You can use custom filters in $args
(not tested)
add_filter( 'posts_where', 'find_by_title_filter', 10, 2 );
function find_by_title_filter( $where, &$wp_query )
{
global $wpdb;
if ( $find_title_arg = $wp_query->get( 'find_title' ) ) {
$where .= ' AND ' . $wpdb->posts . '.post_title LIKE \'%' . esc_sql( $wpdb->esc_like( $find_title_arg ) ) . '';
}
return $where;
}
$custom_var="a";
$paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
$args = array(
'paged' => $paged,
'post_type' => 'post',
'post_status' => 'publish',
'posts_per_page' => 2,
'find_title' => $custom_var
);