Example 1

Transform the PDE by hand. Now proceed.

In[2]:=

  DSolve[u''[x]-s u[x]==-(1+Sin[Pi x]),u[x],x];
  U[x_,s_]:=Evaluate[u[x]/.Flatten[%]]
  U[x,s]

Out[2]=

                                     2
     C[1]       Sqrt[s] x        2 Pi  + 2 s + 2 s Sin[Pi x]
  ---------- + E          C[2] + ---------------------------
   Sqrt[s] x                                   2
  E                                     2 s (Pi  + s)

Include the initial conditions:

In[3]:=

  DSolve[{u''[x]-s u[x]==-(1+Sin[Pi x]),u[0]==1/s,u[1]==1/s},u[x],x];
  U[x_,s_]:=Evaluate[u[x]/.Flatten[%]]
  U[x,s]

Out[3]=

      2
  2 Pi  + 2 s + 2 s Sin[Pi x]
  ---------------------------
                2
         2 s (Pi  + s)

Transform back to get u(x,t).

In[4]:=

  u[x_,t_]:=InverseLaplaceTransform[U[x,s],s,t]
  u=u[x,t]

Out[4]=

      Sin[Pi x]
  1 + ---------
          2
        Pi  t
       E

In[5]:=

  p0=Plot[u/.t->0,{x,0,1},PlotRange->{0,2}];

In[6]:=

  p1=Plot[u/.t->.25,{x,0,1},PlotRange->{0,2}];

In[7]:=

  p2=Plot[u/.t->.5,{x,0,1},PlotRange->{0,2}];

In[8]:=

  Show[p0,p1,p2];

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