# Solving an eigenvalue problem

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


• What errors do you get? At which point? Mar 9, 2021 at 5:54
• I have edited the question and have added the errors. Mar 9, 2021 at 6:14

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