I've never used Bio::DB::Fasta, but having read the man page for it just now, and looking again the OP task description, I think what you want is something like this: 
use strict;
use Bio::DB::Fasta;

my $db = new Bio::DB::Fasta( "Library.fa" )  # this should be the bigg
+er file
    or die "Failed to create Fasta DB object on Library.fa\n";

open( IDLIST, "Name.fa" );  # this is the list of ID strings
while ( <IDLIST> ) {
    my ($id) = (/^>(\S+)/);  # capture the id string (without the init
+ial ">")
    print ">$id\n", $db->seq( $id ), "\n";
}

[download]
As for the code you've just posted, you didn't really try to use the DB::Fasta module, and you don't seem to understand the syntax for a while loop (you need parens around the conditional, and a block of code to be the body of the loop). As for doing a regex match:  /regex/ by itself will test the contents of $_, whereas $var =~ /regex/ tests the contents of $var; what you have,  blahblah && =~ /regex/ is a syntax error.

Also, you need to be aware that "@line1" is a completely different thing (different variable with different storage) from "$line1". To refer to an element of an array, you need square brackets and an array index:  $line1[0] etc.

In reply to Re^3: comparing two fasta files by graff
in thread comparing two fasta files by nemo2

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