On what logical basis are the English letters in the alphabet (a, b, c, ... x, y, z serially arranged? Is there a pattern in the occurrence of vowels (a, e, i, o, and u)? Is the sequence they follow adopted from another language?
The short version is: the English alphabet is based on the Latinised version of the Greek alphabet, which in turn was inspired by the Phoenician alphabet.
The long version has little to do with English, although it can be a very interesting subject. You could start with this Wikipedia article.
It was the Greeks who inserted the vowel signs in their alphabet adopted from semitic alphabets which had no vowel signs. There seems to be some logic in the way they inserted the vowels in the alphabet but I have never read anything about it. They seem to have taken "a" as the first vowel with the fullest mouth opening, then by raising the tongue getting e and i and then by rounding the lips getting o and with more rounding getting u.
But I have no idea why the vowels have their special place in the alphabet. But astonishingly we have kept this order.