It is convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition 13.1.2.2 in the parametric form 13.1.2-2.. Consider the Cauchy problem f
Trang 113.1.2 Cauchy Problem Existence and Uniqueness Theorem
13.1.2-1 Cauchy problem
Consider two formulations of the Cauchy problem
1◦ Generalized Cauchy problem Find a solution w = w(x, y) of equation (13.1.1.1)
satisfying the initial conditions
x = h1(ξ), y = h2(ξ), w = h3(ξ), (13.1.2.1)
where ξ is a parameter (α≤ξ≤β ) and the h k (ξ) are given functions.
Geometric interpretation: find an integral surface of equation (13.1.1.1) passing through the line defined parametrically by equation (13.1.2.1)
2◦ Classical Cauchy problem Find a solution w = w(x, y) of equation (13.1.1.1) satisfying
the initial condition
where ϕ(y) is a given function.
It is convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition (13.1.2.2) in the parametric form
13.1.2-2 Procedure of solving the Cauchy problem
The procedure of solving the Cauchy problem (13.1.1.1), (13.1.2.1) involves several steps First, two independent integrals (13.1.1.2) of the characteristic system (13.1.1.3) are
deter-mined Then, to find the constants of integration C1and C2, the initial data (13.1.2.1) must
be substituted into the integrals (13.1.1.2) to obtain
u1 h1(ξ), h2(ξ), h3(ξ)
= C1, u2 h1(ξ), h2(ξ), h3(ξ)
= C2 (13.1.2.4)
Eliminating C1and C2from (13.1.1.2) and (13.1.2.4) yields
u1(x, y, w) = u1 h1(ξ), h2(ξ), h3(ξ)
,
u2(x, y, w) = u2 h1(ξ), h2(ξ), h3(ξ)
Formulas (13.1.2.5) are a parametric form of the solution of the Cauchy problem (13.1.1.1),
(13.1.2.1) In some cases, one may succeed in eliminating the parameter ξ from relations
(13.1.2.5), thus obtaining the solution in an explicit form
Example 1 Consider the Cauchy problem for linear equation
∂w
subjected to the initial condition (13.1.2.2).
The corresponding characteristic system for equation (13.1.2.6),
dx
1 =
dy
bw, has two independent integrals
y – ax = C1, we–bx = C2 (13 1 2 7 ) Represent the initial condition (13.1.2.2) in parametric form (13.1.2.3) and then substitute the data (13.1.2.3)
into the integrals (13.1.2.7) As a result, for the constants of integration we obtain C1= ξ and C2= ϕ(ξ)
Sub-stituting these expressions into (13.1.2.7), we arrive at the solution of the Cauchy problem (13.1.2.6), (13.1.2.2)
in parametric form:
y – ax = ξ, we–bx = ϕ(ξ).
By eliminating the parameter ξ from these relations, we obtain the solution of the Cauchy problem (13.1.2.6),
(13.1.2.2) in explicit form:
w = e bx ϕ (y – ax).
Trang 213.1.2-3 Existence and uniqueness theorem.
LetG0be a domain in the xy-plane and let G be a cylindrical domain of the xyw-space
obtained fromG0by adding the coordinate w, with the condition|w|< A1being satisfied Let
the coefficients f , g, and h of equation (13.1.1.1) be continuously differentiable functions
of x, y, and w in G and let x = h1(ξ), y = h2(ξ), and w = h3(ξ) be continuously differentiable
functions of ξ for|ξ| < A2defining a curve C in G with a simple projection C0 ontoG0
Suppose that (h 1)2+ (h 2)2 ≠ 0(the prime stands for the derivative with respect to ξ) and
f h
2 – gh 1 ≠ 0 on C Then there exists a subdomain G0 ⊂ G0 containing C0 where
there exists a continuously differentiable function w = w(x, y) satisfying the differential
equation (13.1.1.1) inG0and the initial condition (13.1.2.1) on C0 This function is unique.
It is important to note that this theorem has a local character, i.e., the existence of a
solution is guaranteed in some “sufficiently narrow,” unknown neighborhood of the line C
(see the remark at the end of Example 2)
Example 2 Consider the Cauchy problem for Hopf’s equation
∂w
subject to the initial condition (13.1.2.2).
First, we rewrite the initial condition (13.1.2.2) in the parametric form (13.1.2.3) Solving the characteristic system
dx
1 =
dy
we find two independent integrals,
Using the initial conditions (13.1.2.3), we find that C1= ϕ(ξ) and C2= ξ Substituting these expressions
into (13.1.2.10) yields the solution of the Cauchy problem (13.1.2.8), (13.1.2.2) in the parametric form
The characteristics (13.1.2.12) are straight lines in the xy-plane with slope ϕ(ξ) that intersect the y-axis at the points ξ On each characteristic, the function w has the same value equal to ϕ(ξ) (generally, w takes different
values on different characteristics).
For ϕ (ξ) >0 , different characteristics do not intersect and, hence, formulas (13.1.2.11) and (13.1.2.12) define a unique solution As an example, we consider the initial profile
ϕ (ξ) =
⎧
⎨
⎩
w1 for ξ≤ 0 ,
w2ξ2+ εw1
ξ2+ ε for ξ >0 , (13.1 2 13 )
where w1 < w2 and ε > 0 Formulas (13.1.2.11)–(13.1.2.13) give a unique smooth solution in the entire
half-plane x >0 In the domain filled by the characteristics y = ξ + w1x (for ξ≤ 0 ), the solution is constant, i.e.,
For ξ >0 , the solution is determined by relations (13.1.2.11)–(13.1.2.13).
Let us look how this solution is transformed in the limit case ε →0 , which corresponds to the piecewise-continuous initial profile
ϕ (ξ) =
w1 for ξ≤ 0 ,
w2 for ξ >0 , where w1< w2. (13.1 2 15 )
We further assume that ξ >0[for ξ≤ 0, formula (13.1.2.14) is valid] If ξ = const≠ 0and ε →0 , it follows
from (13.1.2.13) that ϕ(ξ) = w2 Hence, in the domain filled by the characteristics y = ξ + w2x (for ξ >0 ), the solution is constant, i.e., we have
w = w2 for y/x≥w2 (as ε →0 ) (13 1 2 16 )
For ξ →0, the function ϕ can assume any value between w1 and w2 depending on the ratio of the small
parameters ε and ξ; the first term on the right-hand side of equation (13.1.2.12) can be neglected As a result,
Trang 3we find from equations (13.1.2.11) and (13.1.2.12) that the solution has the following asymptotic behavior in explicit form:
w = y/x for w1≤y/x≤w2 (as ε →0 ) (13 1 2 17 )
By combining relations (13.1.2.14), (13.1.2.16), and (13.1.2.17) together, we obtain the solution of the Cauchy problem for equation (13.1.2.8) subject to the initial conditions (13.1.2.15) in the form
w (x, y) =
w1 for y≤w1x,
y/x for w1x≤y≤w2x,
w2 for y≥w2x.
(13 1 2 18 )
Figure 13.1 shows characteristics of equation (13.1.2.8) that satisfy condition (13.1.2.15) with w1 = 12 and
w2= 2 This figure also depicts the dependence of w on y (for x = x0 = 1 ) In applications, such a solution is referred to as a centered rarefaction wave (see also Subsection 13.1.3).
x
y w
y
w2
w1
y2
y1
x0
0
0
y2=w x2 0
y1=w x1 0
Figure 13.1 Characteristics of the Cauchy problem (13.1.2.8), (13.1.2.2) with the initial profile (13.1.2.15)
and the dependence of the unknown w on the coordinate y for w1 = 12, w2 = 2, and x0 = 1
Remark. If there is an interval where ϕ (ξ) < 0 , then the characteristics intersect in some domain According to equation (13.1.2.11), at the point of intersection of two characteristics defined by two distinct
values ξ1and ξ2of the parameter, the function w takes two distinct values equal to ϕ(ξ1) and ϕ(ξ2 ), respectively Therefore, the solution is not unique in the domain of intersecting characteristics This example illustrates the local character of the existence and uniqueness theorem These issues are discussed in Subsections 13.1.3 and 13.1.4 in more detail.
13.1.3 Qualitative Features and Discontinuous Solutions of
Quasilinear Equations
13.1.3-1 Model equation of gas dynamics
Consider a quasilinear equation of the special form*
∂w
∂x + f (w) ∂w
∂y =0, (13.1.3.1)
* Equations of the general form are discussed in Subsection 13.1.4.
Trang 4which represents a conservation law of mass (or another quantity) and is often encountered
in continuum mechanics, gas dynamics, hydrodynamics, wave theory, acoustics, multiphase flows, and chemical engineering This equation is a model for numerous processes of mass transfer: sorption and chromatography, two-phase flows in porous media, flow of water
in river, street traffic development, flow of liquid films along inclined surfaces, etc The
independent variables x and y in equation (13.1.3.1) usually play the role of time and spatial coordinate, respectively, w is the density of the quantity being transferred, and f (w) is the rate of w.
13.1.3-2 Solution of the Cauchy problem Rarefaction wave Wave “overturn.”
1◦ The solution w = w(x, y) of the Cauchy problem for equation (13.1.3.1) subject to the
initial condition
w = ϕ(y) at x=0 (–∞ < y < ∞) (13.1.3.2) can be represented in the parametric form
whereF(ξ) = f ϕ (ξ)
Consider the characteristics y = ξ + F(ξ)x in the yx-plane for various values of the
parameter ξ These are straight lines with slope F(ξ) Along each of these lines, the
unknown function is constant, w = ϕ(ξ) In the special case f = a = const, the equation
in question is linear; solution (13.1.3.3) can be written explicitly as w = ϕ(y – ax), thus representing a traveling wave with a fixed profile The dependence of f on w leads to a
typical nonlinear effect: distortion of the profile of the traveling wave
We further consider the domain x≥ 0and assume* that f >0 for w >0and f w >0
In this case, the greater values of w propagate faster than the smaller values If the initial profile satisfies the condition ϕ (y) >0for all y, then the characteristics in the yx-plane that come from the y-axis inside the domain x >0are divergent lines, and hence there exists a
unique solution for all x >0 In physics, such solutions are referred to as rarefaction waves
Example 1 Figures 13.2 and 13.3 illustrate characteristics and the evolution of a rarefaction wave for
Hopf’s equation [for f (w) = w in (13.1.3.1)] with the initial profile
ϕ (y) = 4
π arctan(y –2 ) + 2 (13 1 3 4 )
It is apparent that the solution is smooth for all x >0
4
x
Figure 13.2 Characteristics for the Hopf’s equation (13.1.2.8) with the initial profile (13.1.3.4).
* By the change x = – 2x the consideration of the domain x≤ 0 can be reduced to that of the domain2x≥ 0
The case f <0can be reduced to the case f >0by the change y = – 2y.
Trang 50 4 8
4
x= 0
y w
Figure 13.3 The evolution of a rarefaction wave for the Hopf’s equation (13.1.2.8) with the initial profile
(13.1.3.4).
2◦ Let us now look at what happens if ϕ (y) <0 on some interval of the y-axis Let y1 and y2be points of this interval such that y1< y2 Then f (y1) > f (y2) It follows from the
first relation in (13.1.3.3) that the characteristics issuing from the points y1and y2intersect
at the “time instant”
x ∗= f (w y2– y1
1) – f (w2), where w1= ϕ(y1), w2= ϕ(y2).
Since w has different values on these characteristics, the solution cannot be continuously extended to x > x ∗ If ϕ (y) <0on a bounded interval, then there exists xmin= min
y1 ,y 2x ∗such that the characteristics intersect in the domain x > xmin(see Fig 13.4) Therefore, the front
part of the wave where its profile is a decreasing function of y will “overturn” with time The time xminwhen the overturning begins is defined by
xmin= – 1
F (ξ0), where ξ0is determined by the condition|F (ξ0)|= max|F (ξ)|forF (ξ) <0, and the wave
is also said to break A formal extension of the solution to the domain x > xmin makes
this solution nonunique The boundary of the uniqueness domain in the yx-plane is the
envelope of the characteristics This boundary can be represented in parametric form as
y = ξ + F(ξ)x, 0=1+F (ξ)x.
0
2
4
x
x = 5
x = 0
Figure 13.4 Characteristics for the Hopf’s equation (13.1.2.8) with the initial profile (13.1.3.5).
Trang 62 4 6
2.0 1.5
w
y
Figure 13.5 The evolution of a solitary wave for the Hopf’s equation (13.1.2.8) with the initial profile (13.1.3.5) Example 2 Figure 13.5 illustrates the evolution of a solitary wave with the initial profile
ϕ (y) = cosh–2(y –2 ) + 1 (13 1 3 5 )
for equation (13.1.3.1) with f (w) = w It is apparent that for x > xmin, where xmin = 34√
3 ≈ 1 3 , the wave
“overturns” (the wave profile becomes triple-valued).
13.1.3-3 Shock waves Jump conditions
In most applications where the equation under consideration is encountered, the unknown
function w(x, y) is the density of a medium and must be unique for its nature In these
cases, one has to deal with a generalized (nonsmooth) solution describing a step-shaped shock wave rather than a continuous smooth solution The many-valued part of the wave profile is replaced by an appropriate discontinuity, as shown in Fig 13.6 It should be
emphasized that a discontinuity can occur for arbitrarily smooth functions f (w) and ϕ(y)
entering equation (13.1.3.1) and the initial condition (13.1.3.2)
w
y
s x( )
Figure 13.6 Replacement of the many-valued part of the wave profile by a discontinuity that cuts off domains
with equal areas (shaded) from the profile of a breaking wave.
In what follows, we assume that w(x, y) experiences a jump discontinuity at the line
y = s(x) in the yx-plane On both sides of the discontinuity the function w(x, y) is smooth and
single-valued; as before, it is described by equations (13.1.3.3) The speed of propagation
of the discontinuity, V , is expressed as V = s (the prime stands for the derivative) and must satisfy the condition
V = F (w2) – F (w1)
f (w) dw, (13.1.3.6)
where the subscript 1 refers to values before the discontinuity and the subscript 2 to those after the discontinuity In applications, relation (13.1.3.6), expressing a conservation law at discontinuity, is conventionally referred to as the Rankine–Hugoniot jump condition (this condition is derived below in Paragraph 13.1.3-4)
Trang 7The continuous wave “overturns” (breaks), thus resulting in a discontinuity if and only
if the propagation velocity f (w) decreases as y increases, i.e., the inequalities
f (w2) < V < f (w1) (13.1.3.7) are satisfied Conditions (13.1.3.7) have the geometric meaning that the characteristics
issuing from the x-axis (these characteristics “carry” information about the initial data) must
intersect the line of discontinuity (see Fig 13.7) In this case, the discontinuous solution
is stable with respect to small perturbations of the initial profile (i.e., the corresponding solution varies only slightly)
y
x
line of discontinuity
characteristics characteristics
Figure 13.7 Mutual arrangement of characteristics and lines of discontinuity in the case of a stable shock
wave.
The position of the point of discontinuity in the yw-plane may be determined
geometri-cally by following Whitham’s rule: the discontinuity must cut off domains with equal areas from the overturning wave profile (these domains are shaded in Fig 13.6) Mathematically, the position of the point of discontinuity can be determined from the equations
s (x) = ξ1+F1x,
s (x) = ξ2+F2x,
w2F2– w1F1= F (w2) – F (w1) + F2–F1
ξ2– ξ1
ξ2
ξ1
w dξ
(13.1.3.8)
Here, w and F are defined as functions of ξ by w = ϕ(ξ) and F = f(w), the function F (w)
is introduced in equation (13.1.3.6), and the subscripts 1 and 2 refer to the values of the
corresponding quantities at ξ = ξ1 and ξ2 Equations (13.1.3.8) permit one to determine
the dependences s = s(x), ξ1= ξ1(x), and ξ2 = ξ2(x) It is possible to show that the jump
condition (13.1.3.6) follows from the last equation in (13.1.3.8)
Example 3 For Hopf’s equation, which corresponds to f (w) = w in equation (13.1.3.1), the jump
condition (13.1.3.6) can be represented as
V = w1+ w2
2 .
Here, we take into account the relation F (w) = 12w2 System (13.1.3.8), which determines the position of the point of discontinuity, becomes
s (x) = ξ1+ ϕ(ξ1)x,
s (x) = ξ2+ ϕ(ξ2)x,
ϕ (ξ1) + ϕ(ξ2 )
2 = ξ21– ξ1
ξ2
ξ1
ϕ (ξ) dξ, where the function ϕ(ξ) specifies the initial wave profile.