Tham khảo tài liệu 'mixed boundary value problems episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 198 Mixed Boundary Value Problems and Problem 2 rc sin kx dk 0 1 x x. 0 Step 3 Fredricks27 showed that dual integral equations of the form A k J sin kx dk f x 0 x a and tt A k sin kx dk 0 a x 0 have the solution A k 2 EE 2m 1 BnJ2m 1 nn J2m 1 ka n 1 m 0 where Bn is given by the Fourier sine series f x 2 Bn sin nnx a n 1 0 x a. Clearly f 0 0. Use this result to show that 4 1 n l _ A k tY N -y--------- 2m 1 Bn J2m 1 nn J2m 1 k . n n n 1 m 0 The figure labeled Problem 2 illustrates this solution u x y . 27 Fredricks R. W. 1958 Solution of a pair of integral equations from elastostatics. Proc. Natl. Acad. Sci. 44 309 312. See also Sneddon I. N. 1962 Dual integral equations with trigonometrical kernels. Proc. Glasgow Math. Assoc. 5 147 152. 2008 by Taylor Francis Group LLC Transform Methods 199 3. Following Example solve Laplace s equation 2 12 0 0 x X 0 y h dx2 dy2 with the boundary conditions Ux 0 y 0 lim u x y 0 0 y h x J uy x 0 g x 0 X a 1 u x 0 0 a x X and u x h 0 0 x X. 1 Step 1 Using separation of variables or transform methods show that the general solution to the problem is 2 fTO u x y A k sinh k h y cos kx dk. n J 0 Step 2 Using boundary condition 1 show that A k satisfies the dual integral equations 2 f kA k 1 M kh sinh kh cos kx dk g x 0 x a n J 0 and 2 A k sinh kh cos kx dk 0 a x X n J 0 where M kh e-kh sinh kh . Step 3 Setting sinh kh A k T h r J0 kr dr 0 show that the second integral equation in Step 2 is identically satisfied. Step 4 Show that the first integral equation in Step 2 leads to the integral equation h t Ị T h r y kM kh j0 kr J0 kt dk dr I j L dx with 0 t a. Step 5 Simplify your results in the case g x 1 and show that they are identical with the results given in Example by Equation through Equation . 2008 by Taylor Francis Group LLC 200 Mixed Boundary Value Problems 4. Following Example solve Laplace s equation28 2 12 0 X x X 0 y h ox2 ay2 with the boundary conditions lim u x y 0 0 y h 1 1 uy x 0 p x h u x
