Handbook of mathematics for engineers and scienteists part 187 pot

1270 LINEAR EQUATIONS AND PROBLEMS OF MATHEM ATICAL PHYSICS T8.1.2-8. 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)= ∞  n=1 1 y n  2 y n (x)y n (ξ)exp(–aμ 2 n t), y n (x)=cos(μ n x)+ k 1 μ n sin(μ n x), y n  2 = k 2 2μ 2 n μ 2 n + k 2 1 μ 2 n + k 2 2 + k 1 2μ 2 n + l 2  1 + k 2 1 μ 2 n  , where μ n are positive roots of the transcendental equation tan(μl) μ = k 1 + k 2 μ 2 – k 1 k 2 . T8.1.3. Equation of the Form ∂w ∂t = a ∂ 2 w ∂x 2 + b ∂w ∂x + cw + Φ(x, t) The substitution w(x, t)=exp(βt + μx)u(x, t), β = c – b 2 4a , μ =– b 2a leads to the nonhomogeneous heat equation ∂u ∂t = a ∂ 2 u ∂x 2 +exp(–βt – μx)Φ(x, t), which is considered in Subsections T8.1.1 and T8.1.2. T8.1.4. Heat Equation with Axial Symmetry ∂w ∂t = a  ∂ 2 w ∂r 2 + 1 r ∂w ∂r  This is a heat (diffusion) equation with axial symmetry, where r =  x 2 + y 2 is the radial coordinate. T8.1.4-1. Particular solutions. w(r)=A + B ln r, w(r, t)=A + B(r 2 + 4at), w(r, t)=A + B(r 4 + 16atr 2 + 32a 2 t 2 ), w(r, t)=A + B  r 2n + n  k=1 4 k [n(n – 1) (n – k + 1)] 2 k! (at) k r 2n–2k  , w(r, t)=A + B  4at ln r + r 2 ln r – r 2  , w(r, t)=A + B t exp  – r 2 4at  , T8.1. PARABOLIC EQUATIONS 1271 w(r, t)=A + B exp(–aμ 2 t)J 0 (μr), w(r, t)=A + B exp(–aμ 2 t)Y 0 (μr), w(r, t)=A + B t exp  – r 2 + μ 2 4t  I 0  μr 2t  , w(r, t)=A + B t exp  – r 2 + μ 2 4t  K 0  μr 2t  , where A, B,andμ are arbitrary constants, n is an arbitrary positive integer, J 0 (z)andY 0 (z) are Bessel functions, and I 0 (z)andK 0 (z) are modified Bessel functions. T8.1.4-2. Formulas allowing the construction of particular solutions. Suppose w = w(r, t) is a solution of the original equation. Then the functions w 1 = Aw( λr, λ 2 t + C)+B, w 2 = A δ + βt exp  – βr 2 4a(δ + βt)  w  r δ + βt , γ + λt δ + βt  , λδ – βγ = 1, where A, B, C, β, δ,andλ are arbitrary constants, are also solutions of this equation. The second formula usually may be encountered with β = 1, γ =–1,andδ = λ = 0. T8.1.4-3. Boundary value problems. For solutions of various boundary value problems, see Subsection T8.1.5 with Φ(r, t) ≡ 0. T8.1.5. Equation of the Form ∂w ∂t = a  ∂ 2 w ∂r 2 + 1 r ∂w ∂r  + Φ(r, t) T8.1.5-1. 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 condition w = f(r)att = 0 (T8.1.5.1) and various homogeneous boundary conditions (the solutions bounded at r = 0 are sought). The solution can be represented in terms of the Green’s function as w(x, t)=  R 0 f(ξ)G(r, ξ, t) dξ +  t 0  R 0 Φ(ξ, τ)G(r, ξ, t – τ)dξ dτ.(T8.1.5.2) T8.1.5-2. Domain: 0 ≤ r ≤ R. First boundary value problem. A boundary condition is prescribed: w = 0 at r = R. 1272 LINEAR EQUATIONS AND PROBLEMS OF MATHEMATIC AL PHYSICS Green’s function: G(r, ξ, t)= ∞  n=1 2ξ R 2 J 2 1 (μ n ) J 0  μ n r R  J 0  μ n ξ R  exp  – aμ 2 n t R 2  , where μ n are positive zeros of the Bessel function, J 0 (μ)=0. Below are the numerical values of the first ten roots: μ 1 = 2.4048, μ 2 = 5.5201, μ 3 = 8.6537, μ 4 = 11.7915, μ 5 = 14.9309, μ 6 = 18.0711, μ 7 = 21.2116, μ 8 = 24.3525, μ 9 = 27.4935, μ 10 = 30.6346. The zeros of the Bessel function J 0 (μ) may be approximated by the formula μ n = 2.4 + 3.13(n – 1)(n = 1, 2, 3, ), which is accurate within 0.3%. As n →∞,wehaveμ n+1 – μ n → π. T8.1.5-3. 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)= 2 R 2 ξ + 2 R 2 ∞  n=1 ξ J 2 0 (μ n ) J 0  μ n r R  J 0  μ n ξ R  exp  – aμ 2 n t R 2  , where μ n are positive zeros of the first-order Bessel function, J 1 (μ)=0. Below are the numerical values of the first ten roots: μ 1 = 3.8317, μ 2 = 7.0156, μ 3 = 10.1735, μ 4 = 13.3237, μ 5 = 16.4706, μ 6 = 19.6159, μ 7 = 22.7601, μ 8 = 25.9037, μ 9 = 29.0468, μ 10 = 32.1897. As n →∞,wehaveμ n+1 – μ n → π. T8.1.6. Heat Equation with Central Symmetry ∂w ∂t = a  ∂ 2 w ∂r 2 + 2 r ∂w ∂r  This is the heat (diffusion) equation with central symmetry; r =  x 2 + y 2 + z 2 is the radial coordinate. T8.1.6-1. Particular solutions. w(r)=A + Br –1 , w(r, t)=A + B(r 2 + 6at), w(r, t)=A + B(r 4 + 20atr 2 + 60a 2 t 2 ), T8.1. PARABOLIC EQUATIONS 1273 w(r, t)=A + B  r 2n + n  k=1 (2n + 1)(2n) (2n – 2k + 2) k! (at) k r 2n–2k  , w(r, t)=A + 2aBtr –1 + Br, w(r, t)=Ar –1 exp(aμ 2 t μr)+B, w(r, t)=A + B t 3/2 exp  – r 2 4at  , w(r, t)=A + B r √ t exp  – r 2 4at  , w(r, t)=Ar –1 exp(–aμ 2 t)cos(μr + B)+C, w(r, t)=Ar –1 exp(–μr)cos(μr – 2aμ 2 t + B)+C, w(r, t)= A r erf  r 2 √ at  + B, where A, B, C,andμ are arbitrary constants, and n is an arbitrary positive integer. T8.1.6-2. Reduction to a constant coefficient equation. Some formulas. 1 ◦ . The substitution u(r, t)=rw(r, t) brings the original equation with variable coefficients to the constant coefficient equation ∂ t u = a∂ ww , which is discussed in Subsection T8.1.1. 2 ◦ . Suppose w = w(r, t) is a solution of the original equation. Then the functions w 1 = Aw( λr, λ 2 t + C)+B, w 2 = A |δ + βt| 3/2 exp  – βr 2 4a(δ + βt)  w  r δ + βt , γ + λt δ + βt  , λδ – βγ = 1, where A, B, C, β, δ,andλ are arbitrary constants, are also solutions of this equation. The second formula may usually be encountered with β = 1, γ =–1,andδ = λ = 0. T8.1.6-3. Boundary value problems. For solutions of various boundary value problems, see Subsection T8.1.7 with Φ(r, t) ≡ 0. T8.1.7. Equation of the Form ∂w ∂t = a  ∂ 2 w ∂r 2 + 2 r ∂w ∂r  + Φ(r, t) T8.1.7-1. 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 condition (T8.1.5.1) and various homogeneous boundary conditions are expressed via the Green’s function by formula (T8.1.5.2). T8.1.7-2. Domain: 0 ≤ r ≤ R. First boundary value problem. A boundary condition is prescribed: w = 0 at r = R. 1274 LINEAR EQUATIONS AND PROBLEMS OF MATHEMATIC AL PHYSICS Green’s function: G(r, ξ, t)= 2ξ Rr ∞  n=1 sin  nπr R  sin  nπξ R  exp  – an 2 π 2 t R 2  . T8.1.7-3. 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)= 3ξ 2 R 3 + 2ξ Rr ∞  n=1 μ 2 n + 1 μ 2 n sin  μ n r R  sin  μ n ξ R  exp  – aμ 2 n t R 2  , where μ n are positive roots of the transcendental equation tan μ – μ = 0.Thefirst five roots are μ 1 = 4.4934, μ 2 = 7.7253, μ 3 = 10.9041, μ 4 = 14.0662, μ 5 = 17.2208. T8.1.8. Equation of the Form ∂w ∂t = ∂ 2 w ∂x 2 + 1–2β x ∂w ∂x This dimensionless equation is encountered in problems of the diffusion boundary layer. For β = 0, β = 1 2 ,orβ =– 1 2 , see the equations in Subsections T8.1.4, T8.1.1, or T8.1.6, respectively. T8.1.8-1. Particular solutions. w(x)=A + Bx 2β , w(x, t)=A + 4(1 – β)Bt + Bx 2 , w(x, t)=A + 16(2 – β)(1 – β)Bt 2 + 8(2 – β)Btx 2 + Bx 4 , w(x, t)=x 2n + n  p=1 4 p p! s n,p s n–β,p t p x 2(n–p) , s q,p = q(q – 1) (q – p + 1), w(x, t)=A + 4(1 + β)Btx 2β + Bx 2β+2 , w(x, t)=A + Bt β–1 exp  – x 2 4t  , w(x, t)=A + B x 2β t β+1 exp  – x 2 4t  , w(x, t)=A + Bγ  β, x 2 4t  , w(x, t)=A + B exp(–μ 2 t)x β J β (μx), w(x, t)=A + B exp(–μ 2 t)x β Y β (μx), w(x, t)=A + B x β t exp  – x 2 + μ 2 4t  I β  μx 2t  , w(x, t)=A + B x β t exp  – x 2 + μ 2 4t  K β  μx 2t  , T8.1. PARABOLIC EQUATIONS 1275 where A, B,andμ are arbitrary constants, n is an arbitrary positive number, γ(β, z)isthe incomplete gamma function, J β (z)andY β (z) are Bessel functions, and I β (z)andK β (z) are modified Bessel functions. T8.1.8-2. Formulas and transformations for constructing particular solutions. 1 ◦ . Suppose w = w(x, t) is a solution of the original equation. Then the functions w 1 = Aw( λx, λ 2 t + C), w 2 = A|a + bt| β–1 exp  – bx 2 4(a + bt)  w  x a + bt , c + kt a + bt  , ak – bc = 1, where A, C, a, b,andc are arbitrary constants, are also solutions of this equation. The second formula usually may be encountered with a = k = 0, b = 1,andc =–1. 2 ◦ . The substitution w = x 2β u(x, t) brings the equation with parameter β to an equation of the same type with parameter –β: ∂u ∂t = ∂ 2 u ∂x 2 + 1 + 2β x ∂u ∂x . T8.1.8-3. Domain: 0 ≤ x < ∞. First boundary value problem. The following initial and boundary conditions are prescribed: w = f(x)att = 0, w = g(t)atx = 0. Solution for 0 < β < 1: w(x, t)= x β 2t  ∞ 0 f(ξ)ξ 1–β exp  – x 2 + ξ 2 4t  I β  ξx 2t  dξ + x 2β 2 2β+1 Γ(β + 1)  t 0 g(τ)exp  – x 2 4(t – τ)  dτ (t – τ) 1+β . T8.1.8-4. Domain: 0 ≤ x < ∞. Second boundary value problem. The following initial and boundary conditions are prescribed: w = f(x)att = 0,(x 1–2β ∂ x w)=g(t)atx = 0. Solution for 0 < β < 1: w(x, t)= x β 2t  ∞ 0 f(ξ)ξ 1–β exp  – x 2 + ξ 2 4t  I –β  ξx 2t  dξ – 2 2β–1 Γ(1 – β)  t 0 g(τ)exp  – x 2 4(t – τ)  dτ (t – τ) 1–β . 1276 LINEAR EQUATIONS AND PROBLEMS OF MATHEMATIC AL PHYSICS T8.1.9. Equations of the Diffusion (Thermal) Boundary Layer 1. f(x) ∂w ∂x + g(x)y ∂w ∂y = ∂ 2 w ∂y 2 . This equation is encountered in diffusion boundary layer problems (mass exchange of drops and bubbles with flow). The transformation (A and B are any numbers) t =  h 2 (x) f(x) dx + A, z = yh(x), where h(x)=B exp  –  g(x) f(x) dx  , leads to a constant coefficient equation, ∂ t w = ∂ zz w, which is considered in Subsection T8.1.1. 2. f(x)y n–1 ∂w ∂x + g(x)y n ∂w ∂y = ∂ 2 w ∂y 2 . This equation is encountered in diffusion boundary layer problems (mass exchange of solid particles, drops, and bubbles with flow). The transformation (A and B are any numbers) t = 1 4 (n+1) 2  h 2 (x) f(x) dx +A, z = h(x)y n+1 2 ,whereh(x)=B exp  – n + 1 2  g(x) f(x) dx  , leads to the simpler equation ∂w ∂t = ∂ 2 w ∂z 2 + 1 – 2k z ∂w ∂z , k = 1 n + 1 , which is considered in Subsection T8.1.8. T8.1.10. Schr ¨ odinger Equation ih ∂w ∂t =– h 2 2m ∂ 2 w ∂x 2 + U(x)w T8.1.10-1. Eigenvalue problem. Cauchy problem. Schr ¨ odinger’s equation is the basic equation of quantum mechanics; w is the wave function, i 2 =–1,  is Planck’s constant, m is the mass of the particle, and U(x) is the potential energy of the particle in the force field. 1 ◦ . In discrete spectrum problems, the particular solutions are sought in the form w(x, t)=exp  – iE n  t  ψ n (x), where the eigenfunctions ψ n and the respective energies E n have to be determined by solving the eigenvalue problem d 2 ψ n dx 2 + 2m  2  E n – U(x)  ψ n = 0, ψ n → 0 at x → ∞,  ∞ –∞ |ψ n | 2 dx = 1. (T8.1.10.1) The last relation is the normalizing condition for ψ n . . basic equation of quantum mechanics; w is the wave function, i 2 =–1,  is Planck’s constant, m is the mass of the particle, and U(x) is the potential energy of the particle in the force field. 1 ◦ I β (z)andK β (z) are modified Bessel functions. T8.1.8-2. Formulas and transformations for constructing particular solutions. 1 ◦ . Suppose w = w(x, t) is a solution of the original equation. Then the functions w 1 =. constants, are also solutions of this equation. The second formula may usually be encountered with β = 1, γ =–1 ,and = λ = 0. T8.1.6-3. Boundary value problems. For solutions of various boundary value