1. Office hours: 3:00-5:00PM, Thursday at Room 523, EECS bldg. 2. e-mail: [email protected]: I should reply every e-mail. 3. Website: For more information and course slides: http://mx.nthu.edu.tw/∼rklee 4. TA hours: 7:00-9:00PM, every Tuesday and Thursday at Room 521, EECS bldg. (2 × 2 hours per week) (a) I-Hong Chen, 2nd-year PhD student, e-mail: [email protected] (b) Chih-Yao Chen, 2nd-year Master student, e-mail: [email protected] EE-2020, Spring 2009 – p. 1/43
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2. e-mail: [email protected]: I should reply every e-mail.
3. Website: For more information and course slides: http://mx.nthu.edu.tw/∼rklee
4. TA hours: 7:00-9:00PM, every Tuesday and Thursday at Room 521, EECS bldg. (2 × 2 hours per week)
(a) I-Hong Chen , 2nd-year PhD student, e-mail: [email protected]
(b) Chih-Yao Chen , 2nd-year Master student, e-mail: [email protected]
EE-2020, Spring 2009 – p. 1/43
Syllabus: for PDE
2. Diffusion-type problems: [Textbook] Ch.12, [Ref.] Ch.2.
Derivation of the Heat equation, (3/2).
Boundary conditions for Diffusion-type problems, (3/5).
Separation of variables, (3/9).
Solving nonhomogeneous PDEs, (3/12).
Integral transforms, (3/16, 3/19).
The Fourier transform, (3/23).
The Laplace Transform, (3/26).
1-D Wave equation, (4/2, 4/6).
D’Alembert solution of the Wave equation, (4/9).
Sturm-Liouville problems, (4/13).
2-D Wave equation in Cartesian and polar coordinates, (4/16, 4/20).
Laplace’s equation in Cartesian, polar, and spherical coordinates, (4/23).
EE-2020, Spring 2009 – p. 2/43
Diffusion equation
Dispersive/Diffractive/Diffused Wave: Linear
. . .
Differential equations
where unknown x is a number or set of numbers.
Differential equation:
f(u, u′, u′′, . . . ) = g(x),
where unknown u is a function. Note that u′, u′′ . . . are not extra unknowns for they can be derived by differentiation once u is determined.
Insteady of differential equation, one can also use the integral equation. E.g. Gauss’s law for the electric field:
∇ · E = ρ
An equation containing derivative(s) of an unknown function (or functions) (dependent variable) u with a single independent variable. E.g.
u′′(x) + u(x) = 0.
Partial Differential equations
Partial Differential Equations, PDEs
An equation containing partial derivative(s) of an unknown function u with two
or more independent variables. E.g.
∂ u(t, x)
Why PDEs?
People sense the real world via four (or multiple) dimensions (x, y, z, t), therefore, physical quantities (e.g. electrical field, temperature, electron distribution in an atom) are fully described by four variables.
Most physical laws are described in terms of PDEs, where the derivatives represent physical quantities.
E.g. Electrostatics (Poisson theory), EM waves (Maxwell’s equations), quantum mechanics (Schrodinger’s equation), heat transfer (heat equation), fluid mechanics.
Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic.
EE-2020, Spring 2009 – p. 6/43
Classification of PDEs
1. Order of PDE: the order of the highest partial derivative. E.g.
ut = uxx, (2nd order);
2. Number of variables: the number of independent variables. E.g.
ut = uxx, (2nd order, two variables: x and t);
ut = urr + 1
r2 uθθ, (2nd order, three variables: r, θ, and t);
3. Linearity: PDEs are either linear or nonlinear,
linear algebraic equation:
.
where coefficients aij and bi are constants independent of unknown x. EE-2020, Spring 2009 – p. 7/43
Classification of PDEs, cont. (linearity)
nonlinear algebraic equation: e.g.
2 + a1 x = b.
a0 a1 a2 · · · aN
= b(x).
where coefficients ai(x) and b(x) are functions of x only and independent of unknown u(x).
nonlinear ODE: e.g. time-independent nonlinear Schrödinger equation,
−1
2m
d2
EE-2020, Spring 2009 – p. 8/43
Classification of PDEs, cont. (linearity)
linear PDE:
[
·
= b(x, y).
where coefficients ai,j(x, y) and b(x, y) are functions of x and y only and independent of unknown u(x, y). E.g.
utt = e−tuxx + Sin t, (linear)
nonlinear PDE: E.g. time-dependent Gross-Pitaevskii equation, i.e ~r = (x, y, z)
i~ ∂
2m Ψ(t, ~r) + V (~r)Ψ(t, ~r) + |Ψ(t, ~r)|2Ψ(t, ~r),
EE-2020, Spring 2009 – p. 9/43
Classification of PDEs, cont.
4. Homogeneity: an equation only containing unknown function u and its derivative(s) is homogeneous. E.g.
ut = uxx, (homogeneous);
ux + xuy = exu, (homogeneous);
ux + xuy = ex, (non-homogeneous).
5. Kinds of Coefficients: if the coefficients ai,j(x, y) are constants, then the PDE is said to have constant coefficients (otherwise, variable coefficients).
6. Second-order linear PDE with two variables:
A ∂2u
∂x2 +B
∂y + Fu(x, y) = G,
where A, B, C, D, E, F , and G can be constants or given functions of x and y.
EE-2020, Spring 2009 – p. 10/43
Big Three PDEs, 2nd-order linear
Second-order linear PDE with two variables: Q: why 2nd-order?
A ∂2u
∂x2 +B
∂y + F u(x, y) = G,
where A, B, C, D, E, F , and G can be constants or given functions of x and y.
∂
∂2
∂x2 +
∂2
∂y2 +
∂2
∂z2 )u,
2. Hyperbolic: B2 − 4AC > 0, describe vibrating systems and wave motion, i.e.
∇2E(x, y, z, t) − µ00 ∂2
∂ t2 E = 0
3. Elliptic: B2 − 4AC < 0, describe steady-state phenomena, i.e. eigenmodes of Laplacian equations,
[ ∂2
∂x2 +
∂2
The purpose of this course, for PDEs
1. Formulate the PDE from the physical problem, (constructing the mathematical model.) You should know what you are working for first (physical picture).
2. Solve the PDE, (along with initial and boundary conditions.) The techniques you learn to attack the problems (tools at hand).
EE-2020, Spring 2009 – p. 12/43
Methods to solve PDEs
1. Separation of Variables: reduce a PDE in n variables to n ODEs.
2. Integral Transforms: reduce a PDE in n variables to one in n− 1 variables.
3. Change of Coordinates: change the original PDE to an ODE or else another PDE (an easier one).
4. Transformation of the Dependent Variable: transform the unknown of a PDE into a new unknown that is easier to find.
5. Eigenfunction Expansion: find the solutions of a PDE as an infinite sum of eigenfunctions.
6. Impulse-response Methods (Green’s function): decompose initial and boundary condtions fo the problem into simple impulse and finds the response to each impulse.
7. Integral Equations: changes a PDE to an integral equation where the unknown is inside the integral.
8. Calculus of Variations Methods: reformulate the equation as a minimization problem.
9. Numerical Methods: change a PDE to a system of difference equations on a computer.
10. Perturbation Methods: change a nonlinear problem into a sequence of linear ones that approximates the nonlinear one.
11. other methods · · ·
A Heat-Flow experiment
Suppose we have a one-dimensional rod of length L for which we make the following assumptions:
1. The rod is made of a single homogeneous conducting material.
2. The rod is laterally insulated (heat flows only in the x-direction).
3. The rod is thin (the temperature at all points of a cross section is constant).
Let u(x, t) represent the temperature of a thin rod governed by the (conduction) heat equation:
ut = α2uxx,
where α2 is the thermal diffusivity (derived by conservation of energy later).
In general, first derivative is for the change rate (gradient); while the second derivative is for the concavity (curvature).
EE-2020, Spring 2009 – p. 14/43
Heat equation in 1D
The heat equation means that the time-rate of change in temperature (ut) is proportional to the concavity (uxx) of the temperature distribution, i.e.
ut ∝ −[u(x, t) − u(x, t)] ≈ −1
x2 [u(x+ x, t) − 2u(x, t) + u(x− x, t)].
EE-2020, Spring 2009 – p. 15/43
Finite difference approximation
u(xj+1) = u(xj)+u ′(xj)(xj+1−xj)+
u′′(xj)
u′(xj) = u(xj+1) − u(xj)
xj+1 − xj − u′′(xj)
2 (xj+1 − xj) −
xj+1 − xj + O(x),
by the same way the Taylor’s expansion for u(x) at xj−1,
u(xj−1) = u(xj)+u ′(xj)(xj−1−xj)+
u′′(xj)
u′(xj) = u(xj) − u(xj−1)
xj − xj−1 − u′′(xj)
2 (xj − xj−1) +
Finite difference approximation
2 (x)2 +
u′′′(xj)
3! (x)3 + . . . ,
2 (x)2 −
u′′′(xj)
3! (x)3 + . . . ,
u′(xj) = u(xj+1) − u(xj−1)
2x − u′′′(xj)
x2 − 2
u′′′′(xj)
4! (x)2 + . . . ,
x2 + O(x2),
Heat equation in 1D
Heat always flows from high-temperature to low-temperature regions.
The flow rate is proportional to the temperature gradient, i.e. ∇tu.
If the temperature at x is higher than its surrounding average,
u > u, then uxx < 0 and ut < 0,
the net flow of heat into x is negative.
If the temperature at x is lower than its surrounding average,
u < u, then uxx > 0 and ut > 0,
the net flow of heat into x is positive.
If the temperature at x is equal to its surrounding average,
u = u, then uxx = 0 and ut = 0,
the net flow of heat into x is zero.
EE-2020, Spring 2009 – p. 18/43
Derivation of the Heat equation
Apply the principle of conservation of heat to the segment[x, x+ x], we obtain
Net change of heat inside[x, x+ x]
= Net flux of heat across the boundaries + Total heat generated inside[x, x+ x].
Total amount of heat (in calories) inside [x, x+ x] at any time t is measured by
Total heat inside[x, x+ x] =
∫ x+x
where
c = thermal capacity of the rod (measures the ability of the rod to store heat).
ρ = density of the rod.
A = cross-section area of the rod.
EE-2020, Spring 2009 – p. 19/43
Derivation of the Heat equation, cont.
Fourier’s law, the flow rate of heat energy q through a surface is proportional to the negative temperature gradient across the surface,
q = κ∇u,
where κ = thermal conductivity of the rod (measure the ability to conduct heat).
In one dimension, the gradient is an ordinary spatial derivative, and so Fourier’s law is
q = κux.
d d t
∫ x+x
∫ x+x
f(x, t) = external heat source (calories per cm per sec).
EE-2020, Spring 2009 – p. 20/43
Derivation of the Heat equation, cont.
If f(x) is a continuous function on [a, b], then there exists at least one number ξ, a < ξ < b that satisfies
∫ b
a f(x)d x = f(ξ)(b− a), mean value theorem.
By apply the mean value theorem above, then
cρAut(ξ1, t)x = κA[ux(x+ x, t) − ux(x, t)] + Af(ξ2, t)x
where x < ξ1,2 < x+ x, or
ut(ξ, t) = κ
x
cρ f(ξ, t)x
Letting x→ 0, we have the heat equation, i.e. ut(ξ, t) = α2uxx(x, t) + F (x, t), where
α2 ≡ κ
F (x, t) = 1
EE-2020, Spring 2009 – p. 21/43
Boundary conditions for Diffusion-type problems
1. Dirichlet BC: temperature is specified on the boundary, i.e.
u(x = 0, t) = g1(t) u(x = L, t) = g2(t).
2. Neumann BC: heat flow across the boundary specified, i.e.
∂u
where ~n is the outward normal direction to the boundary.
3. Mixed BC: temperature of the surrounding medium is specified, i.e.
∂u
∂u
EE-2020, Spring 2009 – p. 22/43
Typical BCs for 1D heat flow
Initial-boundary-value problem:
Suppose we have c copper rod 200cm long that is laterally insulated and has an initial temperature of 0oC.
Suppose the top of the rod (x = 0) is insulated, while the bottom (x = 200) is immersed in moving water that has a constant temperature of g2(t) = 20oC.
The mathematical model for this problem:
PDE: ut = α2uxx, 0 < x < 200, 0 < t < ∞
BCs:
, 0 < t <∞
κ = 0.93cal/cm-secoC, (thermal conductivity of copper)
h = heat exchange coefficient, to determine by the experiment.
EE-2020, Spring 2009 – p. 23/43
More Diffusion-type equations
ut = α2uxx − β(u− u0), β > 0.
Heat loss (u > u0) or gain (u < u0) is proportional to the difference between the temperature u(x, t) of the rod and the surrounding medium u0 (with β the proportionality constant).
Internal heat source:
ut = α2uxx + f(x, t), the nonhomogeneous equation.
The rod is supplied with an internal heat source (everywhere along the rod and for all time t).
Diffusion-convection equation:
ut = α2uxx − νux.
E.g. a pollutant is carried along in a stream moving with velocity ν.
Nonhomogeneous material: ut = α2(x)uxx.
Remarks: think about d
d t f(t) = −βf(t),
∂x f(x, t),
Separation of Variables
The basic idea of separation of variables is to break down the initial conditions of the problem into simple components, find the response to each component, and the add up these individual responses.
Divide and Conquer
Separation of variables applies to problems where
1. The PDE is linear and homogeneous (not necessarily constant coefficients).
2. The boundary conditions are of the form
αux(0, t) + β u(0, t) = 0,
γ ux(1, t) + δ u(1, t) = 0,
where α, β, γ, and δ are constants (boundary conditions of this form are called linear homogeneous BCs ).
EE-2020, Spring 2009 – p. 25/43
Separation of Variables: step 1
PDE: ut = α2uxx, 0 < x < 1, 0 < t < ∞
BCs:
IC: u(x, 0) = φ(x), 0 ≤ x ≤ 1
Find elementary solutions to the PDE:
u(x, t) = X(x)T (t), fundamental solutions
substitute this trial solution into the PDE,
X(x)T ′(t) = α2X′′(x)T (t),
divide each side of this equation by α2X(x)T (t),
T ′(t)
EE-2020, Spring 2009 – p. 26/43
Separation of Variables, setp 1
In this case, x and t are independent of each other, each side must be a fixed constant (say k),
T ′(t)
X′′(x) − kX(x) = 0.
To meet the condition as t→ ∞, k must to be negative, i.e. k = −λ2, where λ is nonzero.
T ′(t) + λ2 α2T (t) = 0,
X′′(x) + λ2X(x) = 0.
ODE Practice
For the two ODEs
X′′(x) + λ2X(x) = 0.
The corresponding solutions are:
u(x, t) = e−λ2α2t[ASin(λx) + B Cos(λx)]
EE-2020, Spring 2009 – p. 29/43
Separation of Variables, setp 2
Find solutions to the BCs
The total solution for u(x, t) = X(x)T (t)
u(x, t) = e−λ2α2t[ASin(λx) + B Cos(λx)]
to satisfy the boundary condtions
u(0, t) = 0,
u(1, t) = 0,
needs to enforce B = 0 and Sinλ = 0 (or λ = ±π,±2π,±3π, . . . ).
We have an infinite number of functions,
un(x, t) = An e −(nπα)2t Sin(nπx), n = 1, 2, . . .
which is called the fundamental solution (an infinite number).
EE-2020, Spring 2009 – p. 30/43
Separation of Variables, setp 3
Find the solutions to the IC
To add the fundamental solutions
u(x, t) = ∞ ∑
n=1
∞ ∑
∫ 1
Am = 2
Separation of Variables: example 1, Dirichlet BCs
PDE: ut = α2uxx, 0 < x < L, 0 < t < ∞
BCs: u(0, t) = 0 and u(L, t) = 0, 0 < t <∞
IC: u(x, 0) =
L− x ; for L/2 ≤ x ≤ L
u(x, t) = ∑
n=1
Ane −(nπ/L)2α2tSin(
Separation of Variables: example 2, Neumann BCs
PDE: ut = α2uxx, 0 < x < L, 0 < t < ∞
BCs: ux(0, t) = 0 and ux(L, t) = 0, 0 < t < ∞
IC: u(x, 0) =
L− x ; for L/2 ≤ x ≤ L
u(x, t) = ∑
n=0
Ane −(nπ/L)2α2tCos(
EE-2020, Spring 2009 – p. 33/43
Separation of Variables: example 3, mixed BCs
PDE: ut = α2uxx, 0 < x < L, 0 < t <∞
BCs: u(0, t) = 0 and ux(L, t) + hu(L, t) = 0, 0 < t <∞
IC: u(x, 0) =
L− x ; for L/2 ≤ x ≤ L
From the BCs:
h .
Separation of Variables: example 3, mixed BCs
u(x, t) = ∞ ∑
n=1
An e −(knα)2t Sin(knx),
By the orthogonality of Xn(x) in [0, L] (for the spatial ODE is a Sturm-Liouville problem):
∫ L
Solutions for heat equations with homogeneous boundaries
ut = κ(uxx + uyy),
PDE: ut = α2uxx, 0 < x < L, 0 < t < ∞
BCs:
IC: u(x, 0) = φ(x), 0 ≤ x ≤ L
Transforming nonhomogeneous BCs to homogeneous ones:
u(x, t) = steady state + transient
= [k1 + x
where
BCs
IC U(x, 0) = φ(x) − [k1 + x
L (k2 − k1)], 0 ≤ x ≤ L
EE-2020, Spring 2009 – p. 37/43
Heat equation with lateral heat loss
PDE: ut = α2uxx − βu, 0 < x < L, 0 < t <∞
BCs: u(0, t) = 0 and u(L, t) = 0, 0 < t <∞
IC: u(x, 0) = φ(x), 0 ≤ x ≤ L
where −βu represents heat flow across the lateral boundary.
By means of the transformation
u(x, t) = e−βtw(x, t),
then the original heat equation with lateral loss becomes,
PDE: wt = α2wxx, 0 < x < L, 0 < t <∞
BCs: w(0, t) = 0 and w(L, t) = 0, 0 < t <∞
IC: w(x, 0) = φ(x), 0 ≤ x ≤ L
with the solutions already known, i.e.
w(x, t) = ∑
n=1
Diffusion-convection equation
BCs: u(0, t) = 0 and u(L, t) = 0, 0 < t < ∞
IC: u(x, 0) = φ(x), 0 ≤ x ≤ L
where ν is a constant, and νux represents the convection component.
By means of the transformation
u(x, t) = eν(x−νt/2)/2α2
PDE: wt = α2wxx, 0 < x < L, 0 < t <∞
BCs: w(0, t) = 0 and w(L, t) = 0, 0 < t <∞
IC: w(x, 0) = φ(x), 0 ≤ x ≤ L
with the solutions already known, i.e.
w(x, t) = ∑
n=1
PDE: ut = α2uxx + f(x, t), 0 < x < 1, 0 < t <∞
BCs: u(0, t) = 0 and u(1, t) = 0, 0 < t <∞
IC: u(x, 0) = φ(x), 0 ≤ x ≤ 1
For f(x, t) = 0, we have solutions for the homogeneous problem, i.e.
u(x, t) = ∑
n=1
ane −(λnα)2tXn(x),
where λn and Xn(x) are the eigenvalues and the eigenfunctions of the Sturm-Liouville problem, i.e.
X′′ + λ2X = 0, and X(0) = 0, X(1) = 0,
For nonhomogeneous problem, f(x, t) 6= 0, we try the slightly more general form,
u(x, t) = ∑
n=1
Nonhomogeneous Heat equation: example
BCs: u(0, t) = 0 and u(1, t) = 0, 0 < t <∞
IC: u(x, 0) = Sin(πx), 0 ≤ x ≤ 1
Since the BCs support Sin(nπx) eigenfunctions,
u(x, t) = ∑
n=1
PDE: ∑
IC: ∑
Nonhomogeneous Heat equation: example
PDE: T ′ n + (nπα)2Tn = 2
∫ 1
IC: Tn(0) = 2
Writing out these equations for n = 1, 2, . . . ,, wee see
(n = 1) T ′ 1 + (πα)2T1 = 0
T1(0) = 1
T2(0) = 0
T3(0) = 0
Nonhomogeneous Heat equation: example
u(x, t) = e−(πα)2tSin(πx) + 1
(3πα)2 [1 − e−(3πα)2t]Sin(3πx)
= transient + steady state
The first term represents transient behavior, due to the initial conditions.
The second ter represents steady state behavior, due to the right-hand side of the PDE (nonhomogeneous term).
EE-2020, Spring 2009 – p. 43/43
hspace {-0.3in}small
hspace {-0.3in}small Diffusion equation
hspace {-0.3in}small Dispersive/Diffractive/Diffused Wave: Linear
hspace {-0.3in}small Differential equations
hspace {-0.3in}small Partial Differential equations
hspace {-0.3in}small Classification of PDEs
hspace {-0.3in}small Classification of PDEs, cont. (linearity)
hspace {-0.3in}small Classification of PDEs, cont. (linearity)
hspace {-0.3in}small Classification of PDEs, cont.
hspace {-0.3in}small Big Three PDEs, 2nd-order linear
hspace {-0.3in}small The purpose of this course, for PDEs
hspace {-0.3in}small Methods to solve PDEs
hspace {-0.3in}small A Heat-Flow experiment
hspace {-0.3in}small Heat equation in 1D
hspace {-0.3in}small Finite difference approximation
hspace {-0.3in}small Finite difference approximation
hspace {-0.3in}small Heat equation in 1D
hspace {-0.3in}small Derivation of the Heat equation
hspace {-0.3in}small Derivation of the Heat equation, cont.
hspace {-0.3in}small Derivation of the Heat equation, cont.
hspace {-0.3in}small Boundary conditions for Diffusion-type problems
hspace {-0.3in}small Typical BCs for 1D heat flow
hspace {-0.3in}small More Diffusion-type equations
hspace {-0.3in}small Separation of Variables
hspace {-0.3in}small Separation of Variables: step 1
hspace {-0.3in}small Separation of Variables, setp 1
hspace {-0.3in}small
hspace {-0.3in}small Separation of Variables: example 1, Dirichlet BCs
hspace {-0.3in}small Separation of Variables: example 2, Neumann BCs
hspace {-0.3in}small Separation of Variables: example 3, mixed BCs
hspace {-0.3in}small Separation of Variables: example 3, mixed BCs
hspace {-0.3in}small Solutions for heat equations with homogeneous boundaries
hspace {-0.3in}small Separation of Variables: nonhomogeneous BCs
hspace {-0.3in}small Heat equation with lateral heat loss
hspace {-0.3in}small Diffusion-convection equation
hspace {-0.3in}small Nonhomogeneous Heat equation: eigenfunction expansions
hspace {-0.3in}small Nonhomogeneous Heat equation: example
hspace {-0.3in}small Nonhomogeneous Heat equation: example
hspace {-0.3in}small Nonhomogeneous Heat equation: example
3. Website: For more information and course slides: http://mx.nthu.edu.tw/∼rklee
4. TA hours: 7:00-9:00PM, every Tuesday and Thursday at Room 521, EECS bldg. (2 × 2 hours per week)
(a) I-Hong Chen , 2nd-year PhD student, e-mail: [email protected]
(b) Chih-Yao Chen , 2nd-year Master student, e-mail: [email protected]
EE-2020, Spring 2009 – p. 1/43
Syllabus: for PDE
2. Diffusion-type problems: [Textbook] Ch.12, [Ref.] Ch.2.
Derivation of the Heat equation, (3/2).
Boundary conditions for Diffusion-type problems, (3/5).
Separation of variables, (3/9).
Solving nonhomogeneous PDEs, (3/12).
Integral transforms, (3/16, 3/19).
The Fourier transform, (3/23).
The Laplace Transform, (3/26).
1-D Wave equation, (4/2, 4/6).
D’Alembert solution of the Wave equation, (4/9).
Sturm-Liouville problems, (4/13).
2-D Wave equation in Cartesian and polar coordinates, (4/16, 4/20).
Laplace’s equation in Cartesian, polar, and spherical coordinates, (4/23).
EE-2020, Spring 2009 – p. 2/43
Diffusion equation
Dispersive/Diffractive/Diffused Wave: Linear
. . .
Differential equations
where unknown x is a number or set of numbers.
Differential equation:
f(u, u′, u′′, . . . ) = g(x),
where unknown u is a function. Note that u′, u′′ . . . are not extra unknowns for they can be derived by differentiation once u is determined.
Insteady of differential equation, one can also use the integral equation. E.g. Gauss’s law for the electric field:
∇ · E = ρ
An equation containing derivative(s) of an unknown function (or functions) (dependent variable) u with a single independent variable. E.g.
u′′(x) + u(x) = 0.
Partial Differential equations
Partial Differential Equations, PDEs
An equation containing partial derivative(s) of an unknown function u with two
or more independent variables. E.g.
∂ u(t, x)
Why PDEs?
People sense the real world via four (or multiple) dimensions (x, y, z, t), therefore, physical quantities (e.g. electrical field, temperature, electron distribution in an atom) are fully described by four variables.
Most physical laws are described in terms of PDEs, where the derivatives represent physical quantities.
E.g. Electrostatics (Poisson theory), EM waves (Maxwell’s equations), quantum mechanics (Schrodinger’s equation), heat transfer (heat equation), fluid mechanics.
Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic.
EE-2020, Spring 2009 – p. 6/43
Classification of PDEs
1. Order of PDE: the order of the highest partial derivative. E.g.
ut = uxx, (2nd order);
2. Number of variables: the number of independent variables. E.g.
ut = uxx, (2nd order, two variables: x and t);
ut = urr + 1
r2 uθθ, (2nd order, three variables: r, θ, and t);
3. Linearity: PDEs are either linear or nonlinear,
linear algebraic equation:
.
where coefficients aij and bi are constants independent of unknown x. EE-2020, Spring 2009 – p. 7/43
Classification of PDEs, cont. (linearity)
nonlinear algebraic equation: e.g.
2 + a1 x = b.
a0 a1 a2 · · · aN
= b(x).
where coefficients ai(x) and b(x) are functions of x only and independent of unknown u(x).
nonlinear ODE: e.g. time-independent nonlinear Schrödinger equation,
−1
2m
d2
EE-2020, Spring 2009 – p. 8/43
Classification of PDEs, cont. (linearity)
linear PDE:
[
·
= b(x, y).
where coefficients ai,j(x, y) and b(x, y) are functions of x and y only and independent of unknown u(x, y). E.g.
utt = e−tuxx + Sin t, (linear)
nonlinear PDE: E.g. time-dependent Gross-Pitaevskii equation, i.e ~r = (x, y, z)
i~ ∂
2m Ψ(t, ~r) + V (~r)Ψ(t, ~r) + |Ψ(t, ~r)|2Ψ(t, ~r),
EE-2020, Spring 2009 – p. 9/43
Classification of PDEs, cont.
4. Homogeneity: an equation only containing unknown function u and its derivative(s) is homogeneous. E.g.
ut = uxx, (homogeneous);
ux + xuy = exu, (homogeneous);
ux + xuy = ex, (non-homogeneous).
5. Kinds of Coefficients: if the coefficients ai,j(x, y) are constants, then the PDE is said to have constant coefficients (otherwise, variable coefficients).
6. Second-order linear PDE with two variables:
A ∂2u
∂x2 +B
∂y + Fu(x, y) = G,
where A, B, C, D, E, F , and G can be constants or given functions of x and y.
EE-2020, Spring 2009 – p. 10/43
Big Three PDEs, 2nd-order linear
Second-order linear PDE with two variables: Q: why 2nd-order?
A ∂2u
∂x2 +B
∂y + F u(x, y) = G,
where A, B, C, D, E, F , and G can be constants or given functions of x and y.
∂
∂2
∂x2 +
∂2
∂y2 +
∂2
∂z2 )u,
2. Hyperbolic: B2 − 4AC > 0, describe vibrating systems and wave motion, i.e.
∇2E(x, y, z, t) − µ00 ∂2
∂ t2 E = 0
3. Elliptic: B2 − 4AC < 0, describe steady-state phenomena, i.e. eigenmodes of Laplacian equations,
[ ∂2
∂x2 +
∂2
The purpose of this course, for PDEs
1. Formulate the PDE from the physical problem, (constructing the mathematical model.) You should know what you are working for first (physical picture).
2. Solve the PDE, (along with initial and boundary conditions.) The techniques you learn to attack the problems (tools at hand).
EE-2020, Spring 2009 – p. 12/43
Methods to solve PDEs
1. Separation of Variables: reduce a PDE in n variables to n ODEs.
2. Integral Transforms: reduce a PDE in n variables to one in n− 1 variables.
3. Change of Coordinates: change the original PDE to an ODE or else another PDE (an easier one).
4. Transformation of the Dependent Variable: transform the unknown of a PDE into a new unknown that is easier to find.
5. Eigenfunction Expansion: find the solutions of a PDE as an infinite sum of eigenfunctions.
6. Impulse-response Methods (Green’s function): decompose initial and boundary condtions fo the problem into simple impulse and finds the response to each impulse.
7. Integral Equations: changes a PDE to an integral equation where the unknown is inside the integral.
8. Calculus of Variations Methods: reformulate the equation as a minimization problem.
9. Numerical Methods: change a PDE to a system of difference equations on a computer.
10. Perturbation Methods: change a nonlinear problem into a sequence of linear ones that approximates the nonlinear one.
11. other methods · · ·
A Heat-Flow experiment
Suppose we have a one-dimensional rod of length L for which we make the following assumptions:
1. The rod is made of a single homogeneous conducting material.
2. The rod is laterally insulated (heat flows only in the x-direction).
3. The rod is thin (the temperature at all points of a cross section is constant).
Let u(x, t) represent the temperature of a thin rod governed by the (conduction) heat equation:
ut = α2uxx,
where α2 is the thermal diffusivity (derived by conservation of energy later).
In general, first derivative is for the change rate (gradient); while the second derivative is for the concavity (curvature).
EE-2020, Spring 2009 – p. 14/43
Heat equation in 1D
The heat equation means that the time-rate of change in temperature (ut) is proportional to the concavity (uxx) of the temperature distribution, i.e.
ut ∝ −[u(x, t) − u(x, t)] ≈ −1
x2 [u(x+ x, t) − 2u(x, t) + u(x− x, t)].
EE-2020, Spring 2009 – p. 15/43
Finite difference approximation
u(xj+1) = u(xj)+u ′(xj)(xj+1−xj)+
u′′(xj)
u′(xj) = u(xj+1) − u(xj)
xj+1 − xj − u′′(xj)
2 (xj+1 − xj) −
xj+1 − xj + O(x),
by the same way the Taylor’s expansion for u(x) at xj−1,
u(xj−1) = u(xj)+u ′(xj)(xj−1−xj)+
u′′(xj)
u′(xj) = u(xj) − u(xj−1)
xj − xj−1 − u′′(xj)
2 (xj − xj−1) +
Finite difference approximation
2 (x)2 +
u′′′(xj)
3! (x)3 + . . . ,
2 (x)2 −
u′′′(xj)
3! (x)3 + . . . ,
u′(xj) = u(xj+1) − u(xj−1)
2x − u′′′(xj)
x2 − 2
u′′′′(xj)
4! (x)2 + . . . ,
x2 + O(x2),
Heat equation in 1D
Heat always flows from high-temperature to low-temperature regions.
The flow rate is proportional to the temperature gradient, i.e. ∇tu.
If the temperature at x is higher than its surrounding average,
u > u, then uxx < 0 and ut < 0,
the net flow of heat into x is negative.
If the temperature at x is lower than its surrounding average,
u < u, then uxx > 0 and ut > 0,
the net flow of heat into x is positive.
If the temperature at x is equal to its surrounding average,
u = u, then uxx = 0 and ut = 0,
the net flow of heat into x is zero.
EE-2020, Spring 2009 – p. 18/43
Derivation of the Heat equation
Apply the principle of conservation of heat to the segment[x, x+ x], we obtain
Net change of heat inside[x, x+ x]
= Net flux of heat across the boundaries + Total heat generated inside[x, x+ x].
Total amount of heat (in calories) inside [x, x+ x] at any time t is measured by
Total heat inside[x, x+ x] =
∫ x+x
where
c = thermal capacity of the rod (measures the ability of the rod to store heat).
ρ = density of the rod.
A = cross-section area of the rod.
EE-2020, Spring 2009 – p. 19/43
Derivation of the Heat equation, cont.
Fourier’s law, the flow rate of heat energy q through a surface is proportional to the negative temperature gradient across the surface,
q = κ∇u,
where κ = thermal conductivity of the rod (measure the ability to conduct heat).
In one dimension, the gradient is an ordinary spatial derivative, and so Fourier’s law is
q = κux.
d d t
∫ x+x
∫ x+x
f(x, t) = external heat source (calories per cm per sec).
EE-2020, Spring 2009 – p. 20/43
Derivation of the Heat equation, cont.
If f(x) is a continuous function on [a, b], then there exists at least one number ξ, a < ξ < b that satisfies
∫ b
a f(x)d x = f(ξ)(b− a), mean value theorem.
By apply the mean value theorem above, then
cρAut(ξ1, t)x = κA[ux(x+ x, t) − ux(x, t)] + Af(ξ2, t)x
where x < ξ1,2 < x+ x, or
ut(ξ, t) = κ
x
cρ f(ξ, t)x
Letting x→ 0, we have the heat equation, i.e. ut(ξ, t) = α2uxx(x, t) + F (x, t), where
α2 ≡ κ
F (x, t) = 1
EE-2020, Spring 2009 – p. 21/43
Boundary conditions for Diffusion-type problems
1. Dirichlet BC: temperature is specified on the boundary, i.e.
u(x = 0, t) = g1(t) u(x = L, t) = g2(t).
2. Neumann BC: heat flow across the boundary specified, i.e.
∂u
where ~n is the outward normal direction to the boundary.
3. Mixed BC: temperature of the surrounding medium is specified, i.e.
∂u
∂u
EE-2020, Spring 2009 – p. 22/43
Typical BCs for 1D heat flow
Initial-boundary-value problem:
Suppose we have c copper rod 200cm long that is laterally insulated and has an initial temperature of 0oC.
Suppose the top of the rod (x = 0) is insulated, while the bottom (x = 200) is immersed in moving water that has a constant temperature of g2(t) = 20oC.
The mathematical model for this problem:
PDE: ut = α2uxx, 0 < x < 200, 0 < t < ∞
BCs:
, 0 < t <∞
κ = 0.93cal/cm-secoC, (thermal conductivity of copper)
h = heat exchange coefficient, to determine by the experiment.
EE-2020, Spring 2009 – p. 23/43
More Diffusion-type equations
ut = α2uxx − β(u− u0), β > 0.
Heat loss (u > u0) or gain (u < u0) is proportional to the difference between the temperature u(x, t) of the rod and the surrounding medium u0 (with β the proportionality constant).
Internal heat source:
ut = α2uxx + f(x, t), the nonhomogeneous equation.
The rod is supplied with an internal heat source (everywhere along the rod and for all time t).
Diffusion-convection equation:
ut = α2uxx − νux.
E.g. a pollutant is carried along in a stream moving with velocity ν.
Nonhomogeneous material: ut = α2(x)uxx.
Remarks: think about d
d t f(t) = −βf(t),
∂x f(x, t),
Separation of Variables
The basic idea of separation of variables is to break down the initial conditions of the problem into simple components, find the response to each component, and the add up these individual responses.
Divide and Conquer
Separation of variables applies to problems where
1. The PDE is linear and homogeneous (not necessarily constant coefficients).
2. The boundary conditions are of the form
αux(0, t) + β u(0, t) = 0,
γ ux(1, t) + δ u(1, t) = 0,
where α, β, γ, and δ are constants (boundary conditions of this form are called linear homogeneous BCs ).
EE-2020, Spring 2009 – p. 25/43
Separation of Variables: step 1
PDE: ut = α2uxx, 0 < x < 1, 0 < t < ∞
BCs:
IC: u(x, 0) = φ(x), 0 ≤ x ≤ 1
Find elementary solutions to the PDE:
u(x, t) = X(x)T (t), fundamental solutions
substitute this trial solution into the PDE,
X(x)T ′(t) = α2X′′(x)T (t),
divide each side of this equation by α2X(x)T (t),
T ′(t)
EE-2020, Spring 2009 – p. 26/43
Separation of Variables, setp 1
In this case, x and t are independent of each other, each side must be a fixed constant (say k),
T ′(t)
X′′(x) − kX(x) = 0.
To meet the condition as t→ ∞, k must to be negative, i.e. k = −λ2, where λ is nonzero.
T ′(t) + λ2 α2T (t) = 0,
X′′(x) + λ2X(x) = 0.
ODE Practice
For the two ODEs
X′′(x) + λ2X(x) = 0.
The corresponding solutions are:
u(x, t) = e−λ2α2t[ASin(λx) + B Cos(λx)]
EE-2020, Spring 2009 – p. 29/43
Separation of Variables, setp 2
Find solutions to the BCs
The total solution for u(x, t) = X(x)T (t)
u(x, t) = e−λ2α2t[ASin(λx) + B Cos(λx)]
to satisfy the boundary condtions
u(0, t) = 0,
u(1, t) = 0,
needs to enforce B = 0 and Sinλ = 0 (or λ = ±π,±2π,±3π, . . . ).
We have an infinite number of functions,
un(x, t) = An e −(nπα)2t Sin(nπx), n = 1, 2, . . .
which is called the fundamental solution (an infinite number).
EE-2020, Spring 2009 – p. 30/43
Separation of Variables, setp 3
Find the solutions to the IC
To add the fundamental solutions
u(x, t) = ∞ ∑
n=1
∞ ∑
∫ 1
Am = 2
Separation of Variables: example 1, Dirichlet BCs
PDE: ut = α2uxx, 0 < x < L, 0 < t < ∞
BCs: u(0, t) = 0 and u(L, t) = 0, 0 < t <∞
IC: u(x, 0) =
L− x ; for L/2 ≤ x ≤ L
u(x, t) = ∑
n=1
Ane −(nπ/L)2α2tSin(
Separation of Variables: example 2, Neumann BCs
PDE: ut = α2uxx, 0 < x < L, 0 < t < ∞
BCs: ux(0, t) = 0 and ux(L, t) = 0, 0 < t < ∞
IC: u(x, 0) =
L− x ; for L/2 ≤ x ≤ L
u(x, t) = ∑
n=0
Ane −(nπ/L)2α2tCos(
EE-2020, Spring 2009 – p. 33/43
Separation of Variables: example 3, mixed BCs
PDE: ut = α2uxx, 0 < x < L, 0 < t <∞
BCs: u(0, t) = 0 and ux(L, t) + hu(L, t) = 0, 0 < t <∞
IC: u(x, 0) =
L− x ; for L/2 ≤ x ≤ L
From the BCs:
h .
Separation of Variables: example 3, mixed BCs
u(x, t) = ∞ ∑
n=1
An e −(knα)2t Sin(knx),
By the orthogonality of Xn(x) in [0, L] (for the spatial ODE is a Sturm-Liouville problem):
∫ L
Solutions for heat equations with homogeneous boundaries
ut = κ(uxx + uyy),
PDE: ut = α2uxx, 0 < x < L, 0 < t < ∞
BCs:
IC: u(x, 0) = φ(x), 0 ≤ x ≤ L
Transforming nonhomogeneous BCs to homogeneous ones:
u(x, t) = steady state + transient
= [k1 + x
where
BCs
IC U(x, 0) = φ(x) − [k1 + x
L (k2 − k1)], 0 ≤ x ≤ L
EE-2020, Spring 2009 – p. 37/43
Heat equation with lateral heat loss
PDE: ut = α2uxx − βu, 0 < x < L, 0 < t <∞
BCs: u(0, t) = 0 and u(L, t) = 0, 0 < t <∞
IC: u(x, 0) = φ(x), 0 ≤ x ≤ L
where −βu represents heat flow across the lateral boundary.
By means of the transformation
u(x, t) = e−βtw(x, t),
then the original heat equation with lateral loss becomes,
PDE: wt = α2wxx, 0 < x < L, 0 < t <∞
BCs: w(0, t) = 0 and w(L, t) = 0, 0 < t <∞
IC: w(x, 0) = φ(x), 0 ≤ x ≤ L
with the solutions already known, i.e.
w(x, t) = ∑
n=1
Diffusion-convection equation
BCs: u(0, t) = 0 and u(L, t) = 0, 0 < t < ∞
IC: u(x, 0) = φ(x), 0 ≤ x ≤ L
where ν is a constant, and νux represents the convection component.
By means of the transformation
u(x, t) = eν(x−νt/2)/2α2
PDE: wt = α2wxx, 0 < x < L, 0 < t <∞
BCs: w(0, t) = 0 and w(L, t) = 0, 0 < t <∞
IC: w(x, 0) = φ(x), 0 ≤ x ≤ L
with the solutions already known, i.e.
w(x, t) = ∑
n=1
PDE: ut = α2uxx + f(x, t), 0 < x < 1, 0 < t <∞
BCs: u(0, t) = 0 and u(1, t) = 0, 0 < t <∞
IC: u(x, 0) = φ(x), 0 ≤ x ≤ 1
For f(x, t) = 0, we have solutions for the homogeneous problem, i.e.
u(x, t) = ∑
n=1
ane −(λnα)2tXn(x),
where λn and Xn(x) are the eigenvalues and the eigenfunctions of the Sturm-Liouville problem, i.e.
X′′ + λ2X = 0, and X(0) = 0, X(1) = 0,
For nonhomogeneous problem, f(x, t) 6= 0, we try the slightly more general form,
u(x, t) = ∑
n=1
Nonhomogeneous Heat equation: example
BCs: u(0, t) = 0 and u(1, t) = 0, 0 < t <∞
IC: u(x, 0) = Sin(πx), 0 ≤ x ≤ 1
Since the BCs support Sin(nπx) eigenfunctions,
u(x, t) = ∑
n=1
PDE: ∑
IC: ∑
Nonhomogeneous Heat equation: example
PDE: T ′ n + (nπα)2Tn = 2
∫ 1
IC: Tn(0) = 2
Writing out these equations for n = 1, 2, . . . ,, wee see
(n = 1) T ′ 1 + (πα)2T1 = 0
T1(0) = 1
T2(0) = 0
T3(0) = 0
Nonhomogeneous Heat equation: example
u(x, t) = e−(πα)2tSin(πx) + 1
(3πα)2 [1 − e−(3πα)2t]Sin(3πx)
= transient + steady state
The first term represents transient behavior, due to the initial conditions.
The second ter represents steady state behavior, due to the right-hand side of the PDE (nonhomogeneous term).
EE-2020, Spring 2009 – p. 43/43
hspace {-0.3in}small
hspace {-0.3in}small Diffusion equation
hspace {-0.3in}small Dispersive/Diffractive/Diffused Wave: Linear
hspace {-0.3in}small Differential equations
hspace {-0.3in}small Partial Differential equations
hspace {-0.3in}small Classification of PDEs
hspace {-0.3in}small Classification of PDEs, cont. (linearity)
hspace {-0.3in}small Classification of PDEs, cont. (linearity)
hspace {-0.3in}small Classification of PDEs, cont.
hspace {-0.3in}small Big Three PDEs, 2nd-order linear
hspace {-0.3in}small The purpose of this course, for PDEs
hspace {-0.3in}small Methods to solve PDEs
hspace {-0.3in}small A Heat-Flow experiment
hspace {-0.3in}small Heat equation in 1D
hspace {-0.3in}small Finite difference approximation
hspace {-0.3in}small Finite difference approximation
hspace {-0.3in}small Heat equation in 1D
hspace {-0.3in}small Derivation of the Heat equation
hspace {-0.3in}small Derivation of the Heat equation, cont.
hspace {-0.3in}small Derivation of the Heat equation, cont.
hspace {-0.3in}small Boundary conditions for Diffusion-type problems
hspace {-0.3in}small Typical BCs for 1D heat flow
hspace {-0.3in}small More Diffusion-type equations
hspace {-0.3in}small Separation of Variables
hspace {-0.3in}small Separation of Variables: step 1
hspace {-0.3in}small Separation of Variables, setp 1
hspace {-0.3in}small
hspace {-0.3in}small Separation of Variables: example 1, Dirichlet BCs
hspace {-0.3in}small Separation of Variables: example 2, Neumann BCs
hspace {-0.3in}small Separation of Variables: example 3, mixed BCs
hspace {-0.3in}small Separation of Variables: example 3, mixed BCs
hspace {-0.3in}small Solutions for heat equations with homogeneous boundaries
hspace {-0.3in}small Separation of Variables: nonhomogeneous BCs
hspace {-0.3in}small Heat equation with lateral heat loss
hspace {-0.3in}small Diffusion-convection equation
hspace {-0.3in}small Nonhomogeneous Heat equation: eigenfunction expansions
hspace {-0.3in}small Nonhomogeneous Heat equation: example
hspace {-0.3in}small Nonhomogeneous Heat equation: example
hspace {-0.3in}small Nonhomogeneous Heat equation: example
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