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Tham khảo tài liệu 'mixed boundary value problems episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 318 Mixed Boundary Value Problems 1 0.9- 0.8- 0.7- 0.6- 0.5- 0.4- 0.3- 0.2 0 r z Problem 1 The figure labeled Problem 1 illustrates this solution when a 0.5. 4.5 JOINT TRANSFORM METHODS In the previous sections we sought to separate problems according to whether the kernels of dual or triple integral equations contained trigonometrical or Bessel functions. Such clear-cut lines of demarcation are not always possible and we conclude with examples where the analysis includes both Fourier and Hankel transforms as well as Fourier and Fourier-Bessel series. Example 4.5.1 In Section 4.1 we solved Laplace s equation on an infinite strip. See Equation 4.1.84 through Equation 4.1.100. Here we again solve101 Laplace s equation but on a semi-infinite domain which contains two regions with different properties Ẽ Ệ 0 0 sub ject to the boundary conditions x m 0 y L 4.5.1 Ux 0 y 0. 0 y L 4.5.2 lim u x y 0. 0 y L 4.5.3 x u x L 0 0 x m 4.5.4 101 See Shindo Y. and A. Atsumi 1975 Thermal stresses in a laminate composite with infinite row of parallel cracks normal to the interfaces. Int. J. Engng. Sci. 13 25 42. 2008 by Taylor Francis Group LLC Transform Methods 319 u h y u h y egux h y Ẽ2ux h y 0 y L 4.5.5 and i Uy x 0 -1 u x 0 0 0 x a a x X 4.5.6 where h a. The effect of these two different regions introduces an interfacial condition Equation 4.5.5. The solution to Equation 4.5.1 to Equation 4.5.4 is u x y Mf aM Jo sinh kL cos kx dk 2 An cosh n 1 for 0 x h and 4.5.7 __ u x y 22 Bn exp sin - LL n 1 4.5.8 for h x X. An interesting aspect of this problem is that Equation 4.5.7 contains both a Fourier cosine transform and a Fourier sine series. Substituting Equation 4.5.7 and Equation 4.5.8 into Equation 4.5.6 yields the dual integral equations í kA k 1 M kL cos kx dk An coshf j 1 0 x a n l L v L 7 4.5.9 and ftt A k cos kx dk 0 a x X 4.5.10 J0 where M p e-vf sinh p . To solve this set of dual integral equations we introduce a A k I g t Jo kt dt. 4.5.11 J0 We chose this definition
