14.12. DUHAMEL’S PRINCIPLES IN NONSTATIONARY PROBLEMS 647 14.12.1-2. Hyperbolic equations with two independent variables. Consider the problem for the homogeneous linear hyperbolic equation ∂ 2 w ∂t 2 + ϕ(x) ∂w ∂t = a(x) ∂ 2 w ∂x 2 + b(x) ∂w ∂x + c(x)w (14.12.1.9) with the homogeneous initial conditions w = 0 at t = 0, ∂ t w = 0 at t = 0, (14.12.1.10) and the boundary conditions (14.12.1.3) and (14.12.1.4). The solution of problem (14.12.1.9), (14.12.1.10), (14.12.1.3), (14.12.1.4) with the non- stationary boundary condition (14.12.1.3) at x = x 1 can be expressed by formula (14.12.1.5) in terms of the solution u(x, t) of the auxiliary problem for equation (14.12.1.9) with the initial conditions (14.12.1.10) and boundary condition (14.12.1.4), for u instead of w,and the simpler stationary boundary condition (14.12.1.6) at x = x 1 . In this case, the remark made in Paragraph 14.12.1-1 remains valid. 14.12.1-3. Second-order equations with several independent variables. Duhamel’s first principle can also be used to solve homogeneous linear equations of the parabolic or hyperbolic type with many space variables, ∂ k w ∂t k = n i,j=1 a ij (x) ∂ 2 w ∂x i ∂x j + n i=1 b i (x) ∂w ∂x i + c(x)w,(14.12.1.11) where k = 1, 2 and x = {x 1 , , x n }. Let V be some bounded domain in R n with a sufficiently smooth surface S = ∂V .The solution of the boundary value problem for equation (14.12.1.11) in V withthe homogeneous initial conditions (14.12.1.2) if k = 1 or (14.12.1.10) if k = 2, and the nonhomogeneous linear boundary condition Γ x [w]=g(t)forx S,(14.12.1.12) is given by w(x, t)= ∂ ∂t t 0 u(x, t – τ) g(τ )dτ = t 0 ∂u ∂t (x, t – τ) g(τ) dτ. Here, u(x, t) is the solution of the auxiliary problem for equation (14.12.1.11) with the same initial conditions, (14.12.1.2) or (14.12.1.10), for u instead of w, and the simpler stationary boundary condition Γ x [u]=1 for x S. Note that (14.12.1.12) can represent a boundary condition of the first, second, or third kind; the coefficients of the operator Γ x are assumed to be independent of t. 648 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 14.12.2. Problems for Nonhomogeneous Linear Equations 14.12.2-1. Parabolic equations. The solution of the nonhomogeneous linear equation ∂w ∂t = n i,j=1 a ij (x) ∂ 2 w ∂x i ∂x j + n i=1 b i (x) ∂w ∂x i + c(x)w + Φ(x, t) with the homogeneous initial condition (14.12.1.2) and the homogeneous boundary condi- tion Γ x [w]=0 for x S (14.12.2.1) can be represented in the form (Duhamel’s second principle) w(x, t)= t 0 U(x, t – τ, τ) dτ.(14.12.2.2) Here, U(x, t, τ) is the solution of the auxiliary problem for the homogeneous equation ∂U ∂t = n i,j=1 a ij (x) ∂ 2 U ∂x i ∂x j + n i=1 b i (x) ∂U ∂x i + c(x)U with the boundary condition (14.12.2.1), in which w must be substituted by U,andthe nonhomogeneous initial condition U = Φ(x, τ)att = 0, where τ is a parameter. Note that (14.12.2.1) can represent a boundary condition of the first, second, or third kind; the coefficients of the operator Γ x are assumed to be independent of t. 14.12.2-2. Hyperbolic equations. The solution of the nonhomogeneous linear equation ∂ 2 w ∂t 2 + ϕ(x) ∂w ∂t = n i,j=1 a ij (x) ∂ 2 w ∂x i ∂x j + n i=1 b i (x) ∂w ∂x i + c(x)w + Φ(x, t) with the homogeneous initial conditions (14.12.1.10) and homogeneous boundary con- dition (14.12.2.1) can be expressed by formula (14.12.2.2) in terms of the solution U = U(x, t, τ) of the auxiliary problem for the homogeneous equation ∂ 2 U ∂t 2 + ϕ(x) ∂U ∂t = n i,j=1 a ij (x) ∂ 2 U ∂x i ∂x j + n i=1 b i (x) ∂U ∂x i + c(x)U with the homogeneous initial and boundary conditions, (14.12.1.2) and (14.12.2.1), where w must be replaced by U, and the nonhomogeneous initial condition ∂ t U = Φ(x, τ)att = 0, where τ is a parameter. Note that (14.12.2.1) can represent a boundary condition of the first, second, or third kind. 14.13. TRANSFORMATIONS SIMPLIFYING INITIAL AND BOUNDARY CONDITIONS 649 14.13. Transformations Simplifying Initial and Boundary Conditions 14.13.1. Transformations That Lead to Homogeneous Boundary Conditions A linear problem with arbitrary nonhomogeneous boundary conditions, Γ (k) x,t [w]=g k (x, t)forx S k ,(14.13.1.1) can be reduced to a linear problem with homogeneous boundary conditions. To this end, one should perform the change of variable w(x, t)=ψ(x, t)+u(x, t), (14.13.1.2) where u is a new unknown function and ψ is any function that satisfies the nonhomogeneous boundary conditions (14.13.1.1), Γ (k) x,t [ψ]=g k (x, t)forx S k .(14.13.1.3) Table 14.14 gives examples of such transformations for linear boundary value problems with one space variable for parabolic and hyperbolic equations. In the third boundary value problem, it is assumed that k 1 < 0 and k 2 > 0. TABLE 14.14 Simple transformations of the form w(x, t)=ψ(x, t)+u(x, t) that lead to homogeneous boundary conditions in problems with one space variables (0 ≤ x ≤ l) No. Problems Boundary conditions Function ψ(x, t) 1 First boundary value problem w = g 1 (t)atx = 0 w = g 2 (t)atx = l ψ(x, t)=g 1 (t)+ x l g 2 (t)–g 1 (t) 2 Second boundary value problem ∂ x w = g 1 (t)atx = 0 ∂ x w = g 2 (t)atx = l ψ(x, t)=xg 1 (t)+ x 2 2l g 2 (t)–g 1 (t) 3 Third boundary value problem ∂ x w + k 1 w = g 1 (t)atx = 0 ∂ x w + k 2 w = g 2 (t)atx = l ψ(x, t)= (k 2 x – 1 – k 2 l)g 1 (t)+(1 – k 1 x)g 2 (t) k 2 – k 1 – k 1 k 2 l 4 Mixed boundary value problem w = g 1 (t)atx = 0 ∂ x w = g 2 (t)atx = l ψ(x, t)=g 1 (t)+xg 2 (t) 5 Mixed boundary value problem ∂ x w = g 1 (t)atx = 0 w = g 2 (t)atx = l ψ(x, t)=(x – l)g 1 (t)+g 2 (t) Note that the selection of the function ψ is of a purely algebraic nature and is not connected with the equation in question; there are infinitely many suitable functions ψ that satisfy condition (14.13.1.3). Transformations of the form (14.13.1.2) can often be used at the first stage of solving boundary value problems. 650 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 14.13.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions A linear problem with nonhomogeneous initial and boundary conditions can be reduced to a linear problem with homogeneous initial and boundary conditions. To this end, one should introduce a new dependent variable u by formula (14.13.1.2), where the function ψ must satisfy nonhomogeneous initial and boundary conditions. Below we specify some simple functions ψ that can be used in transformation (14.13.1.2) to obtain boundary value problems with homogeneous initial and boundary conditions. To be specific, we consider a parabolic equation with one space variable and the general initial condition w = f(x)att = 0.(14.13.2.1) 1. First boundary value problem: the initial condition is (14.13.2.1) and the boundary conditions are given in row 1 of Table 14.14. Suppose that the initial and boundary condi- tions are compatible, i.e., f (0)=g 1 (0)andf(l)=g 2 (0). Then, in transformation (14.13.1.2), one can take ψ(x, t)=f(x)+g 1 (t)–g 1 (0)+ x l g 2 (t)–g 1 (t)+g 1 (0)–g 2 (0) . 2. Second boundary value problem: the initial condition is (14.13.2.1) and the boundary conditions are given in row 2 of Table 14.14. Suppose that the initial and boundary condi- tions are compatible, i.e., f (0)=g 1 (0)andf (l)=g 2 (0). Then, in transformation (14.13.1.2), one can set ψ(x, t)=f(x)+x g 1 (t)–g 1 (0) + x 2 2l g 2 (t)–g 1 (t)+g 1 (0)–g 2 (0) . 3. Third boundary value problem: the initial condition is (14.13.2.1) and the boundary conditions are given in row 3 of Table 14.14. If the initial and boundary conditions are compatible, then, in transformation (14.13.1.2), one can take ψ(x, t)=f(x)+ (k 2 x – 1 – k 2 l)[g 1 (t)–g 1 (0)] + (1 – k 1 x)[g 2 (t)–g 2 (0)] k 2 – k 1 – k 1 k 2 l (k 1 < 0, k 2 > 0). 4. Mixed boundary value problem: the initial condition is (14.13.2.1) and the boundary conditions are given in row 4 of Table 14.14. Suppose that the initial and boundary condi- tions are compatible, i.e., f (0)=g 1 (0)andf (l)=g 2 (0). Then, in transformation (14.13.1.2), one can set ψ(x, t)=f(x)+g 1 (t)–g 1 (0)+x g 2 (t)–g 2 (0) . 5. Mixed boundary value problem: the initial condition is (14.13.2.1) and the boundary conditions are given in row 5 of Table 14.14. Suppose that the initial and boundary condi- tions are compatible, i.e., f (0)=g 1 (0)andf(l)=g 2 (0). Then, in transformation (14.13.1.2), one can take ψ(x, t)=f(x)+(x – l) g 1 (t)–g 1 (0) + g 2 (t)–g 2 (0). References for Chapter 14 Akulenko, L. D. and Nesterov, S. V., High Precision Methods in Eigenvalue Problems and Their Applications, Chapman & Hall/CRC Press, Boca Raton, 2004. Butkovskiy,A.G.,Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons, New York, 1982. REFERENCES FOR CHAPTER 14 651 Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1984. Constanda, C., Solution Techniques for Elementary Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2002. Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 2, Wiley-Interscience, New York, 1989. Dezin, A. A., Partial Differential Equations. An Introduction to a General Theory of Linear Boundary Value Problems, Springer-Verlag, Berlin, 1987. Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965. Duffy,D.G.,Transform Methods for Solving Partial Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2004. Farlow,S.J.,Partial Differential Equations for Scientists and Engineers, John Wiley & Sons, New York, 1982. Guenther,R. B. and Lee, J. W., Partial Differential Equations of Mathematical Physics and Integral Equations, Dover Publications, New York, 1996. Haberman, R., Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Prentice-Hall, Englewood Cliffs, New Jersey, 1987. Hanna, J. R. and Rowland, J. H., Fourier Series, Transforms, and Boundary Value Problems, Wiley- Interscience, New York, 1990. Kanwal, R. P., Generalized Functions. Theory and Technique, Academic Press, Orlando, 1983. Leis, R., Initial-Boundary Value Problems in Mathematical Physics, John Wiley & Sons, Chichester, 1986. Mikhlin, S. G. (Editor), Linear Equations of Mathematical Physics, Holt, Rinehart and Winston, New York, 1967. Miller, W., Jr., Symmetry and Separation of Variables, Addison-Wesley, London, 1977. Moon,P.andSpencer,D.E.,Field Theory Handbook, Including Coordinate Systems, Differential Equations and Their Solutions, 3rd Edition, Springer-Verlag, Berlin, 1988. Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, Vols. 1 and 2, McGraw-Hill, New York, 1953. Petrovsky, I. G., Lectures on Partial Differential Equations, Dover Publications, New York, 1991. Polyanin, A. D., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. Sneddon, I. N., Fourier Transformations, Dover Publications, New York, 1995. Stakgold, I., Boundary Value Problems of Mathematical Physics. Vols. 1 and 2, Society for Industrial & Applied Mathematics, Philadelphia, 2000. Strauss, W. A., Partial Differential Equations. An Introduction, John Wiley & Sons, New York, 1992. Tikhonov, A. N. and Samarskii, A. A., Equations of Mathematical Physics, Dover Publications, New York, 1990. Vladimirov, V. S., Equations of Mathematical Physics, Dekker, New York, 1971. Zauderer, E., Partial Differential Equations of Applied Mathematics, Wiley-Interscience, New York, 1989. Zwillinger, D., Handbook of Differential Equations, 3rd Edition, Academic Press, New York, 1997. Chapter 15 Nonlinear Partial Differential Equations 15.1. Classification of Second-Order Nonlinear Equations 15.1.1. Classification of Semilinear Equations in Two Independent Variables A second-order semilinear partial differential equation in two independent variables has the form a(x, y) ∂ 2 w ∂x 2 + 2b(x, y) ∂ 2 w ∂x∂y + c(x, y) ∂ 2 w ∂y 2 = f x, y, w, ∂w ∂x , ∂w ∂y .(15.1.1.1) This equation is classified according to the sign of the discriminant δ = b 2 – ac,(15.1.1.2) where the arguments of the equation coefficients are omitted for brevity. Given a point (x, y), equation (15.1.1.1) is parabolic if δ = 0, hyperbolic if δ > 0, elliptic if δ < 0. (15.1.1.3) The reduction of equation (15.1.1.1) to a canonical form on the basis of the solution of the characteristic equations entirely coincides with that used for linear equations (see Subsection 14.1.1). The classification of semilinear equations of the form (15.1.1.1) does not depend on their solutions—it is determined solely by the coefficients of the highest derivatives on the left-hand side. 15.1.2. Classification of Nonlinear Equations in Two Independent Variables 15.1.2-1. Nonlinear equations of general form. In general, a second-order nonlinear partial differential equation in two independent vari- ables has the form F x, y, w, ∂w ∂x , ∂w ∂y , ∂ 2 w ∂x 2 , ∂ 2 w ∂x∂y , ∂ 2 w ∂y 2 = 0.(15.1.2.1) 653 . 2004. Farlow,S.J.,Partial Differential Equations for Scientists and Engineers, John Wiley & Sons, New York, 1982 . Guenther,R. B. and Lee, J. W., Partial Differential Equations of Mathematical Physics and. Press, Oxford, 1984 . Constanda, C., Solution Techniques for Elementary Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2002. Courant, R. and Hilbert, D., Methods of Mathematical. 2004. Butkovskiy,A.G.,Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons, New York, 1982 . REFERENCES FOR CHAPTER 14 651 Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids,