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How to solve Laplace's equation in three dimensions? Please anyone suggest me

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- Thread starter danai_pa
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- #1

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How to solve Laplace's equation in three dimensions? Please anyone suggest me

- #2

lurflurf

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You should be more specific.danai_pa said:How to solve Laplace's equation in three dimensions? Please anyone suggest me

-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

- #3

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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?

- #4

StatusX

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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.

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- #5

lurflurf

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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)>

- #6

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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)]

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)]

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