Solving a Simple PDE: Need Assistance!

In summary, the conversation discusses solving a 1D heat equation with specific boundary conditions and the need for an initial condition to obtain a unique solution. The difference between boundary and initial conditions is also explained.
  • #1
Aidyan
180
13
Simple PDE...

I'm trying to solve the PDE:

[itex]\frac{\partial^2 f(x,t)}{\partial x^2}=\frac{\partial f(x,t)}{\partial t}[/itex] with [itex]x \in [-1,1][/itex] and boundary conditions f(1,t)=f(-1,t)=0.

Thought that [itex]e^{i(kx-\omega t)}[/itex] would work, but that obviously does not fit with the boundary conditions. Has anyone an idea?
 
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  • #2


Aidyan said:
I'm trying to solve the PDE:

[itex]\frac{\partial^2 f(x,t)}{\partial x^2}=\frac{\partial f(x,t)}{\partial t}[/itex] with [itex]x \in [-1,1][/itex] and boundary conditions f(1,t)=f(-1,t)=0.

Thought that [itex]e^{i(kx-\omega t)}[/itex] would work, but that obviously does not fit with the boundary conditions. Has anyone an idea?
Your equation is the 1D heat equation, the solutions of which are very well known and understood. A google search should yield what you need.

P.S. You will also need some kind of initial condition.
 
  • #3


Hootenanny said:
Your equation is the 1D heat equation, the solutions of which are very well known and understood. A google search should yield what you need.

P.S. You will also need some kind of initial condition.

Hmm... looks like it isn't just a simple solution, however. It seems I'm lacking the basics ... :confused: I thought this is sufficeint data to solve it uniquely, what is the difference between boundary and initial conditions?
 
  • #4


Aidyan said:
I thought this is sufficeint data to solve it uniquely,
Afraid not, without knowing the temperature distribution at a specific time you aren't going to obtain a (non-trivial) unique solution.
Aidyan said:
what is the difference between boundary and initial conditions?
The former specifies the temperature on the spatial boundaries of the domain (in this case x=-1 and x=1). The latter specifies the temperature distribution at a specific point in time (usually t=0, hence the term initial condition).
 
  • #5


I understand your frustration with trying to solve this PDE. PDEs can be complex and challenging to solve, but with the right approach and tools, it is possible to find a solution.

Firstly, I would recommend checking your boundary conditions. It seems that the boundary conditions you have provided are for a different PDE, as they do not match with the PDE you are trying to solve. It is important to have the correct boundary conditions in order to find a valid solution.

Next, I would suggest looking into different methods for solving PDEs, such as separation of variables, Fourier series, or numerical methods. These methods can help you break down the PDE into simpler equations that are easier to solve.

Additionally, it may be helpful to consult with other experts in the field or look for resources online that provide step-by-step solutions to similar PDEs. Sometimes, a fresh perspective or a different approach can make all the difference in finding a solution.

Don't give up, solving PDEs can be challenging but with determination and the right tools, you can find a solution to this simple PDE. Good luck!
 

1. What is a PDE?

A PDE stands for a partial differential equation. It is a mathematical equation that contains more than one independent variable and their partial derivatives.

2. Why do we need to solve PDEs?

PDEs are used to describe many phenomena in physics, engineering, and other fields. Solving PDEs helps us understand and predict the behavior of these systems.

3. What is the difference between a simple PDE and a complex PDE?

A simple PDE is one that can be solved using basic mathematical techniques, such as separation of variables or the method of characteristics. A complex PDE typically requires more advanced techniques, such as numerical methods, to solve.

4. What are some common techniques for solving PDEs?

Some common techniques for solving PDEs include separation of variables, the method of characteristics, Fourier transforms, and numerical methods such as finite difference and finite element methods.

5. How do I know if my solution to a PDE is correct?

There are several ways to check the correctness of a solution to a PDE. These include verifying that the solution satisfies the original PDE, checking for consistency with any given boundary or initial conditions, and comparing the solution to known exact solutions if available.

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