# Finding the Steady Solution for a PDE Problem

• HalfManHalfAmazing
In summary, the conversation is about finding the steady solution for a PDE involving a slender homogeneous conducting bar with specific boundary conditions. The solution is found by finding the time independent solution and plugging in the given boundary conditions. The conversation ends with the speaker being able to figure out the solution on their own.
HalfManHalfAmazing
I'm wondering if anyone can just run through how this is done. I have the solution so that's now the problem. I just need someone to provide me with the method of finding the steady solution (I can find the transient no problem).

A slender homogeneous conducting bar of uniform cross section lies along the x-axis with ends at x = 0 and x = 1. The lateral surface of the bar radiates heat into the surroundings at temperature zero. The left end is insulated and heat is added through the right end. The initial temperature distributions is u(x,0) = f(x).

Okay so the PDE we have here is:

$$u_t = \alpha^2u_{xx} - u$$
$$u_x(0,t) = 0$$
$$u_x(1,t) = q$$
$$u(x,0) = f(x)$$

And to find the steady solution I do the usual tricks. I know how to get it I just want someone to explain a bit. Thanks.

Okay well I went ahead and did it myself. I figure I'll explain myself in case anyone wishes:

I start by finding the time indepenent solution - where $$u_t$$ vanishes. I find that I have an DE similar to basic differential equations with solution - $$y = c_1e^{\frac{x}{\alpha}}+c_2e^{-\frac{x}{\alpha}}$$

I want to plug in some boundary conditions and for that I need to differentiate and then plug in.

Glad you were able to figure it out!

## 1. What is a PDE problem?

A PDE problem, or partial differential equation problem, involves finding a function that satisfies a given partial differential equation. These equations involve multiple variables and their partial derivatives, and are used to model various physical phenomena in fields such as physics, engineering, and mathematics.

## 2. Why is finding the steady solution for a PDE problem important?

The steady solution, also known as the equilibrium solution, is a solution that remains constant over time. In many applications, such as in fluid dynamics or heat transfer, it is important to find the steady solution to understand the long-term behavior of the system.

## 3. What methods are commonly used to find the steady solution for a PDE problem?

There are several methods that can be used to find the steady solution for a PDE problem, including separation of variables, Fourier series, and numerical methods such as finite difference or finite element methods. The choice of method depends on the specific problem and its complexity.

## 4. What are some challenges in finding the steady solution for a PDE problem?

One of the main challenges in solving PDE problems is the high level of mathematical complexity involved. The equations can be difficult to solve analytically, and numerical methods require careful implementation and validation. Additionally, the physical interpretation of the solution can be challenging and may require further analysis.

## 5. How is the accuracy of the steady solution determined?

The accuracy of the steady solution is determined by comparing it to known solutions or experimental data. Additionally, numerical methods have their own measures of convergence and error, which can be used to assess the accuracy of the solution. In some cases, sensitivity analysis can also be performed to determine the impact of small changes in the solution on the overall accuracy.

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