Mejlbro L.'s Calculus 4c-2, Examples of Eigenvalue Problems PDF

By Mejlbro L.

Show description

Read Online or Download Calculus 4c-2, Examples of Eigenvalue Problems PDF

Similar analysis books

Download e-book for kindle: Risk-Based Reliability Analysis and Generic Principles for by Michael T. Todinov

For a very long time, traditional reliability analyses were orientated in the direction of determining the extra trustworthy method and preoccupied with maximising the reliability of engineering platforms. at the foundation of counterexamples in spite of the fact that, we exhibit that choosing the extra trustworthy process doesn't unavoidably suggest choosing the method with the smaller losses from mess ups!

Read e-book online Analysis and Topology in Nonlinear Differential Equations: A PDF

This quantity is a set of articles awarded on the Workshop for Nonlinear research held in João Pessoa, Brazil, in September 2012. The impact of Bernhard Ruf, to whom this quantity is devoted at the party of his sixtieth birthday, is perceptible in the course of the assortment via the alternative of issues and methods.

Additional info for Calculus 4c-2, Examples of Eigenvalue Problems

Sample text

1) Prove that λ = 0 is not an eigenvalue. 2) Find all the eigenvalues and the corresponding eigenfunctions. 1) If λ = 0, then the equation is reduced to y = 0, the complete solution of which is y = c 1 + c2 x where y = c2 . Then by insertion into the boundary conditions, y(0) − y (0) = c1 − c2 = 0, hence c1 = c2 , and y(π) − y (π) = c1 + c2 π − c2 = 0, so c1 = −(π − 1)c2 . Since c1 = c2 = 0 is the only solution, we conclude that λ = 0 is not an eigenvalue. com 58 Examples of Eigenvalue Problems Eigenvalue problems 2) If λ = 0, then the characteristic polynomial R2 + 2λR + 2λ2 = (R + λ)2 + λ2 , has the simple roots R = −λ ± iλ.

We get from y(0) = 0 that c1 = 0, so we need only in the following to consider functions of the form (2) y(x) = c2 e−x sin(kx). We get from y(1) = 0 that 1 · sin k = 0. e We get proper solutions c2 = 0, when kn = nπ, n ∈ N. com 36 Examples of Eigenvalue Problems Eigenvalue problems • Eigenvalues and eigenfunctions. The eigenvalues are λn = 1 + kn2 = n2 π 2 + 1, n ∈ N. A corresponding eigenfunction is by (2) given by yn (x) = e−x sin(nπx), x ∈ [0, 1], Please click the advert and all eigenfunctions corresponding to λn are given by c · yn (x), where c = 0 is an arbitrary constant.

Now, ϕ(0) = 0, so sinh(k) − k cosh(k) < 0 for alle k > 0, and we only get the solution c2 = 0. Thus, no λ < 0 can be an eigenvalue. Alternatively we see that (1) is equivalent to tanh(k) = k, where a graphical analysis shows that k = 0 is the only solution. com 32 Examples of Eigenvalue Problems Eigenvalue problems 2) Let λ = 0, so the equation is reduced to d2 y = 0. dx2 • The complete solution follows by two integrations, y = c1 x + c 2 where y = c1 . • Insertion into the boundary conditions: y(0) = 0 = c2 , thus y = c1 x where y = c1 .

Download PDF sample

Calculus 4c-2, Examples of Eigenvalue Problems by Mejlbro L.


by William
4.4

Rated 4.56 of 5 – based on 33 votes