In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as
∇
2
f
=
0
or
Δ
f
=
0
,
{\displaystyle \nabla ^{2}\!f=0\qquad {\mbox{or}}\qquad \Delta f=0,}
where
Δ
=
∇
⋅
∇
=
∇
2
{\displaystyle \Delta =\nabla \cdot \nabla =\nabla ^{2}}
is the Laplace operator,
∇
⋅
{\displaystyle \nabla \cdot }
is the divergence operator (also symbolized "div"),
∇
{\displaystyle \nabla }
is the gradient operator (also symbolized "grad"), and
f
(
x
,
y
,
z
)
{\displaystyle f(x,y,z)}
is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
If the right-hand side is specified as a given function,
h
(
x
,
y
,
z
)
{\displaystyle h(x,y,z)}
, we have
Δ
f
=
h
.
{\displaystyle \Delta f=h.}
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.
The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
Homework Statement
A long copper pipe, with it's axis on the z axis, is cut in half and the two halves are insulated. One half is held at 0V, the other at 9V. Find the potential everywhere in space.Homework Equations
\nabla^2V=0The Attempt at a Solution
Alright. This is a laplace's equation...
Hi,
I'm trying to find an analytical solution of Laplace's equation:
\phi_{xx} + \phi_{tt} = 0
with the tricky boundary conditions:
1. \phi(x=0,|t|>\tau)= 0
2. \phi(x\neq0, |t|>>\tau)=0
3. \phi_{x}(x=0, |t|<\tau)=-1
4. \phi_{t}(x, |t|>>\tau)=0
I have the following ansatz(I...
Homework Statement
I have a two part question, the first part involves solving Laplace's equation
u_{xx} + u_{yy} = 0
for the boundary conditions
u_x(0,y) = u_x(2,y) = 0
u(x,0) = 0
u(x,1) = \sin(\pi x)
for
0 < x < 2, 0 < y < 1.
The second part now states a new boundary problem...
Not really Laplace's Equation??
Hi all!
I've been out of school for awhile and so, some of my engineering math is still rusty. While working out a fluids problem, I got stuck on the following PDE:
Y''(y)}Z(z)+Y(y)Z''(z)=-1
\frac{Y''(y)}{Y(y)}+\frac{Z''(z)}{Z(z)}=-\frac{1}{Y(y)Z(z)}
I know...
Homework Statement
Prove the uniqueness of Laplace's equation
Note that V(x,y,z) = X(x) Y(y) Z(z))
Homework Equations
\frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2}+ \frac{d^2 V}{dz^2} = 0
The Attempt at a Solution
Suppose V is a solution of Lapalce's equation then let V1 also be a...
Need To Example Questions Of "laplace's equation in Boundary-value Problem"in 3DSpace
Please Help Me : Need To Example Questions Of "laplace's equation in Boundary-value Problem"in 3D Space.
Can Anyone Give Me a File That Contains This Type Of Questions?
I Need At least 20 Examples...
Homework Statement
Consider a square in the XY plane with corners at (0,0, (a,0), (a,a,) and (0,a). There is no charge nor matter inside the square. The sides perpendicular to the Y axis have potential zero. The side at x=a has constant potentail V0. The side at x=0 has potentail -V0. Find...
E&M: Using Laplace's Equation to solve for a conducting "slit"
Homework Statement
The set up is as follows: You have a conductor at potential 0 along the y-axis at x=0. You have another conductor at potential V=Vo running along the x-axis at y=0. You have a third conductor at potential V=Vo...
I have a problem solving
\nabla^2 T(x,y,z) = 0
T(0,y,z)=T(a,y,z)=0
T(x,0,z)=T(x,b,z)=T_0 \sin{\frac{\pi x}{a}
T(x,y,0)=T(x,y,c)=const.
I use separation of variables and get
X_n (x) = \sin{\frac{n \pi x}{a}
Y_n (y) = \cosh{\sqrt{\frac{n^2 \pi^2}{c^2} + \frac{n^2 \pi^2}{a^2}}y} +...
I need to solve laplace's equation in 2-d polar co-ordinates, and I just get the standard V(r,theta) = A + Blnr + sum to infinity of [An*sin(n*theta) + Bn * cos(n*theta)]*[Cn*(r^-n) + Dn*(r^n)]
by using separation of variables and considering all values of the separation constant which give...
Heat equation
Given the 3-D rectangular solid with sides of length a, b and c in the x, y and z directions, respectively. Find the function T(x,y,z,t) when
Laplace(T)=1/K(dT/dt) subject to the following conditions:
1) Initial conditions: T(x,y,z,0)=0
2) Boundary conditions
a. dT/dx +...
Hi
I am not quit sure I have understand the laplace equation correctly. I hope some one can help me with it.
As far as I understand if we are able to differentiate any function twice, then the function is harmonic.
so we assume V(x,y) is harmonic because of the above.
Does...
I was hesistant wheter to post this in the physics of math section but it's much of math problem I think.
Suppose I have a function V(x,y,z) which obeys Laplace's equation over some path in space. That is to say, for some path parametrized by \vec{r}(t) = x(t)\hat{x} + y(t)\hat{y} +...
Can someone explain how to separate a multivariable differential equation into two independent differential equations? I'm having an issue solving for the potential in spherical co-ordinates in terms of r and theta.