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Mixed Boundary Value Problems Episode 7

Tham khảo tài liệu 'mixed boundary value problems episode 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 168 Mixed Boundary Value Problems for j 1 N xx j j-0.5 dx xxi j j-0.5 dx end the value of x at the interfaces for j 0 N xx_e j 1 j dx xxi_e j 1 j dx end Solve the integral equation Equation 4.1.16 for n 1 N x xx n b n x the right side of Equation 4.1.16 for m 1 N xi xxi m if n m the first term on the left side of Equation 4.1.16 AA n m h else AA n m 0 end if n m NN n m log sqrt 1-x x else NN n m log x-xi x sqrt 1-xi xi -xi sqrt 1-x x end NN n m NN n m 0.5 log x xi Find the non-singular contribution from the integrals in Equation 4.1.16 AA n m AA n m 2 NN n m dx pi . 2 asin x xi dx pi sqrt 1-xi xi end end Add in the contribution from the singular term for n 1 N for m 2 N k n-m 1 psi_0 -1 psi 1 0.25 k k- k-1 k-1 if k 0 psi_0 psi_0 k log abs k psi_1 psi_1 - 0.5 k k log abs k end if k 1 psi_0 psi_0 - k-1 log abs k-1 2008 by Taylor Francis Group LLC Transform Methods 169 psi 1 psi 1 0.5 k-1 k-1 log abs k-1 end psi_1 psi_1 k psi_0 alpha 0.5 dx log dx dx psi_0-psi_1 beta 0.5 dx log dx dx psi_1 AA n m-1 AA n m-1 - alpha pi AA n m AA n m - beta pi end end f AA b Compute f x from Equation 4.1.16 Having found f x from Equation 4.1.16 g t follows from Equation 4.1.15. g 1 0 for n 2 N 1 t xx_e n g n 0 for m 1 n-1 xi xxi m g n g n 2 xi f m dx pi sqrt t t-xi xi end end With g t we can compute A k and u x y . The maximum number of wavenumbers included in the computations is K_max dk. The MATLAB code is Compute A k from Equation 4.1.8. Use Simpson s rule. for n 1 N derivative n g n 1 -g n dx end for k 0 K_max ak k dk t xx 1 A k 1 derivative 1 besselj 0 ak t k 0 term for n 2 N-1 t xx n if mod n 2 0 A k 1 A k 1 4 derivative n besselj 0 ak t else A k 1 A k 1 2 derivative n besselj 0 ak t end end t xx N k kmax term A k 1 A k 1 derivative N besselj 0 ak t A k 1 A k 1 dx 3 2008 by Taylor Francis Group LLC 170 Mixed Boundary Value Problems Figure 4.1.1 The solution to Equation 4.1.1 subject to the mixed boundary conditions given by Equation 4.1.2 through Equation 4.1.4 when h 1. end .