An issue with the built-in solver using ultraspherical spectral method

[Yiting687691]

I'm trying to solve the following PDE over $x\in \left[-10,\frac {13}{\sqrt{\beta}}\right]$ and $\theta\in \left[0,\pi\right]$ using the ultraspherical spectral method:

$$ \frac {\partial H}{\partial x}+\left(\frac {2}{\beta}\sin^{4} \theta\right)\frac {\partial^{2} H}{\partial \theta^{2}}+\left[\left(x+\frac {2}{\beta}\sin 2\theta\right)\sin^{2}\theta-\cos^{2}\theta\right]\frac {\partial H}{\partial \theta}=0, $$

with boundary condition

$$ \begin{align*} H(x,0)=0, \end{align*} $$

and the approximate asymptotic initial condition

$$ \begin{align*} H\left(\frac {13}{\sqrt{\beta}},\theta\right)=\begin{cases} \Phi\left(\frac {\frac {13}{\sqrt{\beta}}-\cot^{2}\theta}{\sqrt{\left(4/\beta\right)\cot \theta}}\right)\quad &0\leq \theta\leq \pi/2,\\ 1\quad &\pi/2\leq \theta. \end{cases} \end{align*} $$

Here $\Phi$ denotes the standard normal distribution function. Following the sample codes in this Notebook, my code for this problem is the following:

β=2;
dθ = 0 ..π; 
dx = -10 ..13/sqrt(β);
d = dθ×dx;
θ,x = Fun(d);
Dθ = Derivative(d,[1,0]);
Dx = Derivative(d,[0,1]);
Φ = xx -> erf.(xx/sqrt(2))/2 + 0.5;
g = (t) -> Φ( (13/sqrt(β) - cot(t).^2)./sqrt.(4/β * cot.(t)) );
h0 = (t) -> t < pi/2 ? g(t) : 1.0;
u0 = Fun(θ->h0(θ),dθ);
u =\([ldirichlet(dx)⊗I; ldirichlet(dθ)⊗I;
Dx+((2/β)*(sin(θ))^4)*(Dθ^2)+((x+(2/β)*sin(2θ))*sin(θ)^2-cos(θ)^2)*Dθ],
[u0; 0; 0];tolerance=1E-4);

When I run the code, I got the following error:

AssertionError: length(order) == 2

@jishnub provides an updated version:

β=2;
dθ = 0 ..π;
dx = -10 ..13/sqrt(β);
d = dθ×dx;
θ = Fun(dθ);
x = Fun(dx);
Dθ = Derivative(space(θ)) ⊗ I;
Dx = I ⊗ Derivative(space(x));
Φ = xx -> erf.(xx/sqrt(2))/2 + 0.5;
g = (t) -> Φ( (13/sqrt(β) - cot(t).^2)./sqrt.(4/β * cot.(t)) );
h0 = (t) -> t < pi/2 ? g(t) : 1.0;
u0 = Fun(θ->h0(θ),dθ);
u =\([ldirichlet(dx)⊗I; ldirichlet(dθ)⊗I;
Dx + (Multiplication((2/β)*(sin(θ))^4) ⊗ I) * (Dθ^2) +
	(((I ⊗ Multiplication(x)) + (Multiplication((2/β)*sin(2θ)) ⊗ I)) * (Multiplication(sin(θ)^2) ⊗ I)
		- (Multiplication(cos(θ)^2) ⊗ I)) * Dθ],
			[u0; 0; 0];tolerance=1E-4);

which generates the following error message:
Implement Conversion from PiecewiseSpace{Tuple{Chebyshev{IntervalSets.ClosedInterval{Float64}, Float64}, Chebyshev{IntervalSets.ClosedInterval{Float64}, Float64}}, DomainSets.UnionDomain{Float64, Tuple{IntervalSets.ClosedInterval{Float64}, IntervalSets.ClosedInterval{Float64}}}, Float64} to Chebyshev{IntervalSets.ClosedInterval{Float64}, Float64}

I wonder if there are bugs in the code or there are some errors in the solve. Thanks.