We use linear algebra to solve the linear DE with constant coefficients.
See [Pon, pp.95-96, Theorem 10; p.98, Theorem 11] for details. [Arn, p.106,
Theorem 15.2] is the final form of the solution. This form suggests some methods
for proving the existence theorems in the general case (see [Arn, p.110, Theorem;
p.215, Example 2] & [Bir, p.173, Theorem 1; p.175, Corollary]).
Stability in
terms of energy [Arn1, p.149, Problem 1]. Level sets of the first integral
[Arn1, p.140, §12.2 &
§12.3] and Liapunov function [Bir,
p.128, l.17] are all derivatives of energy. g-u
[Pon, p.235, Fig.57(b)] is the projection of ΔE
[Arn1, p.150, l.-6] onto the phase plane.
The method in [Inc, §27] is more
effective than that in [Inc,
§24-§26].
[Inc, (26.1)] reduces to [Inc, p.75, l.13]. The reduction justifies the
nomenclature of "p-discriminant
equation".
[Inc, p.75, l.-2; p.77, l.19] show the key difference between a cusp
locus and a tac locus.
[Inc, p.77, l.12] confirms the statement in [Inc, p.74, l.7].
The choice of t in [Inc, p.78, l.-3] is more pertinent than that in
[Inc, §25].
[Inc, (26.1)] is a discriminant in an
algebraic sense [Inc, (27.2)] and is a boundary in
a geometric sense [Inc, p.75
(c) & (d)].
If a method is applied to a weaker condition, the application becomes less
effective (compare [Inc,
§21]
with [Inc, '20]). In other other words, to
obtain a closed-form solution, applying the method to a DE with a weaker
hypothesis will require more drastic cancellations.
In order to find a short cut to solve a DE in practice, [Inc, p.94,
l.10-l.21] uses an idea similar to [Bir, p.156, Corollary 1] when a DE has
enough integrating factors to recover all its distinct first integrals.
Symbolic operations (Mnemonic devices).
Symbolic operations enable us to predict and apply a result immediately
without thinking of its rigorous proof.
Separation of variables. We use infinitesimals to visualize the symbolic
operation and use differential forms to justify its usage [Arn1, p.20, l.5-l.27;
p.42, l.-13-p.43, l.8].
Operating Delta-function as a function [Bir, p.55, l.9-p.56, l.3].
Justification: Distribution theory.
Representing an inverse operator as a reciprocal [Inc, p.125, l.-3].
Justification: [Inc, p.126, l.2-l.11].
If we begin by studying the analytic case, we may strike right to the heart of
the matter and eliminate complications. Very often we may find a more effective
method in the analytic case.
(Rectification) [Arn1, p.89, Theorem 1]. In the analytic case, the
Taylor series of a rectifying diffeomorphism can be determined recursively
[Arn1, p.91, l.-11].
Differentiable dependence of initial conditions: Trivial.
(Perturbation). It is simple to derive the variational equation in the
analytic case [Bir, p.163, l.2-l.15]. In practice, an even simpler method (i.e.,
Newton's series method [Arn1, p.94,
l.17-l.20]) is available.
Effectiveness
Basis for a set of solutions.
Fundamental domain for an individual solution. In order to find the
solution of a PDE in a region R, we first find the solution in a fundamental
domain I Í R. Then the solution in R\I can be derived
from the solution in I by simple algebraic operations [Joh, p.44, Fig.2.5].
Solution's development
Characteristic Manifolds.
The characteristic curves of a second-order PDE with 2 independent variables
[Joh, p.35, (1.12)]
®
The characteristic curves of a first-order system with 2 independent
variables [Joh, p.47, (5.13] (See [Joh, p.46, (5.1), (5.2), (5.3)])
® The characteristic surfaces of a linear first-order system with n
independent variables [Joh, p.59, (2.26)]
® The characteristic surfaces of a mth- order quasi-linear
system with n independent variables [Joh, p.59, (2.24)].
Comments.
A solution has its primitive model. To obtain the key idea of solving a
generalized problem, we have to trace back to the original setting.
After the key idea is generalized to a certain extent, its spirit and
freedom will be developed fully.
Lagrange identities.
[Sne, p.181, (5)] ® [Sne, p.193, (2)]
® [Joh, p.80, (4.8)].
Comments. We concentrate on the development of the solution's
idea rather than the development of its logic. The first step of the idea
development reveals the direction of development, while the first step of the
logic development is not very different from the rest of the steps.
The solution pieces together all the information obtained from various
approaches.
Point-valued function solution: [Sne, p.145, l.-11].
The difficulty at the singular point [Sne, p.145, l.!6]
still remains.
Informal distribution solution: (Potential [Sne, p.145, l.!2]).
Rigorous distribution solution: [Joh, p.92, l.4; p.97, l.-19-l.-15].
(A) is a partial solution of (B): [Joh, p.90, (6.11)].
Existence and uniqueness of the solution of the Dirichlet problem.
A physical and intuitive solution [Sne, p.151, l.19-l.21].
(Not finished).
How the abuse of the supremum property [Bar, p.39, Property 6.4] affects
numerical simulation.
The existence of the function w_{f} [Joh, p.113, (4.9)] comes from
the supremum property. It can not provide any significant numerical
approximation. Therefore, we must correct our attitude toward axioms: Axioms are
designed for understanding a theory's
logic network, not for applying the theory. Indiscriminate use (Example: Perron's
method [Joh, pp.111-116]) of supremum property will result in assuming
that something
unknown is known [Wei, p.v, l.-26-l.-25].
This makes the theory less effective. Confucius said,
"Do not pretend to know something that
you really don't."
If our goal is numerical simulation, we must restrict the use of the supremum
property to cases where the nonempty set is concretely known and the supremum
is numerically approachable.
Solving a PDE involves more than just checking if the given answer is the right
solution. Actually, it is the reverse procedure that is rewarding:
Suppose the solution exists; what form should it take? In other words, we
must use the solution's properties to
eliminate the impossibilities and determine its exact form.
Example 1. For Dirichlet's problem,
[Ru2, p.258, l.10-l.13] makes such a mistake: it
only checks if the given answer
is the solution. We must know the origin of Poisson's
integral formula. That is, the Poisson kernel should be recognized as the normal
derivative of the Green function [Joh, p.106, l.-25-p.107,
l.-8].
Example 2. The 1-dim wave equation: Solution formula [Sne, p.89, l.-22-l.-13];
Reverse [Joh, p.40, l.16-l.27].
Example 2'. The 3-dim wave equation: Solution formula [Pet, p.80, l.-13-p.82,
l.5]; Reverse [Joh, p.128, l.3-p.129, l.10].
Example 3. The diffusion equation [Sne, p.283, (6)]: Solution formula [Sne,
p.283, (5)]; Reverse [Joh, p.208, l.8-p.209, l.4].
The connection among the various concepts of a solution.
Lagrange's identity [Joh, p.97,
(1.17); p.99, (1.23)] simultaneously gives the fundamental solution of the Laplace operator [Joh,
p.97, (1.18)] and the solution of Dirichlet's
problem [Joh, p.107, (3.7)].
(Solidification by Lagrange's
identity [Joh, p.122, l.8; p.123, l.-6])
If a PDE has a point-valued solution, it takes quite an effort [Joh, p.122,
l.4-p.124, l.-1] to recognize these distribution
solutions as functions [Joh, p.123, (5.26b)]. Because the distribution solution
appears in the early stage of solidification, it is quite shallow and vacuous.
However, the distribution method gives room for solving PDE's
which have no point-valued solutions.
By comparing [Joh, p.97, (1.18)] with [Joh, p.100, (1.31a)], we see that
Poisson=s formula [Joh, p.99, (1.28)]
is the precursor of the fundamental solution.
An effective method of proving insolvability.
1-1 correspondence B Reduce to
simpler objects.
Between subfields and subgroups [Her, p.247, Theorem 5.6.6].
Between Lie subgroups and Lie subalgebras [Po3, p.385, Theorem 83 & p.418,
Theorem 91].
A polynomial equation is solvable Þ Its Galois group is solvable [Her, p.255, Theorem 5.7.2].
The spirit of the calculus of variation is to effectively construct a
solution using the completeness relations [Cou, p.97, l.-8; p.178, l.12].
[Cou, pp.97-98, §10.1] and [Cou,
pp.174-175, §2.1] propose two solutions
to the isoperimetric problem. The former gives an effective method to construct
the solution, while the latter does not. [Cou, p.174, l.-12] claims that the set of areas of admissible curves has a least upper bound M
[Ru1, p.11, Theorem 1.36]. However, M in [Cou, p.174, l.-11] cannot be practically implemented. This is because E
¹f [Ru1, p.11, Theorem 1.36] implies that the
existence of elements in E belongs to the type of logical existence [Wan3,
pp.109-110]. We neither know what the elements are nor have enough information
to effectively construct the lub using a specified procedure. Thus, in general, the existence of lub in [Ru1, p.11, l.13]
is not constructive [Wan3, p.109]. However, for a given specific sequence
{a_{n}}, we often have an effective method to construct the lub.
A solution's development (e.g. Green's function associated with a boundary
value problem).
Create a solution from physical consideration
[Bir, p.53, l.3-l.21].
Generalize the problem by good observations [Bir,
p.53, l.-14-l.-7].
Modify the solution in step A accordingly [Bir, p.53, l.-7-p.55,
l.8].
Another example: the forced oscillation [Bir, p.48, Example 6].
Remark:
For the proof of [Bir, p.48, (30)], see [Sym, p.61, (2.207)-(2.210)].
If the method that produces the first solution fails to provide another
independent solution, then how do we construct a second independent solution
from the first solution?
[Bir, p.36, (13)] ® [Har, pp.50-51, Lemma 3.1]. Remark. [Har, pp.50-51, Lemma 3.1] gives a general method
for reducing a linear
system to smaller systems.
(Singularity — using the method of
analytic continuation) The case when the roots of the equation [Guo,
p.53, (8)] are equal: [Guo, p.55, (17)].
Special case (regular singularity). The case when the roots of the indicial equation differ by an integer:
[Guo, p.61, (27)].
Example. [Guo, p.224, l.-3].
The Frobenius Method (regular singularity) [Jef, p.483, l.7-l.20]. The
case when the two roots of the indicial equation differ by an integer:
The case when the two roots are equal: [Guo, p.62, (8)].
The case when the two roots are not equal (using the properties of a double root) : [Guo, p.63, (13)].
Exploiting the
property that the limit of a linear combination of solutions has an
indeterminate form.
Example 1. When u is not an integer, J_{u} and J_{-u} are
linearly independent [Guo, p.348,
l.10]. However, when u is an integer n, J_{u} and J_{-u}
are linearly dependent [Guo, p.348, (8)]. In the latter case, we define Y_{u}
as in [Guo, p.366, (3)] and then let u®n (the limit
of Y_{u} has an indeterminate form).
Example 2. When u is not an integer, I_{u}
and I_{-u} are linearly independent [Guo,
p.374, l.-11]. However, when u is
an integer n, I_{u}
and I_{-u} are linearly dependent [Guo,
p.374, (5)]. In the latter case, we define K_{u}
as in [Guo, p.374, (6)] and then let u®n (the limit
of K_{u} has an indeterminate form).
Example 3. [Guo, p.261, l.8].
Changing the contour of the
first solution's integral representation.
Example. Compare [Guo, p.248, (6); p.251, (1)] with [Guo, p.247, (4)].
By inversion [Chou, p.156, (3.148)].
Using the Wronskian [Bir, p.36, l.-9-p.37, l.7].
Neumann's method: [Wat, p.67, l.8-l.10].
[Guo, §2.1-§2.11]
discusses how to express the solutions in series form; [Guo,
§2.12-§2.14]
discusses how to express the solutions in integral form. According to the way
that [Guo] is written, these two topics seem to be completely independent of
each other. The later topic is mainly based on the method of synthesis. In fact,
the domain of a series solution is usually small. We seek an integral solution
in order to extend the solution's domain. In the case of Bessel functions, [Wat,
p.47, l.-14-l.-13] provides an effective link between a series solution and an
integral solution.
Remark. For a specific-to-general approach (e.g., the series-to-integral
approach in [Wat, §3.33]), the reason why each step leads to the next one can be easily explained. The approach is inspirational. In contrast,
for a general-to-specific approach, which Landau adopts to make his theory of physics compact,
it would very hard to explain why we make a specific choice [Guo, p.352,
l.11].
Solving problems through pattern recognition.
If we want to prove [Mari, p.505, (12.160)] and looking at
this problem
alone, we may not know where to start. However, if we are familiar with
spherical harmonics, we may recognize [Mari, p.505, (12.160)] and [Chou,
p.606, (IV.32)] have the same form. Furthermore, [Jack, p.68, (2.35)] is a
natural generalization of [Chou, p.606, (IV.32)] (see [Chou, p.606, l.-3]).
Therefore, we may prove [Mari, p.505, (12.160)] based on the proof pattern of
[Jack, p.68, (2.35)]. Once we recognize the pattern, the proof of [Mari,
p.505, (12.160)] becomes trivial.
In [Wangs, p.189, Example], Wangsness leaves his solution in a series
form. With this form, we can only evaluate the value of
f. In [Jack, p.75, (2.65)], Jackson expresses his solution more neatly
in a closed form. With this form, we can know more about the shape and the global
properties of the function F.
Green's function for the Poisson equation [Jack, p.35, (1.31)].
Green's functions for the wave equation [Jack,
§6.4 or [1]]. Key idea: We use the Fourier transform to remove the
explicit time dependence and then use "cause and effect" to eliminate the
nonphysical solution.
Remark. (The wave-particle property) In 1924, Louis de Broglie proposed his postulate for matter waves [Eis,
p.56, (3-2)] based on the assumption that matter behaves like radiation. In
1927, G. P. Thomson experimentally discovered electron diffraction and thus
confirmed the postulate. Many physicists believed that the research on the
wave-particle property is complete once the theoretical
assumption is experimentally confirmed. In my opinion, experimental
confirmations is far from complete establishment partly because an experiment
only represents one particular case and partly because a phenomenon can have
many theoretical explanations; one must find the theoretical basis of the
wave-particle property in the right place: the solution of the Schrödinger
equation. For free particles, the Schrödinger equation reduces to the wave
equation [Reif, p.353, l.-3-p.354, l.9]. Only after
its theoretical basis is built in the solution of the wave equation from
multiple perspectives may we say
the wave-particle property is firmly established. The
Fourier transforms in [Jack, p.243, (6.33) & (6.34)] are exactly what we are
looking for. This is because one can easily prove the following theorem
on the wave-particle property by using the argument in [Coh, Complement E_{II}]:
[Q, P] = iħ (from the viewpoint of
commutators or the Poisson brackets)
Û
[Coh, p.190, (23)] (from the viewpoint of gradients)
Û
the wave function y in
p> representation is the Fourier transform of y in
q> representation [Coh, p.191, (27)] (from the viewpoint of Fourier transforms).
One can visualize the wave-particle duality more intuitively using Fourier
transforms than using the Poisson brackets.
Effective methods of finding the strong solutions for a wide scope of differential equations
Modern Mathematicians tend to use the concept of distributions to solve differential equations [Ru3,
chap 8]. Sometimes, the solutions will become too weak to be useful for practical
problems. If we compare the strength of a generalized solution to the precision of a long-range missile,
our goal for solving differential equations should be to aim far and hit the target strongly, precisely and effectively.
That is, we are looking for the most effective way to find the strong
solutions for a wide scope of differential equations.
Effective methods of integration [Inc1, chap. II]
Exact equations of the first order
Separation of variables [Inc1, §2.11]
Remark. Homogeneous equations can be solved by the method of separation of variables.
Linear equations of the first order [Inc1, §2.13]
The homogeneous case can be solved by the method of separation of variables.
Using the form of solution in the homogenous case [Inc1, p.20, l.-9],
we may solve the nonhomogeneous case by the method of variation of
parameters [Inc1, p.20, l.-6].
Remark 1. The Bernoulli equation can be reduced to the linear form [Inc1, p.22, l.23].
Remark 2. The Jacobi equation can be reduced to a Bernoulli equation
[Inc1, p.23, l.-26].
Remark 3. The methods of solving differential equations in [Inc1, §2.42,
§2.43, and §2.44] are similar to the methods (i) and (ii). Consequently, many textbooks
on ordinary differential equations omit these topics even though they have
important applications in theoretical physics [Lan1 or Go2].
(Reducing degrees) The Riccati equation can be reduced to a linear equation of the second
order [Inc1, p.24, l.-1].
The Clairaut equation [Inc1, §2.45;
Sne, chap. 2, §11, (d)] The geometric meaning of singular solutions
is given by [Inc1, p.40, l.14-l.16; Sne, p.60, l.14].
Reducing the order by integration [Inc1, §2.6
& §2.61]
Remark.
The methods of solving differential equations given in [Inc1, §2.62
& §2.63] are similar to those in [Inc1, §2.6
& §2.61].
Updates
For Mercator's projection [Inc1, p.34, l.20], see [Kre, p.191, l.4-l.-1].
For [Inc1, §2.6], see [Wid, p.43, Theorem 1].
Linear differential equations of order n
Homogeneous equations with constant coefficients [Edw,
§3.3; Pon, pp.50-51, Theorem 5; Cod, p.89,
Theorem 6.5]
Remark 1. The formulation for the properties of the Wronskian
determinant given in [Edw, p.162, Theorem 3] is organized, systematic,
complete, and useful.
Remark 2. For the linear independence of [Cod, p.89, (6.20)], the proof given in [Pon, p.54, l.-24-l.-8] is clearer than that given in
[Cod, p.90, l.4-l.20].
Application.
Coddington's
observation regarding similarity [Cod61, p.149, l.9] is inadequate.
Actually, Euler's equation can be transformed into an equation with constant
coefficients [Col, p.110, l.-19-p.120, l.14].
While [Col, chap. II, §6] links Euler's
equation to its indicial equation [Har, p.85, l.14], [Har, p.85, l.5-l.11]
further links Euler's equation to the Jordan normal form of the corresponding
linear system [Har, p.84, (12.3)]. The statement given in [Har, p.85,
l.7-l.8] can be proved by [Har, p.59, (5.18)].
Nonhomogeneous linear differential equations
The method of undetermined coefficients [Edw,
p.191, l.-3-p.201,
l.9].
Remark. Given L[u] = f(x), where L[u] is a linear differential operator.
Let B={b_{a}}
be a basis. If ("b_{a}ÎB,
there exist finite i such that L[b_{a}]
= å_{i }c_{i}b_{i}),
we may find a solution by substituting u(x) =
å_{a }d_{a}b_{a}
into L[u] = f(x). (Proof. Express f(x) as å_{a
}h_{a}b_{a}.)
We shall use the rules of the method of undetermined coefficients to check if
B is an appropriate basis. Examples. We may let B be {x^{n}
| n ³ 0}, {cos
nx, sin nx} or {J_{n}, Y_{n}}.
Similarly, for a regular Sturm-Liouville system (L=D[p(x)D]-q(x)) [Bir, p.256, l.-8],
we may find a solution u(x) =
å_{a }d_{a}b_{a}
(where {b_{a}} are eigenfunctions
of L[u]) of L[u] = f(x).
Homogeneous systems with constant coefficients [Har, chap.
IV, §5]
Remark.
We use [Jaco, vol. II, p.94, (27); p.97, (29)] to choose Q so that J
is in a Jordan normal form [Har, p.58, l.-4].
Linear algebra will reach a dead end after it finishes discussing Jordan
normal forms. Only after we go further into the theory of ordinary
differential equations may we use Jordan normal forms to obtain the solutions of
differential equations.
Application. The case of variable, but periodic, coefficients can theoretically be reduced to the case of constant
coefficients [Cod, p.78, Theorem 5.1; Har, p.60, Theorem 6.1].
Nonhomogeneous systems
(variation of constants): [Har, pp.48-49, Corollary 2.1].
Remark. We discuss first the solutions of linear differential equations of order n,
and then the solution of general linear systems. The former solutions are more
effective than the latter ones. [Cod, chap. 3] and [Har, chap. IV] reverse the
order of presentation. I do not think their approach is appropriate.
(Transformed Bessel Functions) Using the transformation given in [Edw, p.582, (2) and (5)], we may
transform the Bessel equation of order p
z^{2}(d^{2}w/dz^{2})+z(dw/dz)+(z^{2}-p^{2})w=0 into the equation
x^{2}y"+Axy'+(B+Cx^{q})y=0
[Edw, p.582, (3)]. Moreover, using the transformation
[w = x^{(a-1)/2} exp[(b/p)x^{p}]y, z = x^{q},
a = (1-a)/2, g = (2-r+s)/2,
l = 2(|a|)^{1/2}/(2-r+s),
and n = [(1-r)2-4b]1/2/(2-r+s)], we may transform
the Bessel equation
z^{2}(d^{2}w/dz^{2})+z(dw/dz)+(l^{2}z^{2}-n^{2})w=0 into the equation
x^{2}y"+x(a+2bx^{p})y'+[c+dx^{2q}+b(a+p-1)x^{p}+b^{2}x^{2p}]y=0 [Barr, p.586, l.3].
Remark. We must prove [Barr, p.586, Theorem 1] using the proof of [Edw, p.583, Theorem 1] as our model.
In other words, we must be familiar with the technique of solving a simple case
before we attempt to solve a more difficult problem.
Using the continued fraction method to solve the DE's with three-term recurrence relations
Examples: the tidal equation [Jef, §16.08],
Mathieu's equation [Jef, §16.09], and
equations with infinite determinants [Jef, p.489, l.3].
Using the adjoint operator to represent the solution in the complex
integral form [Guo, §2.12]
The Laplace transform [Jef, §16.10; Guo, §2.13].
The main feature of
the Laplace transform method is the integration by parts [Jef, p.489,
l.-4]. The Laplace transform method is useful for finding the asymptotic expansion of
the solution [Guo, p.84, l.-8].
Examples: Bessel's equation [Jef, p.490, (15) & (16)] and the
Hermite equation [Guo, p.84, (17)].
The Euler transform [Guo, §2.14].
The Mellin transform [Guo, p.80, (18)].
Sometimes, the differential equation of the inverse function t(s) is simpler than the equation of motion
for s(t) [Cou2, vol. 1, §4.5.
§4.7.a, §4.7.b].
Green function expansion in cylindrical coordinates [Chou,
§3.14; Jack,
§3.11]
In view of [Chou, p.166, (3.176) & (3.177)], if we replace
d(f-f') and
d(z-z') in [Chou, p.166,
(3.175)] by Q(f)Q*(f')
and Z_{n}(z)Z_{n}*(z')
respectively, then [Chou, (3.175)] will reduce to [Chou, p.167, (3.181)].
Thus, by fixing m and n and using separation of variables we may reduce the 3-dimensional boundary
value problem to a one-dimensional boundary value problem of the Sturm-Liouville
type. Then it is obvious that
G(r, r')= S_{m}
S_{n} g_{mn}(r|r')Q(f)Q*(f')Z_{n}(z)Z_{n}*(z')
will be the solution of [Chou, p.166, (3.175)].
Both Jackson's proof of [Jack, (3.141)] and Choudhury's proof of
[(3.179)-(3.182)] are unnecessarily complicated.
The equivalence of a differential equation and its corresponding integral equation [Coh, p.915,
l.-11-p.916, l.9].
Solutions of partial differential equations
Complete solutions, general solutions, and singular solutions ([Sne, p.49, l.-21-l.-5]
® [Sne, p.59, l.-15-p.60,
l.15])
We try to find the smallest geometric entity which satisfies [Joh,
p.9, (4.2)]. A curve is inadequate as a solution because a line, a typical
curve, has d/dt but cannot have both ¶/¶x
and ¶/¶y. Because a
surface has a higher dimension, we try to use it [Joh, p.9, l.15] as the
building block of our solutions. In view of [Inc1, p.4, l.-16-p.5,
l.17], it is natural to consider the 2-parameter family of surfaces given in [Sne,
p.60, l.4] as complete solutions.
Remark. [Sne, p.59, l.-15-p.60,
l.15] should be supplemented by [Wea1, vol. 1, §15 &
§21] in order to gain a complete
understanding.
Examples [Cou, vol. II, chap. 1, §1.1].
There are two reasons that Courant designed various examples: First, mastering various techniques of solving PDEs will enhance one's ability to solve any PDE. Second, theorems in the
theory of PDEs come from induction based on one's observations when studying
examples. Without examples, it is difficult to find a theorem's origins and
key points.
A necessary and sufficient condition for integrability [Sne, p.21, Theorem
5; Inc1, §2.8].
Remark. The fact that the condition of integrability is a sufficient condition
provides a general method of solving a Pfaffian differential equation [Sne,
p.19, (6)].
Integrability vs. accessibility: Carathéodory's
formulation [Sne, p.33, l.-20-p.37, l.14]; Born's formulation
[Sne, p.37, l.-20-p.38, l.-1].
Remark 1. For a non-integrable equation, we must project the path onto a 2-dim
vector space in order to make the concept of accessibility meaningful [Sne, p.36, Fig.
11; p.38, Fig. 12].
Remark 2. Carathéodory's proof stops at the
equation given in [Sne, p.37, l.13], a condition which is equivalent to
integrability [Sne, p.35, l.-10-l.-5].
Born's proof proceeds further by directly constructing the integral surface.
Effective methods of solving integrable equations [Sne, chap.1, §6].
Remark. Among the methods given in [Sne, chap. 1,
§6], both Method (e) and Method (f) apply to the general case, while
the remaining
methods apply only to special cases.
Carathéodory's version of the laws of thermodynamics: Carathéodory
puts the laws of thermodynamics on a firm mathematical foundation.
(1). The first law [Sne, p.39, l.17-l.21]: The first law defines internal energy [Sne, p.40, (1)] based on [Ru1, p.115, Theorem 6.16] and
also defines the concept of heat [Sne, p.40, (2)]. It shows that for
two variables we do not need the second law [Sne, p.40, l.-6-p.41,
l.6] to find an integrating factor m(p, v)
for DQ [Sne, p.40, (4)]. Thus, in term of
mathematics, integrability is the crucial line dividing the first law and the second law.
(2). The second law [Sne, p.41, l.26-l.27]: The concept of reversible path
given in [Hua, p.17, Fig. 1.7] is not effective
because [Hua] fails
to provide an effective test to check whether the heat in the Pfaffian
differential form is integrable. Carathéodory's
version of the second law [Sne, p.41, l.26-l.27] does provide the most
effective test based on [Sne, p.19, Theorem 2]. In order to define entropy,
the approach using reversible paths to seek an integrating factor, as
Huang employed in [Hua,
§1.4], is good for two independent variables, but is inadequate
for more than two independent variables. The use of accessibility [Sne, p.34,
l.15-l.29] is the only way to make the concept of entropy well-defined [Sne, p.35,
Theorem 8]. The path used to build accessibility runs both ways, so it is
always reversible. Furthermore, there are too many physical versions of the second
law [Reic, §2.D.3]. Carathéodory's version
reveals the mathematical insight which implies all these phenomena. The fact
that heat cannot flow from a cold body to a hotter one is an example of
inaccessibility because the two states are not on the same integral surface. [Zem,
§7-7 & §7-8] fail to prove important theorems [Sne, p.19, Theorem 2; p.35, Theorem 8]
and emphasize only trivial gimmicks. Consequently, after reading these two
sections, readers will remain puzzled about Carathéodory's version of the
second law.
Remark. Textbooks should not be used to show off one's pedantry by giving
new terminology without any references. They should be written with an
audience in mind and should make it easy for readers to understand the essence of the
topic.
Solutions of non-integrable equations: [Inc1,
§2.83; Sne, p.26, Fig. 10].
The first order
Linear
Analytic solutions: [Sne, p.50, Theorem 2; p.53, Theorem 3].
The geometric meaning of general solutions is given by [Inc1,
§2.71; Sne, p.50 Theorem 2]. An integral
surface [Joh, p.9, l.16] represents a general solution of the partial
differential equation [Sne, p.50, (1)]. The general solutions of the corresponding system of ordinary
differential equations [Sne, p.50, (4)] are 2-parameter families of characteristic
curves [Sne, p.51, (5)]. These two solutions are identical [Joh, p.10,
l.7-l.15; Inc1, p.50, l.15-l.28]. [Inc1, .50, l.-9-p.51,
l.-9] proves [Sne, p.50, (3)] which
allows us to form an integral surface [Inc1, p.48, l.-18-p.49,
l.15] by selecting a one-fold infinity of curves of congruence from the
two-parameter family of curves [Sne, p.51, (5)]. Remark. [Joh, p.10, l.-6-p.11, l.3] discusses the
geometric consequence of [Sne, p.50, l.-10-l.-6].
The method of finite difference [Joh, p.7, l.-8-p.8,
l.12] If we use backward differences, the error will not
grow exponentially with the number of steps in the t-direction [Joh, p.7,
(3.17)]. The Courant-Friedrichs-Lewy test is satisfied when [Joh, p.7, (3.16)]
is satisfied.
Nonlinear
Cauchy's method of characteristics
(1).
The initial value problem is a special case of the Cauchy problem [Joh, p.11,
l.-7-p.12, l.2].
(2). Linear [Ches, p.9, l.9-l.21]®
quasi-linear [Joh,
p.12, l.3-p.13, l.18]®
general [Sne, p.65, l.12-l.-4].
If one jumps directly to the third step without going through the first and the
second, it would be difficult to recognize the method's structure, scheme, and
key points.
(3). It is easier to grasp the key idea in a simple case [Ches, p.3, (1-6) or
p.6, (1-16)]. We
choose two natural parameters for the integral surface passing through a given
curve G(s): one from the parameter s of the curve
G and the other
from the parameter t of the characteristic curve passing the fixed point G(s) [Joh, p.9, (4.4)].
[Joh, p.12, l.3-p.13, l.13] proves the local existence of the solution of the
Cauchy problem for the quasi-linear case [Joh, p.9, (4.2)].
The definition of characteristics given in [Joh, p.9, (4.3)] is easily generalized and visualized. In contrast, the definition given in [Ches,
p.4, l.14] is too simple to be generalized, while the definition given in [Sne,
p.64, (18) or p.69, (6)] is too complicated to be visualized. The reason that
five equations are required to determine the characteristics in [Sne, p.69,
(6)] is because the denominators in the first terms in [Sne, p.69, (6)]
involve not only {x,y,z} but also {p,q} [Ches, p.175, l.-16-l.-6;
p.176, l.-2-p.177, l.11].
(4). The characteristic equations given in [Sne, p.64, (18)] reduce the original PDE [Sne, p.62, (2)] to a normal system of first-order ODEs.
(5). The language in [Sne, chap. 2, 8] is very loose, in the following I
try to make the definitions more precise and match their geometrical meanings
more closely:
(i). A plane element satisfying F(x,y,z,p,q)=0 is an integral element.
(ii). A one-parameter (t) family of plane elements [Sne, p.63, (7)] satisfying [Sne, p.63, (8)]
is a strip
containing the curve C(t).
Charpit's method [Sne, chap. 2, §10].
[g(x,y,z,p,q)=0 and f(x,y,z,p,q)=0 are compatible] Û
(¶(f,g)/¶(p,q)
¹ 0 and [f,g]=0) [Sne, p.69, l.-9]. (a). f(p,q)=0 [Sne, p.71, (1)].
(b). f(z,p,q)=0. [Sne, p.71, (5)].
(c). f(x,p)=g(y,q) [Sne, p.72, (7)].
(d). Clairaut equations: z = px+qy+f(p,q) [Sne, p.72, (9)].
Remark 1. [Sne, p.67, (4)] can be derived from the implicit function theorem [Wid,
p.59, Theorem 16].
Remark 2. [f, g] given in [Sne, p.68, (9)] is a generalization of the Poisson bracket [Ches, p.215, l.-1].
Remark 3. Cauchy's method and Clairaut's method reach the same conclusion [Sne,
p.70, l.1-l.3] through different paths.
Remark
1. Any general first-order system of ODEs can be written in the form of a
Hamiltonian system [Ches, §8-19].
Remark 2. [Ches, p.194, Theorem 3] can be proved using envelopes and characteristic curves [Ches, p.194] or using generating functions
F^{(2)} [Ches, p.208, l.-13].
The equation given in [Ches, p.208, (8-177)] is incorrect. It should have been
-H(x,t,p) =
F_{t}^{(2)}.
Solutions satisfying given conditions [Sne, chap. 2,
§12]
An integral surface passing through a given curve C is the envelope of the
set of those members of the family [Sne, p.73, (3)] that touch the curve C.
An integral surface circumscribing a given surface S
is the envelope of the set of those of members of the family [Sne, p.73, (3)]
that touch the surface S.
Properties of solutions of ordinary differential equations (dx/dt = V(t,x), where t
represents time and x represents a point in R^{n})
Uniqueness theorem [Bir, p.142, Theorem 1].
Continuity theorem [Bir, p.143, Corollary (b)].
The global existence theorem for the case that X satisfies the Lipschitz condition globally [Bir, p.152, Theorem 6].
Remark. [Arn1, p.269, Fig.
215; p.274, Fig.221 & Fig.222] give a geometrical interpretation of Picard's
successive approximation. Note that [Arn1, p.269, Fig. 215] provides more
information than [Bir, p.153, Fig. 6.1].
The global existence theorem for linear systems [Bir, p.156, Theorem 7
& Corollary 1].
Remark. In the proof of [Spi, vol.1, p. 228, Proposition 17], Spivak made
one mistake in [Spi, vol.1, p. 228, l.6] and a second in [Spi,
vol.1, p. 229, l.5]. Although the proof of [Arn1,
§27.2] corrects Spivak's mistakes,
Arnold's local approach reduces the solution's effectiveness in [Arn1,
p.101, l.21].
Linear equations with constant coefficients [Arn1, p.164, the
fundamental theorem].
The local existence theorem for the case that V satisfies the Lipschitz condition locally [Bir, p.157, Theorem 8].
Remark. The proof given in [Arn1, chap.4, §31.8,
p.276, Corollary] is well organized: it utilizes mathematical structures neatly.
The local existence theorem for the case that
V is continuous locally [Bir, p.166, Theorem
13].
Smoothness of solutions
If V(t,x) is of class p with
respect to t [Kre,
p.21, l.11-l.19], then any solution g is of class p+1 with respect to t [Pet66, p.54, Theorem].
If V(t,x) is analytic with respect to t, then any solution g is analytic
with respect to t [Bir, p.160, Theorem 9 and its corollaries]. If V(t,x) is
analytic on 0<|t|<a,
then any solution can be represented as g(t) = Z(t)t^{R},
where Z(t) is single-valued and analytic on 0<|t|<a
[Har, p.70, Theorem 10.1].
Remark. The statement given in [Guo, p.136, l.4-l.3] is derived from b.
Continuous dependence on V (local: [Bir, p.147, Corollary])
Differentiability with respect to the parameters (local: [Pon, p.173,
Theorem 14]; global: [Pon, p.197, Theorem 17])
Continuous dependence and differentiability of a solution g with respect
to the initial points x (local:
(i). (VÎC^{1}
in a neighborhood of (t_{0},
x_{0}))
Þ gÎC_{x}^{1} [Pon, pp.179-180, Theorem 15].
(ii). (VÎC^{r}
in a neighborhood of (t_{0},
x_{0}))
Þ (gÎC^{r }
in t and x jointly), where r = 1, 2,
… and ¥ [Arn1,p.285,
l.12]; global:
[Pon, p.198, Theorem 18]).
Remark 1. (ii) is a corollary of (i). See [Arn1, p.285, l.15-l.16] or [Pon,
p.177, l.-20-l.-1]. For
the proof of (i), read [Pon, p.179, l.-13-p.181,
l.11] instead of the confusing proof given in [Arn1, p.285, l.16-p.287, l.-1]
[1].
Remark 2. [Pon,
p.173, (15)] motivates us to define the system of equations of variations
given in [Arn1, p.279, (2)].
Remark 3. [Arn1, p.279, Fig.226] gives a geometrical interpretation of the system of equations of variations.
Solutions represented by phase flows [Arn1, chap.1, §4]
[Spi, vol. 1, p.203, Theorem 5] motivates us to formulate a
clean definition of the phase velocity vector of a phase flow given in
[Arn1, p.63, Definition]. However, the
notations given in [Spi, vol. 1, p.202] are very confusing, so it is difficult to
recognize the conclusion given in [Arn1, p.63, Definition]. Furthermore,
[Arn1, chap.1, §4] provides concrete
examples of phase flows.
The rectification theorem (statement: [Arn1, p.89, Theorem 1]; proof: [Arn1, pp.283-284])
Remark 1. The proof given in [Arn1, pp.283-284] emphasizes the insight. The
proof given in [Spi, vol. 1, pp.205-207] emphasizes calculations. The proof
given in [Bir, p.165, Theorem 1] emphasizes the shortcut to the conclusion.
Consequently, if one reads the third proof without reading the first one, it
would be difficult to understand the proof and catch the essence of the theorem.
Remark 2. Arnold uses the rectification theorem to create a trivial example of
differential equations. Then he uses this trivial example to discuss basic theorems in differential equations.
See [Arn1, chap. 2]. In my opinion, for the basic theorems in differential
equations, we should discard [Arn1, chap. 2] and retain only [Arn1, chap. 4]. This is because Arnold fails to discriminate
among the
effectiveness of various existences. In addition, his construction of a rectifying diffeomorphism
is based on the assumption that the differential equation is solved. Thus, he
fails to discuss all the effective methods of solving differential
equations. In addition, each theorem in differential equation arises from the
need to solve mathematical or physical problems and has its own typical
example to represent its significance. When we have not yet encountered
that problem or example, the discussion of the theorem using a trivial example simply
diminishes or obscures its meaning.
Adjoint equations: Green's formula and the Lagrange identity [Har, p.66,
l.-12-p.67, l.5] [1].
Remark. The proof given in [Har, p.66, l.-6-l.-3]
is simpler than the proofs given in [Cod, p.86, l.8-l.-6].
Stability for plane autonomous systems: [Inc1,
§6.6] ® (linear [Bir, chap. 3,
§5]) ®
(nonlinear [Bir, chap. 5, §7-§9]).
Remark. In order to see the big picture, we must simplify the setting, stress
the main point, use fewer definitions, and ignore minor differences. [Inc1,
§6.6] does not formally define stability,
but it provides a typical and an intuitive case of stability by detaching the
concept of stability from differential equations. [Inc1,
§6.6] also shows us how to produce
stability. In
contrast, [Bir, p.121, Definition] tries to distinguish the nuances among
various stabilities, but ends up only obscuring the big picture of the concept.¬
The order of any determinate system of linear equations with constant coefficients is equal to
the order of its characteristic equation [Inc1, p.150, l.24-l.26].
The solutions of ordinary differential equations that cannot satisfy the
given boundary conditions
Example. [Sag, p.65, l.8].
Formal solutions vs. actual solutions
Singularities of the first kind: The formal solution converges to an actual solution
[Cod, p.117, Theorem 3.1].
Remark 1. z_{0} is an "at most singularity of the first kind" if and only if z_{0} is a regular singularity [Cod, p.124, Theorem 5.1; p.125, Theorem 5.2].
Remark 2. [Was, p.28, l.10-l.-1] shows that the
regular singularity of a single differential equation of nth order is a
special case of the regular singularity of a system of linear differential
equations.
Remark 3. Formal solutions must be inclusive. For formal solutions, the formal
power series given in [Har, p.78, Theorem 11.3] is inadequate for representing
typical solutions. Consequently, we should consider formal logarithmic sums [Cod, p.116,
l.13]. If we examine the proofs of [Cod, p.119, Theorem 4.1] and [Cod, p.121,
Theorem 4.2], it is unnecessary to use [Cod, p.117, Theorem 3.1] to prove that
the above two theorems are valid if we just consider F
as formal solutions rather than actual solutions. With this prerequisite
knowledge, then for the proof of [Cod,
p.117, Theorem 3.1], it will suffice to consider only these reduced formal
solutions given in [Cod, p.121, Theorem 4.2].
Singularities of the second kind: There exists an actual solution such
that it has the formal
solution as its asymptotic expansion [Cod, p.160, Theorem 4.1; chap. 5, sec. 6].
Remark 1. [Col, p.255, l.3-p.256, l.10] provides a complete proof of the
indicial equation.
Remark 2.
[Cod, p.152, l.-15-l.-6]
The essence of [Cod, p.160, Theorem 4.1] lies in [Cod, p.155, Lemma 4.2]. Even
though the
formal solution can be divergent [Cod, p.139, l.11], we may construct an actual solution in the integral form using the variation of constants
[Cod, p.155, l.18-l.21]. Then the error between the actual solution and the
partial sum of the expansion can be estimated by the successive approximations [Cod, p.156, (4.25)].
Remark
3. The second line of [Cod, p.156,
(4.32)] should have been "R(q_{i}(t)-q_{l}(t))
is bounded above" rather than "R(q_{i}(t)-q_{l}(t))
is nonincreasing". Levinson has made a mistake here.
The domain of the solution of a system of ordinary differential equations is closed and connected [Har, p.15, Theorem 4.1].
Peano's existence theorem [Har, p.10, Theorem 2.1]
The definition of y_{e}
given in [Har, p.10, l.-8-l.-1] is clear,
while the definition of x^{n} given in [Bir,
p.166, l.-6-l.-4] is
not. Clarity should be the top priority of a textbook.
When we propose a new theorem that is similar to a theorem
that already exist in a theory, we must describe their sameness as completely as possible and then identify
their crucial differences. The comparison helps us identify the unique feature
of the new theorem. The comparison of the pair [Bir, p.152, Theorem 6;
p.166, Theorem 13] is less complete than that of the pair [Har, p.8, Theorem
1.1; p.10, Theorem 2.1].
Regular singular points
Definitions
(Theoretical definitions) For a linear system, the definition of regular singular
points is given by [Har, p.73, l.7]. A differential equation of dth order [Har,
p.84, (12.1)] has a regular singular point at t=0 if [Har, p.84, (12.3)] has a
regular singular point at t=0. The equation and the system are related by [Har,
p.84, (12.2)].
(Practical definitions) [Har, p.84,
(12.1)] has a regular singular point at t=0
Û The solutions of [Har, p.84, (12.1)] are linear
combinations of t^{l}(log t)^{k}a(t),
where a(t) is analytic for
|t|<a
[1] Û [Har, p.84, (12.3)] has a simple singularity
[Har, p.84, l.13-l.16] at t=0 [Har, p.85, Theorem 12.1].
Remark 1. For singularities at infinity, see [Cod, p.128, Theorem 6.1 & Theorem 6.2].
Remark 2. It would be difficult for one to see the big picture if one studies
only the definition
given in [Bir, p.233, Definition] or [Guo, p.56, l.-8].
The Frobenius Method
When the roots of the indicial equation do not differ by an integer [Bir, p.234, Theorem 5; p.240, Theorem 7]
When the roots of the indicial equation differ by an integer [Bir, p.242, Theorem 8].
Remark 1. Coddington's approach [Cod61, p.158, l.-9-l.-7]
must deal with the minus signs [Cod61, p.146, l.-6-p.147,
l.3] that open Pandora's box. The way out of this mess of entanglement is to treat z^{r}
as exp [r (log z)].¬
Remark 2. [Wat1, §10.32] gives an
alternative method to solve a differential equation with a regular singular
point. Since the equality in [Wat1, p.201, l.4] links to the difference of the
exponents, the solution given in [Wat1, p.201, l.3] is more informative than
the solution given in [Bir, p.242, l.-10].
Fuchsian equations
[Cod, p.129, l.8] The principle of reducing differential equations to a
simple form: A general linear fractional
transformation transforms one set of regular singular points into another set
of regular singular points, and the
indicial equations of the transformed DE coincide with those of the original
DF at corresponding points [Bir, p.251, l.-18-l.-15].
First-order [Bir, p.251, l.6-l.10].
Second-order Fushian equations with two regular singular points can
be reduced to the Euler DE [Bir, p.251, l.-7].
Remark. The solutions given in [Wat1, p.208, l.-14-l.-11]
are derived from [Bir, p.251, l.-8; Col, p.110,
l.12; p.111, l.12].
Second-order Fushian equations with three regular singular points
can be reduced to the hypergeometric DE [Bir, p.253, Corollary; Guo,
§2.9; Cod, p.132, (7.8)]. Example: the associated Legendre
equations [Guo, p.71, (15)].
Remark 1. In their attempt to find the required transformation, [Bir, p.253, l.-15-l.-5] provides the method without
giving the final answer, while [Guo,
p.69, (6); p.70, (11)] provides the final answer without explanation. It is
simpler to derive [Guo, p.70, 12] by means of [Bir, p.251, l.-18-l.-15]
rather than from the general case [Guo, p.70, (11)]. Remark 2. The proof of the Riemann identity
given in [Bir, p.253, (41)] is simpler than that given in [Guo, p.68, (2)]
because the latter considers the general case, while the former considers the
special case, the Riemann DE [Bir, p.252, (40)]. Since the indicial equation
of the general case remains the same as the indicial equation of the special case by a linear fractional transformation,
it is unnecessary to prove the Riemann identity in the general case [Bir, p.251, l.-18-l.-15].
Second-order Fushian equations with five regular singular points [Guo,
§2.8].
Second-order Fushian equations with k regular singular points [Guo,
§2.7] [1].
Remark. The indicial equation at infinity given in [Bir, p.250, (39)] is simpler than
that given in [Guo, p.65, (6)] because the former considers only one regular
singular point z = ¥, while the latter considers the
combination of
the regular singular point z = ¥ and
other k regular singular points.
(Formal solutions near a simple singularity [Har, p.73, l.20]) [Cod, p.117, Theorem 3.1]
(Power series solutions near an ordinary point) [Cod61, p.129, Theorem 12]
Suppose we want to solve a differential equation such as [Edw, p.528, (1)].
Then which of above methods shall we use? If our goal is to show the existence of
a solution, then method A is adequate. Suppose we want to use an effective
method to find the solution. For method A, we must compute integrals, while
for method B and method C all we have to do is compare corresponding
coefficients. Therefore, for computations the latter two methods are simpler.
In addition, for method C we need consider only analytic (ordinary) points,
while for method B we have to expand our consideration to simple
singularities. Furthermore, method B shows that the solution is in the form of
formal logarithmic sums, while method C shows that the solution is in the form
of power series. The latter is more specific. In summary, in order to find
effective solutions, our computations must be simple; our consideration should
be focused on as small a scope as possible; the form of solutions must be specific.¬
In [Wat1, p.478, l.10], the solution of the differential equation given in [Wat1, p.478, l.12] is expressed in terms of the Theta functions.
In [Gon1, p.421, (5.17-1)], the solution of the differential equation given in
[Gon1, p.425, (5.17-7)] is expressed in terms of Tan. In the above two cases,
we use more familiar functions such as Tan and the Theta functions as our
tools to make the solution sn tangible and constructive. The attempt to link
the solution to the functions with which we are familiar leads to many new questions. For example, how do we find t if k is
given? The answer is given in [Wat1, §21.7,
§21.71 & §21.711].
In contrast, sn u in [Guo, p.530, (1)] is defined by the differential equation
given in [Guo, p.931, l.-7]. The solution is less
tangible, but we try to find its meaning [Guo, §10.2]
and establish its properties such as the addition formula without depending on
other familiar functions [Guo,
§10.4].
Remark. Treating differential equations as a isolated subject is a superficial approach. Solving a differential equation involves solving a group of differential equations. In order to fully understand the solution of a differential equation, we must study it from multiple
perspectives.
The introduction of generating functions facilitates the operation of solutions,
e.g., establishing recurrence relations [1].
As another example, the use of generating function simplifies the proof of orthonormality relations for a certain measure (compare [Guo, p.324, l.-2-p.325-l.7] with [Wat1, p.350, l.-11-p.351, l.12]).
Furthermore, we use Laurent's theorem [Wat1,
§5.6] to produce generating functions in
most cases [Guo, p.323, (13); Wat1, p.355, l.16].
Laplace's equation
There are at most 2n+1 independent solutions of degree n [Wat1, p.389, l.12-l.17].
g_{m}(X,Y,Z) is an even function of Y, while h_{m}(X,Y,Z) is an odd function of Y.
Proof. Z + iX cos u + i(-Y) sin u = Z + iX cos u+ iY sin (-u).
g_{m}(0£m£n) and h_{m}(1£m£n) are independent solutions of degree n [Wat1, p.389, l.-11-l.-6]
Remark 1. We may use the same method prove the following theorem about PDE:
Let {f_{n}(u)} be an orthonormal basis and V_{0}(x,u) =
S_{n=1}^{¥} g_{n}(x) f_{n}(u) be a particular solution of P(D)V(x)=0.
Then P(D) g_{n}(x) = 0 (n = 1,2,3,…).
Remark 2. By mathematical induction, cos ^{h }q sin ^{k} q = S _{m=0}^{h+k} (A_{m} cos mq + B_{m} sin mq).
Solving linear equations of the second order using continued fractions [Inc1, §7.5]
Remark. The statement given in [Inc1, p.179, l.-17-l.-10] can be proved by [Perr, p.292, Satz 46D; p.24, Satz 2].
The proof of [Perr, p.292, Satz 46D] uses [Perr, p.285, Satz 41]. The proof of
[Perr, p.285, Satz 41] uses [Perr, pp.285-286, Satz 42; p.276, Satz 38]. The proof of [Perr,
pp.285-286, Satz 42] uses [Perr, p.262, Satz 30] and [Perr, pp.280-281, Satz
40].
Remark. [Inc1, §7.501] provides a constructive method of solving the ODE given in [Inc1, p.179, l.-7].