How to set the default value as the saved value once a form of widget is saved?

You need a corresponding function UPDATE to complete this. See the Widgets API on the codex:

http://codex.wordpress.org/Widgets_API

Below is their example.

/**
 * Adds Foo_Widget widget.
 */
class Foo_Widget extends WP_Widget {

    /**
     * Register widget with WordPress.
     */
    function __construct() {
        parent::__construct(
            'foo_widget', // Base ID
            __('Widget Title', 'text_domain'), // Name
            array( 'description' => __( 'A Foo Widget', 'text_domain' ), ) // Args
        );
    }

    /**
     * Front-end display of widget.
     *
     * @see WP_Widget::widget()
     *
     * @param array $args     Widget arguments.
     * @param array $instance Saved values from database.
     */
    public function widget( $args, $instance ) {
        $title = apply_filters( 'widget_title', $instance['title'] );

        echo $args['before_widget'];
        if ( ! empty( $title ) )
            echo $args['before_title'] . $title . $args['after_title'];
        echo __( 'Hello, World!', 'text_domain' );
        echo $args['after_widget'];
    }

    /**
     * Back-end widget form.
     *
     * @see WP_Widget::form()
     *
     * @param array $instance Previously saved values from database.
     */
    public function form( $instance ) {
        if ( isset( $instance[ 'title' ] ) ) {
            $title = $instance[ 'title' ];
        }
        else {
            $title = __( 'New title', 'text_domain' );
        }
        ?>
        <p>
        <label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?></label> 
        <input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo esc_attr( $title ); ?>">
        </p>
        <?php 
    }

    /**
     * Sanitize widget form values as they are saved.
     *
     * @see WP_Widget::update()
     *
     * @param array $new_instance Values just sent to be saved.
     * @param array $old_instance Previously saved values from database.
     *
     * @return array Updated safe values to be saved.
     */
    public function update( $new_instance, $old_instance ) {
        $instance = array();
        $instance['title'] = ( ! empty( $new_instance['title'] ) ) ? strip_tags( $new_instance['title'] ) : '';

        return $instance;
    }

} // class Foo_Widget