PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth. PDE.pdfEvans L. C. 1998, Partial Differential...

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Transcript of PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth. PDE.pdfEvans L. C. 1998, Partial Differential...

  • 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