The first one is that she is given the task to solve the problem. The second one is that she will solve the problem (under her own free will) in the future.

"Shе іs to solve a very dіffіcult problem." is a very odd construction that you will almost never hear or use. It isn't wrong, but it isn't common. It doesn't just say what she's going to do, but also implies that it's her purpose for existing (or at least for being in the room). If you just want to say what will happen next, then use "She's going to solve a very difficult problem"

"She is to solve..." doesn't sound natural. Unless, you mean: "She's to solve..." a contraction for "She has to solve..." which is correct but may sound like "she is to solve". In that case, "she is going to solve..." and "she has to solve..." would mean two slightly different things: 1. She is going to solve... : She is in the process of solving (or) she will solve it in the future... 2. She has to solve... (She's to solve...) : She has been given a problem that she must solve.

To is a preposition —used to indicate the place, person, or thing that someone or something moves toward —used to indicate the place where someone participates in a particular activity —used to indicate the direction of something Going is a noun : the act of leaving a place : the condition of the ground for walking, running, etc. —used to describe a situation in which you are trying to make progress or do something When using to you are linking words . So probably she is going to solve a very difficult problem is much more appropriate