# PDE Heat Equation Solution with Homogenous Boundary Conditions | PF Discussion

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In summary, the conversation discusses a solution to a PDE with certain boundary conditions. The first method used to solve the problem involved setting a new variable and using separation of variables. However, this approach did not work well with the given initial condition. The conversation then explores a different approach using a different initial condition, which ultimately leads to a successful solution.
member 428835
Hi PF!

I'm wondering if my solution is correct. The PDE is ##h_t = h_{zz}## subject to ##h_z(0,t)=0##, ##h(1,t)=-1##, and let's not worry about the initial condition now. To solve I want homogenous boundary conditions, so let's set ##v = h+1##. Then we have the following: ##v_t = v_{zz}## subject to ##v_z(0,t)=0##, ##v(1,t)=0##. To solve take separation of variables where ##v = T(t)Z(z) \implies T'/T= Z''/Z = -\lambda \implies Z''+\lambda Z = 0##. Guess ##Z = A \cos \sqrt{\lambda} z+B\sin \sqrt{\lambda} z##. Then ##Z'(0)=0 \implies B=0##. Thus ##Z = \cos \sqrt{\lambda} z##. ##Z(1)=0 \implies \cos \sqrt{\lambda} = 0 \implies \sqrt{\lambda} = \pi/2+n\pi:n\in0,1,2...##. Thus, ##Z = \cos((\pi/2+n\pi)z)##. Yet this doesn't look right. Any ideas?

joshmccraney said:
this doesn't look right
In what way?

$$\sqrt{\lambda}=(2n-1)\frac{\pi}{2}\tag{n=1,2,3,...}$$

haruspex said:
In what way?
I was looking for a solution that worked much better with the initial condition I am working with (square wave ##h(z,0) = 1/2 - \sum 2/(\pi n) \sin(n \pi /2) \cos(n \pi z/2)##). The issue is, if I use $$Z=\cos((\pi/2+n\pi)z)\implies\\ T = \exp(-(\pi/2+ n\pi)^2t) \implies\\ h(z,0) = -1 + \sum A_n \left( \cos(n\pi z)\cos(\pi/2z)-\sin(n\pi z)\sin(\pi/2z) \right) \implies \\ -1 + \sum A_n \left( \cos(n\pi z)\cos(\pi/2z)-\sin(n\pi z)\sin(\pi/2z) \right) = 1/2 - \sum 2/(\pi n) \sin(n \pi /2) \cos(n \pi z/2)$$ which doesn't seem to work. However, if I make the ansatz based on the initial condition I can see that a good guess is ##A_n Z = 2/(\pi n) \sin(n \pi /2) \cos(n \pi z/2)##. Then I can superimpose the exponential ##\exp(\lambda_1 t)## and try this guess at the PDE, ultimately giving $$h(z,t) = 1/2 - \sum 2/(\pi n) \sin[n \pi /2] \cos[n \pi z/2] \exp[-n^2 \pi^2 t/4]:n\in 1,2,3...$$ which works out perfectly. But the question remains: why did the first method fail?

Hint: ##\sin(\pi n/2) = 0## for odd n.

Orodruin said:
Hint: ##\sin(\pi n/2) = 0## for odd n.
So, the first method doesn't fail.

I think matching the Fourier coefficients for ##A_n## fails since the constants prior to the sum are unequal: ##1/2\neq-1##. However, after more thinking of the problem, I reformulated the initial condition so this works. I'd specify further if anyone is actually interested but I doubt the details would be of interest. Thank you all for your help and input!

## 1. What is the PDE Heat Equation?

The PDE Heat Equation is a partial differential equation that describes the diffusion of heat in a given medium over time. It is commonly used in physics, engineering, and other scientific fields to model heat transfer and temperature changes in various systems.

## 2. What are homogenous boundary conditions?

Homogenous boundary conditions refer to the conditions at the boundaries of a system that are constant and do not change over time. In the context of the PDE Heat Equation, this means that the temperature at the boundaries of the system remains the same throughout the entire duration of the simulation.

## 3. How is the PDE Heat Equation solved with homogenous boundary conditions?

The PDE Heat Equation with homogenous boundary conditions can be solved using various numerical methods, such as finite difference or finite element methods. These methods involve discretizing the equation and solving it iteratively to approximate the temperature distribution over time.

## 4. What are some applications of the PDE Heat Equation with homogenous boundary conditions?

The PDE Heat Equation with homogenous boundary conditions has many practical applications, such as predicting the temperature distribution in a building or analyzing the cooling of a hot object. It is also used in fields such as meteorology, geophysics, and materials science to model heat transfer in various natural and industrial systems.

## 5. What are the limitations of the PDE Heat Equation with homogenous boundary conditions?

The PDE Heat Equation with homogenous boundary conditions is a simplified model that assumes certain conditions, such as constant material properties and no external heat sources. This may not accurately reflect real-world scenarios, and more complex equations or boundary conditions may be needed for more accurate predictions.

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