14.9. BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES 633 Example. Consider a boundary value problem for the Laplace equation ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = 0 in a strip 0 ≤ x ≤ l,–∞ < y < ∞ with mixed boundary conditions w = f 1 (y)atx = 0, ∂w ∂x = f 2 (y)atx = l. This equation is a special case of equation (14.9.1.1) with a(x)=1 and b(x)=c(x)=Φ(x, t)=0.The corresponding Sturm–Liouville problem (14.9.1.5)–(14.9.1.7) is written as u  xx + λy = 0, u = 0 at x = 0, u  x = 0 at x = l. The eigenfunctions and eigenvalues are found as u n (x)=sin  π(2n – 1)x l  , λ n = π 2 (2n – 1) 2 l 2 , n = 1, 2, Using formulas (14.9.1.3) and (14.9.1.4) and taking into account the identities ρ(ξ)=1 and y n  2 = l/2 (n = 1, 2, ) and the expression for Ψ n from the first row in Table 14.8, we obtain the Green’s function in the form G(x, y, ξ, η)= 1 l ∞  n=1 1 σ n sin(σ n x)sin(σ n ξ)e –σ n |y–η| , σ n = √ λ n = π(2n – 1) l . 14.9.2. Representation of Solutions to Boundary Value Problems via the Green’s Functions 14.9.2-1. First boundary value problem. The solution of the first boundary value problem for equation (14.9.1.1) with the boundary conditions w = f 1 (y)atx = x 1 , w = f 2 (y)atx = x 2 , w = f 3 (x)aty = 0, w = f 4 (x)aty = h is expressed in terms of the Green’s function as w(x, y)=a(x 1 )  h 0 f 1 (η)  ∂ ∂ξ G(x, y, ξ, η)  ξ=x 1 dη – a(x 2 )  h 0 f 2 (η)  ∂ ∂ξ G(x, y, ξ, η)  ξ=x 2 dη +  x 2 x 1 f 3 (ξ)  ∂ ∂η G(x, y, ξ, η)  η=0 dξ –  x 2 x 1 f 4 (ξ)  ∂ ∂η G(x, y, ξ, η)  η=h dξ +  x 2 x 1  h 0 Φ(ξ, η)G(x, y, ξ, η) dη dξ. 14.9.2-2. Second boundary value problem. The solution of the second boundary value problem for equation (14.9.1.1) with boundary conditions ∂ x w = f 1 (y)atx = x 1 , ∂ x w = f 2 (y)atx = x 2 , ∂ y w = f 3 (x)aty = 0, ∂ y w = f 4 (x)aty = h 634 LINEAR PARTIAL DIFFERENTIAL EQUATIONS is expressed in terms of the Green’s function as w(x, y)=–a(x 1 )  h 0 f 1 (η)G(x, y, x 1 , η) dη + a(x 2 )  h 0 f 2 (η)G(x, y, x 2 , η) dη –  x 2 x 1 f 3 (ξ)G(x, y, ξ, 0) dξ +  x 2 x 1 f 4 (ξ)G(x, y, ξ, h) dξ +  x 2 x 1  h 0 Φ(ξ, η)G(x, y, ξ, η) dη dξ. 14.9.2-3. Third boundary value problem. The solution of the third boundary value problem for equation (14.9.1.1) in terms of the Green’s function is represented in the same way as the solution of the second boundary value problem (the Green’s function is now different).  Solutions of various boundary value problems for elliptic equations can be found in Section T8.3. 14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green’s Function 14.10.1. Problems for Parabolic Equations 14.10.1-1. Statement of the problem. In general, a nonhomogeneous linear differential equation of the parabolic type in n space variables has the form ∂w ∂t – L x,t [w]=Φ(x, t), (14.10.1.1) where L x,t [w] ≡ n  i,j=1 a ij (x, t) ∂ 2 w ∂x i ∂x j + n  i=1 b i (x, t) ∂w ∂x i + c(x, t)w, x = {x 1 , , x n }, n  i,j=1 a ij (x, t)ξ i ξ j ≥ σ n  i=1 ξ 2 i , σ > 0. (14.10.1.2) Let V be some simply connected domain in R n with a sufficiently smooth boundary S = ∂V . We consider the nonstationary boundary value problem for equation (14.10.1.1) in the domain V with an arbitrary initial condition, w = f(x)att = 0,(14.10.1.3) and nonhomogeneous linear boundary conditions, Γ x,t [w]=g(x, t)forx S.(14.10.1.4) In the general case, Γ x,t is a fi rst-order linear differential operator in the space coordinates with coefficients dependent on x and t. 14.10. BOUNDARY VALUE PROBLEMS WITH MANY SPACE VARIABLES 635 14.10.1-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem defined by (14.10.1.1)– (14.10.1.4) can be represented as the sum w(x, t)=  t 0  V Φ(y, τ )G(x, y, t, τ)dV y dτ +  V f(y)G(x, y, t, 0) dV y +  t 0  S g(y, τ)H(x, y, t, τ ) dS y dτ,(14.10.1.5) where G(x, y, t, τ) is the Green’s function; for t > τ ≥ 0, it satisfies the homogeneous equation ∂G ∂t – L x,t [G]=0 (14.10.1.6) with the nonhomogeneous initial condition of special form G = δ(x – y)att = τ (14.10.1.7) and the homogeneous boundary condition Γ x,t [G]=0 for x S.(14.10.1.8) The vector y= {y 1 , , y n } appears in problem (14.10.1.6)–(14.10.1.8) as an n-dimensional free parameter (y V ), and δ(x – y)=δ(x 1 – y 1 ) δ(x n – y n )isthen-dimensional Dirac delta function. The Green’s function G is independent of the functions Φ, f,andg that characterize various nonhomogeneities of the boundary value problem. In (14.10.1.5), the integration is performed everywhere with respect to y, with dV y = dy 1 dy n . The function H(x, y, t, τ) involved in the integrand of the last term in solution (14.10.1.5) can be expressed via the Green’s function G(x, y, t, τ). The corresponding formulas for H(x, y, t, τ ) are given in Table 14.9 for the three basic types of boundary value problems; in the third boundary value problem, the coefficient k can depend on x and t. The boundary conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (14.10.1.2); these operators act as follows: ∂G ∂M x ≡ n  i,j=1 a ij (x, t)N j ∂G ∂x i , ∂G ∂M y ≡ n  i,j=1 a ij (y, τ )N j ∂G ∂y i ,(14.10.1.9) where N = {N 1 , , N n } is the unit outward normal to the surface S. In the special case where a ii (x, t)=1 and a ij (x, t)=0 for i ≠ j, operator (14.10.1.9) coincides with the ordinary operator of differentiation along the outward normal to S. TABLE 14.9 The form of the function H(x, y, t, τ) for the basic types of nonstationary boundary value problems Type of problem Form of boundary condition (14.10.1.4) Function H(x, y, t, τ) First boundary value problem w = g(x, t)forx S H(x, y, t, τ)=– ∂G ∂M y (x, y, t, τ) Second boundary value problem ∂w ∂M x = g(x, t)forx S H(x, y, t, τ)=G(x, y, t, τ) Third boundary value problem ∂w ∂M x + kw = g(x, t)forx S H(x, y, t, τ)=G(x, y, t, τ) 636 LINEAR PARTIAL DIFFERENTIAL EQUATIONS If the coefficient of equation (14.10.1.6) and the boundary condition (14.10.1.8) are independent of t, then the Green’s function depends on only three arguments, G(x, y, t, τ)= G(x, y, t – τ). Remark. Let S i (i = 1, , p) be different portions of the surface S such that S = p  i=1 S i and let boundary conditions of various types be set on the S i , Γ (i) x,t [w]=g i (x, t)forx S i , i = 1, , p.(14.10.1.10) Then formula (14.10.1.5) remains valid but the last term in (14.10.1.5) must be replaced by the sum p  i=1  t 0  S i g i (y, τ)H i (x, y, t, τ) dS y dτ.(14.10.1.11) 14.10.2. Problems for Hyperbolic Equations 14.10.2-1. Statement of the problem. The general nonhomogeneous linear differential hyperbolic equation in n space variables can be written as ∂ 2 w ∂t 2 + ϕ(x, t) ∂w ∂t – L x,t [w]=Φ(x, t), (14.10.2.1) where the operator L x,t [w] is explicitly defined in (14.10.1.2). We consider the nonstationary boundary value problem for equation (14.10.2.1) in the domain V with arbitrary initial conditions, w = f 0 (x)att = 0, ∂ t w = f 1 (x)att = 0, (14.10.2.2) (14.10.2.3) and the nonhomogeneous linear boundary condition (14.10.1.4). 14.10.2-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem defined by (14.10.2.1)– (14.10.2.3), (14.10.1.4) can be represented as the sum w(x, t)=  t 0  V Φ(y, τ )G(x, y, t, τ)dV y dτ –  V f 0 (y)  ∂ ∂τ G(x, y, t, τ)  τ=0 dV y +  V  f 1 (y)+f 0 (y)ϕ(y, 0)  G(x, y, t, 0) dV y +  t 0  S g(y, τ)H(x, y, t, τ ) dS y dτ.(14.10.2.4) Here, G(x, y, t, τ) is the Green’s function; for t > τ ≥ 0 it satisfies the homogeneous equation ∂ 2 G ∂t 2 + ϕ(x, t) ∂G ∂t – L x,t [G]=0 (14.10.2.5) with the semihomogeneous initial conditions G = 0 at t = τ, ∂ t G = δ(x – y)att = τ, and the homogeneous boundary condition (14.10.1.8). 14.10. BOUNDARY VALUE PROBLEMS WITH MANY SPACE VARIABLES 637 If the coefficients of equation (14.10.2.5) and the boundary condition (14.10.1.8) are independent of time t, then the Green’s function depends on only three arguments, G(x, y, t, τ)=G(x, y, t – τ). In this case, one can set ∂ ∂τ G(x, y, t, τ)   τ=0 =– ∂ ∂t G(x, y, t)in solution (14.10.2.4). The function H(x, y, t, τ) involved in the integrand of the last term in solution (14.10.2.4) can be expressed via the Green’s function G(x, y, t, τ ). The corresponding formulas for H are given in Table 14.9 for the three basic types of boundary value problems; in the third boundary value problem, the coefficient k can depend on x and t. Remark. Let S i (i = 1, , p) be different portions of the surface S such that S = p  i=1 S i and let boundary conditions of various types (14.10.1.10) be set on the S i . Then formula (14.10.2.4) remains valid but the last term in (14.10.2.4) must be replaced by the sum (14.10.1.11). 14.10.3. Problems for Elliptic Equations 14.10.3-1. Statement of the problem. In general, a nonhomogeneous linear elliptic equation can be written as –L x [w]=Φ(x), (14.10.3.1) where L x [w] ≡ 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.(14.10.3.2) Two-dimensional problems correspond to n = 2 and three-dimensional problems to n = 3. We consider equation (14.10.3.1)–(14.10.3.2) in a domain V and assume that the equa- tion is subject to the general linear boundary condition Γ x [w]=g(x)forx S.(14.10.3.3) The solution of the stationary problem (14.10.3.1)–(14.10.3.3) can be obtained by passing in (14.10.1.5) to the limit as t →∞. To this end, one should start with equa- tion (14.10.1.1), whose coefficients are independent of t, and take the homogeneous initial condition (14.10.1.3), with f(x)=0, and the stationary boundary condition (14.10.1.4). 14.10.3-2. Representation of the problem solution in terms of the Green’s function. The solution of the linear boundary value problem (14.10.3.1)–(14.10.3.3) can be repre- sented as the sum w(x)=  V Φ(y)G(x, y) dV y +  S g(y)H(x, y) dS y .(14.10.3.4) Here, the Green’s function G(x, y) satisfi es the nonhomogeneous equation of special form –L x [G]=δ(x – y)(14.10.3.5) with the homogeneous boundary condition Γ x [G]=0 for x S.(14.10.3.6) 638 LINEAR PARTIAL DIFFERENTIAL EQUATIONS The vector y = {y 1 , , y n } appears in problem (14.10.3.5), (14.10.3.6) as an n-dimensional free parameter (y V ). Note that G is independent of the functions Φ and g characterizing various nonhomogeneities of the original boundary value problem. The function H(x, y) involved in the integrand of the second term in solution (14.10.3.4) can be expressed via the Green’s function G(x, y). The corresponding formulas for H are given in Table 14.10 for the three basic types of boundary value problems. The boundary conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (14.10.3.2); these operators are defined by (14.10.1.9); in this case, the coefficients a ij depend on x only. TABLE 14.10 The form of the function H(x, y) involved in the integrand of the last term in solution (14.10.3.4) for the basic types of stationary boundary value problems Type of problem Form of boundary condition (14.10.3.3) Function H(x, y) First boundary value problem w = g(x)forx S H(x, y)=– ∂G ∂M y (x, y) Second boundary value problem ∂w ∂M x = g(x)forx S H(x, y)=G(x, y) Third boundary value problem ∂w ∂M x + kw = g(x)forx S H(x, y)=G(x, y) Remark. For the second boundary value problem with c(x) ≡ 0, the thus defined Green’s function must not necessarily exist; see Polyanin (2002). 14.10.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types Table 14.11 lists brief formulations of boundary value problems for second-order equations of elliptic, parabolic, and hyperbolic types. The coefficients of the differential operators L x and Γ x in the space variables x 1 , , x n are assumed to be independent of time t;these operators are the same for the problems under consideration. TABLE 14.11 Formulations of boundary value problems for equations of various types Type of equation Form of equation Initial conditions Boundary conditions Elliptic –L x [w]=Φ(x) not set Γ x [w]=g(x)forx S Parabolic ∂ t w – L x [w]=Φ(x, t) w = f(x)att = 0 Γ x [w]=g(x, t)forx S Hyperbolic ∂ tt w – L x [w]=Φ(x, t) w = f 0 (x)att = 0, ∂ t w = f 1 (x)att = 0 Γ x [w]=g(x, t)forx S Below are the respective general formulas defining the solutions of these problems with 14.11. CONSTRUCTION OF THE GREEN’S FUNCTIONS.GENERAL FORMULAS AND RELATIONS 639 zero initial conditions (f = f 0 = f 1 = 0): w 0 (x)=  V Φ(y)G 0 (x, y) dV y +  S g(y)H  G 0 (x, y)  dS y , w 1 (x, t)=  t 0  V Φ(y, τ )G 1 (x, y, t – τ) dV y dτ +  t 0  S g(y, τ)H  G 1 (x, y, t – τ)  dS y dτ, w 2 (x, t)=  t 0  V Φ(y, τ )G 2 (x, y, t – τ) dV y dτ +  t 0  S g(y, τ)H  G 2 (x, y, t – τ)  dS y dτ, where the G n are the Green’s functions, and the subscripts 0, 1,and2 refer to the elliptic, parabolic, and hyperbolic problem, respectively. All solutions involve the same opera- tor H[G]; it is explicitly defined in Subsections 14.10.1–14.10.3 (see also Section 14.7) for different boundary conditions. It is apparent that the solutions of the parabolic and hyperbolic problems with zero initial conditions have the same structure. The structure of the solution to the problem for a parabolic equation differs from that for an elliptic equation by the additional integration with respect to t. 14.11. Construction of the Green’s Functions. General Formulas and Relations 14.11.1. Green’s Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains 14.11.1-1. Expressions of the Green’s function in terms of infinite series. Table 14.12 lists the Green’s functions of boundary value problems for second-order equa- tions of various types in a bounded domain V. It is assumed that L x is a second-order linear self-adjoint differential operator (e.g., see Zwillinger, 1997) in the space variables x 1 , , x n ,andΓ x is a zeroth- or first-order linear boundary operator that can define a boundary condition of the first, second, or third kind; the coefficients of the operators L x and Γ x can depend on the space variables but are independent of time t. The coefficients λ k and the functions u k (x) are determined by solving the homogeneous eigenvalue problem L x [u]+λu = 0,(14.11.1.1) Γ x [u]=0 for x S.(14.11.1.2) It is apparent from Table 14.12 that, given the Green’s function in the problem for a parabolic (or hyperbolic) equation, one can easily construct the Green’s functions of the corresponding problems for elliptic and hyperbolic (or parabolic) equations. In particular, the Green’s function of the problem for an elliptic equation can be expressed via the Green’s function of the problem for a parabolic equation as follows: G 0 (x, y)=  ∞ 0 G 1 (x, y, t) dt.(14.11.1.3) Here, the fact that all λ k are positive is taken into account; for the second boundary value problem, it is assumed that λ = 0 is not an eigenvalue of problem (14.11.1.1)–(14.11.1.2). . Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types Table 14.11 lists brief formulations of boundary value problems for second-order equations of elliptic,. 14.11 Formulations of boundary value problems for equations of various types Type of equation Form of equation Initial conditions Boundary conditions Elliptic –L x [w]=Φ(x) not set Γ x [w]=g(x)forx. corresponding formulas for H are given in Table 14.10 for the three basic types of boundary value problems. The boundary conditions of the second and third kind, as well as the solution of the first