Fork me on GitHub
Math for the people, by the people.

User login

Eigenfunction expansion

Primary tabs

Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution.

u′′=f(x),u(0)=α,u(1)=βformulae-sequencesuperscriptu′′fxformulae-sequencesuperscriptunormal-′0αsuperscriptunormal-′1βu^{{\prime\prime}}=f(x),u^{{\prime}}(0)=\alpha,u^{{\prime}}(1)=\beta

I use the function

ϕ′′+λϕ=0superscriptϕ′′λϕ0\phi^{{\prime\prime}}+\lambda\phi=0

to get the eigenfunction is

ϕ=Acosxλ+BcosxλϕAxλBxλ\phi=A\cos x\sqrt{\lambda}+B\cos x\sqrt{\lambda}

but how should I decide A and B? Is it by system

ϕ(0)=α,ϕ(1)=βformulae-sequencesuperscriptϕnormal-′0αsuperscriptϕnormal-′1β\phi^{{\prime}}(0)=\alpha,\phi^{{\prime}}(1)=\beta

or it should be

ϕ(0)=0,ϕ(1)=0formulae-sequencesuperscriptϕnormal-′00superscriptϕnormal-′10\phi^{{\prime}}(0)=0,\phi^{{\prime}}(1)=0

and why? After getting eigenvalue and eigenfunctions, what should I do? I hope somebody can give me a answer with details. Thanks.


i would say ϕ′⁢(0)=α,ϕ′⁢(1)=β But you made a mistake in that it should be

Asin(... +Bcos(.... ANd not both cos(....

And then you should be able to solve A,B for arbitrary a and beta as long as lambda is not an integer multiple of pi^2 in which case there may not be a solution - also you should have chosen different symbols because a and A could be taken to be the same when they need to be different arbitrary and likewise beta and B.

Subscribe to Comments for "Eigenfunction expansion"