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Mixed Boundary Value Problems Episode 5

Tham khảo tài liệu 'mixed boundary value problems episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 108 Mixed Boundary Value Problems Substituting Equation 3.2.41 into Equation 3.2.35 we obtain the integral equation i h t yx Pn cos V2Jc Ln 1 Pn-1 cos t sin nx dt f x . 3.2.42 Using the results from Problem 3 in Section 1.3 Equation 3.2.42 simplifies to r h t dt cscfxÌ f x Jx ựcos x cos t V 2 J c x n. 3.2.43 From Equation 1.2.11 and Equation 1.2.12 we obtain 2d h t --n dt r f x cos x 2 dx t ựcos t cos x 3.2.44 Using the results from Equations 3.2.22 3.2.28 3.2.34 3.2.35 3.2.41 and 3.2.44 the solution to the dual equations 00 yx nCn sin ny g n y 0 y Y n 1 oo yx Cn sin ny f n y Y y n k n 1 3.2.45 is n Cn -y h t Pn-1 cos t Pn cos t dt y2 J 0 3.2.46 where h t u t D r g 2 dỉ n 2J J0 vcos cos t and h t _ 2 d n ffr- 0 cos 2 h t ndt Jt ựcos t cos e edi 0 t Y Y t n. 3.2.47 3.2.48 Therefore the solution to Equation 3.2.20 and Equation 3.2.21 is Cn h t Pn-1 cos t Pn cos t dt 3.2.49 20 where 2 . t t U0 L v- o n sin 2 n o L du Jo vda cst 0 t Y. 3-2-5U 2008 by Taylor Francis Group LLC Separation of Variables 109 Consequently making the back substitution the dual series 00 7 sin nx f x n n 1 s oo y bn sin nx g x . k n 1 0 x c c x n 3.2.51 has the solution n bn k t Pn-1 cos t Pn cos t dt V 2 J 0 3.2.52 where k t 2d f f ff . n dt J0 ựcos cos t 0 t c 3.2.53 and 2 k t tan n y r gX cos 2 2 d d Jt ựcos t cos c t n. 3.2.54 Using Equation 3.2.51 through Equation 3.2.54 we finally have that An p i k t Pn-1 cos t Pn cos t dt 20 3.2.55 where k t 2 d r U0 LỆ n sin 2 k ndt Jo ựcos e cos t d 0 t ni L. 3.2.56 3.3 DUAL FOURIER-BESSEL SERIES Dual Fourier-Bessel series arise during mixed boundary value problems in cylindrical coordinates where the radial dimension is of finite extent. Here we show a few examples. Example 3.3.1 Let us find16 the potential for Laplace s equation in cylindrical coordinates Ề ÌTT. df 0 0 r 1 0 3.3.1 dr2 r dr dz2 16 Originally solved by Borodachev N. M. and F. N. Borodacheva 1967 Considering the effect of the walls for an impact of a circular disk on liquid. Mech. .