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Second-Order Linear ODEs (Textbook, Chap 2)
Motivation Recall from notes, pp. 58-59, the second example of a
DE that we introduced there.
'11 022
wQdxd
−−=− φλ
φλ
φ (a1)
This equation represents conservation of water mass (actually
conservation of water volume; the model assumes an essentially
constant water density) in a confined, but leaky aquifer. The
unknown is denoted by the letter φ [L], representing hydraulic head
and the independent variable by the letter x [L], representing
location. It is a second order ordinary differential equation
(ODE). The symbols λ, φ0, and Qw’ denote known parameters.
Respectively, they represent aquifer transmissivity divided by a
leakage coefficient (λ [L2] =TB’/K’ , see p. 59 and the paragraph
on dimensional homogeneity), the head [L] in a vertically adjacent
(overlying or underlying) aquifer, and the pumping per unit area
[L/t]. All terms in (a) have units of [1/L]. To avoid confusion
with the notation in the text, which uses the symbol λ for
something else, let’s replace λ with the symbol β (= λ) , or
'11 022
wQdxd
−−=− φβ
φβ
φ (a2)
Another DE presented on p. 46 represents solute mass balance in
a flowing river, with the time rate of change of mass in storage
balanced by advective and dispersive fluxes, or
022
=∂∂
−∂∂
+∂∂
xCD
xCv
tC (b)
Here the unknown is denoted by the letter C, representing solute
concentration. There are “… two independent variables, denoted by
the letters x (space) and t (time). As there are now two (or more)
independent variables, it is a partial differential equation (PDE).
The highest order derivative in x defines it as a second order
equation in space, while the highest order derivative in t defines
it as a first order equation in time. … The symbols v and D denote
parameters, the mean river velocity and a dispersion coefficient.”
Suppose that there is a continuous, unchanging source of solute,
and that the flow rate in the river is constant. Then ∂C/∂t=0, the
solute concentration is in steady-state, and (b) becomes a second
order ODE in x.
022
=−dx
CdDdxdCv (c)
The first term represents solute advection and the second
dispersion, which are in balance. Notice that the partial
differences ∂( ) have become ordinary differences d( )’s, as there
is now
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only one independent variable, x. Suppose further that the
solute is organic and that it can biotransformed by in situ
reactions. The simplest model of this is that the solute
concentration decays exponentially with time, which in (c) is
modeled by adding a so-called first-order decay process,
Cdx
CdDdxdCv α−=− 2
2
(d1)
where α [1/T] is the decay constant. Here advection and
dispersion are balance by decay. We can write this in mathematical
standard form by dividing by the quantity (–D) and gathering all
terms on the left-hand side.
022
=−− CDdx
dCDv
dxCd α (d2)
This is second-order ODE in spatial location x. These are two
common examples of second-order ODEs encountered in hydrology, but
there are many others. These equations are also linear, if their
parameters are independent of the dependent variable. They can vary
with location x, however. Introduction (Textbook, Section 2.1)
These two examples belong to the general class of linear
second-order ODEs which your book writes as (1) )()(')('' xryxqyxpy
=++ where, r is the forcing and, for a linear model, parameters p,
q, and r do NOT depend on the dependent variable, y, but can depend
on the independent variable, x. How do the two examples above fit
this model? Model Translation to eqn. (1)
'11 022
wQdxd
−−=− φβ
φβ
φ '1,1,0,, 0 wQrqqpxxy −−==−==== φββφ
022
=−− CDdx
dCDv
dxCd α 0,,,, =−=−=== r
Dq
DvpxxCy α
Note that p,q, and r can vary with x, and the models remain
linear. Thus β, φ0, and Qw’ in the first model, and v, D, and α, in
the second model, can also vary in x, for linear models.
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Then, from the text, Chapter 2.
Not in the text’s Chapter 2: For a second-order homogeneous
linear ODE (2) a boundary value problem consists of equation (2)
taken over a finite interval, a ≤ x ≤ b, with two boundary
conditions (BCs), one at each end of the interval. The boundary
conditions can be written as
b
a
KbylbylKaykayk
=+=+
)(')()(')(
21
21 (e)
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where the k’s and l’s are parameters, weighing the relative
contribution of y and y’ at each boundary, and the K’s are values.
For example, if you know the value of y at a, then k1 =1, k2 =0,
and Ka is the value y(a). In PDEs this is called a first type or
Dirichlet boundary. If you know the gradient of y at b, then then
l1 =0, l2 =1, and Kb is the value y’(b). In PDEs this is called a
second type or Neumann boundary. Finally, at a, if both k1 and k2
are non-zero it is called a third-type or Cauchy boundary in PDEs.
All three boundary condition types are used to solve equations of
the type given in (1). However, Chapter 2 of your textbook focuses
on initial value problems for 2nd order ODEs. In hydrology these
problems are less common than 2nd order BVPs, but they are very
common in other areas of mathematical physics, like geophysics.
Back to the text …
Or for the BVP, the conditions in eqn. (e) on the previous page
are used to determine the two constants, c1 and c2. Two conditions
are needed as this is a 2nd order equation. In the IVPs these two
initial conditions (4) are taken at the beginning of the domain,
and in BVPs these two boundary conditions (e) are taken at each end
of the domain. In either event, the general solution is given by
(5).
*
*
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Reduction of Order (Section 2.1)
Two
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Second-Order Linear ODEs with Constant Coefficients (Section
2.1)
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Two distinct real roots
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Real double root
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Two complex roots
Summary 2nd order ODE with constant coefficients: ODE 0''' =++
byayy Characteristic eqn. 02 =++ λλλ ba
where 4
22 ab −=ω , and A, B, c1, and c2 are constants.
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Nonhomogeneous ODEs (Section 2.7)
To solve the nonhomogeneous ODE (1), or an IVP or BVP for (1),
we have to solve the homogeneous ODE (2) for a solution, and then
find any solution yp of (1), to obtain a general solution (3) of
equation (1). See the book for details and proofs of why this
works. The method of undetermined coefficients There are more
general methods to find yp, but we will focus on a simple method
that works for most problems that you are likely to encounter. The
more general method is Variation of Parameters, and is discussed in
§2.20 (not covered in lecture). The simpler, but restricted method
is called the Method of Undetermined Coefficients. The basic idea
is to guess all possible terms in the solution yp, then solve for
their coefficients, term by term, to satisfy (1). The result is yp.
The guess consists of sums of terms like those in r(x). This works
only if r(x) is composed of terms involving simple functions, in
particular powers of x, trigonometric functions, and exponentials,
or their products:
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Your book appears to neglect products, but you can handle
products of the basic functions in the first column of Table 2.1.
You just apply the modification and sum rules to the product.
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Equidimensional ODEs (Section 2.5)
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Series Solutions (Chap. 5) So far we’ve examined the general
class of linear second-order ODEs which your book writes as (1)
)()(')('' xryxqyxpy =++ where r is the forcing, and parameters p,
q, and r do NOT depend on the dependent variable, y, but can depend
on the independent variable, x. The methods we’ve examined have
considered relatively simple and special functions for the
parameters, but what happens if the parameters have more
complicated form? If r(x) is more complicated than can be handled
by the Method of Undetermined Coefficients (§2.7) then we typically
resort to the Method of Variation of Parameters (§2.10), which we
have not covered in glass. But there is another method that works
very well in this case called LaPlace Transforms (Chap 6) which we
use on 1st and 2nd order linear equations, and which is very
versatile. It even gives the particular solution right away without
solving for the homogeneous solutions first as an intermediate
step. We’ll come back to it later, when we study PDEs. There is
another method we use if p(x) and q(x) have more complex functional
form, and its particularly valuable for BVPs. In this approach we
resort to series solutions. We know that we can approximate
functions by series. The idea is to assume that the solution is
written as a power series, to plug that series into the equation,
and to equate sums of coefficients of equal power to zero, solving
for the coefficients in the power series. For a second order
equation each coefficient can be written terms of another
coefficient, resulting in a recurrence relation, for the
coefficients, and leaving only two underdetermined coefficients to
be resolved by boundary or initial conditions. Often the series
solutions are represented by special functions, with new functional
names, such as the Bessel Function of first type of order zero.
Certain forms of 2nd order ODEs occur often in mathematical
physics, engineering, and hydrology. The series solutions for these
problems have been worked out in detail, and are typically
represented by the special functions. It is useful to learn to
recognize them or, at least, to know that you should look to see if
the equation you are dealing with is of one of these types. For
example, if you have a problem in radial flow, such as in
groundwater well hydraulics, Bessel Equations are common, while
Legendre Equations are common in spherical flow or solute diffusion
within a particle, like a “spherical” microporous feldspar grain.
Later we’ll see that the methods can be combined to solve PDEs. For
example, consider a transient groundwater flow problem that is
one-dimension in space. It can be LaPlace Transformed in time, to
remove time and reduce it to an ODE in space, which is solved by
one of the methods we have been studying. An example is radial flow
to a pumping well in an infinite, leaky confined aquifer. After the
LaPlace Transform in time, the resulting ODE (in LaPlace Space) is
a Bessel Equation, for which one can look up and apply the
solution.
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The series solution approach, background from the text (p.
168):
The text then goes on in §5.2 to talk about theory. If the
series converges everywhere, the function is said to be analytic.
If it diverges in certain locations those are called singularities.
A
simple example is a divide by zero. The function and its
derivatives must converge to be analytic. Convergence is defined in
the usual ways, e.g., by comparing successive terms in the series.
Series solutions for analytical functions are straight-forward, it
is all the special cases involving singularities that make solving
a problem difficult. For certain of these more complicated
equations the Method of Frobenius (§5.4) is particularly useful. In
any event, we won’t visit these cases or this method. You need to
take a course in ODEs, although you could also self-learn from the
text, which is very clear on these subjects. Let’s take a closer
look at the special case of Bessel’s Equation, which is well known
and often encountered in hydrology. From the text, p. 189
(§5.5):
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Bessel’s equation can be solved by the Method of Frobenius (see
text, p. 189) …
Etc …, leading to (p. 202, §5.6):
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