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.
Highlights
We consider the ordinary differential equation (ODE) initial value problem (IVP): ⎧ ⎪⎨ y = f (y),⎪⎩ y(0) = (0, . . . , 0), (1)n in which f : [−K, K]n → [−M, M ]n is a continuous vector field, for some natural number n ≥ 1, and positive rational numbers K, M ∈ Q+
Given a non-autonomous equation y (t) = f (y(t), t) with y(t) = (y1(t), . . . , yn(t)), the function defined by z(t) = (y(t), t) satisfies the autonomous equation z (t) = g(z(t)) with g(θ1, . . . , θn+1) ··= (f (θ1, . . . , θn+1), 1). It is well-known that the uniqueness of solutions for the non-autonomous IVP is guaranteed if the field f is Lipschitz continuous in its first n arguments
We presented a domain-theoretic framework for solving IVPs using the second-order Euler method
Summary
We consider the ordinary differential equation (ODE) initial value problem (IVP):. n in which f : [−K, K]n → [−M, M ]n is a continuous vector field, for some natural number n ≥ 1, and positive rational numbers K, M ∈ Q+. Picard’s method has quite a large memory footprint, and the first-order Euler’s method—which has a smaller memory footprint—does not use the local Lipschitz properties of the field This is while, in the first-order Euler method [10], Lipschitz continuity of the field is required, in order to guarantee uniqueness of solutions. We present a domain-theoretic secondorder Euler method, which makes use of the local Lipschitz properties of the field. It is well-known that the uniqueness of solutions for the non-autonomous IVP is guaranteed if the field f is Lipschitz continuous in its first n (out of n + 1) arguments. The results of the current paper can be generalized, in a straightforward manner, to similar analysis of any n-th order Euler method. This, in turn, paves the way for a more detailed bit complexity, similar to [15]
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