PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8...

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PARTIAL DIFFERENTIAL EQUATIONS r-order PDE Κ(y (r) , y (r-1) ,... y (1) ,y,x)=0 x=(x 1 , x 2 ,... x n ) n independent Real Variables y (k) = k y ∂x 1 k 1 ∂x 2 k 2 ∂x n k n , k 1 + k 2 +...+ k n = k , 0≤k i ≤k, 1≤k≤r , i=1,2,...,n r-order system of M PDE y is a vector of N variables y= 1 Κ is a vector function = 1 of the variables y i (k) , i=1,2,...N , k=1,2,...,r

Transcript of PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8...

Page 1: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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

∂x1k1 ∂x2

k2…∂xnkn , 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

Page 2: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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

Page 3: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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.

Page 4: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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/

Page 5: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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,

Page 6: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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)

Page 7: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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

Page 8: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

1. Potential Eq = Elliptic Eq L Laplace Eq

ΔΨ = 0 , ΔΨ = ∂2Ψ∂x1

2 + ⋯ + ∂2Ψ∂x𝑛

2

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

Page 9: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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

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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

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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

Page 12: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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)

⟺ 𝑑2𝛹1

𝑑𝑥2 + 𝑘12𝛹1 = 0

𝑑2𝛹2

𝑑𝑦2 + 𝑘22𝛹2 = 0

𝑑

2𝛹3𝑑𝑧2 + 𝑘3

2𝛹3 = 0 The Harmonic Oscillator equation

𝑘12 +𝑘2

2 +𝑘32 = λ = �𝒌��⃗ �

𝟐

Page 13: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Other Solutions

𝛹𝑘1,𝑘2,𝑘3(�⃗�) = 𝑒𝑖𝑘1𝑥𝑒𝑖𝑘2𝑦𝑒𝑖𝑘3𝑧=𝐞+𝒊𝒌��⃗ 𝒙��⃗

𝛹0,𝑘2,𝑘3(�⃗�) = (𝑎 + 𝑏𝑥)𝑒𝑖𝑘2𝑦𝑒𝑖𝑘3𝑧

𝛹0,0,𝑘3(�⃗�) = (𝑎 + 𝑏𝑥)(𝑎 + 𝑏𝑦)𝑒𝑖𝑘3𝑧

Page 14: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Laplacian in Coordinate Systems

In Cartesian coordinates,

In cylindrical coordinates,

In spherical coordinates:

Page 15: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

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

Page 16: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Spectral Geometry

Direct Problem: Given the Manifold, Find the Spectrum of the Laplace Operator

Inverse Problem: Given the Spectrum, Find Manifolds with the same spectrum of their Laplace Operator

Berard P. 1986, Spectral Geometry: Direct and Inverse Problems, Springer, Berlin

Page 17: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Wave Eq = Hyperbolic Eq Linear Wave Propagation, Acoustics, EM Waves ∂2𝜓∂𝑡2 = c Δ2ψ Wave Equation as harmonic oscillations and Helmholtz Eq

∂2𝜓∂𝑡2 = c Δ2ψ with ψ(�⃗�, t) = Q(�⃗�)φ(𝑡)

⟺ 𝑑2𝜑

𝑑𝑡2 + 𝑘2φ = 0 The Harmonic Oscillator equation

–ΔQ = 𝑘2Q The Helmoltz equation

Page 18: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Elementary Solutions:

ψ(x�⃗ ,t) = A𝑒−𝑖(𝜔𝑡−𝑘�⃗ �⃗�) , 𝑘�⃗ = 𝜔𝑐

𝑛�⃗ harmonic waves propagating in the direction 𝑛�⃗

D’Alembert Solutions

ψ(x�⃗ ,t) = A(x�⃗ ) 𝜑(ℎ(x�⃗ ) ± 𝑡)

A(x�⃗ ) the waveform

φ(h(x�⃗ ) ± t) the propagation law

h(x�⃗ ) ± t the phase

Aσκηση

Eπιλυση της Εξισωσης Κυματος σε 1 διασταση με τις 5 Συνοριακες Συνθηκες

Αναλυτικη + Προσομοιωση (0.2+0.2) x 4 =2

Page 19: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Diffusion = Heat Conduction Equation = Parabolic Eq Linear Διάχυση, Αγωγη Θερμότητας, Αγωγη Ρευματος,

∂𝜓∂t

= βΔψ

β = Diffusion coefficient

1 dim solutions 𝜕𝜓𝜕𝑡

= 𝛽𝜕2𝜓𝜕𝑥2

ψ(x, t) = A(x2 + 2βt) + B,

ψ(x, t) = A(x3 + 6βtx) + B,

ψ(x, t) = A(x4 + 12βtx2 + 12 β2 t 2) + B

𝜓(𝑥, 𝑡) = 𝐴√𝑡

exp �−𝑥2

4𝛽𝑡� + 𝛣

Page 20: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Aσκηση

Επαληθευση, Προσομοιωση και Συγκριση των 4 Λυσεων {0.5 x 4 = 2}

Diffusion Equation as linear Relaxation and Helmholtz Eq

Ψ(x,t) = u(x)T(t)

Δψ =−λψ

𝜕𝜓𝜕𝑡

= −𝜆𝛽𝜓

Aσκηση

Eπιλυση της Εξισωσης Διαχυσης σε 1 διασταση με τις 5 Συνοριακες Συνθηκες

Αναλυτικη + Προσομοιωση (0.2+0.2) x 4 =2

Page 21: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Μετασχηματισμος των Λυσεων της Εξισωσης Διαχυσης στις Μονοφορες Μεταθεσεις 𝑽𝒕𝒇(𝒙) = 𝒇(𝒙 − 𝒕)

Νεα Αναπαρασταση της Λυσης μεσω του Τελεστη του Χρονου

Antoniou I., Prigogine I., Sadovnichii V., Shkarin S. 2000,

Time Operator for Diffusion, Chaos, Solitons and Fractals 11, 465-477

Page 22: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Στοχαστικες Διαδικασιες Διαχυσης Black-Scholes

∂𝜓∂t

= − 12 σ2 x2 ∂2𝜓

∂𝑥2 −𝑟 x ∂𝜓∂x

+𝑟 ψ

ψ is the price of the Derivative as a function of time and stock price x

Μετασχηματισμος στην Εξισωση Διαχυσης Ασκηση 1

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Εξισώσεις Αντίδρασης – Διάχυσης Αυτο-οργανωση Bιολογικη Μορφογεννεση

∂𝜓∂t

= βΔψ + R(x)

Turing A. 1952, The Chemical Basis of Morphogenesis, Philosophical Transactions of the Royal Society

of London. Series B, Biological Sciences 237, No. 641, 37-72

Prigogine I. 1980, From Being to Becoming, Freeman, New York.

Murray J. 2002, Mathematical Biology:I. An Introduction, 3d ed. , Springer, New York

Murray J. 2003, Mathematical Biology:II. Spatial Models and Biomedical Applications, 3d ed. Springer, New York

Grindrod P. 1996, The theory and applications of reaction-diffusion equations: Patterns and waves 2nd ed., Clarendon Press, New York

Page 24: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Fisher-Κοlmogorov Equation 1937

∂𝜓∂t

= β∂2ψ∂x2 + 𝑟𝜓(1 −

𝜓𝛫

)

β is the diffusion coefficient

Ψ the population spatial density

r is the Reproduction Rate

K is the Ecosystem Capacity Fisher R. A. 1937,"The wave of advance of advantageous genes", Ann. Eugenics 7:353–369.

Kolmogorov A., Petrovskii I., Piscounov N.1937, Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Moscow Univ, Bull. Math. 1, 1-25, in V. M. Tikhomirov, editor, Selected Works of A. N. Kolmogorov I, pages 248--270. Kluwer 1991

Ablowitz M. , Zeppetella A. 1979, Explicit solutions of Fisher's equation for a special wave speed, Bulletin of Mathematical Biology 41, 835-840

Aσκ 2

Page 25: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Brusselator = (Brussels + Oscillator)

𝛛𝐗𝛛𝐭

= 𝜷𝟏𝛛𝟐𝚾𝛛𝐫𝟐 + 𝜜 − (𝜝 + 𝟏)𝜲 + 𝜲𝟐𝜰

𝛛𝚼𝛛𝐭

= 𝜷𝟐𝛛𝟐𝚼𝛛𝐫𝟐 + 𝜝𝜲 − 𝜲𝟐𝜰

Χ, Υ concentrations of interacting chemicals

Prigogine I. 1978, "Time, Structure, and Fluctuations," (Nobel Lecture in Chemistry) Science 201 , 777-785. Prigogine I. 1980, From Being to Becoming, Freeman, New York.

Ασκ 4

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Chemical Oscillations Waves

Page 27: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Waves with Damping = Hyperbolic-Parabolic Eq Linear

Telegraphist Eq Heat / Diffusion Propagation with finite velocity ⟺ waves with damping

𝜕2𝜓𝜕𝑡2 = 𝑐2∆𝜓 − 𝛾

𝜕𝜓𝜕𝑡

Aρμονικος Ταλαντωτης με Αποσβεση γ + Ηelmoltz Equation Aσκηση 0.5

ψ(x�⃗ ,t)=Ae−γ2𝑡𝑒−𝑖(𝜔𝑡−𝑘�⃗ �⃗�) , 𝑘�⃗ = 𝜔

𝑐𝑛�⃗

waves propagating in the direction 𝑛�⃗ , with damping parameter γ/2.

Page 28: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Schroedinger Eq Linear Quantum Mechanics

∂𝜓∂t

= − Δ2ψ + V(x)ψ

Τhe solution is obtained from the solution to the Eigenvalue problem of the Hamilton Operator:

− Δ2ψ + V(x)ψ = λψ

a n-dim Sturm-Liouville Εquation

a(x) 𝑑2𝑦𝑑𝑥2 + 𝑏(𝑥) 𝑑𝑦

𝑑𝑥= 𝜆𝑦

Special Functions

Page 29: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Fluid Dynamics Euler Eq for ideal Fluid 1755 ∂𝑣∂t

= − v∇v − 1ρ

(−𝛻𝑝 + 𝐹)

Η Εuler είναι Hamilton DE Ασκηση 0.5

Page 30: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Navier-Stokes Eq (Fluid Momentum Balance) 1843

∂𝑣∂t

= − v∇v + 1ρ (−𝛻𝑝 + 𝛻𝕊 + 𝐹)

v(x,t) = �𝑣1(𝑥, 𝑡)𝑣2(𝑥, 𝑡)𝑣3(𝑥, 𝑡)

� in [0,+∞) x 𝔇, 𝔇 ⊆ ℝ3 the Fluid Velocity

v=0 on ∂𝔇

v(x,0) = v0(x)

p the fluid pressure ρ the fluid density 𝕊 the stress tensor (model) F body forces (model)

Continuity Eq (Mass Balance) ∂𝜌∂t

= −𝛻(𝜌𝑣) + σ

σ the fluid source (model)

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Waves follow our boat as we meander across the lake turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. One of the Unsolved Millennium Prize (1 million dollar) Problems stated by the Clay Mathematics Institute http://www.claymath.org/millennium/Navier-Stokes_Equations/

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Soliton Equations Non Linear

BURGERS Equation 𝜕𝜓𝜕𝑡 + 𝜓

𝜕𝜓𝜕𝑥 − 𝛾

𝜕2𝜓𝜕𝑥2 = 0

𝜕𝑡𝜓 + 𝜓𝜕𝑥𝜓 − 𝛾𝜕𝑥2𝜓 = 0

ψ= velocity , γ=viscosity Burgers' equation is a fundamental PDE from fluid mechanics. modeling gas dynamics and traffic flow Burgers J.M. 1948, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1, 171-199.

Theorem

The Burgers Eq can be transformed to the Heat Eq

𝜓 = −2𝛾1𝜑

𝜕𝜑𝜕𝑥 = −

𝜕𝜕𝑥 𝑙𝑛𝜑

If ψ satisfies the Burgers Eq, the φ satisfies the Heat Equation: 𝜕φ𝜕𝑡

= 𝛾 𝜕2φ𝜕𝑥2

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Korteweg-De Vries (KdV) Equation

𝜕𝜓𝜕𝑡

+ 𝜓𝜕3𝜓𝜕𝑥3 + 𝜓

𝜕𝜓𝜕𝑥

= 0

𝜕𝑡𝜓 + 𝜓𝜕𝑥3𝜓 + 𝜓𝜕𝑥𝜓 = 0

PDE Notation 𝜕3𝜓

𝜕𝑥3 = 𝜕𝑥3𝜓 = 𝜓𝑥𝑥𝑥

Theorem

There is no Transformation of KdV to Heat Eq , like the Burgers Eq

Proof Ασκηση {2}

Page 34: PARTIAL DIFFERENTIAL EQUATIONS - cosal.auth.grcosal.auth.gr/iantonio/sites/default/files/M1/8 PDE.pdfEvans L. C. 1998, Partial Differential Equations, American Mathematical Society,

Solution ψ(x,t)=3c sech2 [√c2

(x − ct)] , sechx = 2ex+e−x

The soliton speed is c, the graph moves c units of x in one unit of t

𝜓(𝑥, 𝑡) = 6

ex−1

2 +e−x−12

c=1 , t=1

Ασκηση Soliton Αναλυτικη Λυση και προσομοιωση 0.5 +0.5 =1

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Soliton Waves discovery:

Russell J.S. 1845, "Report on waves",

Proc. of the British Association for the Advancement of Science, London, , p. 311.

"I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height'. Its height gradually diminished, and after a chase of one or two miles I"lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation "

Russell's results were controversial since it was not believed at that time that such a wave could be stable. The Astronomer Royal, Sir John Herschel, dismissed it as "merely half of a common wave that has been cut off." There was also a dispute with Airy, who had developed a shallow-water wave theory in which such waves were not stable. The controversy was resolved in 1895 by Korteweg and de Vries who derived the KdV Equation governing weakly nonlinear shallow-water waves

Korteweg D. and de Vries G. 1895, "On the change of form of long waves advancing in a rectangular

canal, and on a new type of long stationary waves", Phil. Mag. 39, 422.

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the word "soliton" was introduced to characterize waves that do not disperse and preserve their form during propagation and after a collision.

Zabusky N. J., Kruskal M. D. 1965, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240

𝜕𝜓𝜕𝑡

+ 𝑎𝜕3𝜓𝜕𝑥3 + 𝜓𝑛

𝜕𝜓𝜕𝑥

= 0

General References: Drazin, P. G., Johnson, R. S. 1989, Solitons: an introduction (2nd ed.). Cambridge University Press

Manton N.; Sutcliffe, P. 2004, Topological solitons, Cambridge University Press

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Soliton Applications

Particle Models

Rebbi C., Soliani G. 1984, Solitons and Particles, World Scientific

Manton N. 2008, Solitons as elementary particles: a paradigm scrutinized,

Nonlinearity 21, T221–T232 doi:10.1088/0951-7715/21/11/T01

Optical Fibers

Mollenauer, L., James P. 2006, Solitons in optical fibers, Elsevier Academic Press.

Biology

Yakushevich, L. 2004, Nonlinear Physics of DNA (2nd ed.), Wiley-VCH

Sinkala Z. 2006, "Soliton/exciton transport in proteins". J. Theor. Biol. 241 (4): 919–927, doi:10.1016/j.jtbi.2006.01.028