Heat equation on a rectangular domain
Not what you're looking for?
2-1 a, b
Consider the heat equation for a rectangular region, 0 < x < a, 0 < y < b, t > 0
ut = k(uxx + uyy) , 0 < x < a, 0 < y < b, t > 0
subject to the initial conditions: u(x,y) = f(x,y)
a) ux (0, y, t) = 0, ux (a, y, t) = 0, 0 < y < b, t > 0
uy (x, 0, t) = 0, uy (x, b, t) = 0, 0 < x < a, t > 0
b) u (0, y, t) = 0, ux (a, y, t) = 0, 0 < y < b, t > 0
uy (x, 0, t) = 0, uy (x, b, t) = 0, 0 < x < a, t > 0
Purchase this Solution
Solution Summary
The 17-pages solution shows in great detail how to apply the method of separation of variables to the heat equation on a rectangle with mixed boundary conditions, and how to obtain the expansion coefficients using Fourier analysis.
Solution Preview
Hi C
Here it is.
The difference between part (a) and (b) is only in the x-dependent eigenfunction, so most of the analysis of part (a) is valid to part (b).
Part (a)
The equation is:
(1.1)
With Neumann boundary conditions:
(1.2)
And
(1.3)
While the initial condition is:
(1.4)
We use separation of variables:
(1.5)
The boundary conditions become:
(1.6)
And:
(1.7)
Substituting (1.5) into (1.1) we get:
(1.8)
The left hand side is a function of t while the right hand sides is s function of x and y.
Since it must be true for any both sides must be equal a constant:
(1.9)
Furthermore,
(1.10)
The left hand side of (1.10) is a function of x while the right hand side is a function of y, therefore both sides must be equal a constant:
(1.11)
And by the same token we can write:
(1.12)
so again the equation is separated and both sides equal a different constant:
(1.13)
And from (1.9) we see that :
(1.14)
So we start with equation (1.11):
(1.15)
with the boundary conditions
(1.16)
We distinguish between three cases:
Case 1:
The equation is and its solution is
(1.17)
Applying the boundary conditions:
We get the trivial solution
Case ...
Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.