The main topic of this article is to discuss the hyperbolic equation. Now we give some ideal problems which can be represented as a hyperbolic equation in the math-ematical model but we just talk about the Telegrapher’s equation in detail.
Telegrapher’s Equation
Suppose that a transmission lines has a voltage V (x, t) across them and a current I(x, t) at position x and time t. One of the transmission line contains a resistance (R) and a inductance (L). And the two transmission line connected with a capac-itance (C) and a leakage resistance (G). Assume the energy is conserved. Now, if the current passes through an inductor, the voltage across the inductor is directly proportional to the time rate of change V = LdI
dt. By Kirchhoff’s current law and
Kirchhoff’s voltage law, we could get
I(x + ∆x, t) = I(x, t) − GV (x, t)∆x − C∂V (x, t)
∂t ∆x, (1.4)
V (x + ∆x, t) = V (x, t) − RI(x, t)∆x − L∂I(x, t)
∂t ∆x. (1.5)
Let ∆x → 0, then (1.4), (1.5) will become
Ix(x, t) = −GV (x, t) − CVt(x, t), (1.6) Vx(x, t) = −RI(x, t) − LIt(x, t). (1.7) Since we assume that there is no energy lost, we get R = 0 and G = 0. Hence, (1.6) and (1.9) can be deduced as
Ix(x, t) = −CVt(x, t), (1.8)
Vx(x, t) = −LIt(x, t). (1.9)
After differentiating (1.8) and (1.9) with t and x, respectively, we have
Ixt(x, t) = −CVtt(x, t), (1.10) Vxx(x, t) = −LItx(x, t). (1.11) From (1.10) and (1.11), we could get
Vtt(x, t) − k2Vxx(x, t) = 0 where k2 = 1
LC. (1.12)
Similarly, differentiating (1.8) and (1.9) for x and t, respectively, we have
Ixx(x, t) = −CVtx(x, t), (1.13) Vxt(x, t) = −LItt(x, t). (1.14) And we can derive
Itt(x, t) − k2Ixx(x, t) = 0 where k2 = 1
LC. (1.15)
from the equation (1.13) and equation (1.14). The (1.12) and (1.15) shows that the voltage and current will satisfy the wave equation under the ideal ( no energy lost) condition.
Next, we introduce the vibrating string. Its mathematical model is also a hy-perbolic equation under some ideal assumptions and Newton’s second law. The mathematical model can be expressed as the following system:
utt− c2uxx = 0, for 0 < x < l, (1.16)
u(x, 0) = f (x), (1.17)
ut(x, 0) = g(x), (1.18)
u(0, t) = u(l, t) = 0. (1.19)
(1.17) and (1.18) are the initial conditions. They limit the shape of position and ve-locity for the solution at t = 0. (1.19) is called the boundary condition. It represent the states at x = 0 and x = l for all t. The detail of the derivation can read the reference [4].
There are many other problems can be represented as hyperbolic equation like Maxwell’s equation and so on. This shows that the linear hyperbolic equation can be applied in our life. But in most of time, the real problems in our life are correspond-ing to the nonlinear equation. Thus, we will discuss a nonlinear equation in the whole following contents. It is called Sine-Gordon equation, utt− uxx+ sin(u) = 0.
On the other hand, it could be transferred to be a ordinary differential equation, uθθ+ sin(u) = 0, by letting θ = kx − ωt with ω2− k2 = 1.
Consider a pendulum consisting of a light rod of length l to which is attached a ball of mass m. The position of the mass at time t is described by u(θ). By Newton’s law, the mathematical model of the motion of the pendulum could be represented as
mld2u
dθ2 = −bldu
dθ − mgsin(u), (1.20)
where g is the gravitational acceleration and b represents the coefficient of friction with b > 0 [8] [9] [10]. Assume that the pendulum is frictionless (b = 0) with m = 1 and l = g. Then the equation (1.20) will become uθθ+ sin(u) = 0. This implies that the equation uθθ+ sin(u) = 0 could be regarded as the motion of a ideal pendulum with m = 1 and l = g. The whole chapter 3 will discuss the problem in detail.
Chapter 2
Elliptic functions
2.1 Definitions and Properties
Before introducing the elliptic functions, we introduce some definitions. We just talk about some important and interesting parts of the elliptic function. The fol-lowing contents referred to [3] and you could get more information about the elliptic functions from it.
Definition 2. The point z0 is called the singularity ( singular point ) of f (z) if f (z) is not analytic at z = z0. If z0 is a singularity and there exists a neighborhood N(z0) of z0 such that the function f (z) is analytic in N(z0)\{z0}, then z0 is isolated.
Moreover, if there is an analytic function g : N(z0) → C such that g(z) = f (z) on N(z0)\{z0}, the point z0 is called a removable singularity.
Examples
(i) f (z) = ln(z) , z = 0 is a non-isolated singularity.
(ii) f (z) = 1
z , z = 0 is an isolated singularity, but it is not a removable singularity.
(iii) f (z) = z2− z
z is analytic except z =0. z = 0 is a removable singularity since we can define g(x) =
z − 1, if z = 0 z2− z
z , otherwise. Then g(z) = g(x) on NR(0)\{z0}.
Definition 3. The pole z0 of the function f (z) satisfies:
1. z0 is a singularity.
2. z0 is isolated.
3. ∃ min k ∈ N such that (z − z0)kf (z) is analytic at z0.
After knowing the previous definitions, we can define elliptic function now:
Definition 4. Assume that f is a doubly-periodic function with periods 2ω1 and 2ω2.(That is, f (z + 2ω1) = f (z + 2ω2) = f (z).) And f is called an elliptic function if it is analytic (except poles) and has no singularities other than poles in the finite part of the plane.
Remark 1.
a. The constants ω1, ω2 ∈ F (F = R or C) and ω1
ω2 is not purely real number.
b. If there is no ω inside the parallelograms such that f (z + ω) = f (z), ∀z, the parallelogram constructed by z, z + 2ω1, z + 2ω2, z + 2(ω1+ ω2) is called a fundamental period-parallelogram for an elliptic function with period 2ω1, 2ω2. The points z, z + 2ω1, z + 2ω2, z + 2(ω1 + ω2), ... will have the same value after transferring by f since 2ω1 and 2ω2 are periods. And any pair of such points are said to be ”congruent” to one another. The congruence of two points z, z0 is denoted by z ≡ z0 ( mod 2ω1, 2ω2). And the set of poles of an elliptic function in any given cell is called an irreducible set.
Some simple properties of elliptic functions
1. The number of poles of an elliptic function in any cell is finite.
2. The number of zeros of an elliptic function in any cell is finite.
3. The sum of the residue of an elliptic function, f (z), at its poles in any cell is zero.
4. Liouville’s Theorem:
An elliptic function ,f (z), with no poles in a cell is merely a constant.
The order of an elliptic function
f (z) is an elliptic function and the number of the roots of the equation f (z) = c (where c is any constant) which lies in any cell depends only on f (z). Then the number is called the order of the elliptic function.
Remark 2.
a. The order of f (z) is the number of poles in the cell.
b. The order of an elliptic function is = 2.
c. The simplest elliptic function could be divided into two classes. One is the elliptic functions which have a single irreducible double pole with residue = 0.
The other is the elliptic functions which have two single poles and the sum of their residues is 0.
After knowing some basic properties of elliptic function. We will introduce the Weierstrass elliptic function which belongs to the former class and the Jacobian elliptic functions in the following sections.