14.4. METHOD OF SEPARATION OF VARIABLES (FOURIER METHOD) 605 where ε ij (λ)= s i (ϕ j ) x + k i ϕ j x=x i . For system (14.4.1.10) to have nontrivial solutions, its determinant must be zero; we have ε 11 (λ)ε 22 (λ)–ε 12 (λ)ε 21 (λ)=0.(14.4.1.11) Solving the transcendental equation (14.4.1.11) for λ, one obtains the eigenvalues λ = λ n , where n = 1, 2, For these values of λ, equation (14.4.1.6) has nontrivial solutions, ϕ n (x)=ε 12 (λ n )ϕ 1 (x, λ n )–ε 11 (λ n )ϕ 2 (x, λ n ), (14.4.1.12) which are called eigenfunctions (these functions are defined up to a constant multiplier). To facilitate the further analysis, we represent equation (14.4.1.6) in the form [p(x)ϕ x ] x +[λρ(x)–q(x)]ϕ = 0,(14.4.1.13) where p(x)=exp b(x) a(x) dx , q(x)=– c(x) a(x) exp b(x) a(x) dx , ρ(x)= 1 a(x) exp b(x) a(x) dx . (14.4.1.14) It follows from the adopted assumptions (see the end of Paragraph 14.4.1-1) that p(x), p x (x), q(x), and ρ(x) are continuous functions, with p(x)>0 and ρ(x)>0. The eigenvalue problem (14.4.1.13), (14.4.1.8) is known to possess the following prop- erties: 1. All eigenvalues λ 1 , λ 2 , are real, and λ n →∞as n →∞; consequently, the number of negative eigenvalues is finite. 2. The system of eigenfunctions ϕ 1 (x), ϕ 2 (x), is orthogonal on the interval x 1 ≤ x ≤ x 2 with weight ρ(x), i.e., x 2 x 1 ρ(x)ϕ n (x)ϕ m (x) dx = 0 for n ≠ m.(14.4.1.15) 3. If q(x) ≥ 0, s 1 k 1 ≤ 0, s 2 k 2 ≥ 0,(14.4.1.16) there are no negative eigenvalues. If q ≡ 0 and k 1 = k 2 = 0, the least eigenvalue is λ 1 = 0 and the corresponding eigenfunction is ϕ 1 = const. Otherwise, all eigenvalues are positive, provided that conditions (14.4.1.16) are satisfied; the first inequality in (14.4.1.16) is satisfied if c(x) ≤ 0. Subsection 12.2.5 presents some estimates for the eigenvalues λ n and eigenfunc- tions ϕ n (x). The procedure of constructing solutions to nonstationary boundary value problems is further different for parabolic and hyperbolic equations; see Subsections 14.4.2 and 14.4.3 below for results (elliptic equations are treated in Subsection 14.4.4). 14.4.2. Problems for Parabolic Equations: Final Stage of Solution 14.4.2-1. Series solutions of boundary value problems for parabolic equations. Consider the problem for the parabolic equation ∂w ∂t = a(x) ∂ 2 w ∂x 2 + b(x) ∂w ∂x + c(x)+γ(t) w (14.4.2.1) 606 LINEAR PARTIAL DIFFERENTIAL EQUATIONS (this equation is obtained from (14.4.1.1) in the case α(t) ≡ 0 and β(t)=1) with homogeneous linear boundary conditions (14.4.1.2) and initial condition (14.4.1.3). First, one searches for particular solutions to equation (14.4.2.1) in the product form (14.4.1.5), where the function ϕ(x) is obtained by solving an eigenvalue problem for the ordinary differential equation (14.4.1.6) with the boundary conditions (14.4.1.8). The solution of equation (14.4.1.7) with α(t) ≡ 0 and β(t)=1 corresponding to the eigenvalues λ = λ n and satisfying the normalizing conditions ψ n (0)=1 has the form ψ n (t)=exp –λ n t + t 0 γ(ξ)dξ .(14.4.2.2) Then the solution of the nonstationary boundary value problem (14.4.2.1), (14.4.1.2), (14.4.1.3) is sought in the series form w(x, t)= ∞ n=1 A n ϕ n (x) ψ n (t), (14.4.2.3) where the A n are arbitrary constants and the functions w n (x, t)=ϕ n (x) ψ n (t) are particular solutions (14.4.2.1) satisfying the boundary conditions (14.4.1.2). By the principle of linear superposition, series (14.4.2.3) is also a solution of the original partial differential equation that satisfies the boundary conditions. To determine the coefficients A n , we substitute series (14.4.2.3) into the initial condi- tion (14.4.1.3), thus obtaining ∞ n=1 A n ϕ n (x)=f 0 (x). Multiplying this equation by ρ(x)ϕ n (x), where the weight function ρ(x)isdefined in (14.4.1.14), then integrating the resulting relation with respect to x over the interval x 1 ≤ x ≤ x 2 , and taking into account the properties (14.4.1.15), we find A n = 1 ϕ n 2 x 2 x 1 ρ(x)ϕ n (x)f 0 (x) dx, ϕ n 2 = x 2 x 1 ρ(x)ϕ 2 n (x) dx.(14.4.2.4) Relations (14.4.2.3), (14.4.2.2), (14.4.2.4), and (14.4.1.12) give a formal solution of the nonstationary boundary value problem (14.4.2.1), (14.4.1.2), (14.4.1.3). Example. Consider the first (Dirichlet) boundary value problem on the interval 0 ≤ x ≤ l for the heat equation ∂w ∂t = ∂ 2 w ∂x 2 (14.4.2.5) with the general initial condition (14.4.1.3) and the homogeneous boundary conditions w = 0 at x = 0, w = 0 at x = l.(14.4.2.6) The function ψ(t) in the particular solution (14.4.1.5) is found from (14.4.2.2), where γ(t)=0: ψ n (t)=exp(–λ n t). (14.4.2.7) The functions ϕ n (x) are determined by solving the eigenvalue problem (14.4.1.6), (14.4.1.8) with a(x)=1, b(x)=c(x)=0, s 1 = s 2 = 0, k 1 = k 2 = 1, x 1 = 0,andx 2 = l: ϕ xx + λϕ = 0; ϕ = 0 at x = 0, ϕ = 0 at x = l. 14.4. METHOD OF SEPARATION OF VARIABLES (FOURIER METHOD) 607 So we obtain the eigenfunctions and eigenvalues: ϕ n (x)=sin nπx l , λ n = nπ l 2 , n = 1, 2, (14.4.2.8) The solution to problem (14.4.2.5)–(14.4.2.6), (14.4.1.3) is given by formulas (14.4.2.3), (14.4.2.4). Taking into account that ϕ n 2 = l/2, we obtain w(x, t)= ∞ n=1 A n sin nπx l exp – n 2 π 2 t l 2 , A n = 2 l l 0 f 0 (ξ)sin nπξ l dξ.(14.4.2.9) If the function f 0 (x) is twice continuously differentiable and the compatibility conditions (see Para- graph 14.4.2-2) are satisfied, then series (14.4.2.9) is convergent and admits termwise differentiation, once with respect to t and twice with respect to x. In this case, formula (14.4.2.9) gives the classical smooth solution of problem (14.4.2.5)–(14.4.2.6), (14.4.1.3). [If f 0 (x) is not as smooth as indicated or if the compatibility conditions are not met, then series (14.4.2.9) may converge to a discontinuous function, thus giving only a generalized solution.] Remark. For the solution of linear nonhomogeneous parabolic equations with nonhomogeneous boundary conditions, see Section 14.7. 14.4.2-2. Conditions of compatibility of initial and boundary conditions. Suppose the function w has a continuous derivative with respect to t and two continuous derivatives with respect to x and is a solution of problem (14.4.2.1), (14.4.1.2), (14.4.1.3). Then the boundary conditions (14.4.1.2) and the initial condition (14.4.1.3) must be con- sistent; namely, the following compatibility conditions must hold: [s 1 f 0 + k 1 f 0 ] x=x 1 = 0,[s 2 f 0 + k 2 f 0 ] x=x 2 = 0.(14.4.2.10) If s 1 = 0 or s 2 = 0, then the additional compatibility conditions [a(x)f 0 + b(x)f 0 ] x=x 1 = 0 if s 1 = 0, [a(x)f 0 + b(x)f 0 ] x=x 2 = 0 if s 2 = 0 (14.4.2.11) must also hold; the primes denote the derivatives with respect to x. 14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution 14.4.3-1. Series solution of boundary value problems for hyperbolic equations. For hyperbolic equations, the solution of the boundary value problem (14.4.1.1)–(14.4.1.4) is sought in the series form w(x, t)= ∞ n=1 ϕ n (x) A n ψ n1 (t)+B n ψ n2 (t) .(14.4.3.1) Here, A n and B n are arbitrary constants. The functions ψ n1 (t)andψ n2 (t) are particular solutions of the linear equation (14.4.1.7) for ψ (with λ = λ n ) that satisfy the conditions ψ n1 (0)=1, ψ n1 (0)=0; ψ n2 (0)=0, ψ n2 (0)=1.(14.4.3.2) The functions ϕ n (x)andλ n are determined by solving the eigenvalue problem (14.4.1.6), (14.4.1.8). 608 LINEAR PARTIAL DIFFERENTIAL EQUATIONS Substituting solution (14.4.3.1) into the initial conditions (14.4.1.3)–(14.4.1.4) yields ∞ n=1 A n ϕ n (x)=f 0 (x), ∞ n=1 B n ϕ n (x)=f 1 (x). Multiplying these equations by ρ(x)ϕ n (x), where the weight function ρ(x)isdefined in (14.4.1.14), then integrating the resulting relations with respect to x on the interval x 1 ≤ x ≤ x 2 , and taking into account the properties (14.4.1.15), we obtain the coefficients of series (14.4.3.1) in the form A n = 1 ϕ n 2 x 2 x 1 ρ(x)ϕ n (x)f 0 (x) dx, B n = 1 ϕ n 2 x 2 x 1 ρ(x)ϕ n (x)f 1 (x) dx.(14.4.3.3) The quantity ϕ n is defined in (14.4.2.4). Relations (14.4.3.1), (14.4.1.12), and (14.4.3.3) give a formal solution of the nonsta- tionary boundary value problem (14.4.1.1)–(14.4.1.4) for α(t)>0. Example. Consider a mixed boundary value problem on the interval 0 ≤ x ≤ l for the wave equation ∂ 2 w ∂t 2 = ∂ 2 w ∂x 2 (14.4.3.4) with the general initial conditions (14.4.1.3)–(14.4.1.4) and the homogeneous boundary conditions w = 0 at x = 0, ∂ x w = 0 at x = l.(14.4.3.5) The functions ψ n1 (t)andψ n2 (t) are determined by the linear equation [see (14.4.1.7) with α(t)=1, β(t)=γ(t)=0,andλ = λ n ] ψ tt + λψ = 0 with the initial conditions (14.4.3.2). We find ψ n1 (t)=cos √ λ n t , ψ n2 (t)= 1 √ λ n sin √ λ n t .(14.4.3.6) The functions ϕ n (x) are determined by solving the eigenvalue problem (14.4.1.6), (14.4.1.8) with a(x)=1, b(x)=c(x)=0, s 1 = k 2 = 0, s 2 = k 1 = 1, x 1 = 0,andx 2 = l: ϕ xx + λϕ = 0; ϕ = 0 at x = 0, ϕ x = 0 at x = l. So we obtain the eigenfunctions and eigenvalues: ϕ n (x)=sin(μ n x), μ n = √ λ n = π(2n – 1) 2l , n = 1, 2, (14.4.3.7) The solution to problem (14.4.3.4)–(14.4.3.5), (14.4.1.3)–(14.4.1.4) is given by formulas (14.4.3.1) and (14.4.3.3). Taking into account that ϕ n 2 = l/2,wehave w(x, t)= ∞ n=1 A n cos(μ n t)+B n sin(μ n t) sin(μ n x), μ n = π(2n – 1) 2l , A n = 2 l l 0 f 0 (x)sin(μ n x) dx, B n = 2 lμ n l 0 f 1 (x)sin(μ n x) dx. (14.4.3.8) If f 0 (x)andf 1 (x) have three and two continuous derivatives, respectively, and the compatibility conditions are met (see Paragraph 14.4.3-2), then series (14.4.3.8) isconvergent and admits double termwise differentiation. In this case, formula (14.4.3.8) gives the classical smooth solution of problem (14.4.3.4)–(14.4.3.5), (14.4.1.3)– (14.4.1.4). Remark. For the solution of linear nonhomogeneous hyperbolic equations with nonhomogeneous bound- ary conditions, see Section 14.8. 14.4. METHOD OF SEPARATION OF VARIABLES (FOURIER METHOD) 609 14.4.3-2. Conditions of compatibility of initial and boundary conditions. Suppose w is a twice continuously differentiable solution of problem (14.4.1.1)–(14.4.1.4). Then conditions (14.4.2.10) and (14.4.2.11) must hold. In addition, the following conditions of compatibility of the boundary conditions (14.4.1.2) and initial condition (14.4.1.4) must be satisfied: [s 1 f 1 + k 1 f 1 ] x=x 1 = 0,[s 2 f 1 + k 2 f 1 ] x=x 2 = 0. 14.4.4. Solution of Boundary Value Problems for Elliptic Equations 14.4.4-1. Solution of special problem for elliptic equations. Now consider a boundary value problem for the elliptic equation a(x) ∂ 2 w ∂x 2 + α(y) ∂ 2 w ∂y 2 + b(x) ∂w ∂x + β(y) ∂w ∂y + c(x)+γ(y) w = 0 (14.4.4.1) with homogeneous boundary conditions (14.4.1.2) in x and the following mixed (homoge- neous and nonhomogeneous) boundary conditions in y: σ 1 ∂ y w + ν 1 w = 0 at y = y 1 , σ 2 ∂ y w + ν 2 w = f (x)aty = y 2 . (14.4.4.2) We assume that the coefficients of equation (14.4.4.1) and boundary conditions (14.4.1.2) and (14.4.4.2) meet the following requirements: a(x), b(x), c(x) α(y), β(y), and γ(t) are continuous functions, a(x)>0, α(y)>0, |s 1 | + |k 1 | > 0, |s 2 | + |k 2 | > 0, |σ 1 | + |ν 1 | > 0, |σ 2 | + |ν 2 | > 0. The approach is based on searching for particular solutions of equation (14.4.4.1) in the product form w(x, y)=ϕ(x) ψ(y). (14.4.4.3) As before, we first arrive at the eigenvalue problem (14.4.1.6), (14.4.1.8) for the function ϕ = ϕ(x); the solution procedure is detailed in Paragraph 14.4.1-3. Further on, we assume the λ n and ϕ n (x) have been found. The functions ψ n = ψ n (y) are determined by solving the linear ordinary differential equation α(y)ψ yy + β(y)ψ y +[γ(y)–λ n ]ψ = 0 (14.4.4.4) subject to the homogeneous boundary condition σ 1 ∂ y ψ + ν 1 ψ = 0 at y = y 1 ,(14.4.4.5) which is a consequence of the first condition (14.4.4.2). The functions ψ n are determined up to a constant factor. Taking advantage of the principle of linear superposition, we seek the solution to the boundary value problem (14.4.4.1), (14.4.4.2), (14.4.1.2) in the series form w(x, y)= ∞ n=1 A n ϕ n (x)ψ n (y), (14.4.4.6) 610 LINEAR PARTIAL DIFFERENTIAL EQUATIONS where A n are arbitrary constants. By construction, series (14.4.4.6) will satisfy equa- tion (14.4.4.1) with the boundary conditions (14.4.1.2) and the first boundary condition (14.4.4.2). In order to find the series coefficients A n , substitute (14.4.4.6) into the second boundary condition (14.4.4.2) to obtain ∞ n=1 A n B n ϕ n (x)=f(x), B n = σ 2 dψ n dy y=y 2 + ν 2 ψ n (y 2 ). (14.4.4.7) Further, we follow the same procedure as in Paragraph 14.4.2-1. Specifically, multiplying (14.4.4.7) by ρ(x)ϕ n (x), then integrating the resulting relation with respect to x over the interval x 1 ≤ x ≤ x 2 , and taking into account the properties (14.4.1.15), we obtain A n = 1 B n ϕ n 2 x 2 x 1 ρ(x)ϕ n (x)f(x) dx, ϕ n 2 = x 2 x 1 ρ(x)ϕ 2 n (x) dx,(14.4.4.8) where the weight function ρ(x)isdefined in (14.4.1.14). Example. Consider the first (Dirichlet) boundary value problem for the Laplace equation ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = 0 (14.4.4.9) subject to the boundary conditions w = 0 at x = 0, w = 0 at x = l 1 ; w = 0 at y = 0, w = f(x)aty = l 2 (14.4.4.10) in a rectangular domain 0 ≤ x ≤ l 1 , 0 ≤ y ≤ l 2 . Particular solutions to equation (14.4.4.9) are sought in the form (14.4.4.3). We have the following eigenvalue problem for ϕ(x): ϕ xx + λϕ = 0; ϕ = 0 at x = 0, ϕ = 0 at x = l 1 . On solving this problem, we find the eigenfunctions with respective eigenvalues ϕ n (x)=sin(μ n x), μ n = √ λ n = πn l 1 , n = 1, 2, (14.4.4.11) The functions ψ n = ψ n (y) are determined by solving the following problem for a linear ordinary differential equation with homogeneous boundary conditions: ψ yy – λ n ψ = 0; ψ = 0 at y = 0. (14.4.4.12) It is a special case of problem (14.4.4.4)–(14.4.4.5) with α(y)=1, β(y)=γ(y)=0, σ 1 = 0,andν 1 = 1.The nontrivial solutions of problem (14.4.4.12) are expressed as ψ n (y)=sinh(μ n y), μ n = √ λ n = πn l 1 , n = 1, 2, (14.4.4.13) Using formulas (14.4.4.6), (14.4.4.8), (14.4.4.11), (14.4.4.13) and taking into account the relations B n = ψ n (l 2 ) = sinh(μ n l 2 ), ρ(x)=1,andϕ n 2 = l/2,wefind the solution of the original problem (14.4.4.9)– (14.4.4.10) in the form w(x, y)= ∞ n=1 A n sin(μ n x)sinh(μ n y), A n = 2 l 1 sinh(μ n l 2 ) l 1 0 f(x)sin(μ n x) dx, μ n = πn l 1 . 14.5. INTEGRAL TRANSFORMS METHOD 611 TABLE 14.5 Description of auxiliary problems for equation (14.4.4.1) and problems for associated functions ϕ(x)andψ(y) that determine particular solutions of the form (14.4.4.3). The abbreviation HBC below stands for a “homogeneous boundary condition” Auxiliary problem Functions vanishing in the boundary conditions (14.4.4.14) Eigenvalue problem with homogeneous boundary conditions Another problem with one homogeneous boundary condition (for λ n found) Problem 1 f 2 (y)=f 3 (x)=f 4 (x)=0, function f 1 (y) prescribed functions ψ n (y)andvaluesλ n to be determined functions ϕ n (x) satisfy an HBC at x = x 2 Problem 2 f 1 (y)=f 3 (x)=f 4 (x)=0, function f 2 (y) prescribed functions ψ n (y)andvaluesλ n to be determined functions ϕ(x) satisfy an HBC at x = x 1 Problem 3 f 1 (y)=f 2 (y)=f 4 (x)=0, function f 3 (x) prescribed functions ϕ n (x)andvaluesλ n to be determined functions ψ(y) satisfy an HBC at y = y 2 Problem 4 f 1 (y)=f 2 (y)=f 3 (x)=0, function f 4 (x) prescribed functions ϕ n (x)andvaluesλ n to be determined functions ψ(y) satisfy an HBC at y = y 1 14.4.4-2. Generalization to the case of nonhomogeneous boundary conditions. Now consider the linear boundary value problem for the elliptic equation (14.4.4.1) with general nonhomogeneous boundary conditions s 1 ∂ x w + k 1 w = f 1 (y)atx = x 1 , s 2 ∂ x w + k 2 w = f 2 (y)atx = x 2 , σ 1 ∂ y w + ν 1 w = f 3 (x)aty = y 1 , σ 2 ∂ y w + ν 2 w = f 4 (x)aty = y 2 . (14.4.4.14) The solution to this problem is the sum of solutions to four simpler auxiliary problems for equation (14.4.4.1), each corresponding to three homogeneous and one nonhomogeneous boundary conditions in (14.4.4.14); see Table 14.5. Each auxiliary problem is solved using the procedure given in Paragraph 14.4.4-1, beginning with the search for solutions in the form of the product of functions with different arguments (14.4.4.3), determined by equations (14.4.1.6) and (14.4.4.4). The separation parameter λ is determined by the solution of a eigenvalue problem with homogeneous boundary conditions; see Table 14.5. The solution to each of the auxiliary problems is sought in the series form (14.4.4.6). Remark. For the solution of linear nonhomogeneous elliptic equations subject to nonhomogeneous bound- ary conditions, see Section 14.9. 14.5. Integral Transforms Method Various integral transforms are widely used to solve linear problems of mathematical physics. The Laplace transform and the Fourier transform are in most common use (its and other integral transforms are considered in Chapter 11 in detail). 14.5.1. Laplace Transform and Its Application in Mathematical Physics 14.5.1-1. Laplace and inverse Laplace transforms. Laplace transforms for derivatives. The Laplace transform of an arbitrary (complex-valued) function f(t) of a real variable t (t ≥ 0)isdefined by f(p)=L f(t) ,whereL f(t) ≡ ∞ 0 e –pt f(t) dt,(14.5.1.1) where p = s + iσ is a complex variable, i 2 =–1. . INTEGRAL TRANSFORMS METHOD 611 TABLE 14.5 Description of auxiliary problems for equation (14.4.4.1) and problems for associated functions ϕ(x )and (y) that determine particular solutions of the form (14.4.4.3). The. detail). 14.5.1. Laplace Transform and Its Application in Mathematical Physics 14.5.1-1. Laplace and inverse Laplace transforms. Laplace transforms for derivatives. The Laplace transform of an arbitrary (complex-valued). searching for particular solutions of equation (14.4.4.1) in the product form w(x, y)=ϕ(x) ψ(y). (14.4.4.3) As before, we first arrive at the eigenvalue problem (14.4.1.6), (14.4.1.8) for the function ϕ