Yes, it certainly is! How did you get that function?Engineer913 said:Homework Statement
1) What is the Equilibrium temperature distributions if α > 0?
2) Assume α > 0, k=1, and L=1, solve the PDE with initial condition u(x,0) = x(1-x)
Homework Equations
du/dt = k(d^2u/dx^2) - (α*u)
The Attempt at a Solution
I got u(x) = [(α*u*x)/2k]*[x-L] for Part#1 but this was told to be wrong
An "equilibrium" solution is one that no longer changes: du/dt= 0 so the partial differential equation becomes k d^2u/dx^2- au= 0, a linear, homogeneous, second order, ordinary differential equations. If you are taking a course in "partial differential equations", you certainly should know how to solve such an equation.Part #2 I got α*u = -2, Also wrong.
Any insight how to correct this?
A PDE, or partial differential equation, is a type of mathematical equation that involves functions of multiple independent variables and their partial derivatives. It is important to find the equilibrium temperature distribution in order to understand how heat is distributed in a system and to predict how it will change over time.
The equilibrium temperature distribution of a PDE is determined by solving the equation using mathematical techniques such as separation of variables or the method of characteristics. This results in a mathematical expression that describes the temperature distribution at any point in the system.
Yes, the equilibrium temperature distribution of a PDE can change over time if there are factors such as external heat sources or changes in the boundary conditions. In these cases, the PDE must be solved for each time step to determine the changing temperature distribution.
One common application is in studying heat transfer in materials and structures. For example, engineers may use PDEs to determine the temperature distribution in a building to optimize heating and cooling systems. PDEs are also used in modeling weather patterns and predicting climate change.
While PDEs are a powerful tool for modeling temperature distributions, they do have some limitations. In complex systems, it may be difficult to accurately model all of the variables and boundary conditions. Additionally, the equations may become too complex to solve analytically, requiring the use of numerical methods which can introduce some error into the results.