14.2. BASIC PROBLEMS OF MAT H E M AT I C A L PHYSICS 591 many particular solutions. The specific solution that describes the physical phenomenon under study is separated from the set of particular solutions of the given differential equation by means of the initial and boundary conditions. Throughout this section, we consider linear equations in the n-dimensional Euclidean space R n or in an open domain V R n (exclusive of the boundary) with a sufficiently smooth boundary S = ∂V . 14.2.1-1. Parabolic equations. Initial and boundary conditions. In general, a linear second-order partial differential equation of the parabolic type with n independent variables can be written as ∂w ∂t – L x,t [w]=Φ(x, t), (14.2.1.1) 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,(14.2.1.2) 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. Parabolic equations govern unsteady thermal, diffusion, and other phenomena dependent on time t. Equation (14.2.1.1) is called homogeneous if Φ(x, t) ≡ 0. Cauchy problem (t ≥ 0, x R n ). Find a function w that satisfies equation (14.2.1.1) for t > 0 and the initial condition w = f(x)att = 0.(14.2.1.3) Boundary value problem*(t ≥ 0, x V ). Find a function w that satisfies equa- tion (14.2.1.1) for t > 0, the initial condition (14.2.1.3), and the boundary condition Γ x,t [w]=g(x, t)atx S (t > 0). (14.2.1.4) In general, Γ x,t is a first-order linear differential operator in the space variables x with coefficient dependent on x and t. The basic types of boundary conditions are described in Subsection 14.2.2. The initial condition (14.2.1.3) is called homogeneous if f(x) ≡ 0. The boundary condition (14.2.1.4) is called homogeneous if g(x, t) ≡ 0. 14.2.1-2. Hyperbolic equations. Initial and boundary conditions. Consider a second-order linear partial differential equation of the hyperbolic type with n independent variables of the general form ∂ 2 w ∂t 2 + ϕ(x, t) ∂w ∂t – L x,t [w]=Φ(x, t), (14.2.1.5) * Boundary value problems for parabolic and hyperbolic equations are sometimes called mixed or initial- boundary value problems. 592 LINEAR PARTIAL DIFFERENTIAL EQUATIONS where the linear differential operator L x,t is defined by (14.2.1.2). Hyperbolic equations govern unsteady wave processes, which depend on time t. Equation (14.2.1.5) is said to be homogeneous if Φ(x, t) ≡ 0. Cauchy problem (t ≥ 0, x R n ). Find a function w that satisfies equation (14.2.1.5) for t > 0 and the initial conditions w = f 0 (x)att = 0, ∂ t w = f 1 (x)att = 0. (14.2.1.6) Boundary value problem (t≥ 0, x V ). Find a function w that satisfi es equation (14.2.1.5) for t > 0, the initial conditions (14.2.1.6), and boundary condition (14.2.1.4). The initial conditions (14.2.1.6) are called homogeneous if f 0 (x) ≡ 0 and f 1 (x) ≡ 0. Generalized Cauchy problem. In the generalized Cauchy problem for a hyperbolic equation with two independent variables, values of the unknown function and its first derivatives are prescribed on a curve in the (x, t) plane. Alternatively, values of the unknown function and its derivative along the normal to this curve may be prescribed. For more details, see Paragraph 14.8.4-4. Goursat problem. On the characteristics of a hyperbolic equation with two independent variables, the values of the unknown function w are prescribed; for details, see Para- graph 14.8.4-5). 14.2.1-3. Elliptic equations. Boundary conditions. In general, a second-order linear partial differential equation of elliptic type with n inde- pendent variables can be written as –L x [w]=Φ(x), (14.2.1.7) 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,(14.2.1.8) n i,j=1 a ij (x)ξ i ξ j ≥ σ n i=1 ξ 2 i , σ > 0. Elliptic equations govern steady-state thermal, diffusion, and other phenomena independent of time t. Equation (14.2.1.7) is said to be homogeneous if Φ(x) ≡ 0. Boundary value problem. Find a function w that satisfies equation (14.2.1.7) and the boundary condition Γ x [w]=g(x)atx S.(14.2.1.9) In general, Γ x is a first-order linear differential operator in the space variables x. The basic types of boundary conditions are described below in Subsection 14.2.2. The boundary condition (14.2.1.9) is called homogeneous if g(x) ≡ 0. The boundary value problem (14.2.1.7)–(14.2.1.9) is said to be homogeneous if Φ ≡ 0 and g ≡ 0. 14.2. BASIC PROBLEMS OF MAT H E M AT I C A L PHYSICS 593 TABLE 14.3 Boundary conditions for various boundary value problems specified by parabolic and hyperbolic equations in two independent variables (x 1 ≤ x ≤ x 2 ) Type of problem Boundary condition at x = x 1 Boundary condition at x = x 2 First boundary value problem w = g 1 (t) w = g 2 (t) Second boundary value problem ∂ x w = g 1 (t) ∂ x w = g 2 (t) Third boundary value problem ∂ x w + β 1 w = g 1 (t)(β 1 < 0) ∂ x w + β 2 w = g 2 (t)(β 2 > 0) Mixed boundary value problem w = g 1 (t) ∂ x w = g 2 (t) Mixed boundary value problem ∂ x w = g 1 (t) w = g 2 (t) 14.2.2. First, Second, Third, and Mixed Boundary Value Problems For any (parabolic, hyperbolic, and elliptic) second-order partial differential equations, it is conventional to distinguish four basic types of boundary value problems, depending on the form of the boundary conditions (14.2.1.4) [see also the analogous condition (14.2.1.9)]. For simplicity, here we confine ourselves to the case where the coefficients a ij of equations (14.2.1.1) and (14.2.1.5) have the special form a ij (x, t)=a(x, t)δ ij , δ ij = 1 if i = j, 0 if i ≠ j. This situation is rather frequent in applications; such coefficients are used to describe various phenomena (processes) in isotropic media. The function w(x, t) takes prescribed values at the boundary S of the domain: w(x, t)=g 1 (x, t)forx S.(14.2.2.1) Second boundary value problem. The derivative along the (outward) normal is pre- scribed at the boundary S of the domain: ∂w ∂N = g 2 (x, t)forx S.(14.2.2.2) In heat transfer problems, where w is temperature, the left-hand side of the boundary condition (14.2.2.2) is proportional to the heat flux per unit area of the surface S. Third boundary value problem. A linear relationship between the unknown function and its normal derivative is prescribed at the boundary S of the domain: ∂w ∂N + k(x, t)w = g 3 (x, t)forx S.(14.2.2.3) Usually, it is assumed that k(x, t) = const. In mass transfer problems, where w is concen- tration, the boundary condition (14.2.2.3) with g 3 ≡ 0 describes a surface chemical reaction of the first order. Mixed boundary value problems. Conditions of various types, listed above, are set at different portions of the boundary S. If g 1 ≡ 0, g 2 ≡ 0,org 3 ≡ 0, the respective boundary conditions (14.2.2.1), (14.2.2.2), (14.2.2.3) are said to be homogeneous. Boundary conditions for various boundary value problems for parabolic and hyperbolic equations in two independent variables x and t are displayed in Table 14.3. The equation coefficients are assumed to be continuous, with the coefficients of the highest derivatives being nonzero in the range x 1 ≤ x ≤ x 2 considered. Remark. For elliptic equations, the first boundary value problem is often called the Dirichlet problem, and the second boundary value problem is called the Neumann problem. 594 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 14.3. Properties and Exact Solutions of Linear Equations 14.3.1. Homogeneous Linear Equations and Their Particular Solutions 14.3.1-1. Preliminary remarks. For brevity, in this paragraph a homogeneous linear partial differential equation will be written as L[w]=0.(14.3.1.1) For second-order linear parabolic and hyperbolic equations, the linear differential opera- tor L[w]isdefined by the left-hand side of equations (14.2.1.1) and (14.2.1.5), respectively. It is assumed that equation (14.3.1.1) is an arbitrary homogeneous linear partial differential equation of any order in the variables t, x 1 , , x n with sufficiently smooth coefficients. A linear operator L possesses the properties L[w 1 + w 2 ]=L[w 1 ]+L[w 2 ], L[Aw]=AL[w], A = const. An arbitrary homogeneous linear equation (14.3.1.1) has a trivial solution, w ≡ 0. A function w is called a classical solution of equation (14.3.1.1) if w, when substituted into (14.3.1.1), turns the equation into an identity and if all partial derivatives of w that occur in (14.3.1.1) are continuous; the notion of a classical solution is directly linked to the range of the independent variables. In what follows, we usually write “solution” instead of “classical solution” for brevity. 14.3.1-2. Usage of particular solutions for the construction of other solutions. Below are some properties of particular solutions of homogeneous linear equations. 1 ◦ .Letw 1 = w 1 (x, t), w 2 = w 2 (x, t), , w k = w k (x, t) be any particular solutions of the homogeneous equation (14.3.1.1). Then the linear combination w = A 1 w 1 + A 2 w 2 + ···+ A k w k (14.3.1.2) with arbitrary constants A 1 , A 2 , , A k is also a solution of equation (14.3.1.1); in physics, this property is known as the principle of linear superposition. Suppose {w k } is an infinite sequence of solutions of equation (14.3.1.1). Then the series ∞ k=1 w k , irrespective of its convergence, is called a formal solution of (14.3.1.1). If the solutions w k are classical, the series is uniformly convergent, and the sum of the series has all the necessary particular derivatives, then the sum of the series is a classical solution of equation (14.3.1.1). 2 ◦ .Letthecoefficients of the linear differential operator L be independent of time t.If equation (14.3.1.1) has a particular solution w = w(x, t), then the partial derivatives of w with respect to time,* ∂ w ∂t , ∂ 2 w ∂t 2 , , ∂ k w ∂t k , , are also solutions of equation (14.3.1.1). * Here and in what follows, it is assumed that the particular solution w is differentiable sufficiently many times with respect to t and x 1 , , x n (or the parameters). 14.3. PROPERTIES AND EXACT SOLUTIONS OF LINEAR EQUATIONS 595 3 ◦ .Letthecoefficients of the linear differential operator L be independent of the space variables x 1 , , x n . If equation (14.3.1.1) has a particular solution w = w(x, t), then the partial derivatives of w with respect to the space coordinates ∂ w ∂x 1 , ∂ w ∂x 2 , ∂ w ∂x 3 , , ∂ 2 w ∂x 2 1 , ∂ 2 w ∂x 1 ∂x 2 , , ∂ k+m w ∂x k 2 ∂x m 3 , are also solutions of equation (14.3.1.1). If the coefficients of L are independent of only one space coordinate, say x 1 ,and equation (14.3.1.1) has a particular solution w = w(x, t), then the partial derivatives ∂ w ∂x 1 , ∂ 2 w ∂x 2 1 , , ∂ k w ∂x k 1 , are also solutions of equation (14.3.1.1). 4 ◦ .Letthecoefficients of the linear differential operator L be constant and let equa- tion (14.3.1.1) have a particular solution w = w(x, t). Then any particular derivatives of w with respect to time and the space coordinates (inclusive mixed derivatives) ∂ w ∂t , ∂ w ∂x 1 , , ∂ 2 w ∂x 2 2 , ∂ 2 w ∂t∂x 1 , , ∂ k w ∂x k 3 , are solutions of equation (14.3.1.1). 5 ◦ . Suppose equation (14.3.1.1) has a particular solution dependent on a parameter μ, w = w(x, t; μ), and the coefficients of the linear differential operator L are independent of μ (but can depend on time and the space coordinates). Then, by differentiating w with respect to μ, one obtains other solutions of equation (14.3.1.1), ∂ w ∂μ , ∂ 2 w ∂μ 2 , , ∂ k w ∂μ k , Example 1. The linear equation ∂w ∂t = a ∂ 2 w ∂x 2 + bw has a particular solution w(x, t)=exp[μx +(aμ 2 + b)t], where μ is an arbitrary constant. Differentiating this equation with respect to μ yields another solution w(x, t)=(x + 2aμt)exp[μx +(aμ 2 + b)t]. Let some constants μ 1 , , μ k belong to the range of the parameter μ. Then the sum w = A 1 w(x, t; μ 1 )+···+ A k w(x, t; μ k ), (14.3.1.3) where A 1 , , A k are arbitrary constants, is also a solution of the homogeneous linear equation (14.3.1.1). The number of terms in sum (14.3.1.3) can be both finite and infinite. 6 ◦ . Another effective way of constructing solutions involves the following. The particular solution w(x, t; μ), which depends on the parameter μ (as before, it is assumed that the coefficients of the linear differential operator L are independent of μ), is first multiplied by an arbitrary function ϕ(μ). Then the resulting expression is integrated with respect to μ over some interval [α, β]. Thus, one obtains a new function, β α w(x, t; μ)ϕ(μ) dμ, which is also a solution of the original homogeneous linear equation. The properties listed in Items 1 ◦ – 6 ◦ enable one to use known particular solutions to construct other particular solutions of homogeneous linear equations of mathematical physics. 596 LINEAR PARTIAL DIFFERENTIAL EQUATIONS TABLE 14.4 Homogeneous linear partial differential equations that admit multiplicative separable solutions No. Form of equation (14.3.1.1) Form of particular solutions 1 Equation coefficients are constant w(x, t)=A exp(λt + β 1 x 1 + ···+ β n x n ), λ, β 1 , , β n are related by an algebraic equation 2 Equation coefficients are independent of time t w(x, t)=e λt ψ(x), λ is an arbitrary constant, x = {x 1 , , x n } 3 Equation coefficients are independent of the coordinates x 1 , , x n w(x, t)=exp(β 1 x 1 + ···+ β n x n )ψ(t), β 1 , , β n are arbitrary constants 4 Equation coefficients are independent of the coordinates x 1 , , x k w(x, t) = exp(β 1 x 1 +···+β k x k )ψ(t, x k+1 , , x n ), β 1 , , β k are arbitrary constants 5 L t [w]+L x [w]=0, operator L t depends on only t, operator L x depends on only x w(x, t)=ϕ(t)ψ(x), ϕ(t) satisfies the equation L t [ϕ]+λϕ = 0, ψ(x) satisfies the equation L x [ψ]–λψ = 0 6 L t [w]+L 1 [w]+···+ L n [w]=0, operator L t depends on only t, operator L k depends on only x k w(x, t)=ϕ(t)ψ 1 (x 1 ) ψ n (x n ), ϕ(t) satisfies the equation L t [ϕ]+λϕ = 0, ψ k (x k ) satisfies the equation L k [ψ k ]+β k ψ k = 0, λ + β 1 + ···+ β n = 0 7 f 0 (x 1 )L t [w]+ n k=1 f k (x 1 )L k [w]=0, operator L t depends on only t, operator L k depends on only x k w(x, t)=ϕ(t)ψ 1 (x 1 ) ψ n (x n ), L t [ϕ]+λϕ = 0, L k [ψ k ]+β k ψ k = 0, k = 2, , n, f 1 (x 1 )L 1 [ψ 1 ]– λf 0 (x 1 )+ n k=2 β k f k (x 1 ) ψ 1 = 0 8 ∂w ∂t + L 1,t [w]+···+ L n,t [w]=0, where L k,t [w]= m k s=0 f ks (x k , t) ∂ s w ∂x s k w(x, t)=ψ 1 (x 1 , t)ψ 2 (x 2 , t) ψ n (x n , t), ∂ψ k ∂t + L k,t [ψ k ]=λ k (t)ψ k , k = 1, , n, λ 1 (t)+λ 2 (t)+···+ λ n (t)=0 14.3.1-3. Multiplicative and additive separable solutions. 1 ◦ . Many homogeneous linear partial differential equations have solutions that can be represented as the product of functions depending on different arguments. Such solutions are referred to as multiplicative separable solutions; very commonly these solutions are briefly, but less accurately, called just separable solutions. Table 14.4 presents the most commonly encountered types of homogeneous linear dif- ferential equations with many independent variables that admit exact separable solutions. Linear combinations of particular solutions that correspond to different values of the separa- tion parameters, λ, β 1 , , β n , are also solutions of the equations in question. For brevity, the word “operator” is used to denote “linear differential operator.” For a constant coefficient equation (see the first row in Table 14.4), the separation parameters must satisfy the algebraic equation D(λ, β 1 , , β n )=0,(14.3.1.4) which results from substituting the solution into equation (14.3.1.1). In physical applica- tions, equation (14.3.1.4) is usually referred to as a dispersion equation.Anyn of the n + 1 separation parameters in (14.3.1.4) can be treated as arbitrary. 14.3. PROPERTIES AND EXACT SOLUTIONS OF LINEAR EQUATIONS 597 Example 2. Consider the linear equation ∂ 2 w ∂t 2 + k ∂w ∂t = a 2 ∂ 2 w ∂x 2 + b ∂w ∂x + cw. A particular solution is sought in the form w = A exp(βx + λt). This results in the dispersion equation λ 2 + kλ = a 2 β 2 + bβ + c, where one of the two parameters β or λ can be treated as arbitrary. For more complex multiplicative separable solutions to this equation, see Subsection 14.4.1. Note that constant coefficient equations also admit more sophisticated solutions; see the second and third rows, the last column. The eighth row of Table 14.4 presents the case of incomplete separation of variables where the solution is separated with respect to the space variables x 1 , , x n , but is not separated with respect to time t. Remark 1. For stationary equations that do not depend on t, one should set λ = 0, L t [w] ≡ 0,andϕ(t) ≡ 1 in rows 1, 6, and 7 of Table 14.4. Remark 2. Multiplicative separable solutions play an important role in the theory of linear partial differ- ential equations; they are used for finding solutions to stationary and nonstationary boundary value problems; see Sections 14.4 and 14.7–14.9. 2 ◦ . Linear partial differential equations of the form L t [w]+L x [w]=f (x)+g(t), where L t is a linear differential operator that depends on only t and L x is a linear differential operator that depends on only x, have solutions that can be represented as the sum of functions depending on different arguments w = u(x)+v(t). Such solutions are referred to as additive separable solutions. Example 3. The equation from Example 2 admits an exact additive separable solution w = u(x)+v(t) with u(x)andv(t) described by the linear constant-coefficient ordinary differential equations a 2 u xx + bu x + cu = C, v tt + kv t – cv = C, where C is an arbitrary constant, which are easy to integrate. A more general partial differential equation with variable coefficients a = a(x), b = b(x), k = k(t), and c = const also admits an additive separable solution. 14.3.1-4. Solutions in the form of infinite series in t. 1 ◦ . The equation ∂w ∂t = M[w], where M is an arbitrary linear differential operator of the second (or any) order that only depends on the space variables, has the formal series solution w(x, t)=f(x)+ ∞ k=1 t k k! M k [f(x)], M k [f]=M M k–1 [f] , where f(x) is an arbitrary infinitely differentiable function. This solution satisfies the initial condition w(x, 0)=f(x). . the range of the independent variables. In what follows, we usually write “solution” instead of “classical solution” for brevity. 14.3.1-2. Usage of particular solutions for the construction of other. irrespective of its convergence, is called a formal solution of (14.3.1.1). If the solutions w k are classical, the series is uniformly convergent, and the sum of the series has all the necessary particular. LINEAR PARTIAL DIFFERENTIAL EQUATIONS 14.3. Properties and Exact Solutions of Linear Equations 14.3.1. Homogeneous Linear Equations and Their Particular Solutions 14.3.1-1. Preliminary remarks. For