# Magnus expansion for linear operators

I want to learn about Magnus expansion for the time dependent pdes of the form $u_t(t,x)=A(t)u(t,x)$. According to the wikipedia explanation http://en.wikipedia.org/wiki/Magnus_expansion $A$ has to be a matrix. However, I don't want to descritize in $x$ dimension to get a matrix which approximates $A$, I want to keep it as an operator. So, does the theory still holds by replacing a matrices by operators or I might get in trouble? I don't know much about operator theory but I want to make sure I can apply Magnus expansion to the case of operators as well.

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You can take $A$ to be an operator. However, you are likely to run into problems regarding the convergence of the series given by the Magnus expansion. As the Wikipedia article states, the series converges if $\int_0^T \|A(t)\| \,\mathrm{d}t < \pi$. If $A$ is an operator containing spatial derivatives, then it is unbounded, and the series will not converge.
thanks. Assuming $A(t):=a(t)u_x(t,x)+b(t)u_{xx}(t,x)$ and I want to apply Lie splitting to that. In order to calculate the splitting error I do the Taylor expansion and compare the two(for the solution itself and the result of splitting). However, how do boundary conditions come in play? Assume I solve it on $[0,T]\times[x_{m}, x_M]$ and I have $$u(t,x_m)=v(t,x),u(t,x_M)=w(t,x), u(0,x)=f(x)$$This is what I don't find for the Magnus expansion and splitting methods. They have to affect the Taylor expansion in some way, don't they? – Medan Sep 30 '12 at 13:30