PARTIAL DIFFERENTIAL EQUATIONS r-order PDE

Κ(y(r), y(r-1),... y(1),y,x)=0 x=(x1, x2,... xn) n independent Real Variables

y(k)= ∂ ky

∂x1 k1 ∂x2

k2…∂xn kn , k1 + k2 +...+ kn = k , 0≤ki≤k, 1≤k≤r , i=1,2,...,n

r-order system of M PDE

y is a vector of N variables y=� 𝑦1 ⋮

𝑦𝑁 �

Κ is a vector function 𝛫 = � 𝛫1 ⋮

𝛫𝑀 � of the variables yi

(k) , i=1,2,...N , k=1,2,...,r

The PDE is defined by a Formula and Limit Conditions

Initial Conditions = Behaviour at present time and space

Long Term Conditions = Behaviour at the Remote Past, Remote Future

Boundary Conditions = Behaviour at Boundaries Obstacles

Large Distance Conditions = Behaviour Far from a Region of interest

DE describe processes in terms of local rates of change

The Class of Solutions of the PDE

{ψ : K[ψ]=0 , IC[ψ]=true , BC[ψ]=true , LTC[ψ]=true , LDC[ψ]=true}

Notation: ∂y ∂x

= 𝜕𝑥

PDE Τα 3 Προβληματα

1.Προβλημα Cauchy, Εξελιξη, Δυναμικα Συστηματα ,

Ημιομάδες

2.Το Προβλημα των Συνοριακων Συνθήκων.

Οι 3 Συνοριακες Συνθήκες Dirichlet, Neumann, Robin.

3.Το Προβλημα Ιδιοτιμών. Εξίσωση Helmholtz,

Sturm-Liouville, Εξίσωση Schrodinger.

Partial Differential Equations Δασιος Γ. , Κυριακη Κ. 2004, Μερικες Διαφορικες Εξισωσεις, Αθηνα Evans L. C. 1998, Partial Differential Equations, American Mathematical Society, Providence Gustafson K. 1999, Introduction to Partial Differential Equations and Hilbert Space Methods, Dover, New York. Hormander L. 1990, The Analysis of Linear Partial Differential Operators 1: Distribution Theory and Fourier Analysis, Springer

Hormander L. 1999, The Analysis of Linear Partial Differential Operators 2: Differential Operators with Constant Coefficients, Springer. Hormander L. 1985, The Analysis of Linear Partial Differential Operators 3: Pseudo-Differential Operators, Springer Hormander L. 1994, The Analysis of Linear Partial Differential Operators 4: Fourier Integral Operators, Springer. Sobolev S. 1989, Partial Differential Equations of Mathematical Physics, Dover, New York. VVesdsnsky D. 1992, Partial Differential Equations with Mathematica, Addison Wesley, New York. PDF Simulations: Femlab http://www.comsol.com/

http://www.comsol.com/

The theory of partial differential equations hardly existed before 1840. The vibrating string (wave) equation

𝜕2𝜓 𝜕𝑡2

= 𝑐2 𝜕2𝜓 𝜕𝑥2

ψ: ℝ1+1 ⟶ℝ : (x,t) ⟼ ψ(x,t) ψ(x,t) had been solved by d'Alembert, Euler, and Daniel Bernoulli around 1750,

Lagrange asserted in 1800 that all problems of fluid mechanics (Euler Equation) reduced to the mathematical problem of integrating his “general equations” (Newton Eq. Hamilton Equations). Monge (1746-1818), Laplace (1749-1827), Fourier (1768-1830), Poisson (1781-1840), Cauchy (1789-1857), all Members of the Ecole Polytechnique in Paris reduced heat conduction and the bending of elastic solids to the problem of Integrating Linear PDEs obtained solutions by techniques of Fourier Analysis solved the initial value problem for the PDE of the vibrating rod and vibrating plate. The first general existence theorem about PDEs was proved by Cauchy in 1842 Sonia Kowalewski in her thesis (1874), written under Weierstrass generalized Cauchy's results to PDEs containing time derivatives of any finite order r Cauchy's theorem as generalized by Kowalewski has since then been called Cauchy-Kowalewski Theorem.

PDE of Mathematical Physics (Potential, Transport-Diffusion, Propagation)

The 1st order Transport PDE

𝜕𝑡𝜌 = −𝑣1𝜕𝑥1𝜌 − 𝑣2𝜕𝑥2𝜌 − 𝑣3𝜕𝑥3𝜌

𝜕𝑡𝜌 = −𝑣∇𝜌

v(x,t) = 𝑣 the constant transport Velocity

ρ(x,t) the density Theorem ρ(x,t) = f(x−vt) For any smooth real function is a solution of the 1 dimensional linear transport equation 𝜕𝑡𝜌 = −𝑣𝜕𝑥𝜌 Aσκηση 0.5

Find the analogous solution of the 3 dimensional linear transport equation 𝜕𝑡𝜌 = −𝑣∇𝜌

Aσκηση 0.5

1. Potential Eq = Elliptic Eq L Laplace Eq

ΔΨ = 0 , ΔΨ = ∂ 2Ψ

∂x12 + ⋯ + ∂

2Ψ ∂x𝑛2

Poisson Eq Electrostatics Potential in Space

ΔΨ(x) = − f(x) −ΔΨ = − ∂

2Ψ ∂x12

− ⋯ − ∂ 2Ψ

∂x𝑛2 the Laplace Operator

Helmholtz Eq

−ΔΨ(x) = λΨ(x) Eigenvalue problem of Laplace Operator

The Meaning of the Laplacian

The Laplacian of the function Ψ is the flux density of the gradient flow of Ψ.

Laplacian :

ΔΨ = div (gradΨ) = 𝛛 𝟐𝚿

𝛛𝐱𝟐 + 𝛛

𝟐𝚿 𝛛𝒚𝟐

+ 𝛛 𝟐𝚿

𝛛𝐳𝟐

Laplace Operator = the Positive Laplacian :

LΨ = −ΔΨ = − 𝛛 𝟐𝚿

𝛛𝐱𝟐 − 𝛛

𝟐𝚿 𝛛𝒚𝟐

− 𝛛 𝟐𝚿

𝛛𝐳𝟐

ΔΨ(x) = AverageV [Ψ(x)] – Ψ(x),

AverageV [Ψ(x)] = ∫𝑽𝒅𝒗𝜳(𝒚)

𝑽 , V is a neighbourhood of x

Leubner C. 1987, Coordinate-free Interpretation of the Laplacian, Eur. J. Phys. 10-11

Aσκηση 1

http://en.wikipedia.org/wiki/Flux_density http://en.wikipedia.org/wiki/Gradient_flow

Τhe Analysis of the Laplace Operator is extensively used for modeling all kinds of physical phenomena involving spatial changes

electric and gravitational potentials

waves

diffusion

transport

Graph Laplacian Matrix: Network Analysis, WWW Bıyıkoglu T., Leydold J., Stadler P. 2008, Laplacian Eigenvectors of Graphs. Perron-Frobenius and Faber-Krahn Type Theorems, Springer, Berlin Brouwer A., Haemers W. 2011, Spectra of Graphs, Springer, New York

Van Mieghem P. 2011, Graph Spectra for Complex Networks, Cambridge University Press, Cambridge

http://en.wikipedia.org/wiki/Electric_potential http://en.wikipedia.org/wiki/Gravitational_potential

Helmholtz Eq Homogeneous Boundary conditions

Cartesian Coordinates

Elementary Solutions of the Eigenvalue problem: (–Δ)Ψ = λΨ

−∆𝒆−𝒊𝒌��⃗ 𝒙��⃗ = �𝒌��⃗ � 𝟐

𝐞+𝒊𝒌��⃗ 𝒙��⃗

−∆𝒆+𝒊𝒌��⃗ 𝒙��⃗ = �𝒌��⃗ � 𝟐

𝐞−𝒊𝒌��⃗ 𝒙��⃗

𝒌��⃗ 𝒙��⃗ = 𝒌𝟏𝒙𝟏 + 𝒌𝟐𝒙𝟐 + 𝒌𝟑𝒙𝟑

ΔΨ = 𝛛 𝟐𝚿

𝛛𝐱𝟐 + 𝛛

𝟐𝚿 𝛛𝒚𝟐

+ 𝛛 𝟐𝚿

𝛛𝐳𝟐

Solutions as Superpositions of Natural Vibrations Normal Modes

Ψ(x,y,z) = ∑ 𝐴𝒌𝟏,𝒌𝟐,𝒌𝟑𝒌𝟏,𝒌𝟐,𝒙𝟑 𝐞 +𝒊(𝒌𝟏𝒙𝟏+𝒌𝟐𝒙𝟐+𝒌𝟑𝒙𝟑) = ∑ 𝐴𝒌��⃗𝒌��⃗ 𝐞

+𝒊𝒌��⃗ 𝒙��⃗

–ΔΨ = λΨ with Ψ(x,y,z) = 𝛹1(𝑥)𝛹2(𝑦)𝛹3(𝑧)

⟺ 𝑑 2𝛹1

𝑑𝑥2 + 𝑘12𝛹1 = 0

𝑑 2𝛹2

𝑑𝑦2 + 𝑘22𝛹2 = 0

𝑑

2𝛹3 𝑑𝑧2

+ 𝑘32𝛹3 = 0 The Harmonic Oscillator equation

𝑘12 +𝑘22 +𝑘32 = λ = �𝒌��⃗ � 𝟐

Other Solutions

𝛹𝑘1,𝑘2,𝑘3(�⃗�) = 𝑒 𝑖𝑘1𝑥𝑒𝑖𝑘2𝑦𝑒𝑖𝑘3𝑧=𝐞+𝒊𝒌��⃗ 𝒙��⃗

𝛹0,𝑘2,𝑘3(�⃗�) = (𝑎 + 𝑏𝑥)𝑒 𝑖𝑘2𝑦𝑒𝑖𝑘3𝑧

𝛹0,0,𝑘3(�⃗�) = (𝑎 + 𝑏𝑥)(𝑎 + 𝑏𝑦)𝑒 𝑖𝑘3𝑧

Laplacian in Coordinate Systems

In Cartesian coordinates,

In cylindrical coordinates,

In spherical coordinates:

http://en.wikipedia.org/wiki/Cartesian_coordinates http://en.wikipedia.org/wiki/Cylindrical_coordinates http://en.wikipedia.org/wiki/Spherical_coordinates

Can we hear the shape of the Drum?

⟺ Can we infer the shape of the Drum

from the Spectrum of the Laplace Operator on the Drum? Kac M. 1966, "Can one hear the shape of a drum?", American Mathematical Monthly 73 (4, part 2): 1–23

Yes for the square Drum

Not in general

Gordon C. , Webb D., Wolpert S. 1992, "One Cannot Hear the Shape of a Drum",

Bulletin of the American Mathematical Society 27 (1): 134–138

Chapman, S.J. 1995, "Drums that sound the same", American Mathematical Monthly (February): 124–138

Aσκηση Παραδειγμα ΝΑΙ – ΟΧΙ: 3+3=6

Spectral Geometry

Direct Problem: Given the Manifold, Find the Spectrum o