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Dr. Noemi Friedman, 04.11.2015.
Introduction to PDEs and Numerical Methods
Lecture 2:
Analytical solution of ODEs and PDEs
04. 11. 2015. | PDE lecture | Seite 2
Basic information on the course tutorials and exercises
Time:
Friday: 16:00-17:30
Place:
Hans Sommerstr. 65
HS 65.4
First assignment:
Due to next lecture!!!
04. 11. 2015. | PDE lecture | Seite 3
Overview of the course
Introduction (definition of PDEs, classification, basic math,
introductory examples of PDEs)
Analytical solution of elementary PDEs (Fourier series/transform,
separation of variables, Greens function)
Numerical solutions of PDEs:
Finite difference method
Finite element method
04. 11. 2015. | PDE lecture | Seite 4
Overview of this lecture
Classification of PDEs revisited, some examples of PDEs again,
boundary conditions
Solving linear systems solving linear PDEs
Eigenvalues, eigenfunctions, solving linear equations with spectral
method spectral method for PDEs
Fourier series
Separation of variables analytical solution of the heat equation
04. 11. 2015. | PDE lecture | Seite 5
Classification of PDEs
Constant/variable coefficients
Stationary/instationary (not time dependent/time dependent)
Linear/nonlinear
linearity condition:
order
order of the highest derivative
homogeneous/inhomogenous
inhomogeneous: additive terms which do not depend on unknown function
homogenous: = 0 is a solution of the equation elliptic/parabolic/hyperbolic (only for second order PDEs)
+ 2 + + lower order derivatives = 0
2 = 0 parabolic 2 < 0 hyperbolic 2 > 0 elliptic
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Examples of PDEs and their classification
+ = 0Transport equation:
lim
= 0
Heat equation:
(Diffusion/membrane equation/
electrical conduction problem temperature: electric potential, heat flux: electric current)
()()
() =
Steady state of the heat equation:
=
Hanging bar:
(the wawe equation : vibrating string)
=
= 0
(Laplace equation)
= = 0 (Laplace equation)
=
Steady state of the
hanging bar: =
=
(Poissons equation)
04. 11. 2015. | PDE lecture | Seite 7
Examples of PDEs and their classification
= 00Gausss law for magnetism:
=
0
Gausss law: 0: permittivity of free space, electrical field
= 0Faraday law of induction:
: magnetic field
+ = Burgers equation:
: magnetic flux
(fluid mechanics, nonlinear acoustics, gas dynamics, traffic flow)
Inviscid Burgers equation: + = 0
Plate equation: + += 0
Analytical solution of the 2D Laplace equation
04. 11. 2015. | PDE lecture | Seite 8
Analytical solution of the 2D Laplace equation
Solve by seperation of variables
ansatz + seperation:
04. 11. 2015. | PDE lecture | Seite 9
Analytical solution of the 2D Laplace equation
seperation constant
a) Lets assume
= 0Any linear equation would
satisfy this equation:
= 1 + 2
b) Lets assume
Assuming the solution in the form:
=
2 + = 0 (characterisitic equation)
2 = = The solution is the linear combination of the two possible
solutions
= 3 + 4
3, 4 are arbitrary constants
eigenvalue-eigenfunction
problem: =
04. 11. 2015. | PDE lecture | Seite 10
= 3 + 4
=
3 cos( ) + sin( ) + 4 cos sin( ) =
3 + 4 cos + 3 4 sin( )
where , are new constants with3 + 4 = A 3 4 = B
= cos + sin( )
Introducing : = 2
= cos + sin()
Eigenvalue problem Poisson equation with hom. D.B.C
Lets assume
=
The solution:
= =
Lets assume
= 3 + 4
: separation constant,
eigenvalue of L =
2
: eigenfrequency of L =
2
Introducing : = 2 = 3
+ 4
04. 11. 2015. | PDE lecture | Seite 11
Analytical solution of the 2D Laplace equation
2) ansatz + seperation:
seperation constant
04. 11. 2015. | PDE lecture | Seite 12
Analytical solution of the 2D Laplace equation
2) ansatz + seperation:
0, = 0 = 1 = 0
04. 11. 2015. | PDE lecture | Seite 13
Analytical solution of the 2D Laplace equation
BC: 0 = 0
BC: 1 = 0
=
cos + sin < 01 + 2 = 0
3 + 4
> 0
0 =
cos + 00 + 2
30 + 4
0= 0
= 0
2 = 0
3= 4: =
2
=
sin < 01 = 0
2= sinh > 0
1 =
sin1
sinh(1)= 0
= , = 1, 2, 3, . .1 = 0 = 0
leads to () = 0(trivial solution)
0
04. 11. 2015. | PDE lecture | Seite 14
Analytical solution of the 2D Laplace equation
2) ansatz + seperation:
= sin( ) = , = 1, 2, 3, . .
04. 11. 2015. | PDE lecture | Seite 15
Analytical solution of the 2D Laplace equation
2) ansatz + seperation:
0 = 0 0 = 50 + 6
0
0 = 5 + 6 = 0 5= 6: =2
=
2= sinh()
sinh()
04. 11. 2015. | PDE lecture | Seite 16
Analytical solution of the 2D Laplace equation
, = () = sin(x) sinh()
, =
=1
() =
=1
sin(x)sinh()
04. 11. 2015. | PDE lecture | Seite 17
Analytical solution of the 2D Laplace equation
, =
=1
sin(x)sinh()
, 1 =
=1
sin sinh = 2 +
=
write the Fourier series of the initial condition :2 + =
=1
sin
=
sinh
04. 11. 2015. | PDE lecture | Seite 18