R., Numerical Methods for Differential Equations: A Computational Approach, CRC Press, Boca Raton, 1996.. Grimshaw, R., Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton,
Trang 112.7 NONLINEARSYSTEMS OFORDINARYDIFFERENTIALEQUATIONS 549
12.7.3-3 Lyapunov function Theorems of stability and instability.
In the cases where the theorems of stability and instability by first approximation fail to resolve the issue of stability for a specific system of nonlinear differential equations, more subtle methods must be used Such methods are considered below.
A Lyapunov function for system of equations (12.7.3.1) is a differentiable function
V = V (x1, , xn) such that
1 ) V > 0 if
n
k=1
x2
k≠ 0 , V = 0 if x1= · · · = xn = 0 ;
2 ) dV
dt =
n
k=1
fk(t, x1, , xn) ∂x ∂V
k ≤ 0 for t ≥ 0
Remark The derivative with respect to t in the definition of a Lyapunov function is taken along an integral
curve of system (12.7.3.1)
THEOREM(STABILITY, LYAPUNOV) Let system (12.7.3.1) have the trivial solution x1 =
x2= · · · = xn= 0 This solution is stable if there exists a Lyapunov function for the system.
THEOREM(ASYMPTOTIC STABILITY, LYAPUNOV) Let system (12.7.3.1) have the trivial
solution x1= · · · = xn= 0 This solution is asymptotically stable if there exists a Lyapunov function satisfying the additional condition
dV
dt ≤ –β < 0 with
n
k=1
x2
k≥ ε1> 0 , t ≥ ε2≥ 0 ,
where ε1and ε2are any positive numbers.
Example 2 Let us perform a stability analysis of the two-dimensional system
x t = –ay – xϕ(x, y), y t = bx – yψ(x, y), where a >0, b >0, ϕ(x, y)≥ 0, and ψ(x, y)≥ 0(ϕ and ψ are continuous functions).
A Lyapunov function will be sought in the form V = Ax2+ By2, where A and B are constants to be determined The first condition characterizing a Lyapunov function will be satisfied automatically if A >0and
B>0(it will be shown later that these inequalities do hold) To verify the second condition, let us compute the derivative:
dV
dt = f1(x, y) ∂V
∂x + f2(x, y) ∂V
∂y = –2Ax [ay + xϕ(x, y)] +2By [bx – yψ(x, y)]
=2(Bb – Aa)xy –2Ax2ϕ (x, y) –2By2ψ (x, y).
Setting here A = b >0and B = a >0(thus satisfying the first condition), we obtain the inequality
dV
dt = –2bx2ϕ (x, y) –2ay2ψ (x, y)≤ 0 This means that the second condition characterizing a Lyapunov function is also met Hence, the trivial solution
of the system in question is stable
Example 3 Let us perform a stability analysis for the trivial solution of the nonlinear system
x t = –xy2, y t = yx4
Let us show that the V (x, y) = x4+ y2is a Lyapunov function for the system Indeed, both conditions are satisfied:
1) x4+ y2>0 if x2+ y2≠ 0, V(0,0) =0 if x = y =0;
2) dV
dt = –4x4y2+2x4y2= –2x4y2≤ 0 Hence the trivial solution of the system is stable
Trang 2550 ORDINARYDIFFERENTIALEQUATIONS
Remark No stability analysis of the systems considered in Examples 2 and 3 is possible based on the theorem of stability by first approximation
THEOREM(INSTABILITY, CHETAEV) Suppose there exists a differentiable function W =
W (x1, , xn) that possesses the following properties:
1 In an arbitrarily small domain R containing the origin of coordinates, there exists a subdomain R+⊂ R in which W > 0 , with W = 0 on part of the boundary of R+in R.
2 The condition
dW
dt =
n
k=1
fk(t, x1, , xn) ∂W ∂x
k > 0
holds in R+and, moreover, in the domain of the variables where W ≥ α > 0 , the inequality
dW
dt ≥ β > 0 holds.
Then the trivial solution x1 = · · · = xn= 0 of system (12.7.3.1) is unstable.
Example 4 Perform a stability analysis of the nonlinear system
x t = y3ϕ (x, y, t) + x5, y t = x3ϕ (x, y, t) + y5,
where ϕ(x, y, t) is an arbitrary continuous function.
Let us show that the W = x4– y4satisfies the conditions of the Chetaev theorem We have:
1 W >0 for |x|>|y|, W =0 for |x|=|y|
2 dW
dt =4x3[y3ϕ (x, y, t) + x5] –4y3[x3ϕ (x, y, t) + y5] =4(x8– y8) >0 for |x|>|y|
Moreover, if W ≥α>0, we have dW
dt =4α (x4+ y4)≥ 4α2 = β >0 It follows that the equilibrium point
x = y =0of the system in question is unstable
References for Chapter 12
Akulenko, L D and Nesterov, S V., High Precision Methods in Eigenvalue Problems and Their Applications,
Chapman & Hall/CRC Press, Boca Raton, 2005
Arnold, V I., Kozlov, V V., and Neishtadt, A I., Mathematical Aspects of Classical and Celestial Mechanics,
Dynamical System III, Springer-Verlag, Berlin, 1993.
Bakhvalov, N S., Numerical Methods: Analysis, Algebra, Ordinary Differential Equations, Mir Publishers,
Moscow, 1977
Bogolyubov, N N and Mitropol’skii, Yu A., Asymptotic Methods in the Theory of Nonlinear Oscillations
[in Russian], Nauka Publishers, Moscow, 1974
Boyce, W E and DiPrima, R C., Elementary Differential Equations and Boundary Value Problems, 8th
Edition, John Wiley & Sons, New York, 2004.
Braun, M., Differential Equations and Their Applications, 4th Edition, Springer-Verlag, New York, 1993 Cole, G D., Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company, Waltham, MA,
1968
Dormand, J R., Numerical Methods for Differential Equations: A Computational Approach, CRC Press, Boca
Raton, 1996
El’sgol’ts, L E., Differential Equations, Gordon & Breach, New York, 1961.
Fedoryuk, M V., Asymptotic Analysis Linear Ordinary Differential Equations, Springer-Verlag, Berlin, 1993 Finlayson, B A., The Method of Weighted Residuals and Variational Principles, Academic Press, New York,
1972
Grimshaw, R., Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton, 1991.
Gromak, V I., Painlev´e Differential Equations in the Complex Plane, Walter de Gruyter, Berlin, 2002 Gromak, V I and Lukashevich, N A., Analytical Properties of Solutions of Painlev´e Equations [in Russian],
Universitetskoe, Minsk, 1990
Ince, E L., Ordinary Differential Equations, Dover Publications, New York, 1964.
Kamke, E., Differentialgleichungen: L¨osungsmethoden und L ¨osungen, I, Gew¨ohnliche Differentialgleichungen,
B G Teubner, Leipzig, 1977
Kantorovich, L V and Krylov, V I., Approximate Methods of Higher Analysis [in Russian], Fizmatgiz,
Moscow, 1962
Trang 3REFERENCES FORCHAPTER12 551
Keller, H B., Numerical Solutions of Two Point Boundary Value Problems, Society for Industrial & Applied
Mathematics, Philadelphia, 1976
Kevorkian, J and Cole, J D., Multiple Scale and Singular Perturbation Methods, Springer-Verlag, New York,
1996
Korn, G A and Korn, T M., Mathematical Handbook for Scientists and Engineers, 2nd Edition, Dover
Publications, New York, 2000
Lambert, J D., Computational Methods in Ordinary Differential Equations, Cambridge University Press, New
York, 1973
Lee, H J and Schiesser, W E., Ordinary and Partial Differential Equation Routines in C, C++, Fortran,
Java, Maple, and MATLAB, Chapman & Hall/CRC Press, Boca Raton, 2004.
Levitan, B M and Sargsjan, I S., Sturm–Liouville and Dirac Operators, Kluwer Academic, Dordrecht,
1990
Marchenko, V A., Sturm–Liouville Operators and Applications, Birkh¨auser Verlag, Basel, 1986.
Murphy, G M., Ordinary Differential Equations and Their Solutions, D Van Nostrand, New York, 1960 Nayfeh, A H., Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.
Nayfeh, A H., Perturbation Methods, Wiley-Interscience, New York, 1973.
Petrovskii, I G., Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka Publishers,
Moscow, 1970
Polyanin, A D and Zaitsev, V F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd
Edition, Chapman & Hall/CRC Press, Boca Raton, 2003.
Schiesser, W E., Computational Mathematics in Engineering and Applied Science: ODEs, DAEs, and PDEs,
CRC Press, Boca Raton, 1993
Tenenbaum, M and Pollard, H., Ordinary Differential Equations, Dover Publications, New York, 1985 Wasov, W., Asymptotic Expansions for Ordinary Differential Equations, John Wiley & Sons, New York, 1965 Zhuravlev, V Ph and Klimov, D M., Applied Methods in Oscillation Theory [in Russian], Nauka Publishers,
Moscow, 1988
Zwillinger, D., Handbook of Differential Equations, 3rd Edition, Academic Press, New York, 1997.
Trang 5Chapter 13
First-Order Partial Differential Equations
13.1 Linear and Quasilinear Equations
13.1.1 Characteristic System General Solution
13.1.1-1 Equations with two independent variables General solution Examples.
1◦ A first-order quasilinear partial differential equation with two independent variables
has the general form
f (x, y, w) ∂w
∂x + g(x, y, w) ∂w
∂y = h(x, y, w). (13 1 1 1 ) Such equations are encountered in various applications (continuum mechanics, gas dy-namics, hydrodydy-namics, heat and mass transfer, wave theory, acoustics, multiphase flows, chemical engineering, etc.).
If two independent integrals,
u1(x, y, w) = C1, u2(x, y, w) = C2, (13. 1 1 2 )
of the characteristic system
dx
f (x, y, w) =
dy
g (x, y, w) =
dw
h (x, y, w) (13. 1 1 3 )
are known, then the general solution of equation (13.1.1.1) is given by
Φ(u1, u2) = 0 , (13 1 1 4 ) where Φ(u, v) is an arbitrary function of two variables With equation (13.1.1.4) solved for
u1or u2, we often specify the general solution in the form
uk= Ψ(u3 –k),
where k = 1 , 2 and Ψ(u) is an arbitrary function of one variable.
2◦ For linear equations (13.1.1.1) with the functions f , g, and h independent of the
unknown w, the first integrals (13.1.1.2) of the characteristic system (13.1.1.3) have a simple structure (one integral is independent of w and the other is linear in w):
U (x, y) = C1, w – V (x, y) = C2.
In this case the general solution can be written in explicit form
w = V (x, y) + Ψ(U(x, y)),
where Ψ(U) is an arbitrary function of one variable.
553
Trang 6554 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
TABLE 13.1 General solutions to some special types of linear and quasilinear first-order partial differential equations;Ψ(u)
is an arbitrary function The subscripts x and y indicate the corresponding partial derivatives
1 w x + [f (x)y + g(x)]w y=0 w=Ψe–F –7
e–F (x) dx
F =7
f (x) dx
2 w x + [f (x)y + g(x)y k ]w y=0 w=Ψe–F 1–k– (1– k)7
e–F (x) dx
F = (1– k)7
f (x) dx
3 w x + [f (x)e λy + g(x)]w y=0 w=Ψe–λy E + λ7
f (x)E dx
E= exp
λ7
g (x) dx
f(x)–7 dy
g(y)
5 aw x + bw y = f (x)g(w) 7 dw
g(w) = a1 7
f (x) dx + Ψ(bx – ay) solution in implicit form
h(w)=7 dx
f(x)+Ψ(u) u=7 dx
f(x)–7 dy
g(y)
8 w x + [f (w) + ay]w y=0 x= 1alnay + f (w)+Ψ(w), a≠ 0 solution in implicit form
9 w x + [f (w) + g(x)]w y=0 y = xf (w) +7
g (x) dx + Ψ(w) solution in implicit form
Example 1 Consider the linear constant coefficient equation
∂w
∂x + a ∂w
∂y =0 The characteristic system for this equation is
dx
1 =
dy
a = dw
0 .
It has two independent integrals:
y – ax = C1, w = C2 Hence, the general solution of the original equation is given byΦ(y – ax, w) =0 On solving this equation
for w, one obtains the general solution in explicit form
w=Ψ(y – ax).
It is the traveling wave solution
Example 2 Consider the quasilinear equation
∂w
∂x + aw ∂w
∂y =1 The characteristic system
dx
1 =
dy
aw = dw
1
has two independent integrals:
x – w = C1, 2y – aw2= C2 Hence, the general solution of the original equation is given by
Φ(x – w,2y – aw2) =0
3◦ Table 13.1 lists general solutions to some linear and quasilinear first-order partial
differential equations in two independent variables.
In Sections T7.1–T7.2, many more first-order linear and quasilinear partial differential
equations in two independent variables are considered than in Table 13.1.
Trang 713.1 LINEAR ANDQUASILINEAREQUATIONS 555
13.1.1-2 Construction of a quasilinear equations when given its general solution Given a set of functions
w = F x , y, Ψ(G(x, y)) , (13 1 1 5 )
where F (x, y, Ψ) and G(x, y) are prescribed and Ψ(G) is arbitrary, there exists a quasilinear
first-order partial differential equation such that the set of functions (13.1.1.5) is its general
solution To prove this statement, let us differentiate (13.1.1.5) with respect to x and y and
then eliminate the partial derivative ΨGfrom the resulting expression to obtain
wx– Fx
Gx =
wy– Fy
Gy . (13. 1 1 6 )
On solving the relation w = F (x, y, Ψ) [see (13.1.1.5)] for Ψ and substituting the resulting
expression into (13.1.1.6), one arrives at the desired partial differentiable equation.
Example 3 Let us construct a partial differential equation whose general solution is given by
w = x k Ψ(ax n + by m), (13.1.1.7) whereΨ(z) is an arbitrary function.
Differentiating (13.1.1.7) with respect to x and y yields the relations w x = kx k–1 Ψ + anx k+n–1Ψz and
w y = bmx k y m–1Ψz EliminatingΨzfrom them gives
w x – kx k–1Ψ
anx n–1 = w y
bmy m–1 (13.1.1.8) Solving the original relation (13.1.1.7) forΨ, we get Ψ = x–k w Substituting this expression into (13.1.1.8) and rearranging, we arrive at the desired equation
bmxy m–1 ∂w
∂x – anx n ∂w
∂y = bkmy m–1, whose general solution is the function (13.1.1.7)
13.1.1-3 Equations with n independent variables General solution.
A first-order quasilinear partial differential equation with n independent variables has the
general form
f1(x1, , xn, w) ∂x ∂w
1 + · · · + fn (x1, , xn, w) ∂w
∂xn = g(x1, , xn, w). (13. 1 1 9 )
Let n independent integrals,
u1(x1, , xn, w) = C1, . , un(x1, , xn, w) = Cn,
of the characteristic system
dx1
f1(x1, , xn, w) = · · · = dxn
fn(x1, , xn, w) =
dw
g (x1, , xn, w)
be known Then the general solution of equation (13.1.1.9) is given by
Φ(u1, , un) = 0 , where Φ is an arbitrary function of n variables.