tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 4 Part 8

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 4 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | x2 1y Ax2 1y 0 y a y b 0 Now we verify that the Sturm-Liouville properties are satisfied. The eigenvalues 2 An o i . n G Z ln b a are real. There are an infinite number of eigenvalues A1 A2 A3 _ 2 o_ 2 o_ 2 2 I n 2 I 2n 2 I 3n a ln b a J a Ụn b a J a ỵin b a J There is a least eigenvalue _ 2 A1 M 0 but there is no greatest eigenvalue An -TO as n to . For each eigenvalue we found one unique to within a multiplicative constant eigenfunction ộn. We were able to choose the eigenfunctions to be real-valued. The eigenfunction ộn x a sin i ln x a A vn ln b a has exactly n 1 zeros in the open interval a x b. 1454 The eigenfunctions are orthogonal with respect to the weighting function a x x2 1 1. í X. 7 VX. x x dx x a sin nn - x a sin mn - x 2a 1 dx Ắ x a x J a x si nn ln b aj x si ln b aj x dx Ỉ sin n llgal sin mn igf- ai 1dx Ja ln b a ln b a x _ In b a r sin nx sin mx dx n Jo ln b a fn _ T-2 cos n m x cos n m x dx 2n Jo 0 if n m The eigenfunctions are complete. Any piecewise continuous function f x defined on a x b can be expanded in a series of eigenfunctions f x 2 cn n x n 1 where cn f x ộn x ợ x dx O n x x dx The sum converges to 1 f x f x . We do not prove this property. .