Journal article 3 views
Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
Abbas Edalat,
Amin Farjudian,
Mina Cyrus 
,
Dirk Pattinson
,
Dirk Pattinson
Electronic Notes in Theoretical Computer Science, Volume: 352, Pages: 105 - 128
Swansea University Author:
Mina Cyrus
Full text not available from this repository: check for access using links below.
DOI (Published version): 10.1016/j.entcs.2020.09.006
Abstract
A domain-theoretic method for solving initial value problems (IVPs) is presented, together with proofs of soundness, completeness, and some results on the algebraic complexity of the method. While the common fixed-precision interval arithmetic methods are restricted by the precision of the underlyin...
| Published in: | Electronic Notes in Theoretical Computer Science |
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| ISSN: | 1571-0661 |
| Published: |
Elsevier BV
2020
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| Online Access: |
Check full text
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa68136 |
| Abstract: |
A domain-theoretic method for solving initial value problems (IVPs) is presented, together with proofs of soundness, completeness, and some results on the algebraic complexity of the method. While the common fixed-precision interval arithmetic methods are restricted by the precision of the underlying machine architecture, domain-theoretic methods may be complete, i.e., the result may be obtained to any degree of accuracy. Furthermore, unlike methods based on interval arithmetic which require access to the syntactic representation of the vector field, domain-theoretic methods only deal with the semantics of the field, in the sense that the field is assumed to be given via finitely-representable approximations, to within any required accuracy.In contrast to the domain-theoretic first-order Euler method, the second-order method uses the local Lipschitz properties of the field. This is achieved by using a domain for Lipschitz functions, whose elements are consistent pairs that provide approximations of the field and its local Lipschitz properties. In the special case where the field is differentiable, the local Lipschitz properties are exactly the local differential properties of the field. In solving IVPs, Lipschitz continuity of the field is a common assumption, as a sufficient condition for uniqueness of the solution. While the validated methods for solving IVPs commonly impose further restrictions on the vector field, the second-order Euler method requires no further condition. In this sense, the method may be seen as the most general of its kind.To avoid complicated notations and lengthy arguments, the results of the paper are stated for the second-order Euler method. Nonetheless, the framework, and the results, may be extended to any higher-order Euler method, in a straightforward way. |
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| Keywords: |
Domain theory; domain of Lipschitz functions; initial value problem; algebraic complexity; interval arithmetic |
| College: |
Faculty of Science and Engineering |
| Start Page: |
105 |
| End Page: |
128 |