Differential Equations Lecture Work Solutions 144

Differential Equations Lecture Work Solutions 144 - 1a utt...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
1a. u tt c 2 u xx = S ( x, t ) u ( x, 0) = f ( x ) u t ( x, 0) = 0 u (0 , t ) = u ( L, t ) = 0 u ( x, t ) = n =1 u n ( t ) φ n ( x ) S ( x, t ) = n =1 s n ( t ) φ n ( x ) φ n ( x ) = sin n π L x λ n = n π L 2 n = 1 , 2 , · · · s n ( t ) = L 0 S ( x, t ) sin n π L x dx L 0 sin 2 n π L x dx ¨ u n ( t ) + c 2 n π L 2 u n ( t ) = s n ( t ) u n ( t ) = c 1 cos n π L ct + c 2 sin n π L ct + t 0 s n ( τ ) sin c n π L ( t τ ) c n π L u n (0) = c 1 = L 0 f ( x ) sin n π L x dx L 0 sin 2 n π L x dx since u ( x, 0) = f ( x ) ˙ u n (0) = c 2 c n π L = 0
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern