Eliminating dual solutions?

Hi, is there some method for eliminating dual solutions to a given differential equation that are not wanted? Let me explain: Consider the differential equation

where is a linear operator (that doesn't operate in the time dimension). However, since the equation is going to be solved numerically, and the operator is very computationally expensive to be implemented directly, the equation is modified:

where is much computationally cheaper to implement. However, this equation gives dual solutions that do not satisfy the original equation (*), since (**) can be shown to allow any linear combination of solutions to the equations

(*, our original equation)

and to

So the modified equation can't be used solely. Is there some way to eliminate the solutions given by (***) when solving the system numerically, and only obtain solutions given by (*)? Thanks in advance.

Re: Eliminating dual solutions?

Re: Eliminating dual solutions?

I read in these course notes (on page 31) that "if a crazy looking operator like is okay, then the exact problem can be recast into a canonical problem". What is a canonical problem, and does that have anything to do with my question? Can that help me in some way? Thanks.

P.S. The linear operator that I'm using in my problem has the property that if it's applied to a function , the outcome will be which I guess means that can be expressed as , which looks very similar to the one used in the slide show.

**Edit:** Oh, and I forgot to say that the PDE is only supposed to be solved numerically, not analytically.