12.7. NONLINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 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 (x 1 , , x n ) such that 1) V > 0 if n  k=1 x 2 k ≠ 0, V = 0 if x 1 = ···= x n = 0; 2) dV dt = n  k=1 f k (t, x 1 , , x n ) ∂V ∂x 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 x 1 = x 2 = ···= x n = 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 x 1 = ···= x n = 0 . This solution is asymptotically stable if there exists a Lyapunov function satisfying the additional condition dV dt ≤ –β < 0 with n  k=1 x 2 k ≥ ε 1 > 0, t ≥ ε 2 ≥ 0, where ε 1 and ε 2 are 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,andψ(x, y) ≥ 0 (ϕ and ψ are continuous functions). A Lyapunov function will be sought in the form V = Ax 2 + By 2 ,whereA and B are constants to be determined. The first condition characterizing a Lyapunov function will be satisfied automatically if A > 0 and B > 0 (it will be shown later that these inequalities do hold). To verify the second condition, let us compute the derivative: dV dt = f 1 (x, y) ∂V ∂x + f 2 (x, y) ∂V ∂y =–2Ax[ay + xϕ(x, y)] + 2By[bx – yψ(x, y)] = 2(Bb – Aa)xy – 2Ax 2 ϕ(x, y)–2By 2 ψ(x, y). Setting here A = b > 0 and B = a > 0 (thus satisfying the first condition), we obtain the inequality dV dt =–2bx 2 ϕ(x, y)–2ay 2 ψ(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 =–xy 2 , y  t = yx 4 . Let us show that the V (x, y)=x 4 + y 2 is a Lyapunov function for the system. Indeed, both conditions are satisfied: 1) x 4 + y 2 > 0 if x 2 + y 2 ≠ 0, V (0, 0)=0 if x = y = 0; 2) dV dt =–4x 4 y 2 + 2x 4 y 2 =–2x 4 y 2 ≤ 0. Hence the trivial solution of the system is stable. 550 ORDINARY DIFFERENTIAL EQUATIONS 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 (x 1 , , x n ) 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 f k (t, x 1 , , x n ) ∂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 x 1 = ···= x n = 0 of system (12.7.3.1) is unstable. Example 4. Perform a stability analysis of the nonlinear system x  t = y 3 ϕ(x, y, t)+x 5 , y  t = x 3 ϕ(x, y, t)+y 5 , where ϕ(x, y, t) is an arbitrary continuous function. Let us show that the W = x 4 – y 4 satisfies the conditions of the Chetaev theorem. We have: 1. W > 0 for |x| > |y|, W = 0 for |x| = |y|. 2. dW dt = 4x 3 [y 3 ϕ(x, y, t)+x 5 ]–4y 3 [x 3 ϕ(x, y, t)+y 5 ]=4(x 8 – y 8 )>0 for |x| > |y|. Moreover, if W ≥ α > 0,wehave dW dt = 4α(x 4 + y 4 ) ≥ 4α 2 = β > 0. It follows that the equilibrium point x = y = 0 of 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 undL ¨ 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 FOR CHAPTER 12 551 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.andKorn,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 ◦ .Afirst-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, hydrodynamics, heat and mass transfer, wave theory, acoustics, multiphase flows, chemical engineering, etc.). If two independent integrals, u 1 (x, y, w)=C 1 , u 2 (x, y, w)=C 2 ,(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(u 1 , u 2 )=0,(13.1.1.4) where Φ(u, v) is an arbitrary function of two variables. With equation (13.1.1.4) solved for u 1 or u 2 , we often specify the general solution in the form u k = Ψ(u 3–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,andh independent of the unknown w,thefirst 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)=C 1 , w – V (x, y)=C 2 . 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-ORDER PARTIAL DIFFERENTIAL EQUATIONS 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 No. Equations General solutions Notation, remarks 1 w x +[f(x)y + g(x)]w y = 0 w = Ψ  e –F y –  e –F g(x) dx  F =  f(x) dx 2 w x +[f(x)y + g(x)y k ]w y = 0 w = Ψ  e –F y 1–k –(1 – k)  e –F g(x) dx  F =(1 – k)  f(x) dx 3 w x +[f(x)e λy + g(x)]w y = 0 w = Ψ  e –λy E + λ  f(x)Edx  E =exp  λ  g(x) dx  4 f(x)w x + g(y)w y = 0 w = Ψ   dx f(x) –  dy g(y)  5 aw x + bw y = f (x)g(w)  dw g(w) = 1 a  f(x) dx + Ψ(bx – ay) solution in implicit form 6 f(x)w x + g(y)w y = h(w)  dw h(w) =  dx f(x) + Ψ(u) u =  dx f(x) –  dy g(y) 7 w x + f(w)w y = 0 y = xf(w)+Ψ(w) solution in implicit form 8 w x +[f(w)+ay]w y = 0 x = 1 a ln   ay + f(w)   + Ψ(w), a ≠ 0 solution in implicit form 9 w x +[f(w)+g(x)]w y = 0 y = xf(w)+  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 = C 1 , w = C 2 . Hence, the general solution of the original equation is given by Φ(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 = C 1 , 2y – aw 2 = C 2 . Hence, the general solution of the original equation is given by Φ(x – w, 2y – aw 2 )=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 AND QUASILINEAR EQUATIONS 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)andG(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 Ψ G from the resulting expression to obtain w x – F x G x = w y – F y G y .(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) where Ψ(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–1 Ψ z and w y = bmx k y m–1 Ψ z . Eliminating Ψ z fromthemgives w x – kx k–1 Ψ anx n–1 = w y bmy m–1 .(13.1.1.8) Solving the original relation (13.1.1.7) for Ψ,wegetΨ = 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 f 1 (x 1 , , x n , w) ∂w ∂x 1 + ···+ f n (x 1 , , x n , w) ∂w ∂x n = g(x 1 , , x n , w). (13.1.1.9) Let n independent integrals, u 1 (x 1 , , x n , w)=C 1 , , u n (x 1 , , x n , w)=C n , of the characteristic system dx 1 f 1 (x 1 , , x n , w) = ··· = dx n f n (x 1 , , x n , w) = dw g(x 1 , , x n , w) be known. Then the general solution of equation (13.1.1.9) is given by Φ(u 1 , , u n )=0, where Φ is an arbitrary function of n variables. . 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,. 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. solution in the form u k = Ψ(u 3–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,andh independent of the unknown