Download Differential Formulation of Boundary Value Problems

Week 2
Differential Formulation of Boundary Value
Problems
1
Introduction
We are interested in boundary value problems where one must determine a function u representing some meaningful physical variable inside a domain Ω with boundary Γ. We will
focus primarily in BVP governed by either Laplace’s Equation
∇2 u = 0
or Poisson’s Equation
∇2 u + f = 0
both subject to suitable boundary conditions on Γ. Here
∇2 u = ∇ · ∇u
Partial differential equations of the above type are called elliptic. Functions satisfying
Laplace’s equation are called harmonic functions and play a key role in applied mathematics.
Boundary conditions on Γ can be of three types, namely, boundary conditions of first
kind (function values prescribed - Dirichlet problem -)
u=g
conditions of the second kind (values of the derivative normal to Γ prescribed - Neumann
problem -)
∂u
=ψ
∂n
or of third kind (values of a combination of the function and its derivative prescribed - Robin
problem -), i.e.
∂u
+ αu = ψ
∂n
If g or ψ are = 0, the resulting boundary conditions are called homogeneous.
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2
Specific Examples of Elliptic PDEs
2.1
Heat Conduction
A solid heat conducting body is subjected to thermal loading at its boundary. The heat flux
vector q is defined as
q = −k∇u
where k is the thermal conductivity and u is the temperature.
The principle of conservation of energy is expressed by the energy balance equation, which,
for steady state conditions is
−∇ · q = 0
Therefore, the temperature u inside the solid is given by the solution of
−∇ · (−k∇u) = 0
i.e.
∇2 u = 0
subject to the boundary conditions representing the thermal loads at the surface. For instance if the temperature is u1 on the portion of the boundary Γ1 and ∂u/∂n = 0 on Γ2
S
(with Γ = Γ1 Γ2 , the statement of boundary conditions is
u = u0
on Γ1 and
∂u
=0
∂n
on Γ2 .
If the body experiences internal heat generation at a rate f 0 , the temperature is instead
given by the solution to the equation
∇2 u + f = 0
where f = f 0 /k. The solution again is subject to the appropriate form of the thermal loads
at the boundary. For instance, if u = 0 on Γ, the boundary condition becomes
u=0
on Γ.
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2.2
Electromagnetics
The current density J resulting from a specific spatial distribution of electric charge ρ is
given as
J = −λ∇u
where u is the electric potential of the field.
The principle of conservation of electric charge is expressed by the charge balance equation,
which, for steady state conditions is
−∇ · J = 0
Therefore, the potential u is given by the solution of
ρ
−∇ · (−λ∇u) = −
i.e.
ρ
where is the inductive capacity. If there is no charge, the governing equation becomes
simply
∇2 u = −
∇2 u = 0
The above must be solved subject to appropriate boundary conditions on the potential at Γ.
2.3
Potential Flow
In the steady flow of incompressible irrotational fluids, the velocity vector field v is given as
v = −∇u
where u is the velocity potential of the fluid.
The principle of conservation of mass is expressed by the mass balance equation or equation of continuity, which, for steady state conditions and in the absence of sources or sinks
is
−∇ · v = 0
Therefore, the potential u is given by the solution of
−∇ · (−∇u) = 0
i.e.
∇2 u = 0
The above must be solved subject to appropriate boundary conditions on the potential at Γ.
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2.4
Flow in Porous Media
The velocity vector field v associated with the flow of fluid through a porous media is given
by
v = −k∇h = −∇u
where h is the piezometric head, k is the seepage coefficient and u is the velocity potential.
Combining the above with the equation of continuity yields the following governing equation
for the velocity potential
∇2 u = 0
The above must be solved subject to appropriate boundary conditions on the potential at Γ.
2.5
Newtonian Mechanics and Coulombian Electrostatics
A point mass m0 is represented by a Dirac distribution m0 δ. The gravitational potential u
associated with the presence of the point mass is given by the solution of the equation
∇2 u = km0 δ
where k is a proportionality constant. The above must be solved subject to appropriate
boundary conditions on the potential at Γ. An identical formulation is obtained for the
electrostatic potential due to a point charge.
2.6
Solid Mechanics
The static deflection u of a thin plate clamped at the edge and subjected to external force
f is represented by the biharmonic equation
∇ 2 ∇2 u = f
Solutions of the biharmonic equation can always be represented by means of combinations
of two harmonic functions.
2.7
Differential Operator Representation
Note that all the above problems can be represented using differential operator notation; the
goal is to determine the function u satisfying the equation
Au = f
inside Ω and subject to specific boundary conditions on Γ. Here A is a linear differential
operator which transforms u into f . In the case of the Laplacian A = ∇2 .
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3
Laplacian Operator in Curvilinear Coordinates
The Laplacian operator in Cartesian coordinates (x, y, z) is simply
∂2
∂2
∂2
∇ =
+
+
∂x2 ∂y 2 ∂z 2
2
The Laplacian can be readily expressed in other systems of coordinates such as cylindrical
polar, spherical or even any curvilinear orthogonal system of coordinates by using the socalled scale factors h1 , h2 , h3 .
If the new coordinates are (u1 , u2 , u3 ), then, it can be shown that the Laplacian operator
is
∇2 =
1
∂ h2 h3 ∂
∂ h3 h1 ∂
∂ h1 h2 ∂
[
(
)+
(
)+
(
)]
h1 h2 h3 ∂u1 h1 ∂u1
∂u2 h2 ∂u2
∂u3 h3 ∂u3
Specifically, the scale factors h1 , h2 , h3 for a circular cylindrical system of coordinates
(r, θ, z) are hr = 1, hθ = r and hz = 1; and for a spherical coordinates system (r, φ, θ) are
hr = 1, hφ = r and hθ = r sin(φ).
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Relation with Analytic Functions of a Complex Variable
Harmonic functions are intimately involved in the theory of functions of a complex variable.
Specifically, let u(x, y) and v(x, y) be two real values functions of real variables representing
the real and imaginary parts of a function of a complex variable w, i.e.
w = u + iv
An important class of functions of a complex variable is constituted by the so-called analytic
functions, which are those for which a path independent derivative exists, derivatives of
all orders also exist and thus are representable in terms of power series. A necessary and
sufficient condition for analyticity is the satisfaction of the Cauchy-Riemann relations
∂v
∂u
=
∂x
∂y
and
∂v
∂u
=−
∂y
∂x
Differentiating the above yields
∇2 u = 0
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and
∇2 v = 0
Which state that both the real and imaginary parts of an analytic function of a complex
variable satisfy Laplace’s equation. The functions u and v are called conjugate harmonic
functions.
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Some Particular Solutions of Laplace’s Equation
Two solutions of Laplace’s equation are of particular importance. These correspond to
situations where the function u depends only on the distance r from a given point. Solutions
of this form are called radial functions and they satisfy the following form of Laplace’s
equation
1 d n−1 du
(r
)=0
n−1
r
dr
dr
where n is the dimensionality of the problem (i.e. n = 2 for plane problems and n = 3 for
three dimensional problems.
For spherically symmetric situations (i.e. n = 3), Laplace’s equation reduces to
d 2 du
(r
)=0
dr
dr
Integrating twice yields
C1
+ C2
r
making C1 = 1 and C2 = 0 yields the following fundamental solution of Laplace’s equation
1
u=
r
For situations with cylindrical symmetry on the plane (i.e. n = 2), Laplace’s equation
reduces to
d du
(r ) = 0
dr dr
Integrating twice yields
u=
u = C1 ln r + C2
making again C1 = 1 and C2 = 0 yields the following fundamental solution of Laplace’s
equation
u = ln r
Note that the above two solutions satisfy Laplace’s equation everywhere except the origin.
The above fundamental solutions constitute mathematical representations of , respectively,
point and line sources of charge, energy and mass.
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6
General Properties of Harmonic Functions
Harmonic functions posses very special properties. A summary of these properties is presented next without any proofs.
6.1
Condition for the Absence of Sources
Green’s first theorem states that for the any two continuous functions u and v in Ω
possesing continuous second derivatives inside Ω, the following is true
Z
Z
∂v
u∇ vdω = u dγ −
Ω
Γ ∂n
2
Z
Ω
S
Γ
∇u · ∇vdω
Selecting u = 1 and v an harmonic function yields
Z
Γ
∂v
dγ = 0
∂n
This is the condition for the absence of sources (or sinks) inside Ω.
Note that if the boundary condition along Γ is of Neumann type (i.e. ∂u/∂n = f ), the
above requires that
Z
Γ
f dγ = 0
I.e. the fluxes average to zero.
6.2
Mean Value Theorem
Let Γa be a sphere of radius a centered at the point x0 inside a region Ω where u is harmonic
the mean value theorem of harmonic functions states that
1 Z
udγ
u(x0 ) =
4πa2 Γa
Similarly, for a circle of radius a in a planar region Ω where u is harmonic
Z
1
u(x0 ) =
2πa
6.3
Ca
uds
Principle of Maximum Value
S
If the function u is defined and continuous in Ω Γ and harmonic inside Ω then the maximum
and minum values of u are attained at the surface Γ. As a consequence
min(u)|Γ ≤ u ≤ max(u)|Γ
Therefore, only the constant function can have a maximum inside Ω.
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6.4
Uniqueness and Stability
The first boundary value problem for Laplace’s equation consists on the determination of
the function u satisfying
∇2 u = 0
inside Ω and subject to
u=g
at the boundary Γ. It can be shown that this problem has a unique solution even if the
boundary conditions are discontinuous.
Stability requires that small changes in data produce small changes in the associated
solution. Therefore, it can also be shown that if the difference in the given boundary values
of u is smaller than then the difference in the resulted solutions is also smaller than .
The second boundary value problem for Laplace’s equation involves Neumann’s boundary
conditions instead. It can also be shown that the solution to the problem is determined
uniquely within an additive constant.
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