# Question:Unexpected solution to PDE

## Question:Unexpected solution to PDE

Maple 2019

We solve Laplace's equation in the domain
in polar coordinates subject to prescribed Dirichlet data.

Maple produces a solution in the form of an infinite sum,
but that solution fails to satisfy the boundary condition
on the domain's outer arc.  Is this a bug or am I missing
something?

 > restart;
 > kernelopts(version);

 > with(plots):
 > pde := diff(u(r,t),r,r) + diff(u(r,t),r)/r + diff(u(r,t),t,t)/r^2 = 0;

 > a, b, c, d := 1, 2, Pi/6, Pi/2;

 > bc := u(r,c)=c, u(r,d)=0, u(a,t)=0, u(b,t)=t;

We plot the boundary data on the domain's outer arc:

 > p1 := plots:-spacecurve([b*cos(t), b*sin(t), t], t=c..d, color=red, thickness=5);

Solve the PDE:

 > pdsol := pdsolve({pde, bc});

Truncate the infinite sum at 20 terms, and plot the result:

 > eval(rhs(pdsol), infinity=20): value(%): p2 := plot3d([r*cos(t), r*sin(t), %], r=a..b, t=c..d);

Here is the combined plot of the solution and the boundary condition.
We see that the proposed solution completely misses the boundary condition.

 > plots:-display([p1,p2], orientation=[25,72,0]);