1. Homogeneous Differential Equation
Solution of Laplace’s
Homogeneous Differential Equation
SOLO HERMELIN
Updated:

2. Laplace’s Homogeneous Differential Equation
SOLO
Laplace’s Homogeneous Differential Equation
TABLE OF CONTENT
Laplace’s Homogeneous Differential Equation
Green’s Identities
Green’s Function
Solution of the Laplace’s Homogeneous Differential Equation
Boundary Conditions
1. Dirichlet Problem
2. Neumann Problem
Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann
Boundary Conditions
Properties of Irrotational Fluids
References

3. Laplace’s Homogeneous Differential Equation
SOLO
Laplace’s Homogeneous Differential Equation
The Laplace’s Homogeneous Differential Equation for the Potential in a irrotational,
homentropic (constant entropy) fluid is:
Pierre-Simon Laplace
We want to find the Potential Φ at the point F (field) due to all the sources (S) in the volume V, including its boundaries
F inside V
F on the boundary of V
Therefore is the vector from S to F.
Let define the operator
that acts only on the source coordinate

4. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
To find the solution we need to prove the following:
GREEN’s IDENTITY
GREEN’s FUNCTION
This Green’s Function is a particlar solution of the following Poisson’s
Non-homogeneous Differential Equation:
Siméon Denis Poisson

5. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
GREEN’s IDENTITY
Let start from the Gauss’ Identity
where is any vector field (function of position and time) continuous and differentiable in the volume V. Let define
We have
Karl Friederich Gauss
Then
First Green’s Identity
If we interchange G with Φ we obtain
Subtracting the second equation from the first we obtain
Second Green’s Identity

6. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
GREEN’s FUNCTION
Define the Green’s Function is a particlar solution of the following Poisson’s
Non-homogeneous Differential Equation:
where δ (x) is the Dirac function
Let use the Fourier Transformation to write
where

7. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Jean Baptiste Joseph
Fourier
GREEN’s FUNCTION (continue – 1)
Let use the Fourier Transformation to write
Hence or

8. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
GREEN’s FUNCTION (continue – 2)
Let compute:
Therefore:
Because this is true for all k, we obtain

9. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
GREEN’s FUNCTION (continue – 3)
Let use spherical coordinates relative to vector:

10. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
GREEN’s FUNCTION (continue – 4)
where we used (see next slide)
Therefore

11. Poisson’s Non-homogeneous Differential Equation
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Poisson’s Non-homogeneous Differential Equation
GREEN’s FUNCTION (continue – 5)
Let compute:
For this use the integral:
since z = 0 is outside the region of integration
M.R. Spiegel, “Complex Variables”, Schaum’s Outline Series, 1964, pg.184
Therefore:

12. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
GREEN’s FUNCTION (continue – 6)
Hence
a Green’s Function for the Poisson’s Non-homogeneous Differential Equation
This solution is not unique since we can add any function that satisfies the
Laplace’s Equation (1782)
Therefore we have the following Green’s Function

13. Laplace’s Homogeneous Differential Equation
SOLO
Laplace’s Homogeneous Differential Equation
Solution of the Laplace’s Homogeneous Differential Equation
Let return to the Laplace’s Homogeneous Differential Equation for the Potential Φ:
Pierre-Simon Laplace
We want to find the Potential Φ at the point F (field) due to all the sources (S) in the volume V, including its boundaries
Solutions of the Laplace’s equation are known as harmonic functions.
F inside V
F on the boundary of V

14. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Solution of the Laplace’s Homogeneous Differential Equation (continue -1)
is the vector from S to F.
Let define the operator that acts only on the
source coordinate .
Since is no defined at r = 0 we define the volume V’ as the volume V minus a small
sphere of radius and surface around the point F, when F is inside V, or a
semi-sphere of radius and surface around the point F, when F is on the boundary of V.

15. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Using the Green’s Identity
let compute

16. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
We obtain
where
Note
If F is outside V from the Green’s Second Identity we obtain
End Note

17. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
For
we have
where

18. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Physical interpretation of Green’s Functions
1. Point Source
The radial velocity is given by
2. Doublet
The derivative of a source in any direction is called a doublet

19. Johann Peter Gustav Lejeune Dirichlet
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Laplace’s Homogeneous Differential Equation
BOUNDARY CONDITIONS
The General Green Function that is a class of bi-position function, and
contains an arbitrary harmonic function (solution of the Laplace’s Equation)
Let consider th following two simple cases (Dirichlet and Neumann Problems):
Dirichlet Problem
The potential is defined at the boundary S of the volume V.
Let choose such that
In this case
Johann Peter Gustav Lejeune Dirichlet
where

20. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
BOUNDARY CONDITIONS (continues – 1)
2. Neumann Problem
The potential derivative is defined at the boundary S of the volume V.
Let choose such that
In this case
Carl Neumann
where

21. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions
Suppose that we have a solution Φ that satisfies the Laplace Homogeneous Differential Equation:
in the volume V, including its boundaries
Suppose also that Dirichlet or Neumann conditions or a combination of those, are specified. In this case the solution is unique (up to an additive constant).
Proof
Suppose that thee exist two solutions and , and define
We have

22. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Uniqueness of a Laplace Solution that satisfies Dirichlet or Neumann Boundary Conditions
(continue – 1)
Proof (continue)
If Dirichlet conditions are satisfied:
If Neumann conditions are satisfied:
Let use Green’s First Identity (with G = Φ)
We have
End of Proof

23. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Properties of Irrotational Fluids (Karamcheti ,“Principles of Ideal Aerodynamics”,
pp )
1. The potential Φ can neither be a maximum nor a minimum in interior of the fluid
For any point inside the fluid we can choose an infinitesimal sphere δS centered at
this point for which
Since can not be either positive or negative inside the fluid, the maximum or
minimum of the potential Φ can occur only at boundary of motion.
2. The spatial derivative of Φ are also harmonic functions, that is they satisfy Laplace’s equation

24. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
Properties of Irrotational Fluids (Karamcheti “Principles of Ideal Aerodynamics”, pp
(continue – 1)
3. The spatial derivative of Φ can neither be a maximum nor a minimum in interior of the fluid
Follows from (1) and (2).
4. The velocity components can neither be a maximum nor a minimum in interior of the fluid
Follows from (3).
5. The magnitude of the velocity cannot be a maximum in interior of the fluid
Let use Green’s First Identity (with G = Φ)
Since the velocity vector also satisfies the Laplace’s equation
we may choose instead of Φ the components of to obtain
Since the velocity magnitude cannot be maximum inside the fluid.

25. Laplace’s Homogeneous Differential Equation
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Laplace’s Homogeneous Differential Equation
References
H. Lass, “Vector and Tensor Analysis”, McGraw-Hill, 1950, pp.155
Karamcheti,“Principles of Ideal Aerodynamics”, pp

26. Israeli Institute of Technology Israeli Armament Development Authority
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
April 28, 2017