Handbook of mathematics for engineers and scienteists part 84 docx

R., Numerical Methods for Differential Equations: A Computational Approach, CRC Press, Boca Raton, 1996.. Grimshaw, R., Nonlinear Ordinary Differential Equations, CRC Press, Boca Raton,

12.7 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

550 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

REFERENCES 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.

Chapter 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

554 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.

13.1 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.