In the next chapter the methods most commonly used in practice for obtaining solutions to PDEs subject to given boundary conditions will be considered. These methods include the separation of variables, integral transforms and Green’s functions. This division of material is rather arbitrary and has been made only to emphasise the general usefulness of the latter methods. In particular, it will be readily apparent that some of the results of the present chapter are in fact solutions in the form of separated variables, but arrived at by a different approach.

<img src="images/img_677_fig20.1.jpg" alt="image"/>

As in other chapters we will focus our attention on the equations that arise most often in physical situations. We will restrict our discussion, therefore, to linear PDEs, i.e. those of first degree in the dependent variable. Furthermore, we will discuss primarily second-order equations. The solution of first-order PDEs will necessarily be involved in treating these, and some of the methods discussed can be extended without difficulty to third- and higher-order equations. We shall also see that many ideas developed for ordinary differential equations (ODEs) can be carried over directly into the study of PDEs.

In this chapter we will concentrate on general s

<img src="images/img_677_fig20.2.jpg" alt="image"/>

#!C:/Perl/bin/perl.exe -w
#!C:/Perl/bin/perl.exe -w
#!C:/Perl/bin/perl.exe -w
use File::Spec::Functions;
use File::Basename;
open(FH,"<","path1.txt") or die "$!";
$i=1;
$section_flag=1;
while(<FH>)
{
  if($i==1){$path=$_;}
  $i++;
}
close(FH);
opendir(DIR,$path) or die "can't open $path $!";
@files=readdir(DIR);
close(DIR);
foreach(@files)
{
    if (-f $path . "/" . $_ )
    {
       if($_=~m/.html|.xhtml/i)
       {
       $file=catfile($path,$_);
             if($_=~m/toc|content|tableof/i){$toc=$file;}
                else
                {
                   push(@file_list,$file);
                }
       }
    }
}
#print @file_list;
foreach( @file_list)
{
    $f=$_;
   #print "$f\n";
   open(FH,"<", "$f")or die "unable to open toc file $!";
  @content=<FH>;
    foreach(@content)
    {
      $a=$_;
   while($a=~m /figer[ ]([0-9])+.([0-9])+|figer([0-9])+.([0-9])+|fig[ 
+][0-9]+.([0-9])+|fig[0-9]+.([0-9])+|figer[ ]([0-9])+|figer([0-9])+|fi
+g[ ][0-9]+|fig[0-9]+|fig([0-9])+-([0-9])+/i)
     {
     $a= $';
     $match_key=$&;
    #print " --match : -- $match_key \n ";
      $front=$`;
        if($_=~m/<p class="fig-img"|<p class="image"/i)
        {
          $match_key1=$';
          $replace_line=$&."id=$match_key".$';
         #print " $replace_line \n";
          $rep1=$rep1.$replace_line;
          #print" ------- $replace_line \n";
           if($match_key1=~m/fig( |[a-z])*(\d+)(\.(\d+))?/i)
            {

               $key="figure ".$2.'.'.$4;
                a();

            }

        }
         #push(@file2,$a);

     }
     $rep1=$rep1.$a;
   # print " .... $rep1 \n";
     push(@figer,$rep1);
     $rep1="";
  }
close(FH);
}

sub a()
{

 foreach(@file_list)
 {
    $ff=$_;

      my($filename, $directories, $suffix) = fileparse($f);
      $ka=$filename;
     open(FH,"<", "$ff")or die "unable to open toc file $!";
  @content1=<FH>;
    foreach(@content1)
    {
      c();
        # print"11111111111 $rep \n";
      }
close(FH);

 }
   # push(@figer1);
    print "@figer1 \n";

}

sub c()
{
  $b=$_;
         if($b=~m/<p class="fig-img">|<p class="image">|<p class="capt
+ion">|<p class="fig-caption">/i)
       {
       push(@file1,$b);
       }
       else
       {
            while($b=~m/$key/i)
            {
                   $b=$';
               $match_key3=$&;
               $before3=$`;
              $replace_1=$before3.'<a href='.$ka.'"'.$match_key.'"'.'>
+'.$match_key3.'</a>';
              $rep=$rep.$replace_1;
            }


       push(@file1,$b);

       }

         $rep=$rep.$b;
         push(@figer1,$rep);
         $rep="";

}

error
------
1.its print separate separate function

output
----------
1.i need output is the @figer and @figer1 array is merge and update in
+ once only
2.and update that .html file
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