In the case that there isn't an oven;
is it possible to make cake on electric cooktop by using basic kitchenware?
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Yes, but it's a pain (and depends how much kitchenware you have). When I was in the Peace Corps, I learned how to construct an "oven". You'll need two pots of the same diameter that are big enough to enclose your cake pan, hopefully with about at least an inch of room on all sides. (Just one pot is needed if it's tall enough.)
Put one pot on the burner, then put something in it to elevate the cake pan at least 1/2 inch above the bottom of the pot. Three small rocks work well, as will three stacks of coins. Put the cake pan on top of this stand, then put the last pot upside-down on top as a lid. (If your first pot is tall enough, you would only need a regular lid, not a second pot.) It's best to set this up on the burner as moving it can be risky. Use a low setting for the burner and expect a longer baking time.
This method can work on a stove or on a charcoal grill (with very low heat in the case of grill). It will work better if you can add heat from the top as well (e.g. put a few coals on top). It's finicky though and takes some practice. Bread is a pretty safe thing to try, though I would guess that a box-cake would do okay.
Microwave cake is an easier option, just google "microwave cake" and you'll find lots of recipes.
Not really. But you might want to look into English-style steamed puddings. Typically these are cake-like desserts steamed in a porcelain bowl sitting in a covered bath of boiling water. They're a bit moister than cake and don't hold their shape as nicely. Often they are served with a hot rum or caramel sauce.