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Quantum spectral methods for differential equations

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arXiv 2019年

作者： Childs, Andrew M. Liu, Jin-Peng Joint Center for Quantum Information and Computer Science Department of Computer Science Institute for Advanced Computer Studies Applied Mathematics Statistics and Scientific Computation

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a d-dimensional system of linear equations or linear differential equations with complexity poly(log d). While several of these algorithms approximate the solution to within o with complexity poly(log(1/o)), no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity poly(log d, log(1/o)). Copyright © 2019, The Authors. All rights reserved.

关键词： Quantum theory

A truncated V-fractional derivative in Rn

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arXiv 2017年

作者： Sousa, J. Vanterler Da C. De Oliveira, E. Capelas Department of Applied Mathematics Institute of Mathematics Statistics and Scientific Computation University of Campinas - UNICAMP rua Sergio Buarque de Holanda 651 Campinas SP13083-859 Brazil

Using the six parameters truncated Mittag-Leffler function, we introduce a convenient truncated function to define the so-called truncated V-fractional derivative type. After a discussion involving some properties associated with this derivative, we propose the derivative of a vector valued function and define the V-fractional Jacobian matrix whose properties allow us to say that: the multivariable truncated V-fractional derivative type, as proposed here, generalizes the truncated V-fractional derivative type and can bee extended to obtain a truncated V-fractional partial derivative type. As applications we discuss and prove the change of order associated with two index i.e., the commutativity of two truncated V-fractional partial derivative type and propose the truncated V-fractional Green's theorem. Copyright © 2017, The Authors. All rights reserved.

关键词： Jacobian matrices

Leibniz Type Rule: Ψ-Hilfer Fractional Derivative

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arXiv 2018年

作者： Sousa, J. Vanterler Da C. Oliveira, E. Capelas De Department of Applied Mathematics Institute of Mathematics University of Campinas-UNICAMP Statistics and Scientific Computation rua Sérgio Buarque de Holanda 651 Campinas13083-859 Brazil

In this paper, we present the Leibniz rule for the Ψ-Hilfer (Ψ-H) fractional derivative in two versions, the first in relation to Ψ-RL fractional derivative and the second in relation to the Ψ-H fractional derivative. In this sense, we present some particular cases of Leibniz rules and Leibniz type rules from the investigated case.26A33, 33A30 Copyright © 2018, The Authors. All rights reserved.

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Topology Optimization of Structures Using Higher Order Finite Elements in Analysis

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Topology Optimization of Structures Using Higher Order Finit...

International Conference on Modeling, Simulation and Optimization, CoMSO 2020

作者： Mukherjee, Sougata Lu, Dongcheng Dutta, Subhrajit Raghavan, Balaji Breitkopf, Piotr Xiao, Manyu Department of Civil Engineering National Institute of Technology Silchar Assam788010 India Xi’an Key Laboratory of Scientific Computation and Applied Statistics Department of Applied Mathematics School of Mathematics and Statistics Northwestern Polytechnical University Xi’an710072 China Laboratoire de Genie Civil et Genie Mecanique EA 3913 Institut National des Sciences Appliquees de Rennes Rennes France Alliance Sorbonne Universites Universite de Technologie de Compiegne Laboratoire Roberval FRE UTC-CNRS 2012 Compiegne France

ISBN: (纸本)9789811598289

Topology optimization in general progresses with numerical instability i.e. checkerboard patterns if no filtering radius is used i.e. imposement of minimum length scale. Many studies are done in past decades to get rid of numerical instability or checkerboard pattern while optimizing the topology of mechanical structures. The proposed solution in this paper the structure is meshing structure twice (dual mesh) where first mesh defines the material distribution, and another mesh is utilized for computing the stress and displacement on an element which lets us to a stress-based refinement of mesh without modification of optimum design variables. Optimality criteria method is used for optimizing the design variable i.e. relative density over the domain. Another significance of this approach is that imposition minimum radius not needed to optimize design variables over the structural domain, instead the length scale is determined by the design mesh. A comparison is done between the topology obtained, compliance value and time taken after discretizing the structure with 4 noded element, 8 noded element and 9 noded quadrilateral element and a graph is also plotted to show the compliance value. © 2021, The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

关键词： Topology

Ulam-hyers-rassias stability for a class of fractional integro-differential equations

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arXiv 2018年

作者： Sousa, J. Vanterler Da C. De Oliveira, E. Capelas Department of Applied Mathematics Institute of Mathematics Statistics and Scientific Computation University of Campinas - UNICAMP rua Sérgio Buarque de Holanda 651 Campinas SP13083-859 Brazil

By means of the recent Ψ-Hilfer fractional derivative and of the Banach fixedpoint theorem, we investigate stabilities of Ulam-Hyers, Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias on closed intervals [a, b] and [a,∞) for a particular class of fractional integrodifferential equations. Copyright © 2018, The Authors. All rights reserved.

关键词： Stability

Truncated v-fractional taylor's formula with applications

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arXiv 2017年

作者： Da C. Sousa, J. Vanterler De Oliveira, E. Capelas Department of Applied Mathematics Institute of Mathematics Statistics and Scientific Computation University of Campinas - UNICAMP rua Sérgio Buarque de Holanda 651 Campinas SP13083-859 Brazil

In this paper, we present and prove a new truncated V-fractional Taylor's formula using the truncated V-fractional variation of constants formula. In this sense, we present the truncated V-fractional Taylor's remainder by means of V-fractional integral, essential for analyzing and comparing the error, when approaching functions by polynomials. From these new results, some applications were made involving some inequalities, specifically, we generalize the Cauchy-Schwartz inequality. Copyright © 2017, The Authors. All rights reserved.

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Existence of mild solutions to Hilfer fractional evolution equations in Banach space

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arXiv 2018年

作者： Sousa, J.C. Vanterler Da Department of Applied Mathematics Institute of Mathematics Statistics and Scientific Computation University of Campinas – UNICAMP rua Sérgio Buarque de Holanda 651 Campinas SP13083–859 Brazil

In this paper, we investigate the existence of mild solutions to Hilfer fractional equation of semi-linear evolution with non-instantaneous impulses, using the concepts of equicontinuous C0-semigroup and Kuratowski measure of non-compactness in Banach space Ω.MSC Codes 26A33, 34K45, 47D06 Copyright © 2018, The Authors. All rights reserved.

关键词： Banach spaces

A note on the Laplace transform and the variable-order differential operators

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arXiv 2017年

作者： Sousa, Vanterler Da C.J. de Oliveira, E. Capelas Department of Applied Mathematics Institute of Mathematics Statistics and Scientific Computation University of Campinas – UNICAMP rua Sérgio Buarque de Holanda 651 Campinas SP13083–859 Brazil

In this short note, using the variable-order differential operator introduced by means of the inverse Laplace transform [1], we questioned the result obtained by Yang and Tenreiro Machado [2]. Copyright © 2017, T... 详细信息

关键词： Laplace transforms

Stability of the fractional volterra integro-differential equation by means of ψ-hilfer operator

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arXiv 2018年

作者： DA C. SOUSA, J. VANTERLER DE OLIVEIRA, E. CAPELAS Department of Applied Mathematics Institute of Mathematics Statistics and Scientific Computation University of Campinas - UNICAMP rua Sérgio Buarque de Holanda 651 Campinas SP13083-859 Brazil

In this paper, using the Riemann-Liouville fractional integral with respect to another function and the ψ -Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro-differential equation. In this sense, for this new fractional Volterra integro-differential equation, we study the Ulam-Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed-point theorem. As an application, we present the Ulam-Hyers stability using the α-resolvent operator in the Sobolev space W1,1(R+, C).MSC Codes 26A33, 34A08, 34A12, 34K20, 37C25 Copyright © 2018, The Authors. All rights reserved.

关键词： Stability