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Tham khảo tài liệu 'mixed boundary value problems episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 288 Mixed Boundary Value Problems where a 1. Using transform methods or separation of variables the general solution to Equation 4.4.27 Equation 4.4.28 and Equation 4.4.29 is u r z i A k Jo kr e kz dk. 4.4.31 J0 Substituting Equation 4.4.31 into Equation 4.4.30 we find that ỉ A k Jo kr dk V 0 i kA k Jo kr dk 0 0 0 r a a r 1 4.4.32 4.4.33 and poo A k Jo kr dk 0 0 1 r x. 4.4.34 To solve this set of integral equations we Equation 4.4.32 through Equation 4.4.34 can let A k B k D k . Then be rewritten B k J0 kr dk f r 0 r a 4.4.35 0 i kB k J0 kr dk 0 a r x 4.4.36 0 i kD k J0 kr dk 0 0 r 1 4.4.37 0 and poo 0 D k J0 kr dk g r 1 r x 4.4.38 where and f r V -i D k J0 kr dk 0 g r i B k J0 kr dk. 0 4.4.39 4.4.40 Equation 4.4.36 define B k and D k and Equation 4.4.37 are automatically satisfied if we as follows pa P X B k J t cos kt dt D k J ý r sin kr dr. 4.4.41 2008 by Taylor Francis Group LLC Transform Methods 289 If we substitute Equation 4.4.41 into Equation 4.4.35 we have that cos kt J0 kr dk dt f r 4.4.42 after we interchange the order of integration. Using Equation 1.4.14 Equation 4.4.42 simplifies to l ĩỉ 2 dt f r . 4.4.43 Jo yr2 -t2 From Equation 1.2.13 and Equation 1.2.14 we obtain t d f dr . 4.4.44 n dt . t2 - r2 J In a similar manner we find that t 2 Ị dr n dr JT r2 - T2 4.4.45 Next we substitute for D k in Equation 4.4.39 and find that f r V Ị Ị rf T sin kT d T Jo kr dk V Ị rf T y sin kT J0 kr dk dT T t 2 r2 dT 4.4.46 4.4.47 4.4.48 for 0 r a. Here we have used Equation 1.4.13. In a similar manner it is readily shown that g r - Ị r2 t2 dt 1 r x- 4.4.49 Finally we substitute Equation 4.4.48 into Equation 4.4.44 and find that 2 d r r rV 1 0 t --f- -.dr n dt _Jo t2 - r2 _ 2 d j 1 i t d 1 r dr I n dt Jo J1 t2 - r2 _ ỵ t2 - r2 J _ 2V 2 fd rr rdr n nJ 1 dt _ Jo t 2 - r2 t2 - r2 __ 2V 2 r t t n n 1 t2 - T2 d T 4.4.50 4.4.51 4.4.52 2008 by Taylor Francis Group LLC 290 Mixed Boundary Value Problems 1 0.8 0.6 0.4 0.2 0 0 22 Figure 4.4.1 The solution to Equation 4.4.27 .
