CS205A hw8_solutions

# Scientific Computing

This preview shows pages 1–2. Sign up to view the full content.

CS205 Homework #8 Solutions Problem 1 Give a criterion for the well-posedness of the k th order, scalar, homogeneous, constant-coefficient ODE u ( k ) + c k - 1 u ( k - 1) + · · · + c 1 u + c 0 u = 0 (Hint: Transform to a first-order system y = Ay and observe A is a matrix we’ve encountered previously in homework 3 problem 2) Solution Transforming the differential equation into a system of first order equations yields: u 1 u 2 . . . u k - 1 u k = u 2 u 3 . . . u k - k i =1 c i - 1 u i 1 = 0 1 0 · · · 0 0 0 0 1 · · · 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 · · · 1 0 - c 0 - c 1 - c 2 · · · - c k - 2 - c k - 1 u 1 u 2 . . . u k - 1 u k The matrix is a companion matrix as we saw in homework 3. Recall its characteristic polynomial is p ( λ ) = c 0 + c 1 λ + · · · + c k - 1 λ k - 1 + λ k . The eigenvalues of the matrix will be the roots of this polynomial. Thus, if the real parts of the roots are less than zero it is well-posed. If they are all not strictly less than zero then it is asymptotically stable. If any real part is positive then it is ill-posed.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern