $\vec{b}$ is indexed only by one subindex. I know I am not understanding something, but this was my professor's explanation and I am confused.

Thank you so much for your help!

Answer 1

Everything in your OP looks correct.

I think you have just got into a muddle with the notation, and what is known and what is unknown.

$\phi_1$ and $\phi_2$ are known functions, i.e. you choose two sensible functions for the problem you want to solve, to give a "good" approximation to the solution you expect.

You can differentiate them with respect to $x$. Let $\dfrac{d\phi_i}{dx} = \phi'_i$.

If you let $w = \phi_1$, your integral equation becomes $$\int_0^1 (a_1 \phi'_1 + a_2 \phi'_2)\phi'_1\,dx - \int_0^1(a_1\phi_1 + a_2\phi_2)\phi_1\,dx = 20 \phi_1(1).$$ You can evaluate the integrals $\int\phi_1^2$, $\int \phi_2\phi_1$, $\int {\phi'_1}^2$, and $\int \phi'_2\phi'_1$ (either analytically or numerically) since the functions $\phi_i$ and $\phi'_i$ are known.

You then get an equation of the form $$K_{11} a_1 + K_{12} a_2 = b_1$$ where you know the $K$ and $b$ terms.

Similarly letting $w = \phi_2$ gives you the equation $$K_{21} a_1 + K_{22} a_2 = b_2.$$

These are the two rows of the matrix equation $\mathbf{K}\vec a = \vec b$.