## THE CHEBOP SYSTEM FOR AUTOMATIC SOLUTION OF DIFFERENTIAL EQUATIONS (2008)

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Citations: | 7 - 5 self |

### BibTeX

@MISC{Driscoll08thechebop,

author = {Tobin A. Driscoll and Folkmar Bornemann and Lloyd N. Trefethen},

title = {THE CHEBOP SYSTEM FOR AUTOMATIC SOLUTION OF DIFFERENTIAL EQUATIONS },

year = {2008}

}

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### Abstract

In Matlab, it would be good to be able to solve a linear differential equation by typing u = L\f, wheref, u, andLare representations of the right-hand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, at least in one space dimension, based on the previously developed chebfun system in object-oriented Matlab. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution.

### Citations

404 | Chebyshev and Fourier Spectral Methods
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- 1999
(Show Context)
Citation Context ...determined automatically, with no explicit connection to the length of f. The spectral discretizations we employ are standard ones based on polynomial interpolants in Chebyshev points as described in =-=[6, 9, 19]-=-. For example, toTHECHEBOPSYSTEM 709 differentiate a function spectrally, one interpolates it by a polynomial in Chebyshev points, differentiates the interpolant, and evaluates the result at the same... |

256 |
ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods
- Lehoucq, Sorensen, et al.
- 1998
(Show Context)
Citation Context ...R algorithm and related methods as in LAPACK [2], and eigs is the command for sparse problems, where a small number of eigenvalues are computed by the implicitly restarted Arnoldi iteration of ARPACK =-=[12]-=-. In the chebop system the functionality required is that of eigs: an operator will generally have infinitely many eigenvalues, and we want to find a finite number of them. The chebop eigenvalue comma... |

254 | Level-spacing distributions and the Airy kernel
- Tracy, Widom
- 1994
(Show Context)
Citation Context ...eld the Tracy–Widom distribution F2(s) of random matrix theory. This gives, in the large matrix limit, the probability that the rescaled maximum eigenvalue of the Gaussian unitary ensemble is below s =-=[18]-=-: ( F2(s) =exp − >> v = sum(u.^2)-cumsum(u.^2); >> F2 = exp(cumsum(v)-sum(v)); ∫ ∞ ∫ ∞ The mean value ∫ sdF2(s) can now be calculated by >> [d,s] = domain(F2.ends); >> sum(s.*diff(F2)) ans = -1.771086... |

247 |
A Practical Guide to Pseudospectral Methods
- Fornberg
(Show Context)
Citation Context ...determined automatically, with no explicit connection to the length of f. The spectral discretizations we employ are standard ones based on polynomial interpolants in Chebyshev points as described in =-=[6, 9, 19]-=-. For example, toTHECHEBOPSYSTEM 709 differentiate a function spectrally, one interpolates it by a polynomial in Chebyshev points, differentiates the interpolant, and evaluates the result at the same... |

146 | Sparse matrices in MATLAB: Design and implementation
- Gilbert, Moler, et al.
- 1992
(Show Context)
Citation Context ...lently by expansions in rescaled Chebyshev polynomials, either globally or piecewise. For example, the following commands construct a chebfun f corresponding to f(x) =sin(x)+sin(x 2 ) on the interval =-=[0, 10]-=- and plot the image shown on the left in Figure 1.1. >> f = chebfun(’sin(x)+sin(x.^2)’,[0,10]); >> plot(f) Figure 1.1: On the left, a chebfun, realized in this case by a polynomial interpolant through... |

52 | An extension of MATLAB to continuous functions and operators
- Battles, Trefethen
(Show Context)
Citation Context ...of Driscoll and Trefethen was supported by UK EPSRC Grant EP/E045847.702 T. A. DRISCOLL ET AL. The system we shall introduce is built on the chebfun system, whose properties we now briefly summarize =-=[3, 4, 16, 20]-=-. In Matlab, one may start with a vector v and apply operations such as sum(v) (sum of the components), diff(v) (finite differences), or norm(v) (square root of sum of squares). In the chebfun system ... |

25 |
Spectral integration and two-point boundary value problem
- Greengard
- 1991
(Show Context)
Citation Context ...ated to ill-conditioning of the associated matrices. A strategy for avoiding this problem, put forward by Greengard in 1991, is to take the highest order derivative of a variable as the basic unknown =-=[7, 11, 13, 14]-=-. For example, in a second-order differential equation involving the variable u, one can regard v = u ′′ as the unknown and reformulate the differential equation in integral form. This is a powerful i... |

16 | An efficient spectral method for ordinary differential equations with rational function coefficients
- Coutsias, Hagstrom, et al.
- 1996
(Show Context)
Citation Context ...ated to ill-conditioning of the associated matrices. A strategy for avoiding this problem, put forward by Greengard in 1991, is to take the highest order derivative of a variable as the basic unknown =-=[7, 11, 13, 14]-=-. For example, in a second-order differential equation involving the variable u, one can regard v = u ′′ as the unknown and reformulate the differential equation in integral form. This is a powerful i... |

14 | Distribution functions for edge eigenvalues in Orthogonal and Symplectic Ensembles: Painlevé representations
- Dieng
- 2005
(Show Context)
Citation Context ...ze below the required tolerance of 10 −14 ,takingabout 1 second on a workstation. The solution is correct to about 13 digits, which is considerably more accurate than the solution calculated by Dieng =-=[8]-=- using Matlab’s bvp4c with a far more sophisticated initial iterate. Once the boundary-value problem is solved, the chebfun system can easily post-process u to yield the Tracy–Widom distribution F2(s)... |

13 |
Accurate solution of the Orr-Sommerfeld equation
- Orszag
- 1971
(Show Context)
Citation Context ...sociated with the eigenvalue instability of plane Poiseuille fluid flow, and in the code below, R and α are set to their critical values R = 5772.22 and α =1.02056, first determined by Orszag in 1971 =-=[15]-=-. [d,x] = domain(-1,1); I = eye(d); D = diff(d); R = 5772.22; alpha = 1.02056; B = D^2 - alpha^2; A = B^2/R - 1i*alpha*(2+diag(1-x.^2)*B); A.lbc(1) = I; A.lbc(2) = D; A.rbc(1) = I; A.rbc(2) = D; e = e... |

12 | A rational spectral collocation method with adaptively transformed Chebyshev grid points
- Tee, Trefethen
(Show Context)
Citation Context ...ut the refinement is global. There are applications where one would wish for local adaptivity, and we hope to investigate extensions of this kind in the future, making use for example of the ideas of =-=[17]-=-. The current state of the art in adaptive spectral methods, however, is not very advanced. Adaptivity has been carried much further for finite difference and finite element methods, but these are usu... |

11 |
Numerical Linear Algebra for Continuous Functions, D.Phil thesis
- Battles
- 2006
(Show Context)
Citation Context ...of Driscoll and Trefethen was supported by UK EPSRC Grant EP/E045847.702 T. A. DRISCOLL ET AL. The system we shall introduce is built on the chebfun system, whose properties we now briefly summarize =-=[3, 4, 16, 20]-=-. In Matlab, one may start with a vector v and apply operations such as sum(v) (sum of the components), diff(v) (finite differences), or norm(v) (square root of sum of squares). In the chebfun system ... |

11 | On the numerical evaluation of Fredholm determinants - Bornemann |

10 | Computing numerically with functions instead of numbers.Math
- TREFETHEN
- 2007
(Show Context)
Citation Context ...of Driscoll and Trefethen was supported by UK EPSRC Grant EP/E045847.702 T. A. DRISCOLL ET AL. The system we shall introduce is built on the chebfun system, whose properties we now briefly summarize =-=[3, 4, 16, 20]-=-. In Matlab, one may start with a vector v and apply operations such as sum(v) (sum of the components), diff(v) (finite differences), or norm(v) (square root of sum of squares). In the chebfun system ... |

6 | L.N.: Piecewise smooth chebfuns - Pachón, Platte, et al. |

5 |
A spectral collocation method based on integrated Chebyshev polynomials for biharmonic boundary-value problems
- Mai-Duy, Tanner
- 2007
(Show Context)
Citation Context ...ated to ill-conditioning of the associated matrices. A strategy for avoiding this problem, put forward by Greengard in 1991, is to take the highest order derivative of a variable as the basic unknown =-=[7, 11, 13, 14]-=-. For example, in a second-order differential equation involving the variable u, one can regard v = u ′′ as the unknown and reformulate the differential equation in integral form. This is a powerful i... |

1 |
On the numerical evaluation of Fredholm determinants, manuscript
- Bornemann
- 2008
(Show Context)
Citation Context ...2(s) can now be calculated by >> [d,s] = domain(F2.ends); >> sum(s.*diff(F2)) ans = -1.77108680741657 s ξ u(η) 2 ) dηdξ . and this result is correct to 12 digits. For information on this problem, see =-=[5]-=-. 7 Discussion. The algorithms and software system we have described are extraordinarily convenient, at least in their basic setting of linear problems with smooth solutions. They give users interacti... |

1 |
A comparison of Chebyshev methods for solving fourth-order semilinear initial boundary value problems, manuscript
- Muite
- 2007
(Show Context)
Citation Context |