lecture 21: partial differential equations i: laplace...
TRANSCRIPT
1. 1. 2. 2.
Lecture 21: Partial Differential Equations I: Laplace equation
1. Key pointsSeparation of variableBoundary conditions: Dirichlet boundary conditions, Neumann boundary condition
2. Examples of PDE in physics
Laplace's equation: .
Poisson's equation: .
Helmholtz equation: .
Diffusion equation: .
Wave equation: .
Schrödinger equation: .
3. Type of boundary conditionsboundary conditions in space
Dirichlet boundary conditions:
Neumann boundary conditions:
boundary condition in timeInitial condition: This function must satisfy a spatial boundary condition, too.
4. Methods to solve PDESeparation of variablesIntegral transformation (Fourier transform and Laplace transform)
5.1 Example: Steady state temperature profileConsider a temperature profile in an open room with three walls as shown in Figure. Two walls at and have temperature T=0°. The temperature at
is also 0°. Only the wall at has T=100°.
x = x =y =
T =100°
T=0°T=0°
T=0° (y=)
Step 1: Find an appropriate differential equation.
Temperature profile is known to satisfy a Laplace equation
.
Step 2: Find the boundary conditions imposed by the problem.
Step 3: Choose a method to solve the differential equation.
We use the method of variable separation, which converts the partial differential equation to multiple ordinary differential equations.
Substituting this product to the original PDE,
The left hand side depends only on and the right hand side only on . That is impossible! Hence, the neither side depends on or . That means they are constant:
Then, we have two ODEs:
,
.
Step 4: Find general solutions to the ODEs.
These equations can be solved analytically. However, the type of solution depends on if , , or . Although it is automatically determined by boundary conditions, it is convenient to know the type of solutions before applying the boundary conditions. When , is oscillatory. However, it is unlikely since the temperature is expected to decay to zero as increases. When , is linear. Since the left and right walls have the same temperature, only possible solution is
= constant, which is again unlikely. So, we assume that . Then, the solutions are
Step 5: Determine the unknown constants by applying the boundary conditions.
There are five unknowns, , , , , and . On the other hand, there are only four boundary conditions. Don't worry. You will see.
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.
.
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Remark 1: Since there are many 's, we have also many 's.Remark 2: and are combined to a single constant .
Noting that the last boundary condition is a Fourier series, we can determine using the theory of Fourier series.
right hand side
which is simplified to
Now, evalute it from to a sufficiently large integer.
Step 6: Construct the final expression of the solution.
.
We plot it for L=1.
(5.1.1)(5.1.1)
(5.1.2)(5.1.2)
MaplePDE := diff(w(x, y), x, x)+diff(w(x, y), y, y) = 0
ans := pdsolve(PDE)
Here and are arbitrary functions.
5.2 Example: Scalar potential and electric field
Two infinitely long rounded metal plates,at and ,are connected at by metal strips maintained at
a constant potential , as shown in Figure. A thin layer of insulation at eachcomer prevents them from shorting out.Find the potential inside the resulting rectangular pipe. Show also the electric field inside the pipe.
Step 1: Find an appropriate differential equation.
Since the pipe is infinitely long, the potential does not depend on the coordinate . Therefore, the potential satisfies the two-dimensional laplace equation.
Step 2: Find the boundary conditions imposed by the problem.
. .
.
Step 3: Choose a method to solve the differential equation.
We use the method of variable separation, which converts the partial differential equation to two independent ordinary differential equations.
Substituting this product to the original PDE, we obtain
.
The left hand side depends only on and the right hand side only on . That is impossible! Hence, the both sides must depend neither on or . That means they are constant:
Then, we have two differential equations:
,
.
Step 4: Find general solutions to the ODEs.
These equations can be solved analytically. However, the type of solution depends on if , , or . Although it is automatically determined by boundary conditions, it is convenient to know the type of solutions before applying the boundary conditions. Since the potential is zero at the boundaries in the direction, it will be convenient to have basis functions that can take zero. When
, is oscillatory and can take zero. So, we assume that . Then, the solutions to the ODEs are
.
.
Step 5: Determine the unknown constants by applying the boundary conditions.
There are five unknowns, , , , , and . On the other hand, there are only four boundary conditions. Don't worry.
.
.
.
.
The last two bouondary conditions suggest that . Hence, the boundary conditions are simply
,
where ..
Noting that this is a Fourier series, we can determine by
right hand side
which is simplified to
.
Step 6: Construct the final expression of the solution.
.
5.3 Example: Spherical coordinatesA spherical shell of radius with an insulating ring in the plane has its upper hemisphere at potential and its lower hemisphere at . Find the potential inside the sphere.
Step 1: Find an appropriate differential equation.
We use the spherical coordinates.
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Step 2: Find the boundary conditions imposed by the problem.
.
where
Step 3: Separation of variables
Separating variables, the PDE can be written as
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At this point, is separated from and . Introducing a separation constant , thePDE can be split to
.
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Now, we can separate and . Introducing another separation constant [ , the ODE for is given by
,
and for ,
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Step 4: Find general solutions to the ODEs.
The ODE for can be solved immediately and its general solution is
.
The ODE for can be also solved easily. Using Maple ODE solver, =
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Solving the ODE for needs additional steps. Introducing a new variable , The ODE becomes
,
where . This ODE is nothing but a generalized Legendre equation. Since the solution must be real, the general solution is associate Legendre polynomials , and must be a non-negative integer.
Combining general all individual solutions,
Step 5: Determine the unknown constants by applying the boundary conditions.
Boundary condition at .The potential should not diverge at . Therefore, .
Boundary condition at .First we note that the potential does not depends on . Only satisfies the condition. Therefore, .
At this point, the potential is written as
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and the boundary condition is
.
Using the orthogonality of Legendre polynomials
.
.
,
and using , we obtain the final expression
Step 6: Plotting the results:
For plotting purpose, we assume and .
Constructing the expansion coefficients:
Constructing the potential:
Draw the boundaries:
Contour plotting of the potential:
Homework Due: 11/21 11am
2 1 . 1Find the steady-state temperature distribution in a metal plate of 10 cm square if one side is held at 100° and the other three sides at 0°. Find numericallt the temperautre at the center of the plate.