User:Egm6322.s09/Lecture plan

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Contents

References[edit]

Books[edit]

\displaystyle \clubsuit D. Zwillinger, Handbook of Differential Equations, Third Edition, Academic Press, 1998. ISBN-10: 0127843965. ISBN-13: 978-0127843964. UF library QA371.Z88 1989, 2 copies, one for in-library use. Google books Amazon.com

\displaystyle \clubsuit L. Lapidus & G.F. Pinder, Numerical solution of partial differential equations in science and engineering, Wiley, 1982. Google books Amazon.com UF Library Q172 .L36 1982

\displaystyle \clubsuit A.P.S. Selvadurai, Partial Differential Equations in Mechanics 1: Fundamentals, Laplace's Equation, Diffusion Equation, Wave Equation, Springer, 2000. ISBN-10: 3540672834. ISBN-13: 978-3540672838. Google books UF library QA805 .S45 2000 Amazon.com

\displaystyle \clubsuit A.N. Kolmogorov & S.V. Fomin, Introductory Real Analysis, Dover, 1975. Google books Amazon.com Not a major reference for this course; only for concept of linear operators.

\displaystyle \clubsuit M. Abramowitz & I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 1972. Read online, Download Wikipedia

\displaystyle \clubsuit P.M. Morse & H. Feshbach, Methods of mathematical physics, McGraw-Hill, 1953. Two volumes. Google books Amazon.com UF library QC20.M6

\displaystyle \clubsuit Additional references

J.W.S. Rayleigh, The theory of sound, Macmillan, London, Vol.1 1877, Vol.2 1878 (free pdf files), from Gallica, French National Library.

Web references[edit]

\displaystyle \spadesuit Partial differential equation in Scholarpedia. Written many well-known, famous experts (some are Nobel laureates, Fields medalists, etc.) and reviewed by experts; Scholarpedia is highly reliable, unlike Wikipedia. I became aware of Scholarpedia from an e-mail entry in NA Digest, Vol.8, No.6, 8 Feb 2009 by one of the authors of the Scholarpedia article Partial differential equation.

\displaystyle \spadesuit EqWorld, The World of Mathematical Equations. It is a good idea to verify the sources, as the site is not responsible for accuracy and correctness; see Rights and obligations of contributors and website administration.

\displaystyle \spadesuit Partial differential equations, wikipedia Be careful; always verify the sources.

\displaystyle \spadesuit Integrals: Lists of integrals (Wikipedia). David de Bierens de Haan, Nouvelles Tables d'Inte'grales De'finies (Engels, Leiden, 1862).

General nonlinear PDEs[edit]

Lapidus & Pinder 1982, p.1.

Order[edit]

Lapidus & Pinder 1982, p.2.

Linearity[edit]

Definition[edit]

Selvadurai 2000, p.74. More general discussion, lecture.

Kolmogorov & Fomin, p.123, linear functionals.

Linear PDEs[edit]

Lapidus & Pinder 1982, p.2 for linear first-order PDEs. General case, lecture.

Quasilinear PDEs[edit]

Lapidus & Pinder 1982, p.2 for quasilinear first-order PDEs. General case, lecture.

Nonlinear PDEs[edit]

Lapidus & Pinder 1982, p.2 for nonlinear first-order PDEs. General case, lecture.

Homogeneous / non-homogeneous PDEs[edit]

Second order PDEs : From general to particular[edit]

I decided to abandon this order of presentation (from general to particular), but rather present the material from particular to general, since it would be less abstract to students, who would be more comfortable to be anchored in what they already knew (or at least heard of), and then learn to generalize to concepts that are new to them.

So go directly to Alternative presentation from particular to general.

General nonlinear, 2nd order PDEs[edit]

n independent variables[edit]

Two independent variables[edit]

PDEs linear wrt 2nd derivatives, but nonlinear in general[edit]

n independent variables[edit]

Classification[edit]

Two independent variables[edit]

Classification[edit]

PDEs linear in all orders[edit]

Variable coefficients[edit]

Constant coefficients[edit]

Two independent variables[edit]

Classification[edit]
Equation for conics in 2-D[edit]

Second-order PDEs : Alternative presentation, from particular to general[edit]

PDEs linear in all orders (more particular)[edit]

Constant coefficients[edit]

Two independent variables[edit]

Linear transformation of coordinates[edit]
Matrix-operator form[edit]
Diagonalization, eigenvalue problem[edit]
Canonical forms[edit]
Nonlinear transformation of coordinates[edit]
Matrix-operator form, Leibniz rule[edit]
Curvilinear coordinates, separation of variables (1)[edit]

Curvilinear coordinates: Google search Wolfram Wikipedia

Polar coordinates Cylindrical coordinates Spherical coordinates Parabolic coordinates Hyperbolic coordinates

Laplacian in different curvilinear coordinates: Zwillinger 1998, p.187

\displaystyle 
   {\rm div}( {\rm grad} \, u ) 
   = 
   \nabla \cdot ( \nabla u )
   = 
   \nabla^2 u
   = 
   \Delta u
   = 
   u_{xx} + u_{yy}
   = 
   u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta \theta}

\displaystyle {\rm div}( {\rm grad} \, u ) = u_{rr} + \frac{1}{r} u_r + \frac{1}{r^2} u_{\theta \theta} + u_{zz}

\displaystyle {\rm div}( {\rm grad} \, u ) = u_{rr} + \frac{2}{r} u_r + \frac{1}{r^2} u_{\theta \theta} + \frac{1}{r^2 {\rm tan} \theta} u_\theta + \frac{1}{r^2 {\rm sin}^2 \theta} u_\psi

Two methods to obtain the above results

Application: Heat conduction problem (Laplace equation) Selvadurai 2000, p.195.

Diagonalization, eigenvalue problem[edit]
Canonical forms[edit]
Classification of PDEs, canonical forms[edit]
Hyperbolic, 2 canonical forms[edit]
Parabolic, canonical form[edit]
Elliptic, canonical form[edit]
Equation for conics in 2-D[edit]

Conics (Wikipedia)

General equation in 2-D[edit]
Connection to 2nd-order linear PDEs[edit]
Transformation of coordinates, eigenvalue problem[edit]
Canonical forms: Hyperbola, parabola, ellipse[edit]
Classification of PDEs revisited[edit]
Invariance[edit]

First-order linear PDEs for classification of 2nd order PDEs[edit]

EqWorld, exact solution for some 1st-order PDEs

Hyperbolic PDEs, 2nd canonical form[edit]

Method of characteristics (1)[edit]

from Wikipedia

Scott Sara's tutorial, Burger's equation, Java animation.

ODEs for characteristics and unknown function[edit]
Initial conditions[edit]
Solving for characteristics, plots[edit]
Solution along characteristics[edit]
General non-constant solution[edit]
Constant solution[edit]
General solution, perspective plots[edit]
Application to hyperbolic PDEs[edit]

Variable coefficients[edit]

Two independent variables[edit]

Classification[edit]

PDEs linear wrt 2nd derivatives, but nonlinear (more general)[edit]

Two independent variables[edit]

Nonlinear transformation of coordinates[edit]
Curvilinear coordinates (2)[edit]
Canonical form[edit]
Diagonalization, eigenvalue problem[edit]

Classification[edit]

n independent variables[edit]

Classification[edit]

General nonlinear, 2nd order PDEs (even more general)[edit]

n independent variables[edit]

Two independent variables[edit]

Heat conductivity dependence on temperature, from a recent talk (Feb 2009) by Simon Phillpot, Materials Science and Engineering, University of Florida.

Power law (Wikipedia)

For diamond:

\displaystyle \kappa \approx 0.4 \, W/Km at \displaystyle T = 1 ^\circ K

\displaystyle \kappa \approx 1000 \, W/Km at \displaystyle T \approx 10.7 ^\circ K


   \displaystyle
   a 
   \approx
   \frac
   { \log(1000) - \log(0.4) }
   { \log(10.7) - \log(1)}
   \approx
   3.39

   \displaystyle
   \log(\kappa) 
   =
   a
   \log(T)
   +
   \log(b)
   \Longrightarrow
   \kappa(T)
   =
   b
   \,
   T^a

   \displaystyle
   \log(\kappa) 
   =
   3.39
   \log(T)
   +
   \log(0.4)
   \Longrightarrow
   \kappa(T)
   =
   0.4
   \,
   T^{3.39}
   \,
   W / Km
   \ {\rm for} \  
   T \in
   \left[
      1^\circ K ,  10.7^\circ K
   \right]

No classification[edit]

Wave equation[edit]

One-dimensional case: Exact solution[edit]

Applications[edit]

Elastic bar vibration[edit]

String vibration[edit]

J.W.S. Rayleigh, The theory of sound, Macmillan, London, Vol.1 1877, Vol.2 1878 (free pdf files), from Gallica, French National Library.

Telegraph equation, transmission line[edit]

Dissipation[edit]
Dispersion[edit]

Method 1: Application of 1st-order PDE method[edit]

Homogeneous wave equation[edit]

Reduction to system of 1st-order PDEs[edit]

Method of characteristics[edit]

Scalar 1st-order PDEs[edit]
System of 1st-order PDEs[edit]

d'Alembert solution[edit]

Method 2: Application of coordinate transformation[edit]

Homogeneous wave equation[edit]

2nd canonical form[edit]

d'Alembert solution[edit]

Method 3: Coordinate transformation, more general initial data[edit]

2nd canonical form[edit]

Cauchy problem, Cauchy data[edit]

Homogeneous solution[edit]

Particular solution for non-homogeneous problem[edit]

General solution for non-homogeneous problem[edit]

Recovering d'Alembert solution[edit]

Three-dimensional case: Exact solution[edit]

Two-dimensional case: Exact solution[edit]

Hadamard's method of descent[edit]

General n-dimensional case: Exact solution[edit]

Odd dimension[edit]

Even dimension[edit]

Method of descent[edit]

Method of characteristics revisited (2)[edit]

from Wikipedia

Scott Sara's tutorial, Burger's equation, Java animation.

Advection equation[edit]

Inviscid Burgers' equation[edit]

Separation of variables (2)[edit]

Zwillinger 1998, p.441

Laplace equation (linear elliptic PDE)[edit]

Polar coordinates[edit]

See applications in Section Curvilinear coordinates, separation of variables (1).

Fluid mechanics application[edit]

Fluid between rotating cylinders. F.M. White, Fluid mechanics, McGraw-Hill, 2006. Google book, p.276 Amazon.com

Divergence in polar coordinates: F.M. White, p.230

Cartesian coordinates \displaystyle (x,y) = (x_1 , x_2), and basis vectors \displaystyle \{ \mathbf i , \mathbf j \} = \{ \mathbf e_x , \mathbf e_y \} = \{ \mathbf e_1 , \mathbf e_2 \}

Polar coordinates \displaystyle (r, \theta) = (\overline x_1 , \overline x_2), and basis vectors \displaystyle \{ \overline \mathbf i , \overline \mathbf j \} = \{ \mathbf e_r , \mathbf e_\theta \} = \{ \overline \mathbf e_1 , \overline \mathbf e_2 \} such that


   \displaystyle
   \overline \mathbf e_i
   :=
   \frac
   {\partial P}
   {\partial \overline x_i}

where \displaystyle P represents the position vector of point \displaystyle P. Note that in general


   \displaystyle
   \parallel
      \overline \mathbf e_i
   \parallel
   \ne
   1

Consider a vector \displaystyle \mathbf v such that: 
   \displaystyle 
   \mathbf v
   =
   v_i \mathbf e_i
   =
   \overline v_i \overline \mathbf e_i
   =
   v_r
   \mathbf e_r
   +
   v_\theta
   \mathbf e_\theta

Divergence:


   \displaystyle
   {\rm div}
   \,
   \mathbf v
   =
   \frac
   {\partial v_i}
   {\partial x_i}
   =
   \frac
   {1}
   {r}
   \frac
   {\partial}
   {\partial r}
   \left(
      r
      \frac
      {\partial v_r}
      {\partial r}
   \right)
   +
   \frac
   {\partial v_\theta}
   {\partial \theta}

Compare this expression of the divergence in polar coordinates to that in F.M. White, p.230, Eq.(4.8): What is the difference? Explanation? See lecture notes.

Unsteady heat equation (linear parabolic PDE)[edit]

Polar coordinates, Bessel functions[edit]

Zwillinger 1998, p.442

Bessel (or cylindrical) functions: First kind, second kind, also known as "cylindrical harmonics". Wikipedia Mathworld

M. Abramowitz & I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 1972. Read online, Download Wikipedia. Bessel functions starting p.358. Integral representations, p.360.

General linear homogeneous PDEs[edit]

Zwillinger 1998, p.441

Some special nonlinear PDEs[edit]

Zwillinger 1998, p.444

Wave equation (linear hyperbolic PDE) with sinusoidal time dependence[edit]

1-D case: Morse & Feshbach, p.125.

2-D, 3-D cases.

Helmholtz equation (linear elliptic PDE)[edit]

Solution for Helmholtz equation[edit]

Two-D case[edit]

Morse & Feshbach, p.498.

Moore & Spencer, Fields theory for engineers, D. Van Nostrand, 1961.

Three-D case, Stackel determinant[edit]

Morse & Feshbach, p.508.

Moore & Spencer, Fields theory for engineers, D. Van Nostrand, 1961.

Conformal mappings: Special method for Laplace equation[edit]

Zwillinger 1998, p.399.

Application[edit]

Functions of complex variables[edit]

tutorial Trigonometric functions

Complex analysis (Wikipedia)

Analytic functions[edit]

Conformal mappings[edit]

Preservation of Laplace equation[edit]

Schwartz-Christoffel transformation[edit]

Zwillinger 1998, p.399.

J.H. Mathew & R.W. Howell, The Schwarz-Christoffel Transformation, Cal State Fullerton, 2006.

Joukowsky (Zhukovsky) transformation[edit]

J.H. Mathew & R.W. Howell, Joukowsky airfoil, Cal State Fullerton, 2006.


   \displaystyle
   w = F(z) = z + \frac{c^2}{z}

with \displaystyle c being a constant, and \displaystyle w and \displaystyle z are complex variables:


   \displaystyle
   w = \xi + i \eta
   \ , \  {\rm and} \  
   z = x + i y

Nondimensionalization, dimensionless PDEs[edit]

Transform PDEs to dimensionless form[edit]

Heat equation[edit]

Wave equation[edit]

Scaling: Crumpling of thin shells[edit]

"Thin naturally curved shells arise on a range of length scales: from nanometer-sized viruses (1) to carbon nanotubes (2), from the micrometer-sized cell wall (3) to bubbles with colloidal armor (4), and from architectural domes (5) to the megameterscale earths crust (6). ... our everyday experience playing with thin flat and curved sheets of similar materials such as sheets of plastic suggests that the natural geometry of the surface dominates its mechanical response: a surface with positive Gauss curvature (e.g., an empty plastic bottle) has a qualitatively different response from that of a surface that is either flat (e.g., a plastic sheet) or has negative Gauss curvature." Vaziri & Mahadevan (2008)

A. Vaziri and L. Mahadevan, Localized and extended deformations of elastic shells, Proceedings of the National Academy of Sciences (USA), 105, 7913, 2008. Movie

Google for "similarity scaling dimensionless"

Concluding remarks[edit]

We have assumed smoothness in the solution in all that we did above. Often, nature is not smooth, and there are many applications for which the smoothness assumption is not valid.

Fractal geometry, which was introduced by Benoit Mandelbrot, has been used to model roughness in many diverse areas (nature, finance, engineering, medicine, cinematography, etc.); watch the following videos:

Hunting the hidden dimension, PBS NOVA program, aired in Oct 2008. Showed applications of fractals from forestry to biology.

Fractals in Science, Engineering and Finance (Roughness and Beauty) Benoit B. Mandelbrot, November 27, 2001, MIT World.