tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 6

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Solution uxx Uyy f x y 0 x a 0 y b u 0 y u a y 0 uy x 0 uy x b 0 We will solve this problem with an eigenfunction expansion in x. To determine a suitable set of eigenfunctions substitute the separation of variables u x y X x Y y into the homogeneous partial differential equation. we uxx uyy 0 XY xx XY yy 0 X Y X Y -A2 With the boundary conditions at x 0 a we have the regular Sturm-Liouville problem X -A2X X 0 X a 0 which has the solutions A nn Xn sin n G Z . aa We expand u x y in a series of the eigenfunctions. E. . . in .r un y sin I a n 1 We substitute this series into the partial differential equation and boundary conditions at y 0 b. _ _ - u ysin un y sin f x a a a n 1 x 7 E 7n nn 77X nnx u n 0 sin - y u n b sin - 0 na na n 1 n 1 1774 We expand f x y in a Fourier sine series. f x y n y sin X a Uy -J f x y sin a Jo dx We obtain the ordinary differential equations for the coefficients in the expansion. y - uM X . un 0 un b 0 n e Z . a We will solve these ordinary differential equations with Green functions. Consider the Green function problem 9n y n - nn 9n y n ổ y - n g n 0 n 9n b n . The homogeneous solutions cosh r y and coshfnn y - b 1 aa satisfy the left and right boundary conditions respectively. We compute the Wronskian of these two solutions. W y cosh nny a nn sinh nny a cosh nn y - b a Ịn sinh nn y - b a nn fcosh nny sinh f 2dv-bd - sinh nny cosh f ndv- aa a a a nn . cnnb aa The Green function is a cosh nny a cosh nn y - b a 9n y n . 1 ỵ nn sinh nnb a .