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Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems

Abbas Edalat, Amin Farjudian, Mina Cyrus Orcid Logo, Dirk Pattinson

Electronic Notes in Theoretical Computer Science, Volume: 352, Pages: 105 - 128

Swansea University Author: Mina Cyrus Orcid Logo

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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...

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Published in: Electronic Notes in Theoretical Computer Science
ISSN: 1571-0661
Published: Elsevier BV 2020
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URI: https://cronfa.swan.ac.uk/Record/cronfa68136
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spelling 2025-01-06T14:09:00.7836430 v2 68136 2024-11-01 Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems fb5320369ba356b005c93d6e38c94caf 0009-0005-9085-4652 Mina Cyrus Mina Cyrus true false 2024-11-01 MACS 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. Journal Article Electronic Notes in Theoretical Computer Science 352 105 128 Elsevier BV 1571-0661 Domain theory; domain of Lipschitz functions; initial value problem; algebraic complexity; interval arithmetic 22 10 2020 2020-10-22 10.1016/j.entcs.2020.09.006 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University Another institution paid the OA fee 2025-01-06T14:09:00.7836430 2024-11-01T11:08:57.7256036 Faculty of Science and Engineering School of Mathematics and Computer Science - Computer Science Abbas Edalat 1 Amin Farjudian 2 Mina Cyrus 0009-0005-9085-4652 3 Dirk Pattinson 4
title Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
spellingShingle Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
Mina Cyrus
title_short Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
title_full Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
title_fullStr Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
title_full_unstemmed Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
title_sort Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems
author_id_str_mv fb5320369ba356b005c93d6e38c94caf
author_id_fullname_str_mv fb5320369ba356b005c93d6e38c94caf_***_Mina Cyrus
author Mina Cyrus
author2 Abbas Edalat
Amin Farjudian
Mina Cyrus
Dirk Pattinson
format Journal article
container_title Electronic Notes in Theoretical Computer Science
container_volume 352
container_start_page 105
publishDate 2020
institution Swansea University
issn 1571-0661
doi_str_mv 10.1016/j.entcs.2020.09.006
publisher Elsevier BV
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Computer Science{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Computer Science
document_store_str 0
active_str 0
description 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.
published_date 2020-10-22T05:45:42Z
_version_ 1821654953161129984
score 11.047674

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