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edr1c
 3 years ago
any1 can explain the principles in solving laplace equation in rectangular coordinates?
i know i need to solve for 2 2nd order ODE but im not sure how to conclude for whether they are trivial or nontrivial.
edr1c
 3 years ago
any1 can explain the principles in solving laplace equation in rectangular coordinates? i know i need to solve for 2 2nd order ODE but im not sure how to conclude for whether they are trivial or nontrivial.

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edr1c
 3 years ago
Best ResponseYou've already chosen the best response.0i let u(x, y) = F(x) G(y) so Uxx = F'' G Uyy = F G" then the equation become F"G + FG" = 0 i separate the variables and equate them to a separation constant, k so the 2 2nd order ODE are F''kF=0 , G"+kG = 0 how do i proceed with the initial conditions given?

edr1c
 3 years ago
Best ResponseYou've already chosen the best response.0i need to get 3 cases when k=0, k<0 and k>0. but i dont know how to conclude after applying the initial conditions into the cases.

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0do you have relevant links?

TuringTest
 3 years ago
Best ResponseYou've already chosen the best response.0I always forget how to do this, so here is a hopefully helpful link http://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx

edr1c
 3 years ago
Best ResponseYou've already chosen the best response.0let say i have the case where k=0 for F"=0 i got F=Ax+B, and i substitube initial conditions to obtain F be trivial (F=0). for G"=0 i got G=Cx+D G(0)=0 : D=0, G=Cx G(24)=24: C=1, G=x ?? is it possible to obtain nontrivial solution for G when F is trivial?

experimentX
 3 years ago
Best ResponseYou've already chosen the best response.0hold on for a while ... I'll work out few other probs and try to work on my copy.

edr1c
 3 years ago
Best ResponseYou've already chosen the best response.0my lecturer practically skipped this part ==

Mikael
 3 years ago
Best ResponseYou've already chosen the best response.0@edr1c The devil is in the notdetails, in the boundary conditions. Each set of B.C. allows/disallows other solutions

Mikael
 3 years ago
Best ResponseYou've already chosen the best response.0The coordinates is in some sense "our imagination"  the equation is not influenced by the coordinates we choose. It IS influenced by the symmetry of its BOUNDARY, and of its BOUNDARY CONDITIONS

Mikael
 3 years ago
Best ResponseYou've already chosen the best response.0So why do we choose the coordinates ?  To fit the symmetry of the boundary of course !
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