Why does my iPhone 5 battery drain so quickly?

I have iOS 8.1.3. My battery goes from 100% to 1% within an hour. This morning I used it to check the iPhone weather app then spoke to my wife for 10 minutes. When I got to work it had drained to 0%. I plugged it in to charge, and when it turned back on, it jumped to 45%. The battery percentages are quite erratic. It often jumps between 30% straight down to 1%, and on charge, vice versa.

I have checked the battery usage and it is showing a number of apps as having low signal. I am on EE and have constant network coverage issues so wonder whether that drains the battery. My 42 minute train communte from West Byfleet to waterloo is on a really badly covered network area, so I have stopped using the phone during this time.

I have reduced brightness to half way and turned location services off for most apps other than maps etc whilst using.

Things I will try now are turning off background app refresh, but I'm sure there is a more fundamental issue with the battery.

Any insights greatly received.

Rob