question on result of DSolve for wave PDE on string. summation problem

Question

I was using Mathematica 11.2 on windows 7, to verify my hand solution to the following 1D wave PDE.

Mathematica gives the solution, which is almost correct, but the sum it gives is over all $n$, where the result should be over odd $n$ only.

I am posting this question to ask if this is a bug or not. Using the result as given by Mathematica will not even run, since it gives division by zero.

Will show my hand solution and show that Mathematica's output will be correct, if the series it gives, it summed over odd n only. Then it will match my hand solution.

Here is the PDE to solve

Solve wave PDE $u_{tt}=4u_{xx}$ on bounded domain $0<x<\pi,t>0$ with boundary conditions $u\left( 0,t\right) =0,u\left( \pi,t\right) =0$ and initial conditions $u\left( x,0\right) =\sin^{2}x,u_{t}\left( x,0\right) =0$.

This is the result from hand solution:

$$ u\left( x,t\right) =\frac{-8}{\pi}\sum_{n=1,3,5,\cdots}^{\infty}\frac {1}{n^{3}-4n}\cos\left( 2nt\right) \sin\left( nx\right) $$

Here is Mathematica's result

ClearAll[u,t,x,n];
pde  = D[u[x,t],{t,2}]==4D[u[x,t],{x,2}];
ic   = {Derivative[0,1][u][x,0]==0,u[x,0]==Sin[x]^2}
bc   = {u[0,t]==0,u[Pi,t]==0};
sol  = DSolve[{pde,bc,ic},u[x,t],{x,t}];
sol  = sol/.K[1]->n  (*n looks better than K[1] for index *)

One can see that Mathematica result matches hand solution when $n$ is odd $1,3,5,\dots$. When $n$ is even, for $n=2$, there will be a division by zero above. So $n=2$ can't be there. And all even $n$ should not be there.

So I think Mathematica solution above is wrong! Do you agree? If so, is there a way to fix this and Make it generate the correct solution?

hand solution

Putting the PDE in standard form $u_{tt}=a^{2}u_{xx}$ shows that $a=2$

Applying separation of variables gives \begin{align*} T^{\prime\prime}X & =4X^{\prime\prime}T\\ \frac{1}{4}\frac{T^{\prime\prime}}{T} & =\frac{X^{\prime\prime}}{X}=-\lambda \end{align*} The eigenvalue ODE\ is $X^{\prime\prime}+\lambda X=0$ with $X\left( 0\right) =0,X\left( \pi\right) =0$. This has eigenfunctions $\Phi_{n}\left( x\right) =\sin\left( \sqrt{\lambda_{n}}x\right) $ with $\lambda_{n}=n^{2}$ where $n=1,2,3,\cdots$. The time ODE becomes $$ T^{\prime\prime}+4\lambda_{n}T=0 $$ Since $\lambda_{n}>0$, the solution is \begin{align*} T\left( t\right) & =A_{n}\cos\left( \sqrt{4\lambda_{n}}t\right) +B_{n}\sin\left( \sqrt{4\lambda_{n}}t\right) \\ & =A_{n}\cos\left( 2nt\right) +B_{n}\sin\left( 2nt\right) \end{align*} And $$ T^{\prime}=-2nA_{n}\sin\left( 2nt\right) +2nB_{n}\cos\left( 2nt\right) $$ Since $T^{\prime}\left( 0\right) =0$, then the above implies that $B_{n}=0$. Therefore the solution simplifies to $$ T_{n}\left( t\right) =A_{n}\cos\left( 2nt\right) $$ And the fundamental solution becomes \begin{align*} u_{n} & =T_{n}X_{n}\\ & =c_{n}\cos\left( 2nt\right) \sin\left( nx\right) \end{align*} Hence by superposition, the general solution is $$ u\left( x,t\right) =\sum_{n=1}^{\infty}c_{n}\cos\left( 2nt\right) \sin\left( nx\right) $$ At $t=0$, $u\left( x,0\right) =\sin^{2}x$, therefore the above becomes $$ \sin^{2}x=\sum_{n=1}^{\infty}c_{n}\sin\left( nx\right) $$ Applying orthogonality gives \begin{align} \int_{0}^{\pi}\sin^{2}x\sin\left( nx\right) dx & =c_{n}\frac{\pi}{2} \tag{1}\\ \int_{0}^{\pi}\left( \frac{1}{2}-\frac{1}{2}\cos2x\right) \sin\left( nx\right) dx & =c_{n}\frac{\pi}{2}\nonumber \end{align} To evaluate $\int_{0}^{\pi}\left( \frac{1}{2}-\frac{1}{2}\cos2x\right) \sin\left( nx\right) dx$, it is split into $\int_{0}^{\pi}\left( \frac {1}{2}\sin\left( nx\right) -\frac{1}{2}\cos2x\sin\left( nx\right) \right) dx$. But the first part is \begin{align*} \int_{0}^{\pi}\frac{1}{2}\sin\left( nx\right) dx & =-\frac{1}{2n}\left( \cos\left( nx\right) \right) _{0}^{\pi}\\ & =-\frac{1}{2n}\left( \cos\left( n\pi\right) -1\right) \end{align*} For even $n=2,4,\cdots$ the above vanishes. For odd $n=1,3,5,\cdots$ the above becomes $$ \int_{0}^{\pi}\frac{1}{2}\sin\left( nx\right) dx=\frac{1}{n} $$ Now the second integral is evaluated $$ \int_{0}^{\pi}-\frac{1}{2}\cos2x\sin\left( nx\right) dx=-\frac{1}{2}\int _{0}^{\pi}\cos2x\sin\left( nx\right) dx $$ Using $\int_{0}^{\pi}\sin\left( px\right) \cos\left( qx\right) dx=-\frac{\cos\left( p-q\right) x}{2\left( p-q\right) }-\frac{\cos\left( p+q\right) x}{2\left( p+q\right) }$, then the above becomes, where $p=n,q=2$

\begin{align*} -\frac{1}{2}\int_{0}^{\pi}\sin\left( nx\right) \cos2xdx & =-\frac{1} {2}\left( -\frac{\cos\left( n-2\right) x}{2\left( n-2\right) }-\frac {\cos\left( n+2\right) x}{2\left( n+2\right) }\right) _{0}^{\pi}\\ & =\frac{1}{2}\left( \frac{\cos\left( n-2\right) x}{2\left( n-2\right) }+\frac{\cos\left( n+2\right) x}{2\left( n+2\right) }\right) _{0}^{\pi }\\ & =\frac{1}{2}\left( \frac{\cos\left( n-2\right) \pi}{2\left( n-2\right) }+\frac{\cos\left( n+2\right) \pi}{2\left( n+2\right) }-\frac{1}{2\left( n-2\right) }-\frac{1}{2\left( n+2\right) }\right) \end{align*} For even $n=2,4,\cdots$ the above vanishes, since it becomes $\frac{1} {2}\left( \frac{1}{2\left( n-2\right) }+\frac{1}{2\left( n+2\right) }-\frac{1}{2\left( n-2\right) }-\frac{1}{2\left( n+2\right) }\right) $, and for odd $n=1,3,5,\cdots$, the above becomes \begin{align*} -\frac{1}{2}\int_{0}^{\pi}\sin\left( nx\right) \cos2xdx & =\frac{1} {2}\left( \frac{-1}{2\left( n-2\right) }-\frac{1}{2\left( n+2\right) }-\frac{1}{2\left( n-2\right) }-\frac{1}{2\left( n+2\right) }\right) \\ & =\frac{1}{2}\left( \frac{-2}{2\left( n-2\right) }+\frac{-2}{2\left( n+2\right) }\right) \\ & =\frac{-1}{2\left( n-2\right) }+\frac{-1}{2\left( n+2\right) }\\ & =-\frac{n}{n^{2}-4} \end{align*} Therefore, the final result of integration is \begin{align*} \int_{0}^{\pi}\sin^{2}x\sin\left( nx\right) dx & =\frac{1}{n}-\frac {n}{n^{2}-4}\qquad n=1,3,5,\cdots\\ & =-\frac{4}{n\left( n^{2}-4\right) }\qquad n=1,3,5,\cdots \end{align*} Hence from (1), this results in \begin{align*} c_{n} & =-\frac{2}{\pi}\frac{4}{n\left( n^{2}-4\right) }\\ & =-\frac{8}{\pi n\left( n^{2}-4\right) }\qquad n=1,3,5,\cdots \end{align*} Hence the final solution is $$ u\left( x,t\right) =\frac{-8}{\pi}\sum_{n=1,3,5,\cdots}^{\infty}\frac {1}{n^{3}-4n}\cos\left( 2nt\right) \sin\left( nx\right) $$

Answer 1

I believe your solution sol is ok. Taking the n-th coefficient of the MMA solution

cn = (4 (-1 + Cos[n \[Pi]]) )/((-4 n + n^3) \[Pi])

Map[Limit[cn,n -> #] &, {0, 2, 4, 6, 8}]
(* {0, 0, 0, 0, 0} *)

Map[Limit[cn,n -> #] &, {1, 3, 5, 7}]
(* {8/(3 \[Pi]), -(8/(15 \[Pi])), -(8/(105 \[Pi])), -(8/(315 \[Pi]))} *) 

even&odd limits conform well with your handsolution. Obviously the Simplify[] for odd and even n is the problem.