T8.1. PARABOLIC EQUATIONS 1277 2 ◦ . In the cases where the eigenfunctions ψ n (x) form an orthonormal basis in L 2 (R), the solution of the Cauchy problem for Schr ¨ odinger’s equation with the initial condition w = f(x)att = 0 (T8.1.10.2) is given by w(x, t)= ∞ –∞ G(x, ξ, t)f(ξ) dξ, G(x, ξ, t)= ∞ n=0 ψ n (x)ψ n (ξ)exp – iE n t . Various potentials U(x) are considered below and particular solutions of the boundary value problem (T8.1.10.1) or the Cauchy problem for Schr ¨ odinger’s equation are presented. T8.1.10-2. Free particle: U(x)=0. The solution of the Cauchy problem with the initial condition (T8.1.10.2) is given by w(x, t)= 1 2 √ iπτ ∞ –∞ exp – (x – ξ) 2 4iτ f(ξ) dξ, τ = t 2m , √ ia = e πi/4 √ |a| if a > 0, e –πi/4 √ |a| if a < 0. T8.1.10-3. Linear potential (motion in a uniform external field): U(x)=ax. Solution of the Cauchy problem with the initial condition (T8.1.10.2): w(x, t)= 1 2 √ iπτ exp –ibτx – 1 3 ib 2 τ 3 ∞ –∞ exp – (x + bτ 2 – ξ) 2 4iτ f(ξ) dξ, τ = t 2m , b = 2am 2 . T8.1.10-4. Linear harmonic oscillator: U(x)= 1 2 mω 2 x 2 . Eigenvalues: E n = ω n + 1 2 , n = 0, 1, Normalized eigenfunctions: ψ n (x)= 1 π 1/4 √ 2 n n! x 0 exp – 1 2 ξ 2 H n (ξ), ξ = x x 0 , x 0 = mω , where H n (ξ) are the Hermite polynomials. The functions ψ n (x) form an orthonormal basis in L 2 (R). T8.1.10-5. Isotropic free particle: U(x)=a/x 2 . Here, the variable x ≥ 0 plays the role of the radial coordinate, and a > 0. The equation with U(x)=a/x 2 results from Schr ¨ odinger’s equation for a free particle with n space coordinates if one passes to spherical (cylindrical) coordinates and separates the angular variables. The solution of Schr ¨ odinger’s equation satisfying the initial condition (T8.1.10.2) has the form w(x, t)= exp – 1 2 iπ(μ + 1)signt 2|τ| ∞ 0 √ xy exp i x 2 + y 2 4τ J μ xy 2|τ| f(y) dy, τ = t 2m , μ = 2am 2 + 1 4 ≥ 1, where J μ (ξ) is the Bessel function. 1278 LINEAR EQUATIONS AND PROBLEMS OF MAT HE MAT IC AL PHYSICS T8.1.10-6. Morse potential: U(x)=U 0 (e –2x/a – 2e –x/a ). Eigenvalues: E n =–U 0 1 – 1 β (n + 1 2 ) 2 , β = a √ 2mU 0 , 0 ≤ n < β – 2. Eigenfunctions: ψ n (x)=ξ s e –ξ/2 Φ(–n, 2s + 1, ξ), ξ = 2βe –x/a , s = a √ –2mE n , where Φ(a, b, ξ) is the degenerate hypergeometric function. In this case the number of eigenvalues (energy levels) E n and eigenfunctions ψ n is finite: n = 0, 1, , n max . T8.2. Hyperbolic Equations T8.2.1. Wave Equation ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 This equation is also known as the equation of vibration of a string. It is often encountered in elasticity, aerodynamics, acoustics, and electrodynamics. T8.2.1-1. General solution. Some formulas. 1 ◦ . General solution: w(x, t)=ϕ(x + at)+ψ(x – at), where ϕ(x)andψ(x) are arbitrary functions. 2 ◦ .Ifw(x, t) is a solution of the wave equation, then the functions w 1 = Aw( λx + C 1 , λt + C 2 )+B, w 2 = Aw x – vt 1 –(v/a) 2 , t – va –2 x 1 –(v/a) 2 , w 3 = Aw x x 2 – a 2 t 2 , t x 2 – a 2 t 2 are also solutions of the equation everywhere these functions are defined (A, B, C 1 , C 2 , v,andλ are arbitrary constants). The signs at λ’s in the formula for w 1 are taken arbitrarily. The function w 2 results from the invariance of the wave equation under the Lorentz transformations. T8.2.1-2. Domain: –∞ < x < ∞. Cauchy problem. Initial conditions are prescribed: w = f(x)att = 0, ∂ t w = g(x)att = 0. Solution (D’Alembert’s formula): w(x, t)= 1 2 [f(x + at)+f(x – at)] + 1 2a x+at x–at g(ξ) dξ. T8.2. HYPERBOLIC EQUATIONS 1279 T8.2.1-3. Domain: 0 ≤ x < ∞. First boundary value problem. The following two initial and one boundary conditions are prescribed: w = f(x)att = 0, ∂ t w = g(x)att = 0, w = h(t)atx = 0. Solution: w(x, t)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 2 [f(x + at)+f (x – at)] + 1 2a x+at x–at g(ξ) dξ for t < x a , 1 2 [f(x + at)–f (at – x)] + 1 2a x+at at–x g(ξ) dξ + h t – x a for t > x a . In the domain t < x/a the boundary conditions have no effect on the solution and the expression of w(x, t) coincides with D’Alembert’s solution for an infinite line (see Para- graph T8.2.1-2). T8.2.1-4. Domain: 0 ≤ x < ∞. Second boundary value problem. The following two initial and one boundary conditions are prescribed: w = f(x)att = 0, ∂ t w = g(x)att = 0, ∂ x w = h(t)atx = 0. Solution: w(x, t)= ⎧ ⎪ ⎨ ⎪ ⎩ 1 2 [f(x+at)+f(x–at)]+ 1 2a [G(x+at)–G(x–at)] for t < x a , 1 2 [f(x+at)+f(at –x)]+ 1 2a [G(x+at)+G(at–x)]–aH t– x a for t > x a , where G(z)= z 0 g(ξ) dξ and H(z)= z 0 h(ξ) dξ. T8.2.1-5. Domain: 0 ≤ x ≤ l. Boundary value problems. For solutions of various boundary value problems, see Subsection T8.2.2 for Φ(x, t) ≡ 0. T8.2.2. Equation of the Form ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 + Φ(x, t) T8.2.2-1. Solutions of boundary value problems in terms of the Green’s function. We consider boundary value problems on an interval 0 ≤ x ≤ l with the general initial conditions w = f(x)att = 0, ∂ t w = g(x)att = 0 (T8.2.2.1) and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as w(x, t)= ∂ ∂t l 0 f(ξ)G(x, ξ, t) dξ + l 0 g(ξ)G(x, ξ, t) dξ + t 0 l 0 Φ(ξ, τ)G(x, ξ, t–τ) dξ dτ . (T8.2.2.2) Here, the upper limit l can assume any finite values. Paragraphs T8.2.2-2 through T8.2.2-4 present the Green’s functions for various types of homogeneous boundary conditions. Remark. Formulas from Subsections 14.8.1–14.8.2 should be used to obtain solutions to corresponding nonhomogeneous boundary value problems. 1280 LINEAR EQUATIONS AND PROBLEMS OF MAT HE MAT IC AL PHYSICS T8.2.2-2. Domain: 0 ≤ x ≤ l. First boundary value problem. Boundary conditions are prescribed: w = 0 at x = 0, w = 0 at x = l. Green’s function: G(x, ξ, t)= 2 aπ ∞ n=1 1 n sin nπx l sin nπξ l sin nπat l . T8.2.2-3. Domain: 0 ≤ x ≤ l. Second boundary value problem. Boundary conditions are prescribed: ∂ x w = 0 at x = 0, ∂ x w = 0 at x = l. Green’s function: G(x, ξ, t)= t l + 2 aπ ∞ n=1 1 n cos nπx l cos nπξ l sin nπat l . T8.2.2-4. Domain: 0 ≤ x ≤ l. Third boundary value problem (k 1 > 0, k 2 > 0). Boundary conditions are prescribed: ∂ x w – k 1 w = 0 at x = 0, ∂ x w + k 2 w = 0 at x = l. Green’s function: G(x, ξ, t)= 1 a ∞ n=1 1 λ n u n 2 sin(λ n x + ϕ n )sin(λ n ξ + ϕ n )sin(λ n at), ϕ n =arctan λ n k 1 , u n 2 = l 2 + (λ 2 n + k 1 k 2 )(k 1 + k 2 ) 2(λ 2 n + k 2 1 )(λ 2 n + k 2 2 ) ; the λ n are positive roots of the transcendental equation cot(λl)= λ 2 – k 1 k 2 λ(k 1 + k 2 ) . T8.2.3. Klein–Gordon Equation ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 – bw This equation is encountered in quantum field theory and a number of applications. T8.2. HYPERBOLIC EQUATIONS 1281 T8.2.3-1. Particular solutions. w(x, t)=cos(λx)[A cos(μt)+B sin(μt)], b =–a 2 λ 2 + μ 2 , w(x, t)=sin(λx)[A cos(μt)+B sin(μt)], b =–a 2 λ 2 + μ 2 , w(x, t)=exp( μt)[A cos(λx)+B sin(λx)], b =–a 2 λ 2 – μ 2 , w(x, t)=exp( λx)[A cos(μt)+B sin(μt)], b = a 2 λ 2 + μ 2 , w(x, t)=exp( λx)[A exp(μt)+B exp(–μt)], b = a 2 λ 2 – μ 2 , w(x, t)=AJ 0 (ξ)+BY 0 (ξ), ξ = √ b a a 2 (t + C 1 ) 2 –(x + C 2 ) 2 , b > 0, w(x, t)=AI 0 (ξ)+BK 0 (ξ), ξ = √ –b a a 2 (t + C 1 ) 2 –(x + C 2 ) 2 , b < 0, where A, B, C 1 ,andC 2 are arbitrary constants, J 0 (ξ)andY 0 (ξ) are Bessel functions, and I 0 (ξ)andK 0 (ξ) are modified Bessel functions. T8.2.3-2. Formulas allowing the construction of particular solutions. Suppose w = w(x, t) is a solution of the Klein–Gordon equation. Then the functions w 1 = Aw( x + C 1 , t + C 2 )+B, w 2 = Aw x – vt 1 –(v/a) 2 , t – va –2 x 1 –(v/a) 2 , where A, B, C 1 , C 2 ,andv are arbitrary constants, are also solutions of this equation. The signs in the formula for w 1 are taken arbitrarily. T8.2.3-3. Domain: 0 ≤ x ≤ l. Boundary value problems. For solutions of the first and second boundary value problems, see Subsection T8.2.4 for Φ(x, t) ≡ 0. T8.2.4. Equation of the Form ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 – bw + Φ(x, t) T8.2.4-1. Solutions of boundary value problems in terms of the Green’s function. Solutions to boundary value problems on an interval 0 ≤ x ≤ l with the general initial conditions (T8.2.2.1) and various homogeneous boundary conditions are expressed via the Green’s function by formula (T8.2.2.2). T8.2.4-2. Domain: 0 ≤ x ≤ l. First boundary value problem. Boundary conditions are prescribed: w = 0 at x = 0, w = 0 at x = l. Green’s function for b > 0: G(x, ξ, t)= 2 l ∞ n=1 sin(λ n x)sin(λ n ξ) sin t a 2 λ 2 n + b a 2 λ 2 n + b , λ n = πn l . 1282 LINEAR EQUATIONS AND PROBLEMS OF MAT HE MAT IC AL PHYSICS T8.2.4-3. Domain: 0 ≤ x ≤ l. Second boundary value problem. Boundary conditions are prescribed: ∂ x w = 0 at x = 0, ∂ x w = 0 at x = l. Green’s function for b > 0: G(x, ξ, t)= 1 l √ b sin t √ b + 2 l ∞ n=1 cos(λ n x)cos(λ n ξ) sin t a 2 λ 2 n + b a 2 λ 2 n + b , λ n = πn l . T8.2.5. Equation of the Form ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂r 2 + 1 r ∂w ∂r + Φ(r, t) This is the wave equation with axial symmetry, where r = x 2 + y 2 is the radial coordinate. T8.2.5-1. Solutions of boundary value problems in terms of the Green’s function. We consider boundary value problems in domain 0 ≤ r ≤ R with the general initial conditions w = f(r)att = 0, ∂ t w = g(r)att = 0,(T8.2.5.1) and various homogeneous boundary conditions at r = R (the solutions bounded at r = 0 are sought). The solution can be represented in terms of the Green’s function as w(r, t)= ∂ ∂t R 0 f(ξ)G(r, ξ, t) dξ + R 0 g(ξ)G(r, ξ, t) dξ + t 0 R 0 Φ(ξ, τ)G(r, ξ, t – τ) dξ dτ.(T8.2.5.2) T8.2.5-2. Domain: 0 ≤ r ≤ R. First boundary value problem. A boundary condition is prescribed: w = 0 at r = R. Green’s function: G(r, ξ, t)= 2ξ aR ∞ n=1 1 λ n J 2 1 (λ n ) J 0 λ n r R J 0 λ n ξ R sin λ n at R , where λ n are positive zeros of the Bessel function, J 0 (λ)=0. The numerical values of the first ten λ n are specified in Paragraph T8.1.5-2. T8.2.5-3. Domain: 0 ≤ r ≤ R. Second boundary value problem. A boundary condition is prescribed: ∂ r w = 0 at r = R. Green’s function: G(r, ξ, t)= 2tξ R 2 + 2ξ aR ∞ n=1 1 λ n J 2 0 (λ n ) J 0 λ n r R J 0 λ n ξ R sin λ n at R , where λ n are positive zeros of the first-order Bessel function, J 1 (λ)=0. The numerical values of the first ten roots λ n are specifi ed in Paragraph T8.1.5-3. T8.2. HYPERBOLIC EQUATIONS 1283 T8.2.6. Equation of the Form ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂r 2 + 2 r ∂w ∂r + Φ(r, t) This is the equation of vibration of a gas with central symmetry, where r = x 2 + y 2 + z 2 is the radial coordinate. T8.2.6-1. General solution for Φ(r, t) ≡ 0. w(t, r)= ϕ(r + at)+ψ(r – at) r , where ϕ(r 1 )andψ(r 2 ) are arbitrary functions. T8.2.6-2. Reduction to a constant coefficient equation. The substitution u(r, t)=rw(r, t) leads to the nonhomogeneous constant coefficient equa- tion ∂ 2 u ∂t 2 = a 2 ∂ 2 u ∂r 2 + rΦ(r, t), which is discussed in Subsection T8.2.1. T8.2.6-3. Solutions of boundary value problems in terms of the Green’s function. Solutions to boundary value problems on an interval 0 ≤ x ≤ R with the general initial conditions (T8.2.5.1) and various homogeneous boundary conditions are expressed via the Green’s function by formula (T8.2.5.2). T8.2.6-4. Domain: 0 ≤ r ≤ R. First boundary value problem. A boundary condition is prescribed: w = 0 at r = R. Green’s function: G(r, ξ, t)= 2ξ πar ∞ n=1 1 n sin nπr R sin nπξ R sin anπt R . T8.2.6-5. Domain: 0 ≤ r ≤ R. Second boundary value problem. A boundary condition is prescribed: ∂ r w = 0 at r = R. Green’s function: G(r, ξ, t)= 3tξ 2 R 3 + 2ξ ar ∞ n=1 μ 2 n + 1 μ 3 n sin μ n r R sin μ n ξ R sin μ n at R , where μ n are positive roots of the transcendental equation tan μ – μ = 0. The numerical values of the first five roots μ n are specifi ed in Paragraph T8.1.7-3. . A, B, C 1 ,andC 2 are arbitrary constants, J 0 (ξ)andY 0 (ξ) are Bessel functions, and I 0 (ξ)andK 0 (ξ) are modified Bessel functions. T8.2.3-2. Formulas allowing the construction of particular. below and particular solutions of the boundary value problem (T8.1.10.1) or the Cauchy problem for Schr ¨ odinger’s equation are presented. T8.1.10-2. Free particle: U(x)=0. The solution of the. are also solutions of this equation. The signs in the formula for w 1 are taken arbitrarily. T8.2.3-3. Domain: 0 ≤ x ≤ l. Boundary value problems. For solutions of the first and second boundary