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Introduction to PDEs and Numerical Methods Lecture 2...

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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)

    =

    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


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