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 . Figure Domain of dependence for the wave equation. Solution 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 .
