Custom WP_QUERY $args

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
            );