tailieunhanh - Advanced Mathematical Methods for Scientists and Engineers Episode 5 Part 1

Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 5 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Exercise 1. Use the cosine transform to solve y a2y 0 on x 0 with y 0 b y x 0. 2. Use the cosine transform to show that the Green function for the above with b 0 is G x 1 e-a x-i __1 e-a x-i Hint Solution Exercise 1. Use the sine transform to solve y a2y 0 on x 0 with y 0 b y x 0. 2. Try using the Laplace transform on this problem. Why isn t it as convenient as the Fourier transform 3. Use the sine transform to show that the Green function for the above with b 0 is 1 e- x-i e-a z 2a Hint Solution Exercise 1. Find the Green function which solves the equation y 2yy ft2 y2 y fi x y 0 ft 0 in the range X x X with boundary conditions y x y x 0. 1574 2. Use this Green s function to show that the solution of y 2yy p2 ụ2 y g x y 0 p 0 y -TO y 0 with g w 0 in the limit as ụ 0 is 1 i x _ . y 4 g O sin p x - d p tt You may assume that the interchange of limits is permitted. Hint Solution Exercise Using Fourier transforms find the solution u x to the integral equation f -__uis__ d 1 0 a b. . X x - 2 a2 x2 b2 Hint Solution Exercise The Fourer cosine transform is defined by fc u I f x cos wx dx. K J 0 1. From the Fourier theorem show that the inverse cosine transform is given by f x 2 fc w cos wx dw. 0 2. Show that the cosine transform of f x is -f - m K .