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Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Dr. Noemi Friedman, 04.11.2015. Introduction to PDEs and Numerical Methods Lecture 2: Analytical solution of ODEs and PDEs
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• Platzhalter fr Bild, Bild auf Titelfolie hinter das Logo einsetzen

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

• 04. 11. 2015. | PDE lecture | Seite 6

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)

=

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

: 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

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