598 LINEAR PARTIAL DIFFERENTIAL EQUATIONS Example 4. Consider the heat equation ∂w ∂t = a ∂ 2 w ∂x 2 . In this case we have M = a ∂ 2 ∂x 2 . Therefore the formal series solution has the form w(x, t)=f(x)+ ∞ k=1 (at) k k! f (2n) x (x), f (m) x = d m dx m f(x). If the function f(x) is taken as a polynomial of degree n, the solution will also be a polynomial of degree n. For example, setting f(x)=Ax 2 + Bx + C, we obtain the particular solution w(x, t)=A(x 2 + 2at)+Bx + C. 2 ◦ . The equation ∂ 2 w ∂t 2 = M [w], where M is a linear differential operator, just as in Item 1 ◦ , has a formal solution represented by the sum of two series as w(x, t)= ∞ k=0 t 2k (2k)! M k [f(x)] + ∞ k=0 t 2k+1 (2k + 1)! M k [g(x)], where f(x)andg(x) are arbitrary infinitely differentiable functions. This solution satisfies the initial conditions w(x, 0)=f (x)and∂ t w(x, 0)=g(x). 14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions 14.3.2-1. Simplest properties of nonhomogeneous linear equations. For brevity, we write a nonhomogeneous linear partial differential equation in the form L[w]=Φ(x, t), (14.3.2.1) where the linear differential operator L is defined above (see the beginning of Paragraph 14.3.1-1). Below are the simplest properties of particular solutions of the nonhomogeneous equa- tion (14.3.2.1). 1 ◦ .Ifw Φ (x, t) is a particular solution of the nonhomogeneous equation (14.3.2.1) and w 0 (x, t) is a particular solution of the corresponding homogeneous equation (14.3.1.1), then the sum Aw 0 (x, t)+w Φ (x, t), where A is an arbitrary constant, is also a solution of the nonhomogeneous equation (14.3.2.1). The following, more general statement holds: The general solution of the nonhomogeneous equation (14.3.2.1) is the sum of the general solution of the correspond- ing homogeneous equation (14.3.1.1) and any particular solution of the nonhomogeneous equation (14.3.2.1). 2 ◦ . Suppose w 1 and w 2 are solutions of nonhomogeneous linear equations with the same left-hand side and different right-hand sides, i.e., L[w 1 ]=Φ 1 (x, t), L[w 2 ]=Φ 2 (x, t). Then the function w = w 1 + w 2 is a solution of the equation L[w]=Φ 1 (x, t)+Φ 2 (x, t). 14.3. PROPERTIES AND EXACT SOLUTIONS OF LINEAR EQUATIONS 599 14.3.2-2. Fundamental and particular solutions of stationary equations. Consider the second-order linear stationary (time-independent) nonhomogeneous equation L x [w]=–Φ(x). (14.3.2.2) Here, L x is a linear differential operator of the second (or any) order of general form whose coefficients are dependent on x,wherex R n . A distribution = (x, y) that satisfies the equation with a special right-hand side L x [ ]=–δ(x – y)(14.3.2.3) is called a fundamental solution corresponding to the operator L x . In (14.3.2.3), δ(x)is an n-dimensional Dirac delta function and the vector quantity y = {y 1 , , y n } appears in equation (14.3.2.3) as an n-dimensional free parameter. It is assumed that y R n . The n-dimensional Dirac delta function possesses the following basic properties: 1. δ(x)=δ(x 1 )δ(x 2 ) δ(x n ), 2. R n Φ(y)δ(x – y) dy = Φ(x), where δ(x k ) is the one-dimensional Dirac delta function, Φ(x) is an arbitrary continuous function, and dy = dy 1 dy n . For constant coefficient equations, a fundamental solution always exists; it can be found by means of the n-dimensional Fourier transform (see Paragraph 11.4.1-4). The fundamental solution = (x, y) can be used to construct a particular solution of the linear stationary nonhomogeneous equation (14.3.2.2) for arbitrary continuous Φ(x); this particular solution is expressed as follows: w(x)= R n Φ(y) (x, y) dy.(14.3.2.4) Remark 1. The fundamental solution is not unique; it is defined up to an additive term w 0 = w 0 (x), which is an arbitrary solution of the homogeneous equation L x [w 0 ]=0. Remark 2. For constant coefficient differential equations, the fundamental solution possesses the property (x, y)= (x – y). Remark 3. The right-hand sides of equations (14.3.2.2) and (14.3.2.3) are often prefixed with the plus sign. In this case, formula (14.3.2.4) remains valid. Remark 4. Particular solutions of linear nonstationary nonhomogeneous equations can be expressed in terms of the fundamental solution of the Cauchy problem; see Section 14.6. Example 1. For the two- and three-dimensional Poisson equations, the fundamental solutions have the forms Equations Fundamental solutions ∂ 2 w ∂x 2 1 + ∂ 2 w ∂x 2 2 =–Φ(x 1 , x 2 ) =⇒ (x 1 , x 2 , y 1 , y 2 )= 1 2π ln 1 ρ , ∂ 2 w ∂x 2 1 + ∂ 2 w ∂x 2 2 + ∂ 2 w ∂x 2 3 =–Φ(x 1 , x 2 , x 3 ) =⇒ (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 )= 1 4πr , where ρ = (x 1 – y 1 ) 2 +(x 2 – y 2 ) 2 and r = (x 1 – y 1 ) 2 +(x 2 – y 2 ) 2 +(x 3 – y 3 ) 2 . Example 2. The two-dimensional Helmholtz equation ∂ 2 w ∂x 2 1 + ∂ 2 w ∂x 2 2 + λw =–Φ(x 1 , x 2 ) 600 LINEAR PARTIAL DIFFERENTIAL EQUATIONS has the following fundamental solutions: (x 1 , x 2 , y 1 , y 2 )= 1 2π K 0 (kρ)ifλ =–k 2 < 0, (x 1 , x 2 , y 1 , y 2 )=– i 4 H (2) 0 (kρ)ifλ = k 2 > 0, where ρ = (x 1 – y 1 ) 2 +(x 2 – y 2 ) 2 , K 0 (z) is the modified Bessel function of the second kind, H (2) 0 (z)isthe Hankel functions of the second kind of order 0, k > 0,andi 2 =–1. Example 3. The three-dimensional Helmholtz equation ∂ 2 w ∂x 2 1 + ∂ 2 w ∂x 2 2 + ∂ 2 w ∂x 2 3 + λw =–Φ(x 1 , x 2 , x 3 ) has the following fundamental solutions: (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 )= 1 4πr exp(–kr)ifλ =–k 2 < 0, (x 1 , x 2 , x 3 , y 1 , y 2 , y 3 )= 1 4πr exp(–ikr)ifλ = k 2 > 0, where r = (x 1 – y 1 ) 2 +(x 2 – y 2 ) 2 +(x 3 – y 3 ) 2 , k > 0,andi 2 =–1. 14.3.3. General Solutions of Some Hyperbolic Equations 14.3.3-1. D’Alembert’s solution for the wave equation. Thewaveequation ∂ 2 w ∂t 2 – a 2 ∂ 2 w ∂x 2 = 0 (14.3.3.1) has the general solution w = ϕ(x + at)+ψ(x – at), (14.3.3.2) where ϕ(x)andψ(x) are arbitrary twice continuously differentiable functions. Solution (14.3.3.2) has the physical interpretation of two traveling waves of arbitrary shape that propagate to the right and to the left along the x-axis with a constant speed a. 14.3.3-2. Laplace cascade method for hyperbolic equation in two variables. A general linear hyperbolic equation with two independent variables can be reduced to an equation of the form (see Subsection 14.1.1): ∂ 2 w ∂x∂y + a(x, y) ∂w ∂x + b(x, y) ∂w ∂y + c(x, y)w = f (x, y). (14.3.3.3) Sometimes it is possible to obtain formulas determining all solutions to equation (14.3.3.3). First consider two special cases. 1 ◦ . First special case. Suppose the identity g ≡ ∂a ∂x + ab – c ≡ 0 (14.3.3.4) is valid; for brevity, the arguments of the functions are omitted. Then equation (14.3.3.3) can be rewritten in the form ∂u ∂x + bu = f ,(14.3.3.5) 14.3. PROPERTIES AND EXACT SOLUTIONS OF LINEAR EQUATIONS 601 where u = ∂w ∂y + aw.(14.3.3.6) Equation (14.3.3.5) is a linear first-order ordinary differential equation in x for u (the vari- able y appears in the equation as a parameter) and is easy to integrate. Further substituting u into (14.3.3.6) yields a linear first-order ordinary differential equation in y for w (now x appears in the equation as a parameter). On solving this equation, one obtains the general solution of the original equation (14.3.3.3) subject to condition (14.3.3.4): w =exp – ady ϕ(x)+ ψ(y)+ f exp bdx dx exp ady– bdx dy , where ϕ(x)andψ(y) are arbitrary functions. 2 ◦ . Second special case. Suppose the identity h ≡ ∂b ∂y + ab – c ≡ 0 (14.3.3.7) holds true. Proceeding in the same ways as in the first special case, one obtains the general solution to (14.3.3.3): w =exp – bdx ψ(y)+ ϕ(x)+ f exp ady dy exp bdx– ady dx . 3 ◦ . Laplace cascade method. In the case g ≠ 0, consider the new equation of the form (14.3.3.3), L 1 [w 1 ] ≡ ∂ 2 w 1 ∂x∂y + a 1 (x, y) ∂w 1 ∂x + b 1 (x, y) ∂w 1 ∂y + c 1 (x, y)w 1 = f 1 (x, y), (14.3.3.8) where a 1 = a – ∂ ln g ∂y , b 1 = b, c 1 = c – ∂a ∂x + ∂b ∂y – b ∂ ln g ∂y , f 1 = a – ∂ ln g ∂y f. If one manages to find w 1 , the corresponding solution to the original equation (14.3.3.3) can be obtained by the formula w = 1 g ∂w 1 ∂x + bw 1 – f . For equation (14.3.3.8), the functions similar to (14.3.3.4) and (14.3.3.7) are expressed as g 1 = 2g – h – ∂ 2 ln g ∂x∂y , h 1 = h. If g 1 ≡ 0, the function w 1 can be found using the technique described above. If g 1 0, one proceeds to the construction, in the same way as above, of the equation L 2 [w 2 ]=f 2 ,and so on. In the case h 0, a similar chain of equations may be constructed: L ∗ 1 [w ∗ 1 ]=f ∗ 1 , L ∗ 2 [w ∗ 2 ]=f ∗ 2 ,etc. If at some step, g k or h k vanishes, it is possible to obtain the general solution of equation (14.3.3.3). 602 LINEAR PARTIAL DIFFERENTIAL EQUATIONS Example. Consider the Euler–Darboux equation ∂ 2 w ∂x∂y – α x – y ∂w ∂x + β x – y ∂w ∂y = 0. We will show that its general solution can be obtained if at least one of the numbers α or β is integer. With the notation adopted for equation (14.3.3.3) and function (14.3.3.4), we have a(x, y)=– α x – y , b(x, y)= β x – y , c(x, y)=f(x, y)=0, g = α(1 – β) (x – y) 2 , which means that g ≡ 0 if α = 0 or β = 1.Ifg 0, we construct equation (14.3.3.8), where a 1 =– 2 + α x – y , b 1 = β x – y , c 1 =– α + β (x – y) 2 . If follows that g 1 = (1 + α)(2 – β) (x – y) 2 , and hence g 1 ≡ 0 at α =–1 or β = 2. Similarly, it can be shown that g k ≡ 0 at α =–k or β = k +1 (k = 0, 1, 2, ). If we use the other sequence of auxiliary equations, L ∗ k [w ∗ k ]=f ∗ k , it can be shown that the above holds for α = 1, 2, and β = 0,–1,–2, 14.4. Method of Separation of Variables (Fourier Method) 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution 14.4.1-1. Scheme of solving boundary value problems by separation of variables. Many linear problems of mathematical physics can be solved by separation of variables. Figure 14.1 depicts the scheme of application of this method to solve boundary value problems for second-order homogeneous linear equations of the parabolic and hyperbolic type with homogeneous boundary conditions and nonhomogeneous initial conditions. For simplicity, problems with two independent variables x and t are considered, with x 1 ≤ x ≤ x 2 and t ≥ 0. The scheme presented in Fig. 14.1 applies to boundary value problems for second-order linear homogeneous partial differential equations of the form α(t) ∂ 2 w ∂t 2 + β(t) ∂w ∂t = a(x) ∂ 2 w ∂x 2 + b(x) ∂w ∂x + c(x)+γ(t) w (14.4.1.1) with homogeneous linear boundary conditions, s 1 ∂ x w + k 1 w = 0 at x = x 1 , s 2 ∂ x w + k 2 w = 0 at x = x 2 , (14.4.1.2) and arbitrary initial conditions, w = f 0 (x)att = 0,(14.4.1.3) ∂ t w = f 1 (x)att = 0.(14.4.1.4) For parabolic equations, which correspond to α(t) ≡ 0 in (14.4.1.1), only the initial condition (14.4.1.3) is set. 14.4. METHOD OF SEPARATION OF VARIABLES (FOURIER METHOD) 603 Figure 14.1. Scheme of solving linear boundary value problems by separation of variables (for parabolic equations, the function F 2 does not depend on ψ tt ,andallB n = 0). 604 LINEAR PARTIAL DIFFERENTIAL EQUATIONS Below we consider the basic stages of the method of separation of variables in more detail. We assume that the coefficients of equation (14.4.1.1) and boundary conditions (14.4.1.2) meet the following requirements: α(t), β(t), γ(t), a(x), b(x), c(x) are continuous functions, α(t) ≥ 0, a(x)>0, |s 1 | + |k 1 | > 0, |s 2 | + |k 2 | > 0. Remark 1. The method of separation of variables is also used to solve linear boundary value problems for elliptic equations of the form (14.4.1.1), provided that α(t)<0, a(x)>0 and with the boundary conditions (14.4.1.2) in x and similar boundary conditions in t. In this case, all results obtained for the general stage of solution, described in Paragraphs 14.4.1-2 and 14.4.1-3, remain valid; for details, see Subsection 14.4.4. Remark 2. In various applications, equations of the form (14.4.1.1) may arise with the coefficient b(x) going to infinity at the boundary, b(x) →∞as x → x 1 , with the other coefficients being continuous. In this case, the first boundary condition in (14.4.1.2) should be replaced with a condition of boundedness of the solution as x → x 1 . This may occur in spatial problems with central or axial symmetry where the solution depends only on the radial coordinate. 14.4.1-2. Derivation of equations and boundary conditions for particular solutions. The approach is based on searching for particular solutions of equation (14.4.1.1) in the product form w(x, t)=ϕ(x) ψ(t). (14.4.1.5) After separation of the variables and elementary manipulations, one arrives at the following linear ordinary differential equations for the functions ϕ = ϕ(x)andψ = ψ(t): a(x)ϕ xx + b(x)ϕ x +[λ + c(x)]ϕ = 0,(14.4.1.6) α(t)ψ tt + β(t)ψ t +[λ – γ(t)]ψ = 0.(14.4.1.7) These equations contain a free parameter λ called the separation constant. With the nota- tion adopted in Fig. 14.1, equations (14.4.1.6) and (14.4.1.7) can be rewritten as follows: ϕF 1 (x, ϕ, ϕ x , ϕ xx )+λϕ = 0 and ψF 2 (t, ψ, ψ t , ψ tt )+λψ = 0. Substituting (14.4.1.5) into (14.4.1.2) yields the boundary conditions for ϕ = ϕ(x): s 1 ϕ x + k 1 ϕ = 0 at x = x 1 , s 2 ϕ x + k 2 ϕ = 0 at x = x 2 . (14.4.1.8) The homogeneous linear ordinary differential equation (14.4.1.6) in conjunction with the homogeneous linear boundary conditions (14.4.1.8) makes up an eigenvalue problem. 14.4.1-3. Solution of eigenvalue problems. Orthogonality of eigenfunctions. Suppose ϕ 1 = ϕ 1 (x, λ)andϕ 2 = ϕ 2 (x, λ) are linearly independent particular solutions of equation (14.4.1.6). Then the general solution of this equation can be represented as the linear combination ϕ = C 1 ϕ 1 (x, λ)+C 2 ϕ 2 (x, λ), (14.4.1.9) where C 1 and C 2 are arbitrary constants. Substituting solution (14.4.1.9) into the boundary conditions (14.4.1.8) yields the fol- lowing homogeneous linear algebraic system of equations for C 1 and C 2 : ε 11 (λ)C 1 + ε 12 (λ)C 2 = 0, ε 21 (λ)C 1 + ε 22 (λ)C 2 = 0, (14.4.1.10) . the above holds for α = 1, 2, and β = 0,–1,–2, 14.4. Method of Separation of Variables (Fourier Method) 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution 14.4.1-1 properties of particular solutions of the nonhomogeneous equa- tion (14.3.2.1). 1 ◦ .Ifw Φ (x, t) is a particular solution of the nonhomogeneous equation (14.3.2.1) and w 0 (x, t) is a particular. Derivation of equations and boundary conditions for particular solutions. The approach is based on searching for particular solutions of equation (14.4.1.1) in the product form w(x, t)=ϕ(x) ψ(t).