You should be more specific.
-In what coordinate system are you working?
-What is the form of your boundry conditions?
-In what form do you desire your answer?
-How accurate should the answer be?
-Must the answer be used in some further work?
A vaugue answer would be
Laplace's equation in three dimensions is often solved using one or a combination of more than one of the following
-Series methods
separation of variables
-Integral methods
Fourier transfom
Hankel transform
-Numerical methods
Finite differences
Finite elements
monte carlo simulation
Given the 3-D rectangular solid with sides of length a, b and c in the x, y and z direction respectively.
Find T(x,y,z) in the interior of the solid when laplace T = 0
Boundary condition are following conditions:
1) x=0, T=0
2) x=a, dT/dx=0
3) y=0, dT/dy=0
4) y=b, dT/dy=0
5) z=0, T=0
6) z=c, T=f(x,y)
please suggest me, How to solve it?
Seperation of variables. Set T(x,y,z)=X(x)*Y(y)*Z(z), and plug this into laplaces equation. You'll get three new differential equations, each involving one of the variables (If you don't see this right away, remember that if F(x)=G(y), these must both be constant functions). Is this starting to sound familiar? You'll end up with 6 unknowns to be determined by the BC's. If I remember correctlly, 4 of these can be solved for right away, but last 2 will be forced by Laplaces equation, and not necessarily fit the final BC. The way to apply that final BC is to set up a sum over all the solutions that satisfy the rest of the BC's and use fourier series methods. This is a very rough sketch of how to do it, and I hope you have at least been introduced to this method or else you probably won't be able to figure it out.
That problem is commonly solved using seperation of variables with rectangular coordinated. You will proabably need a more detailed treatment, but I will out line the approch.
The idea is to look for solutions of the form
f(x)g(y)h(z)
it is of course unlikely that the general solution is of that form so the hope is (since the pde is linear) that the general solution can be written as a sum of such functions.
The PDE requires
f'(x)g(y)h(z) +f(x)g'(y)h(z) +f(x)g(y)h'(z) =0
equivalently
f'(x)/f(x)+g'(y)/g(y)+h'(z)/h(z)=0
we see that this can only be if each quotient is equal to a constant
f'(x)/f(x)=-s^2
g'(y)/g(y)=-t^2
h'(z)/h(z)=s^2+t^2
we write the constants in this form because f,g need to be periodic
solve each ode and apply conditions 1-5
f(x)g(y)h(z) will depend upon 6 constants 1-5 determin 5 leaving 1
we then have
The possible s and t will be countable and we index them with integers
f(x)g(y)h(z)=fn(x)gm(y)hnm(z)
T(x,y,z)=sum[Anm fn(x)gm(y)hnm(z)]
due to 6 we have
T(x,y,c)=f(x,y)=sum[Anm fn(x)gm(y)hnm(c)]
fn and gm (due to the DE they result form) will have orthogonal properties
<fn(x)|fm(x)>=<fn(x)|fn(x)> if n=m 0 otherwise
<gn(y)|gm(y)>=<gn(y)|gn(y)> if n=m 0 otherwise
where <|> means integrate over the cube the de is to be solved over
thus
Anm is easily found by integration
<fn(x)gm(y)|f(x,y)>=Anm<fn(x)gn(m)|fn(x)gm(y)hnm(c)>
Thank you very much for your suggestion. I can solve it
I have a new problem. Please suggestion to me
Let us cosider steady state heat transfer problem in which laplaceT(r)=0
What is the temparature at the center of a thin disc of radius a
whose average boundary temparatue is 70 degree?
Hint:
1) Assume that the temperature distribution is independent of the direction
along the cylinder
2) Use Laplace equation in cylindrical coordinates
3) the temperature at the center is determined from the temperature
distribution for which r=0
4) The functions sin beta(x) and cos beta(x) have a periodicity if and only if
the values of beta are integer
5) The average boundary temperature at r=a is given by
T(average) = 1/2*Pi intregrate from 0 to 2*Pi [(T(a,seta)d(seta)]