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Week 1
Introduction
1
Introduction
Complex systems in engineering and science are often fruitfully investigated by means of
mathematical models. These models are typically formulated as differential or integral equations that must be solved subject to specific boundary and initial conditions. The methods
of advanced calculus are key to the analysis and solution of the model equations.
The mathematical problems resulting from modeling can be deterministic or stochastic
and they can also be linear or non-linear. In this course we will focus on models leading
to linear deterministic mathematical problems. These problems are fairly well understood,
useful closed form solutions exist and they constitute an excellent foundation for subsequent
study of more sophisticated models.
For the sake of motivation, a few selected examples of mathematical models of systems
of engineering interest are now described.
The solution of two-dimensional problems in elastic media subject to constant body forces
is given by the integral of the following differential equation
∂4φ
∂ 4φ
∂ 4φ
+
2
+
=0
∂x4
∂x2 ∂y 2 ∂y 4
where φ(x, y) is the stress function, expressable as
φ = Re[z ∗ ψ(z) + χ(z)]
where z∗ = x − iy and ψ and χ are suitably chosen analytic functions.
Thin straight fins of triangular profile, length l and thickness b at the base are used in
heat exchangers to enhance heat flow. An energy balance in the fin produces the following
expression for the fin excess temperature over the surrounding environment θ as a function
of distance from its tip x
θ
1 dθ
d2 θ
−
β
=0
+
dx2 x dx
x
1
where β = 2hl
, k is the thermal conductivity of the fin material and h is the heat transfer
kb
coefficient. The above equation is a modified Bessel equation and has the solution
√
I0 (2 βx)
√
θ = θb
I0 (2 βl)
where I0 (z) is the modified Bessel function of the first kind, of order 0 and θb is the temperature at the base of the fin.
The velocity distribution v inside an initially motionless fluid produced by wall suddenly
set in motion can be shown to be given by
x
v(x, t) = V [1 − erf ( √ )]
2 νt
where V is the velocity of the wall and ν is the kinematic viscosity of the fluid.
If the motion of an incompressible fluid is simply a pure rotation about a given axis fixed
in space at a rate of w revolutions per unit time, the velocity vector V of the fluid at any
given point satisfies the following equations
∇·V =0
and
∇ × V = 2w
Now, if the fluid does not rotate at all (i.e. w = 0) one talks about irrotational flow for
which a velocity potential ϕ exists such that
V = ∇ϕ
Since the fluid is incompressible, the above yields Laplace’s equation for the velocity potential
∇2 ϕ = 0
Solutions of this equation subject to specific boundary conditions represent the behavior of
a large number of incompressible irrotational flows.
The shape h(r) of an axisymmetric pendant drop is approximately described by the
following differential equation
d2 h 1 dh
− (h0 − h) = 0
+
dr2
r dr
where h0 is a length parameter. The solution of the above equation is
√
h(r) = h0 [J0 (r) − J0 ( Bo)]
2
0
where J0 (z) is the Bessel function of the first kind, of order 0 and Bo = ρgR
is the Bond
σ
number of the drop with ρ being the density, σ the surface tension and R0 the drop footprint
radius.
The vibrations of an elastic string of length l fixed at its ends and released from an initial
deflection y0 (x) are described by the solution of the wave equation
2
∂ 2y
2∂ y
=
c
∂t2
∂x2
where y is the string deflection and c2 = Tρ where T is the tension and ρ the string density. A
solution to the above equation can be obtained using the method of separation of variables
and is
y(x, t) =
∞
X
An cos(
n=1
nπ
nπ
ct) sin( x)
l
l
with
An =
2
l
Z
0
t
y0 (ξ) sin(
nπ
ξ)dξ
l
which is a linear combination of an infinite number of standing waves.
The process of uni-directional solidification of molten metal in a mold can be modeled
using the heat equation. In the solidified portion, this is
∂Ts
∂ 2 Ts
= αs 2
∂t
∂x
in the liquid portion (convection neglected)
∂ 2 Tl
∂Tl
= αl 2
∂t
∂x
and at the solid-liquid interface X
ks
dX
∂Ts
∂Tl
− kl
=L
∂x
∂x
dt
Here, T is temperature, α thermal diffusivity, k thermal conductivity, and L is the latent
heat of solidification per unit volume. Closed form solutions exist for the semi-infite case
but numerical methods are required in most situations.
The self-propagating high temperature synthesis (SHS) process has been modeled by the
expression
ρCp
∂ ∂T
Q
∂T
=
(k
) + Hr K0 (1 − φ)n exp(−
)
∂t
∂x ∂x
RT
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where Hr is the heat released by the synthesis reaction, K0 is a reaction rate constant, φ is
the fraction reacted and Q is the activation energy of the reaction. Again, although a few
closed form solutions are available, most cases of interest require numerical computation.
The electric current j and the electric field E inside a current carrying medium subject
to an electric potential gradient satisfy the following simplified form of Maxwell’s equations
∇·j=0
and
E=
j
κ
where κ is the electric conductivity. The potential φ in turn obeys Laplace’s equation
∇2 φ = 0
The solution of Laplace’s equation for the potential subject to specifically stated boundary
conditions allows determination of the current and electric field distributions inside the
conductor.
A popular model for the price f of a stock option is the Black-Scholes equation
∂f
1
∂2f
∂f
+ rS
+ σ 2 S 2 2 = rf
∂t
∂S 2
∂S
where r is the risk-free interest rate and S is the stock price, commonly represented by an
Ito process of the form
√
dS
= µdt + σ dt
S
Here µ is the expected rate of return of the stock, σ 2 is the variance rate of the proportional
change in the stock price and is a random drawing from a standarized normal distribution.
Many more examples could be given but the above should suffice to show that in order
to obtain a quantitative understanding of real world problems one must know the basic tools
and techniques of advanced calculus.
2
Variables: Real and Complex
Real variables x are quantities who adopt values from the set of real numbers while complex
variables z adopt √
values from the set of complex numbers x + iy where x and y are real
variables and i = −1. x is called the real part and y the imaginary part of the complex
number x + iy. Real numbers can be represented as points along the real axis while complex
numbers must √be represented using a (complex) plane. Useful related concepts are the
modulus |z| = x2 + y 2 and the complex conjugate z∗ = x − iy.
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If z1 = x1 + iy1 and z2 = x2 + iy2 then the basic algebraic operations are:
z1 ± z2 = (x1 ± x2 ) + i(y1 ± y2 )
z1 z2 = (x1 + iy1 )(x2 + iy2 )
z2 /z1 =
x1 x2 + y1 y2
x1 y2 − x2 y1
+i
2
2
x1 + y1
x21 + y12
Introducing polar coordinates with radius r and amplitude θ then x = r cos θ and y =
r sin θ and a useful alternative representation of z is
z = x + iy = r(cos θ + i sin θ) = reiθ
The factor eiθ represents a rotation of the real vector r through an angle θ in the complex
plane.
3
Elementary Functions
Functions whose arguments are real variables are called functions of a real variable, f (x),
while functions whose arguments are complex variables are called functions of a complex
variable, f (z). In general, since z = x + iy then f (z) = u(x, y) + iv(x, y).
For instance, the integral power function (n = integer) is
f (z) = z n = (x + iy)n = rn (cos θ + i sin θ)n = rn einθ = rn (cos nθ + i sin nθ)
The polynomial is just a linear combination of the above, i.e.
N
X
f (z) =
An z n
n=0
The power series about a is
f (z) =
∞
X
An (z − a)n
n=0
which converges whenever
|z − a| <
1
1
=
L
limn→∞ | AAn+1
|
n
The circular and hyperbolic functions are
sin z =
eiz − e−iz
2i
5
cos z =
eiz + e−iz
2
sinh z =
ez − e−z
2
ez + e−z
cosh z =
2
If the complex variable z = ew where w is complex, then the complex logarithm is
w = Logz = log |z| + iθ
Note that this function is multivalued having infinitely many branches and a single branch
point at z = 0. The principal value of θ is restricted to 0 ≤ θP < 2π.
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Analytic Functions of a Complex Variable
The derivative of a function of a complex variable is defined as
f (z + ∆z) − f (z)
df
= f 0 (z) = lim
∆z→0
dz
∆z
If a function f (z) has a finite derivative (regardless the direction of approach) and is
single valued at each point in a region it is called analytic and its derivative is continuous.
Let z = x + iy and w = f (z) = u(x, y) + iv(x, y) then, if f (z) is analytic the CauchyRiemann equations hold
∂v
∂u
=
∂x
∂y
∂v
∂u
=−
∂y
∂x
A consequence of the analyticity of w is that its real and imaginary parts satisfy Laplace’s
equation and are called harmonic functions.
∇2 u = ∇2 v = 0
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5
Line Integrals
R
The integral of a function of a real variable ab f (x)dx generalizes to the line integral (over a
curve C from z0 to z1 ) of a function of a complex variable f (z) = u + iv as follows
Z
Z
f (z)dz =
C
C
(udx − vdy) + i
Z
(vdx + udy)
C
If the curve C can be enclosed in a simple connected region where f (z) is analytic the integral
is path independent. Also, if f (z) = dF (z)/dz then
Z
Z
z1
f (z)dz =
C
z0
dF (z) = F (z1 ) − F (z0 )
Furthermore, if z0 = z1
I
f (z)dz = 0
C
which is Cauchy’s integral theorem. The positive direction of integration is the one that
maintains the enclosed area to the left.
Example: Study the line integration of the function f (z) = 1/z.
Finally, if M is an upper bound for |f (z)| and L is the length of C
|
Z
C
f (z)dz| ≤ M L
If C is a curve in the complex plane and C a smaller circular contour (center = α, radius
= ) completely inside C
I
C
I
f (z)
dz =
z−α
C
f (z)
dz
z−α
and in the limit as → 0
1
f (z) =
2πi
I
C
f (α)
dα
α−z
which is Cauchy’s integral formula.
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Ordinary Differential Equations
An ordinary differential equation is an equation relating two variables in terms of derivatives.
The linear ODE of order n is
a0 (x)
dn y
dn−1 y
dy
+
a
(x)
+ ... + an−1 (x) + an (x)y = Ly = h(x)
1
n
n−1
dx
dx
dx
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where L is the linear differential operator given by
a0 (x)
dn
dn−1
d
+
a
(x)
+ ... + an−1 (x) + an (x)
1
n
n−1
dx
dx
dx
and represents the mathematical operation which produces h(x) when applied onto y(x).
Linear operators satisfy basic properties such as commutativity and distributivity and they
can often be factored.
The function y = u(x) which when substituted above makes left and right hand sides
equal is called a solution of the differential equation. If h(x) = 0 the differential equation is
called homogeneous.
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Linear Dependence
The set of functions u1 (x), u2 (x), ..., un (x) is called linearly independent if none of the functions in the set can be expressed as a linear combination of the others. To check that the given
functions are linearly independent it is sufficient to show that their Wronskian determinant
u1 u2 .....un
du
du
du
n
1
2
..... dx
dx
dx
W (u1 , u2 , ..., un ) = ................................. n−1
dn−1 u1 dn−1 u2
d
un n−1
n−1 .....
n−1
dx
dx
dx
is not equal to zero.
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Complete Solutions
If u1 (x), u2 (x), ..., un (x) are linearly independent solutions of the n-th order linear ODE
Ly = 0, the general solution of the equation is
yH (x) =
n
X
ck uk (x)
k=1
Now, if for the associated non-homogeneous equation Ly = h(x) with h(x) 6= 0, a
particular solution is yP (x), then the complete solution becomes
y(x) = yH (x) + yP (x) =
n
X
k=1
8
ck uk (x) + yP (x)
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Linear Differential Equations
The standard form of a first order linear differential equation is
dy
+ a1 (x)y = h(x)
dx
R
Introducing the integrating factor p(x) = e
a1 (x)dx
, the solution is
C
1Z
phdx +
y(x) =
p
p
where C is an integration constant.
Specifically, if in addition to the differential equation one is given the initial condition
y(x0 ) = y0 , the solution becomes
Z
x
y(x) =
x0
p(ξ)
p(x0 )
h(ξ)dξ + y0
p(x)
p(x)
Example. Solve xy 0 + (1 − x)y = xex .
The equation
dn−1 y
dy
dn y
+ an y = Ly = h(x)
+
a
+ ... + an−1
1
n
n−1
dx
dx
dx
is called the n−th order linear ODE with constant coefficients. Note that the symbol L
stands here for
L=
dn−1
d
dn
+ an
+
a
+ ... + an−1
1
n
n−1
dx
dx
dx
The general homogeneous solution of the above is
yH (x) =
n
X
ck erk x
k=1
and the associated characteristic equation is
rn + a1 rn−1 + ... + an−1 r + an = 0
which may involve imaginary roots. Also, sometimes roots of the characteristic equation are
repeated. In this case less than n independent solutions result but the missing solution can
be obtained from the condition of the vanishing of the partial derivatives with respect to the
repeated roots. Therefore, the part of the homogeneous solution corresponding to an m-fold
root r1 is
er1 x (c1 + c2 x + c3 x2 + ... + cm xm−1 )
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A commonly encountered equation is the n-th order equidimensional linear equation
xn
n−1
dy
dn y
y
n−1 d
+
b
x
+ ... + bn−1 x + bn y = f (x)
1
n
n−1
dx
dx
dx
The above is easily solved by introducing a new independent variable z such that x = ez
which transforms it into a simple linear equation with constant coefficients.
Example. Solve x2 y 00 − 2xy 0 + 2y = x2 + 2
An even simpler procedure is available in the case of f (x) = 0.
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Particular Solutions by Variation of Parameters
Suppose that the general solution of the homogeneous equation
Ly =
dn−1 y
dy
dn y
+
a
(x)
+ ... + an−1 (x) + an (x)y = 0
1
n
n−1
dx
dx
dx
P
has been obtained and it is of the form yH (x) = nk=1 ck uk (x). A particular solution of the
associated non-homogeneous equation Ly = h(x) can be obtained by replacing the constant
parameters ck is the above solution by certain functions of x satisfying certain conditions,
i.e.
n
X
yP (x) =
Ck (x)uk (x)
k=1
Succesive differentiation and manipulation of yP (x) produces two expressions which are used
to obtain the Ck ’s.
For instance, for the equation
dy
d2 y
+ a1 (x) + a2 (x)y = h(x)
2
dx
dx
where a1 , a2 and h(x) are continuous in the domain of interest, the solution obtained by the
method of variation of parameters is
yP (x) = C1 (x)u1 (x) + C2 (x)u2 (x)
with
C1 (x) = −
Z
x
u2 (ξ)h(ξ)
dξ + c1
W (ξ)
and
Z
C2 (x) =
x
u1 (ξ)h(ξ)
dξ + c2
W (ξ)
10
where u1 (x) and u2 (x) are linearly independent solutions of the associated homogeneous
equation and
u u
W (x) = W (u1 (x), u2 (x)) = du11 du22
dx dx
is the Wronskian of u1 (x) and u2 (x).
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Initial and Boundary Value Problems
The general solution of an n-th order ODE involves n arbitrary constants which must be
determined by n suitably prescribed supplementary conditions. For example, the value of
the function and of its first n − 1 derivatives may be prescribed at the single point x = a. If
this is the case one talks about an initial value problem and if the coefficients and h(x) are
continuous there exists a unique solution
y(x) =
n
X
ck uk (x) + yP (x)
k=1
On the other hand, if the values of the function and/or certain of the derivatives are
prescribed at two points x = a and x = b, othe talks about a boundary value problem. In
this case the solution may or may not exist and may or may not be unique.
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