Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems
The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differe...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-07-01
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| Series: | Computation |
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| Online Access: | https://www.mdpi.com/2079-3197/10/7/116 |
| _version_ | 1827597569778253824 |
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| author | Călin-Ioan Gheorghiu |
| author_facet | Călin-Ioan Gheorghiu |
| author_sort | Călin-Ioan Gheorghiu |
| collection | DOAJ |
| description | The Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness). |
| first_indexed | 2024-03-09T03:33:11Z |
| format | Article |
| id | doaj.art-21a51fad8f9f46fa8b7384becf588af0 |
| institution | Directory Open Access Journal |
| issn | 2079-3197 |
| language | English |
| last_indexed | 2024-03-09T03:33:11Z |
| publishDate | 2022-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Computation |
| spelling | doaj.art-21a51fad8f9f46fa8b7384becf588af02023-12-03T14:51:52ZengMDPI AGComputation2079-31972022-07-0110711610.3390/computation10070116Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value ProblemsCălin-Ioan Gheorghiu0Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O. Box 68-1, 401010 Cluj-Napoca, RomaniaThe Chebyshev collocation method implemented in Chebfun is used in order to solve a class of second order one-dimensional singular and genuinely nonlinear boundary value problems. Efforts to solve these problems with conventional ChC have generally failed, and the outcomes obtained by finite differences or finite elements are seldom satisfactory. We try to fix this situation using the new Chebfun programming environment. However, for tough problems, we have to loosen the default Chebfun tolerance in Newton’s solver as the ChC runs into trouble with ill-conditioning of the spectral differentiation matrices. Although in such cases the convergence is not quadratic, the Newton updates decrease monotonically. This fact, along with the decreasing behaviour of Chebyshev coefficients of solutions, suggests that the outcomes are trustworthy, i.e., the collocation method has exponential (geometric) rate of convergence or at least an algebraic rate. We consider first a set of problems that have exact solutions or prime integrals and then another set of benchmark problems that do not possess these properties. Actually, for each test problem carried out we have determined how the Chebfun solution converges, its length, the accuracy of the Newton method and especially how well the numerical results overlap with the analytical ones (existence and uniqueness).https://www.mdpi.com/2079-3197/10/7/116Chebfundifferential equationnon-linearitysingularityconvergenceBernstein growth |
| spellingShingle | Călin-Ioan Gheorghiu Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems Computation Chebfun differential equation non-linearity singularity convergence Bernstein growth |
| title | Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems |
| title_full | Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems |
| title_fullStr | Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems |
| title_full_unstemmed | Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems |
| title_short | Chebfun Solutions to a Class of 1D Singular and Nonlinear Boundary Value Problems |
| title_sort | chebfun solutions to a class of 1d singular and nonlinear boundary value problems |
| topic | Chebfun differential equation non-linearity singularity convergence Bernstein growth |
| url | https://www.mdpi.com/2079-3197/10/7/116 |
| work_keys_str_mv | AT calinioangheorghiu chebfunsolutionstoaclassof1dsingularandnonlinearboundaryvalueproblems |