Important equations

Heat equation

Wave equation

Laplace’s equation

Overall methodology





Boundary conditions (2 each) ϕ(0) = ϕ(L) = 0 dϕ
dx-(0) = dϕ
dx-(L) = 0 PIC




Eigenvalues λn (  )
 nπL2; n = 1,2, (  )
 nLπ2; n = 0,1,2, (  )
 nπL2; n = 0,1,2,




Eigenfunctions sinnπx
L cosnπx
 L sinnπx-
L and cos        nπx
         L




Series f(x) = n=1Bn sinnπx
L-- f(x) = n=0An cosnπx
-L- PIC




Coefficients Bn = 2L- 0Lf(x)sinnπLxdxPICPIC







# of solutionsOrthogonality of forcing func w/homog soln (weight = 1): abf(x)ϕh(x)dx






ϕh = 0 (λ0) 1 0 (orthogonal)



ϕh0 (λ = 0) 0 (orthogonal)



ϕh0 (λ = 0) 0 0 (not orthogonal)



Particular methods

Math reference