Mixed Boundary Value Problems Episode 4 doc

... Group, LLC 90 Mixed Boundary Value Problems 0 0.2 0 .4 0.6 0.8 1 0 0.1 0.2 0.3 0 .4 0.5 0 .4 0.5 0.6 0.7 0.8 0.9 1 x/π y/π u(x,y) Figure 3.1.2:Thesolutionu(x, y)tothe mixed boundary value problem ... Group, LLC 86 Mixed Boundary Value Problems 0 0.2 0 .4 0.6 0.8 1 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0 .4 0.5 0.6 0.7 0.8 0.9 1 x/π y u(x,y) Figure 3.1.1:Thesolutionu(x, y)toth...

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... 143 −2 −1 0 1 2 −2 −1 0 1 2 −0.2 0 0.2 0 .4 0.6 0.8 1 1.2 x y u(x,y) Figure 3 .4. 4:Thesolutionu(x, y)tothe mixed boundary value problem governed by Equation 3 .4. 72 and Equation 3 .4. 75 when α = π/4andβ =3π /4. with Equation 3 .4. 85 ... Francis Group, LLC 140 Mixed Boundary Value Problems −2 −1 0 1 2 −2 −1 0 1 2 0.2 0.3 0 .4 0.5 0.6 0.7 0.8 0.9 1 x/a y/a u(x,y) Figure 3...

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... Taylor & Francis Group, LLC 178 Mixed Boundary Value Problems 4. 1 .47 , and Equation 4. 1 .48 , while modifying Equation 4. 1 .49 to read u(x, h)=0, −∞ <x<∞. (4. 1.67) We begin once again using ... 1 .4. 14 since |x| > 1and0≤ τ ≤ 1. Thus, Equation 4. 1. 143 satisfies Equation 4. 1. 141 identically. We now turn our attention to Equation4.1. 142 . Substituting Equatio...

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... solution to Equation 4. 4. 142 , Equation 4. 4. 143 , and Equation 4. 4. 144 is u(r, z)=  ∞ 0 A(k)J 1 (kr)e −kz dk k . (4. 4. 146 ) Substituting Equation 4. 4. 146 into Equation 4. 4. 145 , we find that  ∞ 0 A(k)J 1 (kr) ... G 2 (ξ)=  ∞ 0 sin(ξη/h) cosh(η) dη. (4. 4.193) In deriving Equation 4. 4.191 and Equation 4. 4.192, we used Equation 1 .4. 14 and J 0 (kr)= 2 π  r 0 cos(kx) √...

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... 366 Mixed Boundary Value Problems 2b z r 2a Figure 5.1 .4: Schematicof the geometry in Example 5.1 .4. u(b, z)=0,u(a − ,z)=u(a + ,z), −∞ <z<∞, ... tanh(β)φ(0,b), (5.1. 144 ) and Φ  − (α, b − ) −α tanh(β)Φ − (α, b − )=i tanh(β)φ(0,b), (5.1. 145 ) because φ(0,b − )=φ(0,b + )=φ(0,b). Using Equation 5.1. 144 and Equation 5.1. 145 in conjunction with ... ik  e ikz dk =0, 0 <z&l...

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... commonly used to solve these types of boundary value problems where the nature of the boundary condition changes along a given boundary —theso-called mixed boundary value problem. We begin by introducing ... with mixed- type boundary conditions. J. Heat Transfer, 107, 48 9 49 1. © 2008 by Taylor & Francis Group, LLC 390 Mixed Boundary Valu e Problems −1.5 −0.5 0....

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... ρ 2 dρ  (6.3.28) = ∞  n=−∞   2 1 (η) rη 2  |n| e inθ  1 2π  2π 0 e −inϑ w 0 dϑ  = w 0 . © 2008 by Taylor & Francis Group, LLC 44 4 Mixed Boundary Value Problems 4 −3 −2 −1 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 3.5 4 − 0 0 0.5 1 1.5 0.005 0.025 0.1 0.3 1 1.5 2 3 x y a ... <η<∞, (7.3 .4) with the boundary conditions lim |ξ|→∞ |u(ξ, η)| < ∞, −∞ <η<∞, (7.3.5)...

Ngày tải lên: 13/08/2014, 05:21

Ngày tải lên: 13/08/2014, 05:22

... (1 .4. 56) 24 Ibid., Section 14. 4. See also MacRobert, T. M., 1931: Fourier integrals. Proc. R.Soc. Edinburgh, Ser. A, 51, 116–126. © 2008 by Taylor & Francis Group, LLC 44 Mixed Boundary Value ... =  2π−ϕ −ϕ e −irρ cos(λ) dλ (1 .4. 44) =  2π 0 e −irρ cos(λ) dλ (1 .4. 45) =2πJ 0 (ρr). (1 .4. 46) Equation 1 .4. 45 is equivalent to Equation 1 .4. 44 because the integral of a p...

Ngày tải lên: 13/08/2014, 05:22

... LLC 60 Mixed Boundary Value Problems 0 0.5 1 1.5 2 0 0.2 0 .4 0.6 0.8 1 0 0.1 0.2 0.3 0 .4 0.5 0.6 0.7 0.8 0.9 1 r/a z/a u(r,z)/a Figure 2.5.1:Thesolutionu(r, z)/a to the mixed boundary value problem ... =2ω, 0 ≤ r<a, (2.6. 14) 24 Gradshteyn and Ryzhik, op. cit., Formulas 8 .47 2.3 and 8 .47 3 .4 with z = kr. © 2008 by Taylor & Francis Group, LLC 56 Mixed Boundar...

Ngày tải lên: 13/08/2014, 05:22