Exponentially Convergent Method For An Abstract Nonlocal Problem With Integral Nonlinearity

Vasylyk, Vitalii; Макаров, В. Л.

doi:10.1007/s11253-017-1332-8

Cited by 1 publication

(1 citation statement)

References 19 publications

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“…This would allows us to study (1), ( 25) by means of the technique developed in Sections 5 and 6 with modifications that would account for an approximation error introduced by the quadrature. Hence, the results for such nonlocal integral conditions are readily obtainable, where the existence conditions for (1), (25), that would imitate (5), can be established following the ideas of earlier works (see, e.g., [189,190]). In other areas, the interest to nonlocal integral initial conditions and their modified versions is also motivated by mathematical models based on fractional differential equations and inclusions, as well as by problems involving dynamic control with state-dependent requirements [191,192].…”

Section: Lemma 3 All Zeros Of P(u) Satisfy the Double Estimate Max{vmentioning

confidence: 90%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

Melnik

2021

MCA

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Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.

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Section: Lemma 3 All Zeros Of P(u) Satisfy the Double Estimate Max{vmentioning

confidence: 90%

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Sytnyk

Melnik

2021

MCA

View full text Add to dashboard Cite

show abstract

Exponentially Convergent Method For An Abstract Nonlocal Problem With Integral Nonlinearity

Cited by 1 publication

References 19 publications

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

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