Solving an eigenvalue problem

Question

I am computing the eigenvalues of Laplacian-type operator on the unit square $\Omega = [0, 1]^2$

Consider the eigenvalue problem on the unit square $\Omega$, $$-L u = \lambda u$$ where $$L = e^{2y} \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2} {\partial y^2} - \frac{\partial}{\partial y}$$

with the Dirichlet boundary condition $u = 0$ at $\partial \Omega$.

I have attempted the following.

First I have specified the Laplacian

ℒ = Exp[2 y]*D[u[x, y], {x, 2}] +  D[u[x, y], {y, 2}] -  D[u[x, y], {y, 1}] &;

Then I have specified the Dirichlet boundary condition

ℬ = DirichletCondition[u[x, y] == 0., True];

Computed the five smallest eigenvalues in the unit square.

{vals, funs} = 
  NDEigensystem[{ℒ, ℬ}, 
   u[x, y], {x, 0, 1},{y, 0, 1},  5]; 

Then I have inspected the eigenvalues

vals

But I am facing difficulties as it gives error.

NDEigensystem:The dependent variables specification(​{u[x,y]}) does not match the differential operator dependent variables

Please help me.

Thanking in advanced.

Answer 1

Three modifications: add dependence on $x$ and $y$ (add [x_,y_] to the definitions and [x,y] to where the script letters are referenced), remove the & (this defines ℒ as an unnamed function which shouldn't be) and you probably mean {y,0,1}.

Clear@ℒ
Clear@ℬ
ℒ[x_,y_]=Exp[2y]*D[u[x,y],{x,2}]+D[u[x,y],{y,2}]-D[u[x,y],{y,1}];
ℬ[x_,y_]=DirichletCondition[u[x,y]==0.,True];
{vals,funs}=NDEigensystem[{ℒ[x,y],ℬ[x,y]},u[x,y],{x,0,1},{y,0,1},5];
vals