Partial DFQ Dirichlet Problem

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In summary, by separating the variables and solving the resulting Y-equation, the potential solution for the given problem is u(x,y)=∑(b_n sin(nx)sinh(ny)), which satisfies three of the four boundary conditions. The remaining boundary condition can be solved for by using the coefficients b_n and finding the appropriate value to satisfy the equation.
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jake2
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Homework Statement



∇[itex]^{2}[/itex]u=0 on 0<x<∏, 0<y<2∏
subject to u(0,y)=u(∏,y)=0
and u(x,0)=0, u(x,2∏)=1

Homework Equations



--

The Attempt at a Solution



I've solved the SLP, and now I am trying to solve the Y-equation that results from separation of variables:

Y''-λY=0, Y(0)=0
Y[itex]_{n}[/itex](y)=Acosh(ny)+Bsinh(ny)
Y(0)=(0)=Acosh(n*0)+Bsinh(n*0)[itex]\Rightarrow[/itex]A=0

Doesn't this effectively "kill" the problem? Or is this the solution:
Y[itex]_{n}[/itex](y)=Bsinh(ny)
 
Last edited:
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  • #2
jake2 said:

Homework Statement



∇[itex]^{2}[/itex]u=0 on 0<x<∏, 0<y<2∏
subject to u(0,y)=u(∏,y)=0
and u(x,0)=0, u(x,2∏)=1

Homework Equations



--

The Attempt at a Solution



I've solved the SLP, and now I am trying to solve the Y-equation that results from separation of variables:

Y''-λY=0, Y(0)=0
Y[itex]_{n}[/itex](y)=Acosh(ny)+Bsinh(ny)
Y(0)=(0)=Acosh(n*0)+Bsinh(n*0)[itex]\Rightarrow[/itex]A=0

Doesn't this effectively "kill" the problem? Or is this the solution:
Y[itex]_{n}[/itex](y)=Bsinh(ny)

Well, so far you have ##X_n(x) = \sin(nx)## and ##Y_n(y) = \sinh ny## so your potential solution is $$
u(x,y)=\sum_{n=1}^\infty b_n\sin(nx)\sinh(ny)$$ which satisfies 3 of the four boundary conditions. You still have all the ##b_n## to use. So you need$$
u(x,2\pi)=\sum_{n=1}^\infty b_n\sin(nx)\sinh(2n\pi)=1$$I'm guessing you know how to do that.
 
Last edited:

1. What is a Partial DFQ Dirichlet Problem?

A Partial DFQ Dirichlet Problem is a type of boundary value problem in mathematics that involves solving a partial differential equation (PDE) on a given domain with specified boundary conditions. This type of problem is named after the mathematician Johann Peter Gustav Lejeune Dirichlet.

2. What is the difference between a partial DFQ Dirichlet Problem and a regular DFQ Dirichlet Problem?

The main difference between a partial DFQ Dirichlet Problem and a regular DFQ Dirichlet Problem is the type of differential equation being solved. A partial DFQ Dirichlet Problem involves solving a partial differential equation, while a regular DFQ Dirichlet Problem involves solving a ordinary differential equation.

3. What are some applications of Partial DFQ Dirichlet Problems?

Partial DFQ Dirichlet Problems have various applications in mathematics and physics, including heat transfer, fluid dynamics, population dynamics, and signal processing. They are also used in engineering fields such as structural analysis and control systems.

4. How are Partial DFQ Dirichlet Problems typically solved?

Partial DFQ Dirichlet Problems can be solved using various methods, including the method of separation of variables, numerical methods such as finite difference or finite element methods, and series solutions. The choice of method depends on the specific problem and its boundary conditions.

5. What are the challenges in solving Partial DFQ Dirichlet Problems?

Solving Partial DFQ Dirichlet Problems can be challenging due to the complexity of the PDE and the specific boundary conditions. Some problems may require advanced mathematical techniques and numerical methods, and may not have closed-form solutions. Additionally, the accuracy and convergence of numerical methods can also be a challenge in solving these types of problems.

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