A nonlinear first-order partial differential equation with two independent variables has the general form F x, y, w, p, q =0, where p= ∂w ∂x, q = ∂w ∂y.. In a sense, the general integral
Trang 1Here, η and ξ are arbitrary numbers and τ > 0 We assume that problem (13.1.4.10), (13.1.4.12) has a unique bounded solution
The stable generalized solution of the Cauchy problem (13.1.4.8), (13.1.4.9) is given by
w (x, y –0) = W x , x, y, ξ–(x, y)
,
w (x, y +0) = W x , x, y, ξ+(x, y)
where ξ–(x, y) and ξ+(x, y) denote, respectively, the greatest lower bound and the least upper
bound of the set of values{ξ= ξ n}for which the function
I (x, y, ξ) =
ξ 0
ϕ (η) – W (0, x, y, η)
dη (13.1.4.14)
takes the minimum value for fixed x and y (x >0) If function (13.1.4.14) takes the minimum
value for a single ξ = ξ1, then ξ–= ξ+and relation (13.1.4.14) describes the classical smooth solution
13.2 Nonlinear Equations
13.2.1 Solution Methods
13.2.1-1 Complete, general, and singular integrals
A nonlinear first-order partial differential equation with two independent variables has the
general form
F (x, y, w, p, q) =0, where p= ∂w
∂x, q = ∂w
∂y (13.2.1.1) Such equations are encountered in analytical mechanics, calculus of variations, optimal control, differential games, dynamic programming, geometric optics, differential geometry, and other fields
In this subsection, we consider only smooth solutions w = w(x, y) of equation (13.2.1.1),
which are continuously differentiable with respect to both arguments (Subsection 13.2.3 deals with nonsmooth solutions)
1◦ Let a particular solution of equation (13.2.1.1),
depending on two parameters C1 and C2, be known The two-parameter family of
so-lutions (13.2.1.2) is called a complete integral of equation (13.2.1.1) if the rank of the
matrix
M =
Ξ1 Ξx1 Ξy
Ξ2 Ξx2 Ξy
(13.2.1.3)
is equal to two in the domain being considered (for example, this is valid if Ξx1Ξy –
Ξx2Ξy ≠ 0) In equation (13.2.1.3), Ξn denotes the partial derivative of Ξ with respect
to C n (n =1,2),Ξxn is the second partial derivative with respect to x and C n, andΞyn is
the second partial derivative with respect to y and C n
In some cases, a complete integral can be found using the method of undetermined coefficients by presetting an appropriate structure of the particular solution sought (The complete integral is determined by the differential equation nonuniquely.)
Trang 2Example 1 Consider the equation
∂w
∂x = a
∂w
∂y
n
+ b.
We seek a particular solution as the sum w = C1 y + C2 + C3 x Substituting this expression into the equation
yields the relation C3 = aC1n + b for the coefficients C1 and C3 With this relation, we find a complete integral
in the form w = C1 y+ aC1n + b
x + C2.
A complete integral of equation (13.2.1.1) is often written in implicit form:*
Ξ(x, y, w, C1, C2) =0 (13.2.1.4)
2◦ The general integral of equation (13.2.1.1) can be represented in parametric form by
using the complete integral (13.2.1.2) [or (13.2.1.4)] and the two equations
C2= f (C1),
∂Ξ
∂C1 +
∂Ξ
∂C2f
(C
where f is an arbitrary function and the prime stands for the derivative In a sense, the
general integral plays the role of the general solution depending on an arbitrary function (the questions whether it describes all solutions calls for further analysis)
Example 2 For the equation considered in the first example, the general integral can be written in
parametric form by using the relations
w = C1 y+ aC1n + b
x + C2, C2= f (C1), y + anC1n–1x + f (C1) =0
Eliminating C2 from these relations and renaming C1 by C, one can represent the general integral in a more
graphic manner in the form
w = Cy + aC n + b
x + f (C),
y = –anC n–1x + f (C).
3◦ Singular integrals of equation (13.2.1.1) can be found without invoking a complete inte-gral by eliminating p and q from the following system of three algebraic (or transcendental)
equations:
F =0, F p=0, F q =0, where the first equation coincides with equation (13.2.1.1)
13.2.1-2 Method of separation of variables Equations of special form
The method of separation of variables implies searching for a complete integral as the sum or
product of functions of various arguments Such solutions are called additive separable and multiplicative separable, respectively Presented below are structures of complete integrals
for some classes of nonlinear equations admitting separation of variables
1◦ If the equation does not depend explicitly on y and w, i.e.,
F (x, w x , w y) =0, then one can seek a complete integral in the form of the sum of two functions with different arguments
w = C1y + C2+ u(x).
The new unknown function u is determined by solving the following ordinary differential
equation:
F (x, u x , C1) =0
Expressing u x from this equation in terms of x, one arrives at a separable differential equation for u = u(x).
* In equations (13.2.1.2) and (13.2.1.4), the symbol Ξ denotes different functions.
Trang 32◦ Consider an equation with separated variables
F1(x, w x ) = F2(y, w y).
Then one can seek a complete integral as the sum of two functions with different arguments,
w = u(x) + v(y) + C1, which are determined by the following two ordinary differential equations:
F1(x, u x ) = C2,
F2(y, v y ) = C2
3◦ Let the equation have the form (generalizes the equation of Item2◦)
F1(x, w x ) + F2(y, w y ) = aw.
Then one can seek a complete integral as the sum of two functions with different arguments,
w = u(x) + v(y) + C1, which are determined by the following two ordinary differential equations:
F1(x, u x ) – au = aC1+ C2,
F2(y, v y ) – av = –C2,
where C1is an arbitrary constant
4◦ Suppose the equation can be rewritten in the form
F ϕ (x, w x ), y, w y
=0 Then one can seek a complete integral as the sum of two functions with different arguments,
w (x, y) = u(x) + v(y) + C1, which are determined by the following two ordinary differential equations:
ϕ (x, u x ) = C2,
F (C2, y, v y ) =0,
where C2is an arbitrary constant
5◦ Let the equation have the form
F1(x, w x /w ) = w k F2(y, w y /w).
Then one can seek a complete integral in the form of the product of two functions with different arguments,
w = u(x)v(y),
which are determined by the following two ordinary differential equations:
F1(x, u x /u ) = C1u k,
F2(y, v y /v ) = C1v–k,
where C1is an arbitrary constant
6◦ Table 13.2 lists complete integrals of the above and some other nonlinear equations of
general form involving arbitrary functions with several arguments
Section T7.3 presents complete integrals for many more nonlinear first-order partial differential equations with two independent variables than in Table 13.2
Trang 4TABLE 13.2 Complete integrals for some special types of nonlinear first-order
partial differential equations; C1 and C2 are arbitrary constants
No Equations and comments Complete integrals Auxiliary equations
1 F (w x , w y) =0 ,
does not depend on x, y, and w implicitly w = C1 + C2 x + C3 y F (C2 , C3) =0
2 F (x, w x , w y) =0 ,
does not depend on y and w implicitly w = C1y + C2+ u(x) F (x, u x , C1) =0
3 F (w, w x , w y) =0 ,
does not depend on x and y implicitly w = u(z), z = C1x + C2y F (u, C1 u z , C2 u z) = 0
4 F1separated equation(x, w x ) = F2(y, w y), w = u(x) + v(y) + C1 F1(x, u x ) = C2,
F2(y, v y ) = C2
5 F1(x, wgeneralizes equation 4x ) + F2 (y, w y ) = aw, w = u(x) + v(y) F1(x, u x ) – au = C1,
F2(y, vy ) – av = –C1
6 F1(x, w x ) = e
aw F
2(y, w y), generalizes equation 4 w = u(x) + v(y)
F1(x, u x ) = C1e au,
F2(y, v y ) = C1e–av
7
F1(x, wx /w ) = w k F2(y, wy /w),
can be reduced to equation 6
by the change of variable w = e z
w = u(x)v(y) F1(x, u x /u ) = C1 u k,
F2(y, v y /v ) = C1v–k
8 w = xwClairaut equationx + yw y + F (w x , w y), w = C1 x + C2 y + F (C1, C2) —
9 F (x, w x , w y , w – yw y) =0 ,
generalizes equation 2 w = C1y + u(x) F x , u x , C1, u
= 0
10 F (w, w x , w y , xw x + yw y) =0 ,
generalizes equations 3 and 8 w = u(z), z = C1x + C2y F (u, C1 u z , C2 u z , zu z) = 0
11 F ϕ (x, w x ), y, w y
= 0 , generalizes equation 4 w = u(x) + v(y) + C1
ϕ (x, u x ) = C2,
F (C2, y, v y ) = 0
13.2.1-3 Lagrange–Charpit method
Suppose that a first integral,
of the characteristic system of ordinary differential equations
dx
F p =
dy
F q =
dw
pF p + qF q = –
dp
F x + pF w = –
dq
F y + qF w (13.2.1.7)
is known Here,
p= ∂w
∂x , q = ∂w
∂y , F x = ∂F
∂x , F y = ∂F
∂y , F w = ∂F
∂w , F p = ∂F
∂p , F q= ∂F
∂q
We assume that solution (13.2.1.6) and equation (13.2.1.1) can be solved for the
deriva-tives p and q, i.e.,
p = ϕ1(x, y, w, C1), q = ϕ2(x, y, w, C1) (13.2.1.8)
Trang 5The first equation of this system can be treated as an ordinary differential equation with
independent variable x and parameter y On finding the solution of this equation depending
on an arbitrary function ψ(y), one substitutes this solution into the second equation to arrive
at an ordinary differential equation for ψ On determining ψ(y) and on substituting it into
the general solution of the first equation of (13.2.1.8), one finds a complete integral of equation (13.2.1.1) In a similar manner, one can start solving system (13.2.1.8) with the
second equation, treating it as an ordinary differential equation with independent variable y and parameter x.
Example 3 Consider the equation
ywp2– q =0 , where p= ∂w
∂x, q= ∂w
∂y.
In this case, the characteristic system (13.2.1.7) has the form
dx
2ywp = –dy
1 =
dw
2ywp2– q = –
dp
yp3 = – dq
wp2+ yp2q.
By making use of the original equation, we simplify the denominator of the third ratio to obtain an integrable
combination: dw/(ywp2) = –dp/(yp3) This yields the first integral p = C1 /w Solving the original equation
for q, we obtain the system
p= C1
w, q= C
2y
w .
The general solution of the first equation has the form w2= 2C1x + ψ(y), where ψ(y) is an arbitrary function With this solution, it follows from the second equation of the system that ψ (y) =2C2y Thus, ψ(y) = C2y2+C2.
Finally, we arrive at a complete integral of the form
w2= 2C1x + C2y2+ C2.
Note that the general solution of the completely integrable Pfaff equation (see Subsec-tion 15.14.2)
dw = ϕ1(x, y, w, C1) dx + ϕ2(x, y, w, C1) dy (13.2.1.9)
is a complete integral of equation (13.2.1.1) Here, the functions ϕ1and ϕ2are the same as
in system (13.2.1.8)
Remark. The relation F (x, y, w, p, q) = C is an obvious first integral of the characteristic system (13.2.1.7).
Hence, the functionΦ determining the integral (13.2.1.6) must differ from F However, the use of
rela-tion (13.2.1.1) makes it possible to reduce the order of system (13.2.1.7) by one.
13.2.1-4 Construction of a complete integral with the aid of two first integrals
Suppose two independent first integrals,
Φ(x, y, w, p, q) = C1, Ψ(x, y, w, p, q) = C2, (13.2.1.10)
of the characteristic system of ordinary differential equations (13.2.1.7) are known Assume
that the functions F ,Φ, and Ψ determining equation (13.2.1.1) and the integrals (13.2.1.10) satisfy the two conditions
(a) J ≡ ∂ (F ,Φ, Ψ)
∂ (w, p, q) 0, (b) [Φ, Ψ]≡
Φp Φx + pΦw
Ψp Ψx + pΨ w
+Φq Φy + qΦw
Ψq Ψy + qΨ w
≡ 0,
(13.2.1.11)
where J is the Jacobian of F , Φ, and Ψ with respect to w, p, and q, and [Φ, Ψ] is the Jacobi–Mayer bracket In this case, relations (13.2.1.1) and (13.2.1.10) form a parametric representation of the complete integral of equation (13.2.1.1) (p and q are considered to
be parameters) Eliminating p and q from equations (13.2.1.1) and (13.2.1.10) followed
by solving the obtained relation for w yields a complete integral in an explicit form w =
w (x, y, C1, C2)
Trang 6Example 4 Consider the equation
pq – aw =0 , where p= ∂w
∂x, q= ∂w
∂y The characteristic system (13.2.1.7) has the form
dx
q = dy
p = dw
2pq = dp
ap = dq
aq Equating the first ratio with the fifth one and the second ratio with the fourth one, we obtain the first integrals
q – ax = C1, p – ay = C2.
Thus, F = pq – aw, Φ = q – ax, and Ψ = p – ax These functions satisfy conditions (13.2.1.11) Solving the
equation and the first integrals for w yields a complete integral of the form
w= 1
a (ax + C1)(ay + C2).
13.2.1-5 Case where the equation does not depend on w explicitly.
Suppose the original equation does not contain the unknown first explicitly, i.e., it has the form
1◦ Given a one-parameter family of solutions w = Ξ(x, y, C1) such that Ξ1 const, a
complete integral is given by w = Ξ(x, y, C1) + C2
2◦ The first integral may be sought in the form
Φ(x, y, p, q) = C1 similar to that of equation (13.2.1.12) In this case, the characteristic system (13.2.1.7) is represented as
dx
F p =
dy
F q = –
dp
F x = –
dq
F y.
The corresponding Pfaff equation (13.2.1.9) becomes
dw = ϕ1(x, y, C1) dx + ϕ2(x, y, C1) dy.
One may integrate this equation in quadrature, thus arriving at the following expression for the complete integral:
w=
x
x0
ϕ1(t, y, C1) dt +
y
y0
ϕ2(x0, s, C1) ds + C2, (13.2.1.13)
where x0and y0are arbitrary numbers
3◦ Suppose that equation (13.2.1.12) can be solved for p or q, for example,
p= –H(x, y, q).
Then, by differentiating this relation with respect to y, we obtain a quasilinear equation for the derivative q in the form
∂q
∂x + ∂
∂y H(x, y, q) =0, q = ∂w
∂y This equation is simpler than the original one; qualitative features of it and solution methods can be found in Section 13.1.1
Trang 713.2.1-6 Hamilton–Jacobi equation.
Equation (13.2.1.1) solved for one of the derivatives, e.g.,
p+H(x, y, w, q) =0, where p= ∂w
∂x, q = ∂w
∂y, (13.2.1.14)
is commonly referred to as the Hamilton–Jacobi equation* and the function H as the
Hamiltonian Equations of the form (13.2.1.14) are frequently encountered in various fields
of mechanics, control theory, and differential games, where the variable x usually plays the role of time and the variable y the role of the spatial coordinate To the Hamilton–Jacobi equation (13.2.1.14) there corresponds the function F (x, y, w, p, q) = p + H(x, y, w, q) in
equation (13.2.1.1)
The characteristic system (13.2.1.7) for equation (13.2.1.14) can be reduced, by taking
into account the relation p = – H, to a simpler system consisting of three differential
equations,
y
x=H q, w
x = q H q–H, q x = –q H w–H y, (13.2.1.15)
which are independent of p; the left-hand sides of these equations are derivatives with respect to x.
13.2.2 Cauchy Problem Existence and Uniqueness Theorem
13.2.2-1 Statement of the problem Solution procedure
Consider the Cauchy problem for equation (13.2.1.1) subject to the initial conditions
x = h1(ξ), y = h2(ξ), w = h3(ξ), (13.2.2.1)
where ξ is a parameter (α≤ξ≤β ) and the h k (ξ) are given functions.
The solution of this problem is carried out in several steps:
1◦ First, one determines additional initial conditions for the derivatives,
p = p0(ξ), q = q0(ξ). (13.2.2.2)
To this end, one must solve the algebraic (or transcendental) system of equations
F h1(ξ), h2(ξ), h3(ξ), p0, q0
p0h
1(ξ) + q0h
2(ξ) – h 3(ξ) =0 (13.2.2.4)
for p0 and q0 Equation (13.2.2.3) results from substituting the initial data (13.2.2.1) into the original equation (13.2.1.1) Equation (13.2.2.4) is a consequence of the dependence
of w on x and y and the relation dw = p dx + q dy, where dx, dy, and dw are calculated in
accordance with the initial data (13.2.2.1)
2◦ One solves the autonomous system
dx
F p =
dy
F q =
dw
pF p + qF q = –
dp
F x + pF w = –
dq
F y + qF w = dτ , (13.2.2.5)
which is obtained from (13.2.1.7) by introducing the additional variable τ (playing the role
of time)
* The Hamilton–Jacobi equation often means equation (13.2.1.14) that does not depend on w explicitly, i.e., the equation p + H(x, y, q) =0