682 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Linear separable equations of mathematical physics admit exact solutions in the form w(x, y)=ϕ 1 (x)ψ 1 (y)+ϕ 2 (x)ψ 2 (y)+···+ ϕ n (x)ψ n (y), (15.5.1.1) where w i = ϕ i (x)ψ i (y) are particular solutions; the functions ϕ i (x), as well as the functions ψ i (y), with different numbers i, are not related to one another. Many nonlinear partial differential equations with quadratic or power nonlinearities, f 1 (x)g 1 (y)Π 1 [w]+f 2 (x)g 2 (y)Π 2 [w]+···+ f m (x)g m (y)Π m [w]=0,(15.5.1.2) also have exact solutions of the form (15.5.1.1). In (15.5.1.2), the Π i [w] are differential forms that are the products of nonnegative integer powers of the function w and its partial derivatives ∂ x w, ∂ y w, ∂ xx w, ∂ xy w, ∂ yy w, ∂ xxx w, etc. We will refer to solutions (15.5.1.1) of nonlinear equations (15.5.1.2) as generalized separable solutions. Unlike linear equations, in nonlinear equations the functions ϕ i (x) with different subscripts i are usually related to one another [and to functions ψ j (y)]. In general, the functions ϕ i (x)andψ j (y) in (15.5.1.1) are not known in advance and are to be identified. Subsection 15.4.2 gives simple examples of exact solutions of the form (15.5.1.1) with n = 1 and n = 2 (for ψ 1 = ϕ 2 = 1)tosome nonlinear equations. Note that most common of the generalized separable solutions are solutions of the special form w(x, y)=ϕ(x)ψ(y)+χ(x); the independent variables on the right-hand side can be swapped. In the special case of χ(x)=0, this is a multiplicative separable solution, and if ϕ(x)=1, this is an additive separable solution. Remark. Expressions of the form (15.5.1.1) are often used in applied and computational mathematics for constructing approximate solutions to differential equations by the Galerkin method (and its modifications). 15.5.1-2. General form of functional differential equations. In general, on substituting expression (15.5.1.1) into the differential equation (15.5.1.2), one arrives at a functional differential equation Φ 1 (X)Ψ 1 (Y )+Φ 2 (X)Ψ 2 (Y )+···+ Φ k (X)Ψ k (Y )=0 (15.5.1.3) for the ϕ i (x)andψ i (y). The functionals Φ j (X)andΨ j (Y ) depend only on x and y, respectively, Φ j (X) ≡ Φ j x, ϕ 1 , ϕ 1 , ϕ 1 , , ϕ n , ϕ n , ϕ n , Ψ j (Y ) ≡ Ψ j y, ψ 1 , ψ 1 , ψ 1 , , ψ n , ψ n , ψ n . (15.5.1.4) Here, for simplicity, the formulas are written out for the case of a second-order equa- tion (15.5.1.2); for higher-order equations, the right-hand sides of relations (15.5.1.4) will contain higher-order derivatives of ϕ i and ψ j . Subsections 15.5.3 and 15.5.4 outline two different methods for solving functional differential equations of the form (15.5.1.3)–(15.5.1.4). Remark. Unlike ordinary differential equations, equation (15.5.1.3)–(15.5.1.4) involves several functions (and their derivatives) with different arguments. 15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 683 15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System of Coordinate Functions 15.5.2-1. Description of the simplified scheme. To construct exact solutions of equations (15.5.1.2) with quadratic or power nonlinearities that do not depend explicitly on x (all f i constant), it is reasonable to use the following simplified approach. As before, we seek solutions in the form of finite sums (15.5.1.1). We assume that the system of coordinate functions {ϕ i (x)} is governed by linear differential equations with constant coefficients. The most common solutions of such equations are of the forms ϕ i (x)=x i , ϕ i (x)=e λ i x , ϕ i (x)=sin(α i x), ϕ i (x)=cos(β i x). (15.5.2.1) Finite chains of these functions (in various combinations) can be used to search for separable solutions (15.5.1.1), where the quantities λ i , α i ,andβ i are regarded as free parameters. The other system of functions {ψ i (y)} is determined by solving the nonlinear equations resulting from substituting (15.5.1.1) into the equation under consideration [or into equation (15.5.1.3)–(15.5.1.4)]. This simplified approach lacks the generality of the methods outlined in Subsections 15.5.3–15.5.4. However, specifying one of the systems of coordinate functions, {ϕ i (x)}, simplifies the procedure of finding exact solutions substantially. The drawback of this approach is that some solutions of the form (15.5.1.1) can be overlooked. It is significant that the overwhelming majority of generalized separable solutions known to date, for partial differential equations with quadratic nonlinearities, are determined by coordinate functions (15.5.2.1) (usually with n = 2). 15.5.2-2. Examples of finding exact solutions of second- and third-order equations. Below we consider specific examples that illustrate the application of the above simplified scheme to the construction of generalized separable solutions of second- and third-order nonlinear equations. Example 1. Consider a nonhomogeneous Monge–Amp ` ere equation of the form ∂ 2 w ∂x∂y 2 – ∂ 2 w ∂x 2 ∂ 2 w ∂y 2 = f(x). (15.5.2.2) We look for generalized separable solutions with the form w(x, y)=ϕ(x)y k + ψ(x), k ≠ 0.(15.5.2.3) On substituting (15.5.2.3) into (15.5.2.2) and collecting terms, we obtain [k 2 (ϕ x ) 2 – k(k – 1)ϕϕ xx ]y 2k–2 – k(k – 1)ϕψ xx y k–2 – f(x)=0.(15.5.2.4) This equation can be satisfied only if k = 1 or k = 2. First case.Ifk = 1, (15.5.2.4) reduces one equation (ϕ x ) 2 – f(x)=0. It has two solutions: ϕ(x)= √ f(x) dx. They generate two solutions of equation (15.5.2.2) in the form (15.5.2.3): w(x, y)= y f(x) dx + ψ(x), where ψ(x) is an arbitrary function. 684 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Second case.Ifk = 2, equating the functional coefficients of the different powers of y to zero, we obtain two equations: 2(ϕ x ) 2 – ϕϕ xx = 0, 2ϕψ xx – f(x)=0. Their general solutions are given by ϕ(x)= 1 C 1 x + C 2 , ψ(x)= 1 2 x 0 (x – t)(C 1 t + C 2 )f(t) dt + C 3 x + C 4 . Here, C 1 , C 2 , C 3 ,andC 4 are arbitrary constants. Table 15.4 lists generalized separable solutions of various nonhomogeneous Monge– Amp ` ere equations of the form ∂ 2 w ∂x∂y 2 – ∂ 2 w ∂x 2 ∂ 2 w ∂y 2 = F (x, y). (15.5.2.5) Equations of this form are encountered in differential geometry, gas dynamics, and mete- orology. Example 2. Consider the third-order nonlinear equation ∂ 2 w ∂x∂t + ∂w ∂x 2 – w ∂ 2 w ∂x 2 = a ∂ 3 w ∂x 3 ,(15.5.2.6) which is encountered in hydrodynamics. We look for exact solutions of the form w = ϕ(t)e λx + ψ(t), λ ≠ 0.(15.5.2.7) On substituting (15.5.2.7) into (15.5.2.6), we have ϕ t – λϕψ = aλ 2 ϕ. We now solve this equation for ψ and substitute the resulting expression into (15.5.2.7) to obtain a solution of equation (15.5.2.6) in the form w = ϕ(t)e λx + 1 λ ϕ t (t) ϕ(t) – aλ, where ϕ(t) is an arbitrary function and λ is an arbitrary constant. 15.5.3. Solution of Functional Differential Equations by Differentiation 15.5.3-1. Description of the method. Below we describe a procedure for constructing solutions to functional differential equations of the form (15.5.1.3)–(15.5.1.4). It involves three successive stages. 1 ◦ . Assume that Ψ k 0. We divide equation (15.5.1.3) by Ψ k and differentiate with respect to y. This results in a similar equation but with fewer terms: Φ 1 (X) Ψ 1 (Y )+ Φ 2 (X) Ψ 2 (Y )+···+ Φ k–1 (X) Ψ k–1 (Y )=0, Φ j (X)=Φ j (X), Ψ j (Y )=[Ψ j (Y )/Ψ k (Y )] y . We repeat the above procedure (k – 3) times more to obtain the separable two-term equation Φ 1 (X) Ψ 1 (Y )+ Φ 2 (X) Ψ 2 (Y )=0.(15.5.3.1) 15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 685 TABLE 15.4 Exact solutions of a nonhomogeneous Monge–Amp ` ere equation of the form (15.5.2.5); f(x), g(x), and h(x) are arbitrary functions; C 1 , C 2 , C 3 ,andβ are arbitrary constants; a, b, k,andλ are some numbers (k ≠ –1,–2) No. Function F (x, y) Solution w(x, y) 1 f(x) C 1 y 2 + C 2 xy + C 2 2 4C 1 x 2 – 1 2C 1 x a (x – t)f(t) dt 2 f(x) 1 x + C 1 C 2 y 2 + C 3 y + C 2 3 4C 2 – 1 2C 2 x a (x – t)(t + C 1 )f(t) dt 3 f(x)y C 1 y 2 + ϕ(x)y + ψ(x)(ϕ and ψ are expressed as quadratures) 4 f(x)y 1 x + C 1 y 2 + ϕ(x)y + ψ(x)(ϕ and ψ are expressed as quadratures) 5 f(x)y C 1 y 3 – 1 6C 1 x a (x – t)f(t) dt 6 f(x)y y 3 (x + C 1 ) 2 – 1 6 x a (x – t)(t + C 1 ) 2 f(t) dt 7 f(x)y 2 ϕ(x)y 2 + ψ(x)y + χ(x)(ϕ, ψ,andχ are determined by ODEs) 8 f(x)y 2 C 1 y 4 – 1 12C 1 x a (x – t)f(t) dt 9 f(x)y 2 y 4 (x + C 1 ) 3 – 1 12 x a (x – t)(t + C 1 ) 3 f(t) dt 10 f(x)y 2 + g(x)y + h(x) ϕ(x)y 2 + ψ(x)y + χ(x)(ϕ, ψ,andχ are determined by ODEs) 11 f(x)y k C 1 y k+2 (k + 1)(k + 2) – 1 C 1 x a (x – t)f(t) dt 12 f(x)y k y k+2 (x + C 1 ) k+1 – 1 (k + 1)(k + 2) x a (x – t)(t + C 1 ) k+1 f(t) dt 13 f(x)y 2k+2 + g(x)y k ϕ(x)y k+2 – 1 (k + 1)(k + 2) x a (x – t) g(t) ϕ(t) dt (ϕ is determined by an ODE) 14 f(x)e λy C 1 e βx+λy – 1 C 1 λ 2 x a (x – t)e –βt f(t) dt 15 f(x)e 2λy + g(x)e λy ϕ(x)e λy – 1 λ 2 x a (x – t) g(t) ϕ(t) dt (ϕ is determined by an ODE) 16 f(x)g(y)+λ 2 C 1 x a (x – t)f(t) dt – 1 C 1 y b (y – ξ)g(ξ)dξ λxy Three cases must be considered. Nondegenerate case: | Φ 1 (X)| + | Φ 2 (X)| 0 and | Ψ 1 (Y )| + | Ψ 2 (Y )| 0. Then the solutions of equation (15.5.3.1) are determined by the ordinary differential equations Φ 1 (X)+C Φ 2 (X)=0, C Ψ 1 (Y )– Ψ 2 (Y )=0, where C is an arbitrary constant. The equations Φ 2 = 0 and Ψ 1 = 0 correspond to the limit case C = ∞. 686 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Two degenerate cases: Φ 1 (X) ≡ 0, Φ 2 (X) ≡ 0 =⇒ Ψ 1,2 (Y ) are any functions; Ψ 1 (Y ) ≡ 0, Ψ 2 (Y ) ≡ 0 =⇒ Φ 1,2 (X) are any functions. 2 ◦ . The solutions of the two-term equation (15.5.3.1) should be substituted into the original functional differential equation (15.5.1.3) to “remove” redundant constants of integration [these arise because equation (15.5.3.1) is obtained from (15.5.1.3) by differentiation]. 3 ◦ . The case Ψ k ≡ 0 should be treated separately (since we divided the equation by Ψ k at the first stage). Likewise, we have to study all other cases where the functionals by which the intermediate functional differential equations were divided vanish. Remark 1. The functional differential equation (15.5.1.3) happens to have no solutions. Remark 2. At each subsequent stage, the number of terms in the functional differential equation can be reduced by differentiation with respect to either y or x. For example, we can assume at the first stage that Φ k 0. On dividing equation (15.5.1.3) by Φ k and differentiating with respect to x, we again obtain a similar equation that has fewer terms. 15.5.3-2. Examples of constructing exact generalized separable solutions. Below we consider specific examples illustrating the application of the above method of constructing exact generalized separable solutions of nonlinear equations. Example 1. The equations of a laminar boundary layer on a fl at plate are reduced to a single third-order nonlinear equation for the stream function (see Schlichting, 1981, and Loitsyanskiy, 1996): ∂w ∂y ∂ 2 w ∂x∂y – ∂w ∂x ∂ 2 w ∂y 2 = a ∂ 3 w ∂y 3 .(15.5.3.2) We look for generalized separable solutions to equation (15.5.3.2) in the form w(x, y)=ϕ(x)ψ(y)+χ(x). (15.5.3.3) On substituting (15.5.3.3) into (15.5.3.2) and canceling by ϕ, we arrive at the functional differential equation ϕ x [(ψ y ) 2 – ψψ yy ]–χ x ψ yy = aψ yyy .(15.5.3.4) We differentiate (15.5.3.4) with respect to x to obtain ϕ xx [(ψ y ) 2 – ψψ yy ]=χ xx ψ yy .(15.5.3.5) Nondegenerate case. On separating the variables in (15.5.3.5), we get χ xx = C 1 ϕ xx , (ψ y ) 2 – ψψ yy – C 1 ψ yy = 0. Integrating yields ψ(y)=C 4 e λy – C 1 , ϕ(x) is any function, χ(x)=C 1 ϕ(x)+C 2 x + C 3 ,(15.5.3.6) where C 1 , , C 4 ,andλ are constants of integration. On substituting (15.5.3.6) into (15.5.3.4), we establish the relationship between constants to obtain C 2 =–aλ. Ultimately, taking into account the aforesaid and formulas (15.5.3.3) and (15.5.3.6), we arrive at a solution of equation (15.5.3.2) of the form (15.5.3.3): w(x, y)=ϕ(x)e λy – aλx + C, where ϕ(x) is an arbitrary function and C and λ are arbitrary constants (C = C 3 , C 4 = 1). Degenerate case. It follows from (15.5.3.5) that ϕ xx = 0, χ xx = 0, ψ(y) is any function. (15.5.3.7) 15.5. METHOD OF GENERALIZED SEPARATION OF VARIABLES 687 Integrating the first two equations in (15.5.3.7) twice yields ϕ(x)=C 1 x + C 2 , χ(x)=C 3 x + C 4 ,(15.5.3.8) where C 1 , , C 4 are arbitrary constants. Substituting (15.5.3.8) into (15.5.3.4), we arrive at an ordinary differential equation for ψ = ψ(y): C 1 (ψ y ) 2 –(C 1 ψ + C 3 )ψ yy = aψ yyy .(15.5.3.9) Formulas (15.5.3.3) and (15.5.3.8) together with equation (15.5.3.9) determine an exact solution of equa- tion (15.5.3.2). Example 2. The two-dimensional stationary equations of motion of a viscous incompressible fluid are reduced to a single fourth-order nonlinear equation for the stream function (see Loitsyanskiy, 1996): ∂w ∂y ∂ ∂x (Δw)– ∂w ∂x ∂ ∂y (Δw)=aΔΔw, Δw = ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 . (15.5.3.10) Here, a is the kinematic viscosity of the fluid and x, y are Cartesian coordinates. We seek exact separable solutions of equation (15.5.3.10) in the form w = f(x)+g(y). (15.5.3.11) Substituting (15.5.3.11) into (15.5.3.10) yields g y f xxx – f x g yyy = af xxxx + ag yyyy . (15.5.3.12) Differentiating (15.5.3.12) with respect to x and y, we obtain g yy f xxxx – f xx g yyyy = 0. (15.5.3.13) Nondegenerate case. If f xx 0 and g yy 0, we separate the variables in (15.5.3.13) to obtain the ordinary differential equations f xxxx = Cf xx , (15.5.3.14) g yyyy = Cg yy , (15.5.3.15) which have different solutions depending on the value of the integration constant C. 1 ◦ . Solutions of equations (15.5.3.14) and (15.5.3.15) for C = 0: f(x)=A 1 + A 2 x + A 3 x 2 + A 4 x 3 , g(y)=B 1 + B 2 y + B 3 y 2 + B 4 y 3 , (15.5.3.16) where A k and B k are arbitrary constants (k = 1, 2, 3, 4). On substituting (15.5.3.16) into (15.5.3.12), we evaluate the integration constants. Three cases are possible: A 4 = B 4 = 0, A n , B n are any numbers (n = 1, 2, 3); A k = 0, B k are any numbers (k = 1, 2, 3, 4); B k = 0, A k are any numbers (k = 1, 2, 3, 4). The first two sets of constants determine two simple solutions (15.5.3.11) of equation (15.5.3.10): w = C 1 x 2 + C 2 x + C 3 y 2 + C 4 y + C 5 , w = C 1 y 3 + C 2 y 2 + C 3 y + C 4 , where C 1 , , C 5 are arbitrary constants. 2 ◦ . Solutions of equations (15.5.3.14) and (15.5.3.15) for C = λ 2 > 0: f(x)=A 1 + A 2 x + A 3 e λx + A 4 e –λx , g(y)=B 1 + B 2 y + B 3 e λy + B 4 e –λy . (15.5.3.17) Substituting (15.5.3.17) into (15.5.3.12), dividing by λ 3 , and collecting terms, we obtain A 3 (aλ – B 2 )e λx + A 4 (aλ + B 2 )e –λx + B 3 (aλ + A 2 )e λy + B 4 (aλ – A 2 )e –λy = 0. 688 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Equating the coefficients of the exponentials to zero, we find A 3 = A 4 = B 3 = 0, A 2 = aλ (case 1), A 3 = B 3 = 0, A 2 = aλ, B 2 =–aλ (case 2), A 3 = B 4 = 0, A 2 =–aλ, B 2 =–aλ (case 3). (The other constants are arbitrary.) These sets of constants determine three solutions of the form (15.5.3.11) for equation (15.5.3.10): w = C 1 e –λy + C 2 y + C 3 + aλx, w = C 1 e –λx + aλx + C 2 e –λy – aλy + C 3 , w = C 1 e –λx – aλx + C 2 e λy – aλy + C 3 , where C 1 , C 2 , C 3 ,andλ are arbitrary constants. 3 ◦ . Solution of equations (15.5.3.14) and (15.5.3.15) for C =–λ 2 < 0: f(x)=A 1 + A 2 x + A 3 cos(λx)+A 4 sin(λx), g(y)=B 1 + B 2 y + B 3 cos(λy)+B 4 sin(λy). (15.5.3.18) Substituting (15.5.3.18) into (15.5.3.12) does not yield new real solutions. Degenerate cases. If f xx ≡ 0 or g yy ≡ 0, equation (15.5.3.13) becomes an identity for any g = g(y)or f = f (x), respectively. These cases should be treated separately from the nondegenerate case. For example, if f xx ≡ 0,wehavef(x)=Ax +B,whereA and B are arbitrary numbers. Substituting this f into (15.5.3.12), we arrive at the equation –Ag yyy = ag yyyy . Its general solution is given by g(y)=C 1 exp(–Ay/a)+C 2 y 2 +C 3 y+C 4 . Thus, we obtain another solution of the form (15.5.3.11) for equation (15.5.3.10): w = C 1 e –λy + C 2 y 2 + C 3 y + C 4 + aλx (A = aλ, B = 0). 15.5.4. Solution of Functional-Differential Equations by Splitting 15.5.4-1. Preliminary remarks. Description of the method. As one reduces the number of terms in the functional differential equation (15.5.1.3)– (15.5.1.4) by differentiation, redundant constants of integration arise. These constants must be “removed” at the final stage. Furthermore, the resulting equation can be of a higher order than the original equation. To avoid these difficulties, it is convenient to reduce the solution of the functional differential equation to the solution of a bilinear functional equation of a standard form and solution of a system of ordinary differential equations. Thus, the original problem splits into two simpler problems. Below we outline the basic stages of the splitting method. 1 ◦ .Atthefirst stage, we treat equation (15.5.1.3) as a purely functional equation that depends on two variables X and Y ,whereΦ n = Φ n (X)andΨ n = Ψ n (Y ) are unknown quantities (n = 1, , k). It can be shown* that the bilinear functional equation (15.5.1.3) has k – 1 different solutions: Φ i (X)=C i,1 Φ m+1 (X)+C i,2 Φ m+2 (X)+···+ C i,k–m Φ k (X), i = 1, , m; Ψ m+j (Y )=–C 1,j Ψ 1 (Y )–C 2,j Ψ 2 (Y )–···– C m,j Ψ m (Y ), j = 1, , k – m; m = 1, 2, , k – 1; (15.5.4.1) where C i,j are arbitrary constants. The functions Φ m+1 (X), , Φ k (X), Ψ 1 (Y ), , Ψ m (Y ) on the right-hand sides of formulas (15.5.4.1) are defined arbitrarily. It is apparent that for fixed m, solution (15.5.4.1) contains m(k – m) arbitrary constants. * These solutions can be obtained by differentiation following the procedure outlined in Subsection 15.5.3, and by induction. Another simple method for finding solutions is described in Paragraph 15.5.4-2, Item 3 ◦ . . aforesaid and formulas (15.5.3.3) and (15.5.3.6), we arrive at a solution of equation (15.5.3.2) of the form (15.5.3.3): w(x, y)=ϕ(x)e λy – aλx + C, where ϕ(x) is an arbitrary function and C and λ. solutions of the form (15.5.1.1) with n = 1 and n = 2 (for ψ 1 = ϕ 2 = 1)tosome nonlinear equations. Note that most common of the generalized separable solutions are solutions of the special form w(x,. are often used in applied and computational mathematics for constructing approximate solutions to differential equations by the Galerkin method (and its modifications). 15.5.1-2. General form of