# Solve Nonhomogenous PDE: Equilibrium Temp Distribution, B Value

• physstudent.4
In summary, the conversation discusses finding an equilibrium temperature distribution for the equation du/dt=(d^2 u)/dx^2+1, with initial conditions u(x,0)=f(x) and boundary conditions du/dx (0,t)=1 and du/dx (L,t)=B. The approach involves separating variables and solving for the constants, resulting in the general solution u(x,t)=-lm^2*x^2/2+ x + lm*C. The question also asks for the value of B for which there is an equilibrium solution.
physstudent.4

## Homework Statement

du/dt=(d^2 u)/dx^2+1
u(x,0)=f(x)
du/dx (0,t)=1
du/dx (L,t)=B
du/dt=0
Determine an equilibrium temperature distribution. For what value of B is there a solution?

## Homework Equations

Not really sure what to put here.

## The Attempt at a Solution

I started by trying to separate variables, with u(x,t)=phi(x)*g(t), and got to
g'(t)/g(t)=phi''(x)/phi(x)+1/(phi(x)g(t))=0.
So g(t) is constant based on the above, but then I get a little lost while trying to solve for phi. I tried letting g(t)=lamda (abbreviated lm from now on), and got

phi''(t)+1/lm=0, which yields a quadratic solution of [(-lm*x^2)/2+x/lm+C/lm] after using the condition du/dx (0,t)=1. Then, since this is phi(x),
u(x,t)= -lm^2*x^2/2+ x + lm*C.
Using the other condition du/dx (L,t)=B, and assuming some quick mental algebra was correct,
B=-lm^2*L+1.
First off, is all of the above a correct approach as far as you can tell? And secondly, do I need to find u(x,t) or a specific value of lm?

Last edited:
I am confused by the question. You say "Determine an equilibrium temperature distribution. For what value of B is there a solution?"

It is very easy to determine the general equilibrium solution. Is the second independent of that or is it asking for a value of B for which there is an equilibrium solution?

I found the equilibrium solution easily enough after posting that, I realized I could solve for lm. Another student and I clarified with our professor today, he wants both, which I now have. Thanks though!

## 1. What is a nonhomogenous PDE?

A nonhomogenous PDE, or partial differential equation, is an equation that involves multiple variables and their partial derivatives. It is considered nonhomogenous because it includes terms that are not dependent on the variables or their derivatives. These terms are often referred to as source terms, forcing terms, or inhomogenous terms.

## 2. What is the equilibrium temperature distribution?

The equilibrium temperature distribution refers to the steady-state temperature distribution in a system, where the temperature does not change over time. This is often calculated using a nonhomogenous PDE, where the source terms represent heat sources or sinks.

## 3. What is the B value in the nonhomogenous PDE?

The B value in a nonhomogenous PDE represents the coefficient of the forcing term, or the term that is not dependent on the variables or their derivatives. It is often referred to as a boundary condition and can affect the overall solution to the equation.

## 4. How is a nonhomogenous PDE solved?

Solving a nonhomogenous PDE typically involves finding a general solution, which includes both the homogeneous and nonhomogeneous parts, and then using boundary or initial conditions to determine the specific solution. This can be done analytically or numerically using various methods such as separation of variables, Fourier series, or finite difference methods.

## 5. What are some real-world applications of solving nonhomogenous PDEs?

Nonhomogenous PDEs have many applications in the fields of physics, engineering, and mathematics. They are commonly used to model heat transfer, fluid dynamics, and electromagnetic fields. They can also be applied in economics and finance to model and predict stock prices and market trends.

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