Small Engine Troubleshooting Problems? Ask a Mechanic Now.
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It sounds like you aren't getting a constant supply of fuel to keep the engine running. First thing that I would try is to remove the fuel cap and see if it will run any longer, if is does then the fuel tank vent is not working that is built into the fuel cap it may be plugged. The other thing that you can check is to see if you still have fuel in the carburetor bowl when the engine dies. The float or float valve may be sticking causing you to not get fuel to the carburetor in order to keep the engine running. When you first start the machine you have enough fuel in the bowl of the carburetor to run around 15 - 20 minutes even if the fuel tank were empty so it leads me to believe that you have a poor fuel supply coming to the carburetor.
No, sorry the diesel will not have a carburetor. But you could still have a fuel tank vent stopped up that would cause the problem. Can you tell what is happening at the fuel solenoid when the machine is running. I wonder if you might be loosing voltage to the fuel solenoid which would cause the solenoid to close and kill the engine. You might want to take a look at the fuel solenoid as well.
Yes, it is a separate item or part. On most diesels you will have an injector pump that pumps the fuel to the engine and through the injectors but the solenoid actually is attached to part of the fuel controls and holds controls open in order to run, if the solenoid doesn't hold the controls open then the engine will not run it can't get the fuel it needs.