I solved the equation by the method of characteristics and found x(t)=sinh-1(ektsinhx0) and so the solution is u(x,t)=u0{sinh-1[e-ktsihnx]}. Now my real hustle is the following(Typed exactly as the lecturer asked it): What you really want to see is how a ‘solitary’ initial state such as u0(x)={ a0x2(1-x)2 for 0

Question

I solved the equation by the method of characteristics and found x(t)=sinh-1(ektsinhx0) and so the solution is u(x,t)=u0{sinh-1[e-ktsihnx]}.
Now my real hustle is the following(Typed exactly as the lecturer asked it):

What you really want to see is how a ‘solitary’ initial state such as
                                                                                        u0(x)={ a0x2(1-x)2 for 0<=x<=1
                                                                                                 { 0 for x>1
                                                                    
will de