Đang chuẩn bị liên kết để tải về tài liệu:
Advanced Mathematical Methods for Scientists and Engineers Episode 6 Part 2

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 6 part 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | This solution demonstrates the domain of dependence of the solution. The first term is an integral over the triangle domain T 0 T t x CT Ệ x CT . The second term involves only the points x ct 0 . The third term is an integral on the line segment 0 x ct Ệ x ct . In totallity this is just the triangle domain. This is shown graphically in Figure 45.4. Figure 45.4 Domain of dependence for the wave equation. Solution 45.18 Single Sum Representation. First we find the eigenfunctions of the homogeneous problem Au k2u 0. We substitute the separation of variables u x y X x Y y into the partial differential equation. X Y XY k2XY 0 X k2 Y Y A2 We have the regular Sturm-Liouville eigenvalue problem X A2X X 0 X a 0 2014 which has the solutions Xn Xn sin n G N. a a We expand the solution u in a series of these eigenfunctions. E Ấnnx cn y sin a n 1 We substitute this series into the partial differential equation to find equations for the cn y . Cn y cn y - k2Cn y sin nnx ---- o x a - 0 y - VO The series expansion of the right side is E . . . innx dn y sin a n 1 dn y 2 i o x - o y - VO sin fnnx dx a J 0 V a J dn y 2 sin nn o y - . aa The the equations for the cn y are cCn y - Cn 0 Cn b 0. The homogeneous solutions are cosh any sinh any where ơn ựk2 nn a 2. The solutions that satisfy the boundary conditions at y 0 and y b are sinh any and sinh an y - b respectively. The Wronskian of these .