| dc.contributor |
Department of Atmospheric and Oceanic Science and Laboratory for Advanced Scientific Computation, University of Michigan, 48109, Ann Arbor, Michigan |
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| dc.contributor |
Ann Arbor |
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| dc.creator |
Boyd, John P. |
|
| dc.date |
2006-09-11T15:31:10Z |
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| dc.date |
2006-09-11T15:31:10Z |
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| dc.date |
1986-09 |
|
| dc.date.accessioned |
2022-05-19T13:30:03Z |
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| dc.date.available |
2022-05-19T13:30:03Z |
|
| dc.identifier |
Boyd, John P.; (1986). "An analytical and numerical study of the two-dimensional Bratu equation." Journal of Scientific Computing 1(2): 183-206. <http://hdl.handle.net/2027.42/44977> |
|
| dc.identifier |
0885-7474 |
|
| dc.identifier |
1573-7691 |
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| dc.identifier |
http://hdl.handle.net/2027.42/44977 |
|
| dc.identifier |
http://dx.doi.org/10.1007/BF01061392 |
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| dc.identifier |
Journal of Scientific Computing |
|
| dc.identifier.uri |
http://localhost:8080/xmlui/handle/CUHPOERS/117317 |
|
| dc.description |
Bratu's problem, which is the nonlinear eigenvalue equation Δu+λ exp( u )=0 with u =0 on the walls of the unit square and λ as the eigenvalue, is used to develop several themes on applications of Chebyshev pseudospectral methods. The first is the importance of symmetry : because of invariance under the C 4 rotation group and parity in both x and y , one can slash the size of the basis set by a factor of eight and reduce the CPU time by three orders of magnitude. Second, the pseudospectral method is an analytical as well as a numerical tool: the simple approximation λ ≈3.2A exp(−0.64 A ), where A is the maximum value of u(x, y) , is derived via collocation with but a single interpolation point, but is quantitatively accurate for small and moderate A . Third, the Newton-Kantorovich/Chebyshev pseudospectral algorithm is so efficient that it is possible to compute good numerical solutions—five decimal places—on a microcomputer in basic . Fourth, asymptotic estimates of the Chebyshev coefficients can be very misleading: the coefficients for moderately or strongly nonlinear solutions to Bratu's equations fall off exponentially rather than algebraically with v until v is so large that one has already obtained several decimal places of accuracy. The corner singularities, which dominate the behavior of the Chebyshev coefficients in the limit v →∞, are so weak as to be irrelevant, and replacing Bratu's problem by a more complicated and realistic equation would merely exaggerate the unimportance of the corner branch points even more. |
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| dc.description |
Peer Reviewed |
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| dc.description |
http://deepblue.lib.umich.edu/bitstream/2027.42/44977/1/10915_2005_Article_BF01061392.pdf |
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| dc.format |
1086454 bytes |
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| dc.format |
3115 bytes |
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| dc.format |
application/pdf |
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| dc.format |
text/plain |
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| dc.format |
application/pdf |
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| dc.language |
en_US |
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| dc.publisher |
Kluwer Academic Publishers-Plenum Publishers; Plenum Publishing Corporation ; Springer Science+Business Media |
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| dc.subject |
Bratu's Problem |
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| dc.subject |
Computational Mathematics and Numerical Analysis |
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| dc.subject |
Mathematical and Computational Physics |
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| dc.subject |
Mathematics |
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| dc.subject |
Algorithms |
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| dc.subject |
Appl.Mathematics/Computational Methods of Engineering |
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| dc.subject |
Nonlinear Eigenvalue Problem |
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| dc.subject |
Spectral Methods |
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| dc.subject |
Science (General) |
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| dc.subject |
Education |
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| dc.subject |
Science |
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| dc.subject |
Social Sciences |
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| dc.title |
An analytical and numerical study of the two-dimensional Bratu equation |
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| dc.type |
Article |
|