0
$\begingroup$

Hi I know roughly how to use the Trapezoidal method for ODEs but am coming stuck when presents with a system, as follows

$$ \bf y'= \begin{bmatrix}\frac{1}{4}x & 1+x^2\\1/2 & -30x\end{bmatrix}+ \begin{bmatrix} cos\pi x \\ sin \pi x \end{bmatrix}$$ also $0<=x<=1$ and $ \bf y(0)=\begin{bmatrix} 0\\1 \end{bmatrix} $

So i know what the general case can be expressed , but problems arises when you consider $$ \bf W_{i+1} = \frac{2+hl(x_i)}{2-hl(x_{i+1})}W_i+ \frac{g(x_{i+1})+g(x_i)}{2-hl(x_{i+1})} $$ considering $l(x_{i+1})$ is a matrix i am guessing it would have to be a matrix multiplited by another matrix inverse but am struggling to see what that would look like and how to continue on ?

thanks

$\endgroup$
0
$\begingroup$

You just transform $$ y_+=y+\frac h2(f(x,y)+f(x_+,y_+)) $$ with $f(x,y)=A(x)y+b(x)$ to first get all of $y_+$ on one side and then isolate it $$ \left(I-\frac h2 A(x_+)\right)y_+ = \left(I+\frac h2 A(x)\right)y+\frac h2 (b(x)+b(x_+)) \\\iff\\ y_+ = \left(2I - h A(x_+)\right)^{-1}\left[\left(2I + h A(x)\right)y + h (b(x) + b(x_+))\right] $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.