Chapter 11 ??????(Partial Differential Equations)

1
Chapter 11 ??????(Partial Differential
Equations)
1. ???? 2. ????(???????) 3. ???? ??????? 4.
DAlemberts Solution of the Wave Equation 5.
Heat Equation Solution by Fourier Series 6. Heat
Equation Solution by Fourier Integrals and
Transforms 7. Modeling Membrane, Two-Dimensional
Wave Equation 8. Rectangular Membrane Use of
Double Fourier Series 9. Laplacian in Polar
Coordinate 10. Circular Membrane Use of
Fourier-Bessel Series 11. Laplaces Equation in
Cylindrical and Spherical Coordinates.
Potential 12. Solution by Laplace Transforms
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Basic Concepts
Partial Differential Equation (PDE)

• ???????????(Exist more than one independent
  variables)
• ?????????

u ????? x, y ??
3
Basic Concepts
Consider a function of two or more variables e.g.
f(x,y). We can talk about derivatives of such a
function with respect to each of its
variables The higher order partial
derivatives are defined recursively and include
the mixed x,y derivatives

4
Basic Concepts
5
Basic Concepts
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General Forms of second-order P.D.E. (2 variables)
??
??
??
7
Hyperbolic (propagation)
8
Parabolic (Time- or space- marching)
???????
9
Elliptic (Diffusion, Equilibrium Problems)
10
System of Coupled P.D.E.s
11
Boundary and Initial Conditions

• Dirichlet condition specify
• Neumann condition specify
• Robin condition specify

At Boundary
or or both are prescribed at t 0
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Modeling Vibrating String, Wave Equation
???????

• Assumptions
• Homogeneous and perfectly elastic string.
• Neglect the action of gravitational force.
• Small vertical displacements

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Modeling Vibrating String, Wave Equation
???????
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Separation of Variables Use of Fourier Series
???????
For all t
Dirichlet boundary conditions
Initial deflection
Initial conditions
Initial velocity
Method of separating variables (Product
method)
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Separation of Variables Use of Fourier Series
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Separation of Variables Use of Fourier Series
For all t
Dirichlet boundary conditions
For all t
X
For k 0
For positive k µ2
X
For negative k -p2
? B 1
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Separation of Variables Use of Fourier Series
??
Eigenfunction (Characteristic function)
Spectrum)
Eigenvalues (Characteristic values)
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Separation of Variables Use of Fourier Series
Initial deflection
Initial conditions
Initial velocity
?????n? un (x,t) ??????????????,??????????
??,???????,????????
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Separation of Variables Use of Fourier Series
?? Bn ? f(x)???????????
?? ? g(x)???????????
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Separation of Variables Use of Fourier Series
Suppose g(x) 0 (?????)
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Separation of Variables Use of Fourier Series
f(x) initial deflection
f ? f ??? 2L??????
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Separation of Variables Use of Fourier Series
??????
??????
??
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DAlemberts Solution of the Wave Equation
Introduce the new independent variables
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DAlemberts Solution of the Wave Equation
??
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DAlemberts Solution of the Wave Equation
DAlemberts Solution
Initial deflection
Initial conditions
Initial velocity
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DAlemberts Solution of the Wave Equation
Initial conditions
? x ??
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DAlemberts Solution of the Wave Equation
If g(x) 0 (?????)
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Heat Equation Solution by Fourier Series
??????
u(x,y,z,t) ????????????????
c2 ????????(thermal diffusivity)
K ????????(thermal conductivity)
s ??????(specific heat)
? ??????(density)
29
Heat Equation Solution by Fourier Series
????????
Dirichlet boundary conditions
For all t
Initial conditions
Initial temperature
Method of separating variables (Product
method)
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Heat Equation Solution by Fourier Series
? B 1
??
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Heat Equation Solution by Fourier Series
?? Bn ? f(x)???????????
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Heat Equation Solution by Fourier Series
Steady-State Two-Dimensional Heat Flow
Steady-State ? ????????? ?
Dirichlet boundary conditions
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Heat Equation Solution by Fourier Series
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Heat Equation Solution by Fourier Series
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Heat Equation Solution by Fourier Series
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Heat Equation Solution by Fourier Integrals and
Transforms
????????????,??????????? ????????,??????
Initial temperature
Method of separating variables (Product
method)
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Heat Equation Solution by Fourier Integrals and
Transforms
??A?B?????,???p???
Initial conditions
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Heat Equation Solution by Fourier Integrals and
Transforms
?????(Fourier Integrals)
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Heat Equation Solution by Fourier Integrals and
Transforms
????
??
??
?
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Heat Equation Solution by Fourier Integrals and
Transforms
??
???????? f(x), ??????????? u(x,t)
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Heat Equation Solution by Fourier Integrals and
Transforms
-1 lt v lt 1
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Heat Equation Solution by Fourier Integrals and
Transforms
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Heat Equation Solution by Fourier Integrals and
Transforms
? ? u ??????,? u ? x ???
Initial conditions
44
Heat Equation Solution by Fourier Integrals and
Transforms
?w????
?w????
45
Modeling Membrane, Two-Dimensional Wave Equation
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Rectangular Membrane Use of Double Fourier Series
Dirichlet boundary conditions at
boundaries
For all t
Initial displacement
Initial conditions
Initial velocity
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Rectangular Membrane Use of Double Fourier Series
?????G(t)???????
?????F(x,y)???????
?????Helmholtz???
??Helmholtz?????????
48
Rectangular Membrane Use of Double Fourier Series
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Rectangular Membrane Use of Double Fourier Series
Dirichlet boundary conditions at
boundaries
For all t
???
???
m n 1,2,3,.
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Rectangular Membrane Use of Double Fourier Series
m n 1,2,3,.
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Rectangular Membrane Use of Double Fourier Series
Initial displacement
Initial conditions
Initial velocity
Double Fourier Series
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Rectangular Membrane Use of Double Fourier Series
?? Bmn ? km???????????
??
?? km ? f(x,y)???????????
??????(generalized Euler formula)
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Rectangular Membrane Use of Double Fourier Series
Initial velocity
Double Fourier Series
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Laplacian in Polar Coordinate
(x,y) ? (r,?)
?
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Laplacian in Polar Coordinate
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Laplacian in Polar Coordinate
????????????
???????????
??????????
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Circular Membrane Use of Fourier-Bessel Series
??????????? u(r.t),????????????
????
????
????
For all t ? 0
????
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Circular Membrane Use of Fourier-Bessel Series
? s kr
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Circular Membrane Use of Fourier-Bessel Series
???????(Bessels differential equation)
?? 0 ????????
W(r)???????????????J0?Y0,?Y0?0?????,??????J0
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Circular Membrane Use of Fourier-Bessel Series
??? r R ?,W(R) J0(kR) 0, J0????????
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Circular Membrane Use of Fourier-Bessel Series
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Circular Membrane Use of Fourier-Bessel Series
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Circular Membrane Use of Fourier-Bessel Series
????
????
????
am????-???????
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Circular Membrane Use of Fourier-Bessel Series
Now n 0
????
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Circular Membrane Use of Fourier-Bessel Series
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Circular Membrane Use of Fourier-Bessel Series
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Circular Membrane Use of Fourier-Bessel Series
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Circular Membrane Use of Fourier-Bessel Series
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