Exponentially Convergent Duhamel-Like Algorithms for Differential Equations with an Operator Coefficient Possessing a Variable Domain in a Banach Space

Abstract

A suitable abstract setting of an initial value problem for the first order operator differential equation is introduced for the case when both the unbounded operator coefficient and its domain can depend on the differentiation variable t. A new exponentially convergent algorithm is proposed. The algorithm is based on a generalization of the Duhamel integral for vector-valued functions. The Duhamel-like technique makes it possible to translate the initial problem to an integral equation and then approximate it with exponential accuracy. The theoretical results are illustrated by examples associated with the heat transfer boundary value problems.Key words. first order differential equations in Banach space, operator coefficient with a variable domain, Duhamel's integral, operator exponential, exponentially convergent algorithms

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

“…Remark 2. The results of Proposition 2 remain valid if, instead of the boundedness of v(z), we assume that the integrand from (38) belongs to the class L a,b (D d ). In such case, the parameters , δ from (37) should be determined by = min{a, b}/a, δ = max{a, b}/b.…”

“…Proof. When t = T, the function te z /(1 + e z ) maps the infinite horizontal strip D d of half-height d into the "eye-shaped" D 2 d (see [50] (38) belongs to the class of functions L 1,α (D d ) for any t ∈ (0, T]. Then, the results regarding the convergence of ( 37) to (38), as well as the form of (37) itself, and the error estimate stated in (39) follow from Theorem 4.2.6 of [50].…”

“…The integrals from the above formula for u(t) cannot be evaluated explicitly for arbitrary α. Thus, we rely upon the numerical evaluation of u(t) via exponentially convergent quadrature formulas (38) and [50] (Theorem 4.2.6) with discretization parameters N J and N I , correspondingly. The analysis conducted in the Proof of Proposition 2 suggests to set N J = N I / min{1, α}.…”

“…In both cases, the method will be free of the issues with approximating ∂ α t u in the vicinity of t = 0, provided that the proposed approximation of (5) converges uniformly. Such application scheme also justifies the use of a moderate in size final time T. If, more generally, we assume that Au = A(t, u), then the problem in question can be reduced to (1), (4) using collocation [2,38] or a similar in nature time-stepping scheme, inspired by [39]. In such scenario, A(t, u) is approximated by A(t k , u k ) having spectral characteristics that may vary drastically with k (see Cahn-Hilliard equation from [40], for instance), and the right-hand side in the form A(t k , u k ) − A(t, u), which makes sense only locally.…”