tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 10

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | v xT 2- ỉ n x - T ỗ x - -T 2n J 1 1 f z z . _ _ _ g n H x - T - H x - - T d C 2n J v x T 1 f x T f x - 1 p x t T 2c - d Finally we make the change of variables t t c u x t v x T to obtain D Alembert s solution of the wave equation 1 1 p x ct u x t 2 ỉ x - Ct f x Ct 2cj 9 2 d2- Solution With the change of variables d dtd 1 d T Ct- dT dTdt Cat- v x T u x t the problem becomes Vtt Vxx v x 0 ỉ x Vt x 0 - g x . C We take the Laplace transform in T of the equation we consider x to be a parameter s2V x s - sv x 0 - Vt x 0 Vxx x s Vxx x s - s2V x s -sf x - Cg x Now we have an ordinary differential equation for V x s now we consider s to be a parameter . We impose the boundary conditions that the solution is bounded at x TO. Consider the Green s function problem 9xx x - s29 x ỗ x - 9 to bounded. 1934 esx is a homogeneous solution that is bounded at x x . e sx is a homogeneous solution that is bounded at x to. The Wronskian of these solutions is W x esx s esx e sx s e-sx 2s. Thus the Green s function is J_ esx e for x e_s x-ị 2s es e-sx for x i 2s The solution for V x s is V x s 2s p e-i - i sf i SK di 1 p 1 p V x s -2J -- - l f i di -XJ e-s x- 1 di 1 f 1 f e-s il V x s 2j e-s i f x di -c J x 0 c Now we take the inverse Laplace transform and interchange the order of integration. v x T 1L 1 e-s i f x i di _J- 1L-1 i s g x i di 2c J- s 1 p 1 p v x T 2j L- e-s I f x i di --J L-1 e-s i s g x Ỵ di 1 r . 1 r . v x T - J s t 1 I f x di -cj H t Iii g x di .1 v x T - f x T f x T 2cJ g x i di .