how to display recent post by category on home page

hi I m new in WordPress i want to display recent posts based on categories like this

I found the following code but it seems not to work for me. here is the code

function wpb_postsbycategory() {
// the query
$the_query = new WP_Query( array( 
'category_name' => 'news', 
'posts_per_page' => 5 
) ); 

// The Loop
if ( $the_query->have_posts() ) {
$string .= '<ul class="postsbycategory widget_recent_entries">';
while ( $the_query->have_posts() ) {
    $the_query->the_post();
        if ( has_post_thumbnail() ) {
        $string .= '<li>';
        $string .= '<a href="' . get_the_permalink() .'" rel="bookmark">' . 
       get_the_post_thumbnail($post_id, array( 50, 50) ) . get_the_title() .'</a></li>';
        } else { 
        // if no featured image is found
        $string .= '<li><a href="' . get_the_permalink() .'" rel="bookmark">' . 
   get_the_title() .'</a></li>';
        }
        }
} else {
// no posts found
}
$string .= '</ul>';

return $string;

/* Restore original Post Data */
wp_reset_postdata();
}
 // Add a shortcode

  add_shortcode('categoryposts', 'wpb_postsbycategory');`