Driving a low resistance load with PWM
Your 1 ohm heating element will want 5 amps from a 5 volt supply.  That drawing indicates the circuit is only good for 500 mA output with an input voltage less than 8 volts.

If the circuit could deliver the required 5 volts, it would attempt to draw about 6.75 Amps from the 3.7 volt battery.  Can the battery deliver that current?