Laplace's equation
The Laplace equation (after Pierre- Simon Laplace ) is the elliptic partial differential equation of second order
For a scalar function in a region where is the Laplace operator. This makes it the homogeneous Poisson equation, that is, the right side is zero. The Laplace equation is the prototype of an elliptic partial differential equation.
- 7.1 Fundamental solution
- 7.2 solution using Green's function
- 7.3 solution in two dimensions
Definition
The mathematical problem is to find a scalar, twice continuously differentiable function which equation
Met. The solutions of this differential equation are called harmonic functions.
Is the Laplace operator is generally defined for a scalar function as:
Coordinate representations
Is a specific coordinate system, where it is possible to compute the representation of the Laplace's equation in these coordinates. In the most commonly used coordinate systems the Laplace equation written as:
Resulting in three-dimensional space according to:
Results.
Importance in physics
The importance of the Laplace equation or potential equation, as it is often called in physics, covers several areas of physics. Divine can this possibly the following examples:
A time-constant temperature gradient can satisfy the Laplace equation.
The Laplace equation can also be obtained from the heat conduction equation. In the stationary case, ie, in equilibrium state, the time derivative of the heat conduction equation is zero. This equation is the Poisson equation. Now, are still no sources or sinks available, so there is no further heat exchange - for example with the environment - as the place looked, the heat conduction equation for the Laplace equation.
Example of this is a metal rod, with which a plug at one end and the other end is cooled by means of ice water. At the bar a temporally constant temperature gradient will form after some time, which satisfies the Laplace equation (temperature exchange with the environment is neglected). The same example something practical can be found in the insulation of houses. The heater inside is the candle and the cold outside air, the ice water.
In electrostatics, the electric potential in the uncharged space of the Laplace equation is sufficient.
For example, a conductive ball made in an external electric field, the electrons to arrange on the surface. Result of this rearrangement is that the potential on the spherical surface is constant. According to the min-max principle ( see below) Thus, the potential within the sphere constant.
This is the working principle of the Faraday shield. Since the voltage is defined as the potential difference and the potential is constant as just said, you're safe inside of electrical shock.
Boundary value problems
We can distinguish three types of boundary value problems. The Dirichlet problem, the Neumann problem and the mixed problem. These differ by the type of additional boundary conditions.
It is generally a bounded domain and the edge of.
Dirichlet problem
In the Dirichlet problem, the continuous mapping on the boundary is specified. Will be given, in other words, the values which is to take the solution of the Laplace equation on the boundary.
The Dirichlet problem can be formulated it in the following way:
The solution of the Dirichlet problem is unique.
Neumann problem
Neumann problem in the normal discharge is set on the edge, which is to take the solution of the Laplace equation.
The Neumann problem can be formulated it in the following way:
Wherein the normal derivative of, ie, the normal component of the gradient of the surface of referred.
The solution of the Neumann problem is unique up to an additive constant.
Mixed problem
The mixed boundary value problem is a combination of the Dirichlet and the Neumann problem,
By a constant, where the solution to this problem other conditions, such as initial values are needed.
The mixed problem has no known additional conditions, such as Initial values, not uniquely solvable. The uniqueness of this problem requires the unique solvability of the differential equation of the values on the boundary:
If this differential equation but on the basis of other information uniquely solvable, then the mixed problem can be transferred to a Dirichlet problem which has a unique solution.
Mean value theorem of Gauss
Is the area harmoniously, so function value at the point is the mean of the surface of each ball to with radius, provided that the ball is in and values of the function on the surface are continuous,
Here, the spherical surface of the sphere with center and radius
With the area of the surface of the dimensional unit sphere
Here is the gamma function, the analytic extension of the faculty to non-natural numbers as they occur for each non-straight.
Minimum-maximum principle
From the mean value theorem of Gauss implies that the solution of the Laplace equation in a bounded domain adopts neither its minimum nor maximum, provided that the values on the boundary are continuous and not constant. This means:
Thus, the function values are always in between the minimum and the maximum of the values on the boundary:
Except from above principle is the trivial case where the boundary values are constant, because in this case the solution is generally constant.
Solution of the Laplace 's equation
Fundamental solution
To find the fundamental solution of the Laplace equation, it makes sense to the rotational invariance of the Laplacian exploit. It is to this purpose, with the Euclidean norm of designated. Using the chain rule for the Laplace equation is transformed into an ordinary differential equation of second order. Obtained for only dependent function then the following dimension-dependent formula:
With the area of the surface of the dimensional unit sphere
Here is the gamma function, the analytic extension of the faculty to non-natural numbers as they occur for each non-straight.
It must be noted that the fundamental solution is not a real solution of the Laplace equation, if the origin is in, because it has a singularity at this point.
In the following, the solution of the Dirichlet problem is discussed. It should be noted that the Neumann problem and the mixed problem can be converted by solution of the equation of the boundary values in a Dirichlet problem.
Solution by means of Green's function
Central problem is the construction of the Green's function, which does not have to exist in each case. The discovery of this is generally difficult, especially since the Green's function of the area on which the Laplace equation is satisfied, depends. Is known, however, the Green's function, as can be clearly with its help the solution of the Dirichlet problem.
Based on the determination of the Green's function is the fundamental solution of the Laplace equation.
In addition, an auxiliary function must be constructed which is continuously differentiable in duplicate and steadily on with which fulfills the following conditions:
Finding these auxiliary function is the central step in the determination of the Green's function.
The Green's function is obtained according to:
From which can be the solution to the Dirichlet problem in calculate:
Solution in two dimensions
Basis of this solution is the Fourier method. The Dirichlet problem is considered here in polar coordinates
And the unknown function by means of the separation of variables split into two independent functions. The approach is thus:
The establishment of this approach in the Laplace equation and using a separation approach the problem leads back to two ordinary differential equations.
The solutions of these ordinary differential equations are:
Here,, ,, constants, and wherein the - separation approach is constant from the positive and real, whereby (in the case of obtaining the solutions ) the periodicity of the angle is satisfied. This periodicity can be interpreted as the continuity of the values of on the edge.
If, as would be present in a singularity, which contradicts again the continuity requirement in. Thus is.
If these solutions used in the separation of variables selected above and summed up according to the principle of superposition of all possible solutions, we obtain the solution of the Laplace equation:
Where, and the Fourier coefficients of the values of are.