Operator methods for boundary value problems

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Linear Differential Operators

This is new in Chebfun Version 4. A differential or integral operator normally has infinitely many eigenvalues, so one could not expect an overload of EIG for chebops. EIGS, however, has been overloaded. Here's an example involving sine waves. By default, eigs tries to find the six eigenvalues whose eigenmodes are "most readily converged to", which approximately means the smoothest ones.

You can change the number sought and tell eigs where to look for them.

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Note, however, that you can easily confuse eigs if you ask for something unreasonable, like the largest eigenvalues of a differential operator. Here we compute 10 eigenvalues of the Mathieu equation and plot the 9th and 10th corresponding eigenfunctions, known as an elliptic cosine and sine. Note the imposition of periodic boundary conditions.

For these one must specify two linear chebops A and B, with the boundary conditions all attached to A. This is a fourth-order generalized eigenvalue problem, requiring two conditions at each boundary. In Matlab, EXPM computes the exponential of a matrix, and this command has been overloaded in Chebfun to compute the exponential of a linear operator.

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Here is a more fanciful analogous computation with a complex initial function obtained from the "scribble" command introduced in Chapter 5. As it happens, expm does not map discontinuous data with the usual Chebfun accuracy, so warning messages are generated. We'll say a word, just a word, about how Chebfun carries out these computations. The methods involved are Chebyshev spectral methods on adaptive grids.


Elliptic differential operators on Lipschitz domains and abstract boundary value problems

The basic idea is that linear differential or integral operators are disretized by spectral differentiation or integration matrices. Such a matrix applies the desired operator to polynomials via interpolation at Chebyshev points, with certain rows of the matrix modified to impose boundary conditions.

Boundary-value Problems and Finite-difference Equations

When a differential equation is solved in Chebshev, the problem is solved on a sequence of grids of size 9, 17, 33, Much more than just this is really going on, however, including the decomposition of intervals into subintervals to handle coefficients that are only piecewise smooth. One matter you might not guess was challenging is the determination of whether or not an operator is linear!

In Chebfun the operator is defined by an anonymous function, but if it is linear, special actions should be possible such as application of EIGS and EXPM and solution of differential equations in a single step without iteration. Chebfun includes special devices to determine whether a chebop is linear so that these effects can be realized. As mentioned, the discretization length of a Chebfun solution is chosen automatically according to the instrinsic resolution requirements.

However, the matrices that arise in Chebyshev spectral methods are notoriously ill-conditioned. Thus the final accuracy in solving differential equations in Chebfun is rarely close to machine precision. Typically one loses two or three digits for second-order differential equations and five or six for fourth-order problems.

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Some problems involve several variables coupled together. In Chebfun, these are treated with the use of quasimatrices, that is, chebfuns with several columns. We can solve the problem like this:. User Account Log in Register Help. Search Close Advanced Search Help. My Content 1 Recently viewed 1 Boundary value problem Show Summary Details.

Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples

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    Operator methods for boundary value problems Operator methods for boundary value problems
    Operator methods for boundary value problems Operator methods for boundary value problems
    Operator methods for boundary value problems Operator methods for boundary value problems
    Operator methods for boundary value problems Operator methods for boundary value problems
    Operator methods for boundary value problems Operator methods for boundary value problems

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