Inhomogeneous Dirichlet problem in a rectangle

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In summary, the conversation discusses solving a problem involving a steady-state solution and separation of variables. The answer is obtained through finding the appropriate eigenfunctions and using the hyperbolic sine addition rule to transform the solution into a linear combination of these eigenfunctions.
  • #1
Incand
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Homework Statement


Solve the problem
\begin{cases}
u_{xx} + u_{yy} = y, \; \; 0 < x < 2, \; \; 0 < y < 1\\
u(x,0)=0, \; \; u(x,1)=0\\
u(0,y)=y-y^3, \; \; y(2,y)=0.
\end{cases}

Homework Equations


N/A

The Attempt at a Solution


I start by trying to find a steady-state solution ##u_0(y)## (or I guess a solution independent of ##x## since we don't have a time variable). We have
##u_0''=y \Longrightarrow y = \frac{y^3}{6}+ay+b##. The boundary values ##u(x,0)=0, \; \; u(x,1)=0## give us that ##u_0 = \frac{y^3-y}{6}##.

Setting ##u = v+u_0## we end up with the new homogeneous problem
\begin{cases}v_{xx}+v_{yy}=0\\
v(x,0) = 0, \; \; v(x,1)=0\\
v(0,y) = \frac{7}{6}(y-y^3), \; \; v(2,y)=\frac{y-y^3}{6}\end{cases}
Separation of variables give us
##X(x) = c_1\cosh \lambda x + c_2 \sinh \lambda x## and
##Y(y) = c_3\cos \lambda y + c_3\sin \lambda y##.
The boundary conditions ##v(x,0) = 0, \; \; v(x,1)=0## give us that ##c_3=0## and ##\lambda = n\pi##. We then have solutions of the form
##v(x,y) = \sum_1^\infty \left( a_n\cosh n\pi x + b_n \sinh n\pi x \right) \sin n\pi y##.
Imposing the boundary condition ##v(0,y) = \frac{7}{6}(y-y^3)## we have that
##\frac{7}{6}(y-y^3) = \sum_1^\infty a_n \sin n\pi y##. Calculating the Fourier coefficients we have that
##a_n = \frac{7}{3}\int_0^1 (y-y^3)\sin n\pi y dy = \frac{14 (-1)^{n-1}}{\pi^3 n^3}##.
So we have
##v(x,y) = \sum_1^\infty \left( \frac{14 (-1)^{n-1}}{\pi^3 n^3}\cosh n\pi x + b_n \sinh n\pi x \right) \sin n\pi y##. Imposing the other boundary condition ##v(2,y)=\frac{y-y^3}{6}## we get
##\frac{y-y^3}{6} =\sum_1^\infty \left( \frac{14 (-1)^{n-1}}{\pi^3 n^3}\cosh n\pi 2 + b_n \sinh n\pi 2 \right) \sin n\pi y## at which point I'm stuck not knowing how to proceed.

The answer should be
##u(x,y) = \frac{1}{6}(y^3-y)+\frac{2}{\pi^3}\sum_1^\infty \frac{(-1)^{n-1}}{n^3\sinh 2n \pi} (\sinh n\pi x + 7 \sinh n\pi(2-x) )\sin n\pi y##.
I guess they choose the eigenfunction basis to the Sturm-Liouville problem as
##\sinh n\pi x## and ##\sinh n\pi (2-x)## instead which work just as well. Is there a way I can get the answer in this form or do I have to redo every calculation with these eigenfunctions?
 
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  • #2
Well, you know that [tex]
\frac{14(-1)^{n-1}}{n^3 \pi^3}\cosh 2n\pi + b_n \sinh 2n\pi[/tex] must be equal to the [itex]\sin n\pi y[/itex] coefficient in the Fourier series for [itex](y - y^3)/6[/itex] - which you know, because you've just calculated the Fourier series of [itex]7(y - y^3)/6[/itex]. Then, if you wish, use the hyperbolic sine addition rule [tex]\sinh (a + b) = \sinh a \cosh b + \cosh a \sinh b[/tex] to express your linear combination of hyperbolic sines and cosines as a linear combination of [itex]\sinh n\pi x[/itex] and [itex]\sinh n\pi(2-x)[/itex].

But the natural choice of eigenfunctions here is indeed [itex]\sinh n\pi x \sin n\pi y[/itex] and [itex]\sinh n\pi (2 - x) \sin n\pi y[/itex]: The boundary condition on [itex]x = 2[/itex] is just [itex]\frac17[/itex] times that on [itex]x = 0[/itex]. Thus if [itex]X_{n,1}[/itex] and [itex]X_{n,2}[/itex] are linear combinations of [itex]e^{n\pi x}[/itex] and [itex]e^{-n\pi x}[/itex] subject to [itex]X_{n,1}(0) = 1[/itex], [itex]X_{n,1}(2) = 0[/itex] and [itex]X_{n,2}(0) = 0[/itex], [itex]X_{n,2}(1) = 1[/itex] then [tex]
v(x,y) = \sum_{n=1}^\infty a_n (X_{n,1}(x) + \tfrac17 X_{n,2}(x)) \sin n\pi y[/tex] where [itex]a_n[/itex] is obtained from the Fourier expansion on [itex]x = 0[/itex]. And on investigation we find [tex]
X_{n,1}(x) = \frac{\sinh n\pi (2 - x)}{\sinh 2n\pi}, \\
X_{n,2}(x) = \frac{\sinh n\pi x}{\sinh 2n\pi}. [/tex]
 
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  • #3
The alternative eigenfunctions really made it a lot easier. How would I go about finding them in the first place? or are they just something you get from experience where someone just happened upon them? Our book does indeed suggest them for a similar example problem but doesn't explain how they got them in the first place.

As for transforming the the ##\cosh ## and ##\sinh## I think I got it right
##\frac{2(-1)^{n-1}}{\pi^3 n^3} = \frac{14 (-1)^{n-1}}{n^3\pi^3}\cosh 2n\pi + b_n \sinh 2n\pi##
Rearranging we have
##b_n= \frac{2(-1)^{n-1}(1-7\cosh 2n\pi)}{\pi^3 n^3 \sinh 2n\pi}##. Our expansion is then
##v(x,y) = \frac{2}{\pi^3}\sum_1^\infty \frac{(-1)^{n-1}\sin n\pi y}{n^3 \sinh 2n\pi} \left(7 \sinh 2n\pi \cosh n\pi x + (1-7\cosh 2n\pi)\sinh n\pi x \right)##
So I want to show that ##(\sinh 2n\pi \cosh n\pi x-\cosh 2n\pi \sinh n\pi x) = \sinh n\pi (2-x)## which follow immediately from your addition rule.
 

What is an inhomogeneous Dirichlet problem in a rectangle?

An inhomogeneous Dirichlet problem in a rectangle is a mathematical problem that involves finding a function that satisfies a specified boundary condition on the boundary of a rectangular region, while also satisfying a given differential equation inside the region.

What are the applications of solving an inhomogeneous Dirichlet problem in a rectangle?

Solving an inhomogeneous Dirichlet problem in a rectangle has many practical applications, such as in heat transfer, fluid dynamics, and electromagnetism. It is also used in various engineering and scientific fields to model and solve real-world problems.

What are the steps involved in solving an inhomogeneous Dirichlet problem in a rectangle?

The first step is to define the problem and determine the boundary conditions. Next, a differential equation is formulated based on the problem. Then, the general solution to the differential equation is found. Finally, the boundary conditions are applied to determine the specific solution to the problem.

What is the difference between a homogeneous and inhomogeneous Dirichlet problem in a rectangle?

A homogeneous Dirichlet problem has the same boundary condition on all sides of a rectangular region, while an inhomogeneous Dirichlet problem has different boundary conditions on each side. This difference affects the formulation and solution of the problem.

What are some techniques used to solve inhomogeneous Dirichlet problems in a rectangle?

Some common techniques used to solve inhomogeneous Dirichlet problems in a rectangle include separation of variables, the method of eigenfunction expansions, and the method of Green's functions. These techniques involve breaking down the problem into simpler parts and using mathematical tools to solve each part separately.

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