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Dr. Noemi Friedman, 31.10.2014.

Introduction to PDEs and Numerical Methods Tutorial 2: Analytical solution to PDE’s

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 2

Overview of this tutorial

Introduction: Differential operators Classification of PDEs Examples of ODEs/PDEs Eigenvalues, eigenvectors

Analytical solution of ODEs (1D – Poisson equation) Some important notations Heat equation Analytical solution to the Poisson equation by integration Solution with the Green’s function – the fundamental solution

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 3

Some important notations Important vector spaces • Euclidian n-space:

• C[a, b] : set of all continuous, real-valued functions defined on the interval [a, b]. • C1 [a, b]: set of all real-valued, continuously differentiable functions defined on

the interval [a, b]. (A function is continuously differentiable if its derivative exists and is continuous.)

• Ck [a, b]: space of real-valued functions defined on [a, b] that have k continuous derivatives

Dirichlet

Subspaces

Neumann

M.S. Gockenbach: Partial Differential Equations: Analytical and Numerical Methods, Chapter 3: Essential linear algebra!!!

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Heat equation 1D

Derivation of heat equation: Lecture Script, Chapter 1.1

homogeneous (no external heat source)

rhs/source term

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 5

Heat equation 1D

Derivation of heat equation: Lecture Script, Chapter 1.1

homogeneous (no external heat source)

rhs/source term

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 6

Heat equation Poisson equation

tenders towards an equilibrium state:

(Laplace equation)

𝑓 𝑓 (Poisson equation)

1D Poisson equation:

Dirichlet B.C.

Neumann B.C.

𝑝(𝑥)

𝑙

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Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)

𝑙

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 8

Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)

𝑙

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 9

Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)

𝑙

𝑥3

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 10

Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)

𝑙

𝑥3

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 11

Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)

𝑙

𝑥3

31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 12

Analytical solution of the Poisson equation

𝑥3 + cos(π𝑥)

We have already solve the PDEs:

𝑔 𝑥 = 𝑥3

ℎ 𝑥 = cos(π𝑥)

𝑢1 𝑥 =1

20 (−𝑥3 + 𝑥)

𝑢2 𝑥 =1π2

(cos π𝑥 + 2𝑥 − 1)

𝑢 𝑥 =1

20 −𝑥3 + 𝑥 +

1π2

(cos π𝑥 + 2𝑥 − 1)