  We present two new algorithms for solving norm equations over global function
fields with at least one infinite place of degree 1 and no wild ramification.
The first of these is a substantial improvement of a method due to Ga\'{a}l and
Pohst, while the second approach uses index calculus techniques and is
significantly faster asymptotically and in practice. Both algorithms
incorporate compact representations of field elements which results in a
significant gain in performance compared to the Ga\'{a}l-Pohst approach. We
provide Magma implementations, analyze the complexity of all three algorithms
under varying asymptotics on the field parameters, and provide empirical data
on their performance.
