Domain theoretic second-order Euler’s method for solving initial value
problems
problems
File(s)
Published version
Author(s)
Edalat, Abbas
Farjudian, Amin
Mohammadian, Mina
Patinson, Dirk
Type
Conference Paper
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.
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.
Date Issued
2020-10-01
Date Acceptance
2020-05-09
Citation
Electronic Notes in Theoretical Computer Science, 2020, 352, pp.105-128
ISSN
1571-0661
Publisher
Elsevier
Start Page
105
End Page
128
Journal / Book Title
Electronic Notes in Theoretical Computer Science
Volume
352
Copyright Statement
© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
License URL
Source
Mathematical Foundations of Programming Semantics
Subjects
Computation Theory & Mathematics
0802 Computation Theory and Mathematics
0803 Computer Software
1702 Cognitive Sciences
Publication Status
Published
Start Date
2020-06-02
Coverage Spatial
Paris
Date Publish Online
2020-10-22