and Jensen, R., Generalized viscosity solutions for Hamilton–Jacobi equations with time-measurable Hamiltonians, J.. Hopf, E., Generalized solutions of nonlinear equations of first orde
Trang 1584 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
Example 2 Consider the terminal value problems for the more general Hamilton–Jacobi equation
∂w
∂x +H
∂w
∂y
= 0 ( 0 ≤x≤L) (13 2 3 18 ) with an arbitrary initial condition
The following two statements hold:
1◦ Let the Hamiltonian satisfy the Lipschitz condition
|H(q2 ) –H(q1 ) | ≤β|q2– q1 | for any q1, q2 R, (13 2 3 20 )
and let the function ϕ(y) be convex Then the function
w (x, y) = sup
q R
qy + (L – x) H(q) – ϕ ∗ (q)
is the viscosity solution of problem (13.2.3.18), (13.2.3.19) The function ϕ ∗ is the conjugate of ϕ, i.e.,
ϕ ∗ (q) = sup
x R
qx – ϕ(x)
.
2◦ Let the HamiltonianH be convex and satisfy the Lipschitz condition (13.2.3.20) Let the function ϕ(y) be
continuous Then the function
w (x, y) = sup
t R
ϕ (y + (L – x)t) – (L – x) H ∗ (t)
is the viscosity solution of problem (13.2.3.18), (13.2.3.19) The function
H ∗ (t) = sup
q R
qt–H(q)
is the conjugate of the Hamiltonian.
References for Chapter 13
Bardi, M and Dolcetta, I C., Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman
Equa-tions, Birkh¨auser Verlag, Boston, 1998.
Barron, E N and Jensen, R., Generalized viscosity solutions for Hamilton–Jacobi equations with
time-measurable Hamiltonians, J Different Equations, Vol 68, No 1, pp 10–21, 1987.
Courant, R and Hilbert, D., Methods of Mathematical Physics, Vol 2, Wiley-Interscience, New York, 1989.
Crandall, M G., Evans, L C., and Lions, P.-L., Some properties of viscosity solutions of Hamilton–Jacobi
equations, Trans Amer Math Soc., Vol 283, No 2, pp 487–502, 1984.
Gelfand, I M., Some problems of the theory of quasi-linear equations, Uspekhi Matem Nauk, Vol 14, No 2,
pp 87–158, 1959 [Amer Math Soc Translation, Series 2, pp 295–381, 1963].
Hopf, E., Generalized solutions of nonlinear equations of first order, J Math Mech., Vol 14, pp 951–973,
1965.
Jeffery, A., Quasilinear Hyperbolic Systems and Waves, Pitman, London, 1976.
Kamke, E., Differentialgleichungen: L¨osungsmethoden und L ¨osungen, II, Partielle Differentialgleichungen
Erster Ordnung f¨ur eine gesuchte Funktion, Akad Verlagsgesellschaft Geest & Portig, Leipzig, 1965.
Kruzhkov, S N., Generalized solutions of nonlinear first order equations with several variables [in Russian],
Mat Sbornik, Vol 70, pp 394–415, 1966.
Lax, P D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves [reprint from
the classical paper of 1957], Society for Industrial & Applied Mathematics, Philadelphia, 1997.
Melikyan, A A., Generalized Characteristics of First Order PDEs: Applications in Optimal Control and
Differential Games, Birkh¨auser Verlag, Boston, 1998.
Oleinik, O A., Discontinuous solutions of nonlinear differential equations, Uspekhi Matem Nauk, Vol 12,
No 3, pp 3–73, 1957 [Amer Math Soc Translation, Series 2, Vol 26, pp 95–172, 1963].
Petrovskii, I G., Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka Publishers,
Moscow, 1970.
Polyanin, A D., Zaitsev, V F., and Moussiaux, A., Handbook of First Order Partial Differential Equations,
Taylor & Francis, London, 2002.
Rhee, H., Aris, R., and Amundson, N R., First Order Partial Differential Equations, Vols 1 and 2, Prentice
Hall, Englewood Cliffs, New Jersey, 1986 and 1989.
Rozhdestvenskii, B L and Yanenko, N N., Systems of Quasilinear Equations and Their Applications to Gas
Dynamics, American Mathematical Society, Providence, Rhode Island, 1983.
Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994.
Subbotin, A I., Generalized Solutions of First Order PDEs: the Dynamical Optimization Perspective,
Birkh¨auser Verlag, Boston, 1995.
Whitham, G B., Linear and Nonlinear Waves, Wiley, New York, 1974.
Trang 2Linear Partial Differential Equations
14.1 Classification of Second-Order Partial Differential
Equations
14.1.1 Equations with Two Independent Variables
14.1.1-1 Examples of equations encountered in applications
Three basic types of partial differential equations are distinguished—parabolic, hyperbolic, and elliptic The solutions of the equations pertaining to each of the types have their own
characteristic qualitative differences
The simplest example of a parabolic equation is the heat equation
∂w
∂t – ∂2w
∂x2 =0, (14.1.1.1)
where the variables t and x play the role of time and the spatial coordinate, respectively Note
that equation (14.1.1.1) contains only one highest derivative term Frequently encountered particular solutions of equation (14.1.1.1) can be found in Paragraph T8.1.1-1
The simplest example of a hyperbolic equation is the wave equation
∂2w
∂t2 –
∂2w
∂x2 =0, (14.1.1.2)
where the variables t and x play the role of time and the spatial coordinate, respectively.
Note that the highest derivative terms in equation (14.1.1.2) differ in sign
The simplest example of an elliptic equation is the Laplace equation
∂2w
∂x2 +
∂2w
∂y2 =0, (14.1.1.3)
where x and y play the role of the spatial coordinates Note that the highest derivative
terms in equation (14.1.1.3) have like signs Frequently encountered particular solutions of equation (14.1.1.3) can be found in Paragraph T8.3.1-1
Any linear partial differential equation of the second-order with two independent vari-ables can be reduced, by appropriate manipulations, to a simpler equation that has one of the three highest derivative combinations specified above in examples (14.1.1.1), (14.1.1.2), and (14.1.1.3)
14.1.1-2 Types of equations Characteristic equations
Consider a second-order partial differential equation with two independent variables that has the general form
a (x, y) ∂
2w
∂x2 +2b (x, y) ∂
2w
∂x∂y + c(x, y) ∂
2w
∂y2 = F
x , y, w, ∂w
∂x,∂w
∂y
, (14.1.1.4)
585
Trang 3586 LINEARPARTIALDIFFERENTIALEQUATIONS
where a, b, c are some functions of x and y that have continuous derivatives up to the
second-order inclusive.*
Given a point (x, y), equation (14.1.1.4) is said to be
parabolic if b2– ac =0,
hyperbolic if b2– ac >0,
elliptic if b2– ac <0
at this point
In order to reduce equation (14.1.1.4) to a canonical form, one should first write out the characteristic equation
a (dy)2–2b dx dy + c (dx)2=0, which splits into two equations
a dy– b+√
b2– ac
dx=0 (14.1.1.5) and
a dy– b–√
b2– ac
dx=0, (14.1.1.6) and then find their general integrals
Remark. The characteristic equations (14.1.1.5)–(14.1.1.5) may be used if a 0 If a≡ 0 , the simpler equations
dx= 0 ,
2b dy – c dx =0
should be used; the first equation has the obvious general solution x = C.
14.1.1-3 Canonical form of parabolic equations (case b2– ac =0)
In this case, equations (14.1.1.5) and (14.1.1.6) coincide and have a common general integral,
ϕ (x, y) = C.
By passing from x, y to new independent variables ξ, η in accordance with the relations
ξ = ϕ(x, y), η = η(x, y), where η = η(x, y) is any twice differentiable function that satisfies the condition of
nonde-generacy of the Jacobian D(x,y) D(ξ,η) in the given domain, we reduce equation (14.1.1.4) to the canonical form
∂2w
∂η2 = F1
ξ , η, w, ∂w
∂ξ ,∂w
∂η
As η, one can take η = x or η = y.
It is apparent that the transformed equation (14.1.1.7) has only one highest-derivative term, just as the heat equation (14.1.1.1)
Remark. In the degenerate case where the function F1 does not depend on the derivative ∂ ξ w,
equa-tion (14.1.1.7) is an ordinary differential equaequa-tion for the variable η, in which ξ serves as a parameter.
* The right-hand side of equation (14.1.1.4) may be nonlinear The classification and the procedure of reducing such equations to a canonical form are only determined by the left-hand side of the equation.
Trang 414.1.1-4 Canonical forms of hyperbolic equations (case b2– ac >0).
The general integrals
ϕ (x, y) = C1, ψ (x, y) = C2
of equations (14.1.1.5) and (14.1.1.6) are real and different These integrals determine two different families of real characteristics
By passing from x, y to new independent variables ξ, η in accordance with the relations
ξ = ϕ(x, y), η = ψ(x, y),
we reduce equation (14.1.1.4) to
∂2w
∂ξ∂η = F2
ξ , η, w, ∂w
∂ξ, ∂w
∂η
This is the so-called first canonical form of a hyperbolic equation.
The transformation
ξ = t + z, η = t – z
brings the above equation to another canonical form,
∂2w
∂t2 –
∂2w
∂z2 = F3
t , z, w, ∂w
∂t , ∂w
∂z
,
where F3 = 4F2 This is the so-called second canonical form of a hyperbolic equation.
Apart from notation, the left-hand side of the last equation coincides with that of the wave equation (14.1.1.2)
In some cases, reduction of an equation to a canonical form allows finding its general solution
Example The equation
kx ∂
2w
∂x2 + ∂
2w
∂x∂y = 0
is a special case of equation (14.1.1.4) with a = kx, b = 12, c =0, and F =0 The characteristic equations
kx dy – dx =0 ,
dy= 0
have the general integrals ky – ln|x|= C1 and y = C2 Switching to the new independent variables
ξ = ky – ln|x| , η = y
reduces the original equation to the canonical form
∂2w
∂ξ∂η = k ∂w
∂ξ.
Integrating with respect to ξ yields the linear first-order equation
∂w
∂η = kw + f (η) where f (η) is an arbitrary function Its general solution is expressed as
w = e kη g (ξ) + e kη
e–kη f (η) dη, where g(ξ) is an arbitrary function.
Trang 5588 LINEARPARTIALDIFFERENTIALEQUATIONS
14.1.1-5 Canonical form of elliptic equations (case b2– ac <0)
In this case the general integrals of equations (14.1.1.5) and (14.1.1.6) are complex conju-gate; these determine two families of complex characteristics
Let the general integral of equation (14.1.1.5) have the form
ϕ (x, y) + iψ(x, y) = C, i2= –1,
where ϕ(x, y) and ψ(x, y) are real-valued functions.
By passing from x, y to new independent variables ξ, η in accordance with the relations
ξ = ϕ(x, y), η = ψ(x, y),
we reduce equation (14.1.1.4) to the canonical form
∂2w
∂ξ2 +
∂2w
∂η2 = F4
ξ , η, w, ∂w
∂ξ ,∂w
∂η
Apart from notation, the left-hand side of the last equation coincides with that of the Laplace equation (14.1.1.3)
14.1.1-6 Linear constant-coefficient partial differential equations
1◦ When reduced to a canonical form, linear homogeneous constant-coefficient partial
differential equations
a ∂
2w
∂x2 +2b ∂
2w
∂x∂y + c ∂
2w
∂y2 + p
∂w
∂x + q ∂w
admit further simplifications In general, the substitution
w (x, y) = exp(β1ξ + β2η )u(ξ, η) (14.1.1.9)
can be used Here, ξ and η are new variables used to reduce equation (14.1.1.8) to a canonical form (see Paragraphs 14.1.1-3 to 14.1.1-5); the coefficients β1and β2in (14.1.1.9) are chosen so that there is only one first derivative remaining in a parabolic equation or both first derivatives vanish in a hyperbolic or an elliptic equation For final results, see Table 14.1
2◦ The coefficients k and k1in the reduced hyperbolic and elliptic equations (see the third,
fourth, and fifth rows in Table 14.1) are expressed as
k= 2bpq – aq2– cp2
16a (b2– ac)2 –
s
4a (b2– ac), k1=
s
4b2 +
cp2–2bpq
16b4 . (14.1.1.10)
If the coefficients in equation (14.1.1.8) satisfy the relation
2bpq – aq2– cp2–4s (b2– ac) =0,
then k =0; in this case with a≠ 0, the general solution of the corresponding hyperbolic equation has the form
w (x, y) = exp(β1ξ + β2η)
f (ξ) + g(η)
, D = b2– ac >0,
ξ = ay – b+√
D
x, η = ay – b–√
D
x,
β1= aq4aD – bp + p
4a √
D, β2= aq4aD – bp – p
4a √
D,
where f (ξ) and g(ξ) are arbitrary functions.
Trang 6TABLE 14.1 Reduction of linear homogeneous constant-coefficient partial differential equations (14.1.1.8) using
transformation (14.1.1.9); the constants k and k1 are given by formulas (14.1.1.10)
Type of equation,
conditions on coefficients
Variables ξ and η in
transformation (14.1.1.9)
Coefficients β1and β2 in transformation (14.1.1.9)
Reduced equation Parabolic equation,
a = b =0, c≠ 0, p≠ 0 ξ= –
c
p x , η = y β1=4cs – q2
4c2 , β2= –q
2c u ξ – u ηη= 0
Parabolic equation, b2– ac =0
(aq – bp≠ 0 , |a| + |b| ≠ 0 ) ξ=
a (ay – bx)
bp – aq , η = x β1=
4as – p2
4a2 , β2= –p
2a u ξ – u ηη= 0
Hyperbolic equation,
a≠ 0, D = b2– ac >0
ξ = ay – b+√
D
x,
η = ay – b–√
D
x β1,2=aq4aD – bp
p
4a √
D u ξη + ku =0
Hyperbolic equation,
a= 0, b≠ 0 η ξ== x,2by – cx β1=
cp– 2bq
4b2 , β2 = – 4p b2 u ξη + k1u= 0
Elliptic equation,
D = b2– ac <0 ξ η = ay – bx,=√|
D|x β1=aq – bp
2aD , β2= – p
2a √
|D| u ξξ + u ηη+4ku=0
Ordinary differential equation,
b2– ac =0, aq – bp =0 ξ η = ay – bx, = x β1= β2=0 aw ηη + pw η + sw =0
3◦ In the degenerate case b2– ac =0, aq – bp =0(where the original equation is reduced
to an ordinary differential equation; see the last row in Table 14.1), the general solution of equation (14.1.1.8) is expressed as
w= exp
–px
2a
f (ay – bx) exp
x √ λ
2a
+ g(ay – bx) exp
–x
√ λ
2a
if λ = p2– 4as> 0 ,
w= exp
–px
2a
f (ay – bx) sin
x √|
λ|
2a
+ g(ay – bx) cos
x √|
λ|
2a
if λ = p2– 4as< 0 ,
w= exp
–px2a
f (ay – bx) + xg(ay – bx)
if 4as – p2= 0 ,
where f (z) and g(z) are arbitrary functions.
14.1.2 Equations with Many Independent Variables
Let us consider a second-order partial differential equation with n independent variables
x1, , x nthat has the form
n
i,j=1
a ij(x) ∂
2w
∂x i ∂x j = F
x, w, ∂w
∂x1, ,
∂w
∂x n
, (14.1.2.1)
where the a ij are some functions that have continuous derivatives with respect to all
variables to the second-order inclusive, and x ={x1, , x n} [The right-hand side of equa-tion (14.1.2.1) may be nonlinear The left-hand side only is required for the classificaequa-tion
of this equation.]
At a point x = x0, the following quadratic form is assigned to equation (14.1.2.1):
Q=
n
i,j=1
a ij(x0)ξ i ξ j. (14.1.2.2)
Trang 7590 LINEARPARTIALDIFFERENTIALEQUATIONS
TABLE 14.2 Classification of equations with many independent variables
Type of equation (14.1.2.1) at a point x = x0 Coefficients of the canonical form (14.1.2.4) Parabolic (in the broad sense) At least one coefficient of the c iis zero Hyperbolic (in the broad sense) All c i are nonzero and some c idiffer in sign
By an appropriate linear nondegenerate transformation
ξ i =
n
k=1
β ik η k (i =1, , n) (14.1.2.3) the quadratic form (14.1.2.2) can be reduced to the canonical form
Q=
n
i=1
c i η2
i (14.1.2.4)
where the coefficients c iassume the values1, –1, and0 The number of negative and zero coefficients in (14.1.2.4) does not depend on the way in which the quadratic form is reduced
to the canonical form
Table 14.2 presents the basic criteria according to which the equations with many independent variables are classified
Suppose all coefficients of the highest derivatives in (14.1.2.1) are constant, a ij = const
By introducing the new independent variables y1, , y nin accordance with the formulas
y i=
n
k=1β ik x k , where the β ikare the coefficients of the linear transformation (14.1.2.3), we reduce equation (14.1.2.1) to the canonical form
n
i=1
c i ∂
2w
∂y2
i
= F1
y, w, ∂w
∂y1, ,
∂w
∂y n
Here, the coefficients c iare the same as in the quadratic form (14.1.2.4), and y ={y1, , y n}
Remark 1 Among the parabolic equations, it is conventional to distinguish the parabolic equations in
the narrow sense, i.e., the equations for which only one of the coefficients, c k , is zero, while the other c iis the same, and in this case the right-hand side of equation (14.1.2.5) must contain the first-order partial derivative
with respect to y k.
Remark 2 In turn, the hyperbolic equations are divided into normal hyperbolic equations—for which
all c i but one have like signs—and ultrahyperbolic equations—for which there are two or more positive c iand
two or more negative c i.
Specific equations of parabolic, elliptic, and hyperbolic types will be discussed further
in Section 14.2
14.2 Basic Problems of Mathematical Physics
14.2.1 Initial and Boundary Conditions Cauchy Problem.
Boundary Value Problems
Every equation of mathematical physics governs infinitely many qualitatively similar phe-nomena or processes This follows from the fact that differential equations have infinitely