Handbook of mathematics for engineers and scienteists part 94 docx

14.7. BOUNDARY VALUE PROBLEMS FOR PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE 619 where L x,t [w] ≡ a(x, t) ∂ 2 w ∂x 2 + b(x, t) ∂w ∂x + c(x, t)w, a(x, t)>0.(14.7.1.2) Consider the nonstationary boundary value problem for equation (14.7.1.1) with an initial condition of general form w = f(x)att = 0,(14.7.1.3) and arbitrary nonhomogeneous linear boundary conditions α 1 ∂w ∂x + β 1 w = g 1 (t)atx = x 1 ,(14.7.1.4) α 2 ∂w ∂x + β 2 w = g 2 (t)atx = x 2 .(14.7.1.5) By appropriately choosing the coefficients α 1 , α 2 , β 1 ,andβ 2 in (14.7.1.4) and (14.7.1.5), we obtain the first, second, third, and mixed boundary value problems for equation (14.7.1.1). 14.7.1-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem (14.7.1.1)–(14.7.1.5) can be represented as w(x, t)=  t 0  x 2 x 1 Φ(y, τ)G(x, y, t, τ) dy dτ +  x 2 x 1 f(y)G(x, y, t, 0) dy +  t 0 g 1 (τ)a(x 1 , τ)Λ 1 (x, t, τ ) dτ +  t 0 g 2 (τ)a(x 2 , τ)Λ 2 (x, t, τ ) dτ .(14.7.1.6) Here, G(x, y, t, τ ) is the Green’s function that satisfies, for t > τ ≥ 0, the homogeneous equation ∂G ∂t – L x,t [G]=0 (14.7.1.7) with the nonhomogeneous initial condition of special form G = δ(x – y)att = τ (14.7.1.8) and the homogeneous boundary conditions α 1 ∂G ∂x + β 1 G = 0 at x = x 1 ,(14.7.1.9) α 2 ∂G ∂x + β 2 G = 0 at x = x 2 .(14.7.1.10) The quantities y and τ appear in problem (14.7.1.7)–(14.7.1.10) as free parameters, with x 1 ≤ y ≤ x 2 ,andδ(x) is the Dirac delta function. The initial condition (14.7.1.8) implies the limit relation f(x) = lim t→τ  x 2 x 1 f(y)G(x, y, t, τ ) dy for any continuous function f = f(x). 620 LINEAR PARTIAL DIFFERENTIAL EQUATIONS TABLE 14.6 Expressions of the functions Λ 1 (x, t, τ)andΛ 2 (x, t, τ)involved in the integrands of the last two terms in solution (14.7.1.6) Type of problem Form of boundary conditions Functions Λ m (x, t, τ ) First boundary value problem (α 1 = α 2 = 0, β 1 = β 2 = 1) w = g 1 (t)atx = x 1 w = g 2 (t)atx = x 2 Λ 1 (x, t, τ)=∂ y G(x, y, t, τ )   y=x 1 Λ 2 (x, t, τ)=–∂ y G(x, y, t, τ )   y=x 2 Second boundary value problem (α 1 = α 2 = 1, β 1 = β 2 = 0) ∂ x w = g 1 (t)atx = x 1 ∂ x w = g 2 (t)atx = x 2 Λ 1 (x, t, τ)=–G(x, x 1 , t, τ) Λ 2 (x, t, τ)=G(x, x 2 , t, τ ) Third boundary value problem (α 1 = α 2 = 1, β 1 < 0, β 2 > 0) ∂ x w + β 1 w = g 1 (t)atx = x 1 ∂ x w + β 2 w = g 2 (t)atx = x 2 Λ 1 (x, t, τ)=–G(x, x 1 , t, τ) Λ 2 (x, t, τ)=G(x, x 2 , t, τ ) Mixed boundary value problem (α 1 = β 2 = 0, α 2 = β 1 = 1) w = g 1 (t)atx = x 1 ∂ x w = g 2 (t)atx = x 2 Λ 1 (x, t, τ)=∂ y G(x, y, t, τ )   y=x 1 Λ 2 (x, t, τ)=G(x, x 2 , t, τ) Mixed boundary value problem (α 1 = β 2 = 1, α 2 = β 1 = 0) ∂ x w = g 1 (t)atx = x 1 w = g 2 (t)atx = x 2 Λ 1 (x, t, τ)=–G(x, x 1 , t, τ ) Λ 2 (x, t, τ)=–∂ y G(x, y, t, τ )   y=x 2 The functions Λ 1 (x, t, τ )andΛ 2 (x, t, τ ) involved in the integrands of the last two terms in solution (14.7.1.6) can be expressed in terms of the Green’s function G(x, y, t, τ). The corresponding formulas for Λ m (x, t, τ ) are given in Table 14.6 for the basic types of boundary value problems. It is significant that the Green’s function G and the functions Λ 1 , Λ 2 are independent of the functions Φ, f , g 1 ,andg 2 that characterize various nonhomogeneities of the boundary value problem. If the coefficients of equation (14.7.1.1)–(14.7.1.2) are independent of time t, i.e., the conditions a = a(x), b = b(x), c = c(x)(14.7.1.11) hold, then the Green’s function depends on only three arguments, G(x, y, t, τ)=G(x, y, t – τ). In this case, the functions Λ m depend on only two arguments, Λ m = Λ m (x, t – τ ), m = 1, 2. Formula (14.7.1.6) also remains valid for the problem with boundary conditions of the third kind if β 1 = β 1 (t)andβ 2 = β 2 (t). Here, the relation between Λ m (m = 1, 2)and the Green’s function G is the same as that in the case of constants β 1 and β 2 ; the Green’s function itself is now different. The condition that the solution must vanish at infinity, w → 0 as x →∞,isoftenset for the first, second, and third boundary value problems that are considered on the interval x 1 ≤ x < ∞. In this case, the solution is calculated by formula (14.7.1.6) with Λ 2 = 0 and Λ 1 specified in Table 14.6. 14.7.2. Problems for Equation s(x) ∂w ∂t = ∂ ∂x  p(x) ∂w ∂x  – q(x)w + Φ(x, t) 14.7.2-1. General formulas for solving nonhomogeneous boundary value problems. Consider linear equations of the special form s(x) ∂w ∂t = ∂ ∂x  p(x) ∂w ∂x  – q(x)w + Φ(x, t). (14.7.2.1) 14.7. BOUNDARY VALUE PROBLEMS FOR PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE 621 TABLE 14.7 Expressions of the functions Λ 1 (x, t)andΛ 2 (x, t) involved in the integrands of the last two terms in solutions (14.7.2.2) and (14.8.2.2); the modified Green’s function G(x, ξ, t) for parabolic equations of the form (14.7.2.1) are found by formula (14.7.2.3), and that for hyperbolic equations of the form (14.8.2.1), by formula (14.8.2.3) Type of problem Form of boundary conditions Functions Λ m (x, t) First boundary value problem (α 1 = α 2 = 0, β 1 = β 2 = 1) w = g 1 (t)atx = x 1 w = g 2 (t)atx = x 2 Λ 1 (x, t)=∂ ξ G(x, ξ, t)   ξ=x 1 Λ 2 (x, t)=–∂ ξ G(x, ξ, t)   ξ=x 2 Second boundary value problem (α 1 = α 2 = 1, β 1 = β 2 = 0) ∂ x w = g 1 (t)atx = x 1 ∂ x w = g 2 (t)atx = x 2 Λ 1 (x, t)=–G(x, x 1 , t) Λ 2 (x, t)=G(x, x 2 , t) Third boundary value problem (α 1 = α 2 = 1, β 1 < 0, β 2 > 0) ∂ x w + β 1 w = g 1 (t)atx = x 1 ∂ x w + β 2 w = g 2 (t)atx = x 2 Λ 1 (x, t)=–G(x, x 1 , t) Λ 2 (x, t)=G(x, x 2 , t) Mixed boundary value problem (α 1 = β 2 = 0, α 2 = β 1 = 1) w = g 1 (t)atx = x 1 ∂ x w = g 2 (t)atx = x 2 Λ 1 (x, t)=∂ ξ G(x, ξ, t)   ξ=x 1 Λ 2 (x, t)=G(x, x 2 , t) Mixed boundary value problem (α 1 = β 2 = 1, α 2 = β 1 = 0) ∂ x w = g 1 (t)atx = x 1 w = g 2 (t)atx = x 2 Λ 1 (x, t)=–G(x, x 1 , t) Λ 2 (x, t)=–∂ ξ G(x, ξ, t)   ξ=x 2 They are often encountered in heat and mass transfer theory and chemical engineering sciences. Throughout this subsection, we assume that the functions s, p, p  x ,andq are continuous and s > 0, p > 0,andx 1 ≤ x ≤ x 2 . The solution of equation (14.7.2.1) under the initial condition (14.7.1.3) and the arbitrary linear nonhomogeneous boundary conditions (14.7.1.4)–(14.7.1.5) can be represented as the sum w(x, t)=  t 0  x 2 x 1 Φ(ξ, τ )G(x, ξ, t – τ)dξ dτ +  x 2 x 1 s(ξ)f(ξ)G(x, ξ, t) dξ + p(x 1 )  t 0 g 1 (τ)Λ 1 (x, t – τ)dτ + p(x 2 )  t 0 g 2 (τ)Λ 2 (x, t – τ) dτ. (14.7.2.2) Here, the modified Green’s function is given by G(x, ξ, t)= ∞  n=1 y n (x)y n (ξ) y n  2 exp(–λ n t), y n  2 =  x 2 x 1 s(x)y 2 n (x) dx,(14.7.2.3) where the λ n and y n (x) are the eigenvalues and corresponding eigenfunctions of the fol- lowing Sturm–Liouville problem for a second-order linear ordinary differential equation: [p(x)y  x ]  x +[λs(x)–q(x)]y = 0, α 1 y  x + β 1 y = 0 at x = x 1 , α 2 y  x + β 2 y = 0 at x = x 2 . (14.7.2.4) The functions Λ 1 (x, t)andΛ 2 (x, t) that occur in the integrands of the last two terms in solution (14.7.2.2) are expressed via the Green’s function (14.7.2.3). The corresponding formulas for Λ m (x, t) are given in Table 14.7 for the basic types of boundary value problems. 622 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 14.7.2-2. Properties of Sturm–Liouville problem (14.7.2.4). Heat equation example. 1 ◦ .Thereareinfinitely many eigenvalues. All eigenvalues are real and different and can be ordered so that λ 1 < λ 2 < λ 3 < ···, with λ n →∞as n →∞(therefore, there can exist only finitely many negative eigenvalues). Each eigenvalue is of multiplicity 1. 2 ◦ . The different eigenfunctions y n (x)andy m (x) are orthogonal with weight s(x)onthe interval x 1 ≤ x ≤ x 2 :  x 2 x 1 s(x)y n (x)y m (x) dx = 0 for n ≠ m. 3 ◦ . If the conditions q(x) ≥ 0, α 1 β 1 ≤ 0, α 2 β 2 ≥ 0 (14.7.2.5) are satisfied, there are no negative eigenvalues. If q ≡ 0 and β 1 = β 2 = 0,thenλ 1 = 0 is the least eigenvalue, to which there corresponds the eigenfunction ϕ 1 = const. Otherwise, all eigenvalues are positive, provided that conditions (14.7.2.5) are satisfied. Other general and special properties of the Sturm–Liouville problem (14.7.2.4) are given in Subsection 12.2.5; various asymptotic and approximate formulas for eigenvalues and eigenfunctions can also be found there. Example. Consider the first boundary value problem in the domain 0 ≤ x ≤ l for the heat equation with a source ∂w ∂t = a ∂ 2 w ∂x 2 – bw under the initial condition (14.7.1.3) and boundary conditions w = g 1 (t)atx = 0, w = g 2 (t)atx = l. (14.7.2.6) The above equation is a special case of equation (14.7.2.1) with s(x)=1, p(x)=a, q(x)=b,andΦ(x, t)=0. The corresponding Sturm–Liouville problem (14.7.2.4) has the form ay  xx +(λ – b)y = 0, y = 0 at x = 0, y = 0 at x = l. The eigenfunctions and eigenvalues are found to be y n (x)=sin  πnx l  , λ n = b + aπ 2 n 2 l 2 , n = 1, 2, Using formula (14.7.2.3) and taking into account that y n  2 = l/2, we obtain the Green’s function G(x, ξ, t)= 2 l e –bt ∞  n=1 sin  πnx l  sin  πnξ l  exp  – aπ 2 n 2 l 2 t  . Substituting this expression into (14.7.2.2) with p(x 1 )=p(x 2 )=s(ξ)=1, x 1 = 0,andx 2 = l and taking into account the formulas Λ 1 (x, t)=∂ ξ G(x, ξ, t)   ξ=x 1 , Λ 2 (x, t)=–∂ ξ G(x, ξ, t)   ξ=x 2 (see the first row in Table 14.7), one obtains the solution to the problem in question.  Solutions to various boundary value problems for parabolic equations can be found in Section T8.1. 14.8. BOUNDARY VALUE PROBLEMS FOR HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE 623 14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s Function. Goursat Problem 14.8.1. Representation of Solutions via the Green’s Function 14.8.1-1. Statement of the problem (t ≥ 0, x 1 ≤ x ≤ x 2 ). In general, a one-dimensional nonhomogeneous linear differential equation of hyperbolic type with variable coefficients is written as ∂ 2 w ∂t 2 + ϕ(x, t) ∂w ∂t – L x,t [w]=Φ(x, t), (14.8.1.1) where the operator L x,t [w]isdefined by (14.7.1.2). Consider the nonstationary boundary value problem for equation (14.8.1.1) with the initial conditions w = f 0 (x)att = 0, ∂ t w = f 1 (x)att = 0 (14.8.1.2) and arbitrary nonhomogeneous linear boundary conditions α 1 ∂w ∂x + β 1 w = g 1 (t)atx = x 1 ,(14.8.1.3) α 2 ∂w ∂x + β 2 w = g 2 (t)atx = x 2 .(14.8.1.4) 14.8.1-2. Representation of the problem solution in terms of the Green’s function. The solution of problem (14.8.1.1), (14.8.1.2), (14.8.1.3), (14.8.1.4) can be represented as the sum w(x, t)=  t 0  x 2 x 1 Φ(y, τ )G(x, y, t, τ) dy dτ –  x 2 x 1 f 0 (y)  ∂ ∂τ G(x, y, t, τ)  τ=0 dy +  x 2 x 1  f 1 (y)+f 0 (y)ϕ(y, 0)  G(x, y, t, 0) dy +  t 0 g 1 (τ)a(x 1 , τ)Λ 1 (x, t, τ ) dτ +  t 0 g 2 (τ)a(x 2 , τ)Λ 2 (x, t, τ ) dτ .(14.8.1.5) Here, the Green’s function G(x, y, t, τ) is determined by solving the homogeneous equation ∂ 2 G ∂t 2 + ϕ(x, t) ∂G ∂t – L x,t [G]=0 (14.8.1.6) with the semihomogeneous initial conditions G = 0 at t = τ, ∂ t G = δ(x – y)att = τ , (14.8.1.7) (14.8.1.8) 624 LINEAR PARTIAL DIFFERENTIAL EQUATIONS and the homogeneous boundary conditions α 1 ∂G ∂x + β 1 G = 0 at x = x 1 ,(14.8.1.9) α 2 ∂G ∂x + β 2 G = 0 at x = x 2 .(14.8.1.10) The quantities y and τ appear in problem (14.8.1.6)–(14.8.1.8), (14.8.1.9), (14.8.1.10) as free parameters (x 1 ≤ y ≤ x 2 ), and δ(x) is the Dirac delta function. The functions Λ 1 (x, t, τ )andΛ 2 (x, t, τ ) involved in the integrands of the last two terms in solution (14.8.1.5) can be expressed via the Green’s function G(x, y, t, τ). The corresponding formulas for Λ m (x, t, τ ) are given in Table 14.6 for the basic types of boundary value problems. It is significant that the Green’s function G and Λ 1 , Λ 2 are independent of the functions Φ, f 0 , f 1 , g 1 ,andg 2 that characterize various nonhomogeneities of the boundary value problem. If the coefficients of equation (14.8.1.1) 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 (14.8.1.5). 14.8.2. Problems for Equation s(x) ∂ 2 w ∂t 2 = ∂ ∂x  p(x) ∂w ∂x  – q(x)w + Φ(x, t) 14.8.2-1. General relations for solving nonhomogeneous boundary value problems. Consider linear equations of the special form s(x) ∂ 2 w ∂t 2 = ∂ ∂x  p(x) ∂w ∂x  – q(x)w + Φ(x, t). (14.8.2.1) It is assumed that the functions s, p, p  x ,andq are continuous and the inequalities s > 0, p > 0 hold for x 1 ≤ x ≤ x 2 . The solution of equation (14.8.2.1) under the general initial conditions (14.8.1.2) and the arbitrary linear nonhomogeneous boundary conditions (14.8.1.3)–(14.8.1.4) can be represented as the sum w(x, t)=  t 0  x 2 x 1 Φ(ξ, τ )G(x, ξ, t – τ)dξ dτ + ∂ ∂t  x 2 x 1 s(ξ)f 0 (ξ)G(x, ξ, t) dξ +  x 2 x 1 s(ξ)f 1 (ξ)G(x, ξ, t) dξ + p(x 1 )  t 0 g 1 (τ)Λ 1 (x, t – τ) dτ + p(x 2 )  t 0 g 2 (τ)Λ 2 (x, t – τ) dτ.(14.8.2.2) Here, the modified Green’s function is determined by G(x, ξ, t)= ∞  n=1 y n (x)y n (ξ)sin  t √ λ n  y n  2 √ λ n , y n  2 =  x 2 x 1 s(x)y 2 n (x) dx,(14.8.2.3) 14.8. BOUNDARY VALUE PROBLEMS FOR HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE 625 where the λ n and y n (x) are the eigenvalues and corresponding eigenfunctions of the Sturm– Liouville problem for the second-order linear ordinary differential equation [p(x)y  x ]  x +[λs(x)–q(x)]y = 0, α 1 y  x + β 1 y = 0 at x = x 1 , α 2 y  x + β 2 y = 0 at x = x 2 . (14.8.2.4) The functions Λ 1 (x, t)andΛ 2 (x, t) that occur in the integrands of the last two terms in solution (14.8.2.2) are expressed in terms of the Green’s function of (14.8.2.3). The corresponding formulas for Λ m (x, t) are given in Table 14.7 for the basic types of boundary value problems. 14.8.2-2. Properties of the Sturm–Liouville problem. The Klein–Gordon equation. The general and special properties of the Sturm–Liouville problem (14.8.2.4) are given in Subsection 12.2.5; various asymptotic and approximate formulas for eigenvalues and eigenfunctions can also be found there. Example. Consider the second boundary value problem in the domain 0 ≤ x ≤ l for the Klein–Gordon equation ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 – bw, under the initial conditions (14.8.1.2) and boundary conditions ∂ x w = g 1 (t)atx = 0, ∂ x w = g 2 (t)atx = l. The Klein–Gordon equation is a special case of equation (14.8.2.1) with s(x)=1, p(x)=a 2 , q(x)=b,and Φ(x, t)=0. The corresponding Sturm–Liouville problem (14.8.2.4) has the form a 2 y  xx +(λ – b)y = 0, y  x = 0 at x = 0, y  x = 0 at x = l. The eigenfunctions and eigenvalues are found to be y n+1 (x)=cos  πnx l  , λ n+1 = b + aπ 2 n 2 l 2 , n = 0, 1, Using formula (14.8.2.4) and taking into account that y 1  2 = l and y n  2 = l/2 (n = 1, 2, ), we find the Green’s function: G(x, ξ, t)= 1 l √ b sin  t √ b  + 2 l ∞  n=1 cos  πnx l  cos  πnξ l  sin  t  (aπn/l) 2 + b   (aπn/l) 2 + b . Substituting this expression into (14.8.2.3) with p(x 1 )=p(x 2 )=s(ξ)=1, x 1 = 0,andx 2 = l and taking into account the formulas Λ 1 (x, t)=–G(x, x 1 , t), Λ 2 (x, t)=G(x, x 2 , t) (see the second row in Table 14.7), one obtains the solution to the problem in question.  Solutions to various boundary value problems for hyperbolic equations can be found in Section T8.2. . of the form (14.7.2.1) are found by formula (14.7.2.3), and that for hyperbolic equations of the form (14.8.2.1), by formula (14.8.2.3) Type of problem Form of boundary conditions Functions Λ m (x,. Λ 1 (x, t )and 2 (x, t) involved in the integrands of the last two terms in solutions (14.7.2.2) and (14.8.2.2); the modified Green’s function G(x, ξ, t) for parabolic equations of the form (14.7.2.1). Λ 1 (x, t )and 2 (x, t) that occur in the integrands of the last two terms in solution (14.8.2.2) are expressed in terms of the Green’s function of (14.8.2.3). The corresponding formulas for Λ m (x,