The two dimensional heat equation - Trinity University

More documents

Recommendations

Info

$Name: Math 1311 Test 2 Fall 2003 1. Find the derivative of (a) y = x ...$

$Name: Math 4364: Theory of Complex Variables Take Home Test 1 ...$

$SYLLABUS Fall 2004 CALCULUS I (Math 1311-5 ... - Trinity University$

$Math 1311 Test 1 Fall 2004 SOLUTIONS 1. Apply the limit laws to ...$

$A Guide to Presentations in LaTeX-beamer - with ... - Trinity University$

$A Guide to Presentations in LaTeX-beamer - with ... - Trinity University$

The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsSeparation of variablesSetting u(x,y) = X(x)Y(y) leads toX ′′ +kX = 0 , Y ′′ −kY = 0,X(0) = X(a) = 0 , Y(0) = 0.We know the nontrivial solutions for X:X(x) = X n (x) = sinµ n x, µ n = nπ a , k = µ2 n .for n ∈ N = {1,2,3,...}. The corresponding solutions for Y areY(y) = Y n (y) = A n e µny +B n e −µny .DailedaThe 2D heat equation
The 2D heat equation Homogeneous Dirichlet boundary conditions Steady state solutionsThe condition Y(0) = 0 yields A n = −B n . Choosing A n = 1/2 wefind thatY n (y) = sinhµ n y.This gives the separated solutionsu n (x,y) = X n (x)Y n (y) = sinµ n x sinhµ n y,and superposition gives the general solutionu(x,y) =∞∑B n sinµ n x sinhµ n y.n=1DailedaThe 2D heat equation
Page 1 and 2: The 2D heat equation Homogeneous Di
Page 17: The 2D heat equation Homogeneous Di
Page 25: The 2D heat equation Homogeneous Di

The two dimensional heat equation - Trinity University

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?