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Journal articles on the topic 'Nonlocal equations in time'

Author: Grafiati

Published: 15 March 2025

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1

Dong, Hongjie, Chulan Zeng, and Qi S. Zhang. "Time Analyticity for Nonlocal Parabolic Equations." SIAM Journal on Mathematical Analysis 55, no. 3 (June 7, 2023): 1883–915. http://dx.doi.org/10.1137/22m1490740.

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2

Ding, Xiao-Li, and Juan J. Nieto. "Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms." Fractional Calculus and Applied Analysis 21, no. 2 (April 25, 2018): 312–35. http://dx.doi.org/10.1515/fca-2018-0019.

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AbstractIn this paper, we consider the analytical solutions of multi-term time-space fractional partial differential equations with nonlocal damping terms for general mixed Robin boundary conditions on a finite domain. Firstly, method of reduction to integral equations is used to obtain the analytical solutions of multi-term time fractional differential equations with integral terms. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the multi-term time-space fractional partial differential equations with nonlocal damping terms to the multi-term time fractional differential equations with integral terms. By applying the obtained analytical solutions to the resulting multi-term time fractional differential equations with integral terms, the desired analytical solutions of the multi-term time-space fractional partial differential equations with nonlocal damping terms are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

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3

Ablowitz, Mark J., and Ziad H. Musslimani. "Integrable space-time shifted nonlocal nonlinear equations." Physics Letters A 409 (September 2021): 127516. http://dx.doi.org/10.1016/j.physleta.2021.127516.

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4

Morawetz, Klaus, Pavel Lipavský, and Václav Špička. "Retarded versus Time-Nonlocal Quantum Kinetic Equations." Annals of Physics 294, no. 2 (December 2001): 135–64. http://dx.doi.org/10.1006/aphy.2001.6197.

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5

Lv, Cong, Deqin Qiu, and Q. P. Liu. "Riemann–Hilbert approach to two-component modified short-pulse system and its nonlocal reductions." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 9 (September 2022): 093120. http://dx.doi.org/10.1063/5.0088293.

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In this paper, a Riemann–Hilbert approach to a two-component modified short-pulse (mSP) system on the line with zero boundary conditions is developed. A parametric representation of the solution to the related Cauchy problem is obtained. Four nonlocal integrable reductions, namely, the real reverse space-time nonlocal focusing and defocusing mSP equations and the complex reverse space-time nonlocal focusing and defocusing mSP equations, are studied in detail. For each case, soliton solutions are presented, and, unlike their local counterparts, the nonlocal equations exhibit certain novel properties induced by the impact of nonlocality.

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6

Cichoń, Mieczysław, Bianca Satco, and Aneta Sikorska-Nowak. "Impulsive nonlocal differential equations through differential equations on time scales." Applied Mathematics and Computation 218, no. 6 (November 2011): 2449–58. http://dx.doi.org/10.1016/j.amc.2011.07.057.

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7

Ludu, Andrei. "Nonlocal Symmetries for Time-Dependent Order Differential Equations." Symmetry 10, no. 12 (December 19, 2018): 771. http://dx.doi.org/10.3390/sym10120771.

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A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

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8

Feng, Wei, and Song-Lin Zhao. "Soliton solutions to the nonlocal non-isospectral nonlinear Schrödinger equation." International Journal of Modern Physics B 34, no. 25 (September 9, 2020): 2050219. http://dx.doi.org/10.1142/s0217979220502197.

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In this paper we study the nonlocal reductions for the non-isospectral Ablowitz-Kaup-Newell-Segur equation. By imposing the real and complex nonlocal reductions on the non-isospectral Ablowitz-Kaup-Newell-Segur equation, we derive two types of nonlocal non-isospectral nonlinear Schrödinger equations, in which one is real nonlocal non-isospectral nonlinear Schrödinger equation and the other is complex nonlocal non-isospectral nonlinear Schrödinger equation. Of both of these two equations, there are the reverse time nonlocal type and the reverse space nonlocal type. Soliton solutions in terms of double Wronskian to the reduced equations are obtained by imposing constraint conditions on the double Wronskian solutions of the non-isospectral Ablowitz-Kaup-Newell-Segur equation. Dynamics of the one-soliton solutions are analyzed and illustrated by asymptotic analysis.

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9

Ashurov, Ravshan, and Yusuf Fayziev. "On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations." Fractal and Fractional 6, no. 1 (January 12, 2022): 41. http://dx.doi.org/10.3390/fractalfract6010041.

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The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

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10

Ma, Wen-Xiu. "Integrable nonlocal nonlinear Schrödinger equations associated with 𝑠𝑜(3,ℝ)." Proceedings of the American Mathematical Society, Series B 9, no. 1 (January 14, 2022): 1–11. http://dx.doi.org/10.1090/bproc/116.

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We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) . The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal reverse-space, reverse-time and reverse-spacetime nonlinear Schrödinger equations associated with so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) .

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11

Zhang, Li-Qin, and Wen-Xiu Ma. "Nonlocal PT-Symmetric Integrable Equations of Fourth-Order Associated with so(3, ℝ)." Mathematics 9, no. 17 (September 2, 2021): 2130. http://dx.doi.org/10.3390/math9172130.

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The paper aims to construct nonlocal PT-symmetric integrable equations of fourth-order, from nonlocal integrable reductions of a fourth-order integrable system associated with the Lie algebra so(3,R). The nonlocalities involved are reverse-space, reverse-time, and reverse-spacetime. All of the resulting nonlocal integrable equations possess infinitely many symmetries and conservation laws.

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12

Shankar, Ravi. "Nonlocal Extensions of First Order Initial Value Problems." Axioms 13, no. 8 (August 21, 2024): 567. http://dx.doi.org/10.3390/axioms13080567.

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We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial “volume” problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities.

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13

Liu, Shi-min, Jing Wang, and Da-jun Zhang. "Solutions to Integrable Space-Time Shifted Nonlocal Equations." Reports on Mathematical Physics 89, no. 2 (April 2022): 199–220. http://dx.doi.org/10.1016/s0034-4877(22)00023-4.

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14

Tarasov, Vasily E. "General Fractional Dynamics." Mathematics 9, no. 13 (June 22, 2021): 1464. http://dx.doi.org/10.3390/math9131464.

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General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

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15

Khare, Avinash, and Avadh Saxena. "Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 122903. http://dx.doi.org/10.1063/5.0109384.

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We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.

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16

Carrillo, José A., and Young-Pil Choi. "Mean-Field Limits: From Particle Descriptions to Macroscopic Equations." Archive for Rational Mechanics and Analysis 241, no. 3 (June 1, 2021): 1529–73. http://dx.doi.org/10.1007/s00205-021-01676-x.

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AbstractWe rigorously derive pressureless Euler-type equations with nonlocal dissipative terms in velocity and aggregation equations with nonlocal velocity fields from Newton-type particle descriptions of swarming models with alignment interactions. Crucially, we make use of a discrete version of a modulated kinetic energy together with the bounded Lipschitz distance for measures in order to control terms in its time derivative due to the nonlocal interactions.

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17

Li, Linrui, and Shu Wang. "The Singularity Formation on the Coupled Burgers–Constantin–Lax–Majda System with the Nonlocal Term." Discrete Dynamics in Nature and Society 2020 (July 17, 2020): 1–8. http://dx.doi.org/10.1155/2020/2757398.

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In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

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18

Umurzakhova, Zh B., K. R. Yesmakhanova, A. A. Naizagarayeva, and U. Meirambek. "Nonlocal Schrödinger-Maxwell-Bloch Equations." Journal of Physics: Conference Series 2090, no. 1 (November 1, 2021): 012061. http://dx.doi.org/10.1088/1742-6596/2090/1/012061.

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Abstract In this paper we research the (1+1)-dimensional system of Schrodinger-Maxwell-Bloch equations (NLS-MBE), which describes the optical pulse propagation in an erbium doped fiber and find PT-symmetric and reverse space-time Schrodinger-Maxwell-Bloch equations, i.e. the kinds of nonlocal Schrodinger-Maxwell-Bloch equations. In particular case, the system of Schrödinger-Maxwell-Bloch equations is integrable by the Inverse Scattering Method as shown in the work of M.A blowitz and Z. Musslimani. Following this method we prove the integrability of the nonlocal system of Schröodinger-Maxwell-Bloch equations by Lax pairs. Also the explicit and different seed solutions are constructed by using Darboux transformation.

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19

HUNTER, JOHN K. "SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES." Journal of Hyperbolic Differential Equations 03, no. 02 (June 2006): 247–67. http://dx.doi.org/10.1142/s0219891606000781.

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We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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20

Wang, Xiaohuan. "Blow-up solutions of the stochastic nonlocal heat equations." Stochastics and Dynamics 19, no. 02 (March 27, 2019): 1950014. http://dx.doi.org/10.1142/s021949371950014x.

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This paper is concerned with the blow-up phenomenon of stochastic nonlocal heat equations. We first establish the sufficient condition to ensure that the stochastic nonlocal heat equations have a unique non-negative solution. Then the problem of blow-up solutions in finite time is considered.

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21

Wu, Chih-Ping, and Yen-Jung Chen. "Cylindrical Bending Vibration of Multiple Graphene Sheet Systems Embedded in an Elastic Medium." International Journal of Structural Stability and Dynamics 19, no. 04 (April 2019): 1950035. http://dx.doi.org/10.1142/s0219455419500354.

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Based on the Eringen nonlocal elasticity theory and multiple time scale method, an asymptotic nonlocal elasticity theory is developed for cylindrical bending vibration analysis of simply-supported, [Formula: see text]-layered, and uniformly or nonuniformly-spaced, graphene sheet (GS) systems embedded in an elastic medium. Both the interactions between the top and bottom GSs and their surrounding medium and the interactions between each pair of adjacent GSs are modeled as one-parameter Winkler models with different stiffness coefficients. In the formulation, the small length scale effect is introduced to the nonlocal constitutive equations by using a nonlocal parameter. The nondimensionalization, asymptotic expansion, and successive integration mathematical processes are performed for a typical GS. After assembling the motion equations for each individual GS to form those of the multiple GS system, recurrent sets of motion equations can be obtained for various order problems. Nonlocal multiple classical plate theory (CPT) is derived as a first-order approximation of the current nonlocal plane strain problem, and the motion equations for higher-order problems retain the same differential operators as those of nonlocal multiple CPT, although with different nonhomogeneous terms. Some nonlocal plane strain solutions for the natural frequency parameters of the multiple GS system with and without being embedded in the elastic medium and their corresponding mode shapes are presented to demonstrate the performance of the asymptotic nonlocal elasticity theory.

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22

Leal, Claudio, Carlos Lizama, and Marina Murillo-Arcila. "Lebesgue regularity for nonlocal time-discrete equations with delays." Fractional Calculus and Applied Analysis 21, no. 3 (June 26, 2018): 696–715. http://dx.doi.org/10.1515/fca-2018-0037.

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Abstract In this work we provide a new and effective characterization for the existence and uniqueness of solutions for nonlocal time-discrete equations with delays, in the setting of vector-valued Lebesgue spaces of sequences. This characterization is given solely in terms of the R-boundedness of the data of the problem, and in the context of the class of UMD Banach spaces.

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23

Fedorov, V. E., N. D. Ivanova, and Yu Yu Fedorova. "On a time nonlocal problem for inhomogeneous evolution equations." Siberian Mathematical Journal 55, no. 4 (July 2014): 721–33. http://dx.doi.org/10.1134/s0037446614040144.

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24

Wang, Guanying, Xingchun Wang, and Yongjin Wang. "Long time behavior for nonlocal stochastic Kuramoto–Sivashinsky equations." Statistics & Probability Letters 87 (April 2014): 54–60. http://dx.doi.org/10.1016/j.spl.2013.12.022.

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25

Ma, Wen-Xiu. "Inverse scattering for nonlocal reverse-time nonlinear Schrödinger equations." Applied Mathematics Letters 102 (April 2020): 106161. http://dx.doi.org/10.1016/j.aml.2019.106161.

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26

Chen, An, Qiang Du, Changpin Li, and Zhi Zhou. "Asymptotically compatible schemes for space-time nonlocal diffusion equations." Chaos, Solitons & Fractals 102 (September 2017): 361–71. http://dx.doi.org/10.1016/j.chaos.2017.03.061.

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27

Sin, Chung-Sik, Hyong-Chol O, and Sang-Mun Kim. "Diffusion equations with general nonlocal time and space derivatives." Computers & Mathematics with Applications 78, no. 10 (November 2019): 3268–84. http://dx.doi.org/10.1016/j.camwa.2019.04.025.

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28

Chang-Lara, Héctor A., and Dennis Kriventsov. "Further Time Regularity for Nonlocal, Fully Nonlinear Parabolic Equations." Communications on Pure and Applied Mathematics 70, no. 5 (October 3, 2016): 950–77. http://dx.doi.org/10.1002/cpa.21671.

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29

Banchuin, Rawid. "Nonlocal fractal calculus based analyses of electrical circuits on fractal set." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 41, no. 1 (November 24, 2021): 528–49. http://dx.doi.org/10.1108/compel-06-2021-0210.

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Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

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30

Liu, Wei, Zhenyun Qin, Kwok Wing Chow, and Senyue Lou. "Families of Rational and Semirational Solutions of the Partial Reverse Space-Time Nonlocal Mel′nikov Equation." Complexity 2020 (May 22, 2020): 1–18. http://dx.doi.org/10.1155/2020/2642654.

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Exact periodic and localized solutions of a nonlocal Mel′nikov equation are derived by the Hirota bilinear method. Many conventional nonlocal operators involve integration over a spatial or temporal domain. However, the present class of nonlocal equations depends on properties at selected far field points which result in a potential satisfying parity time symmetry. The present system of nonlocal partial differential equations consists of two dependent variables in two spatial dimensions and time, where the dependent variables physically represent a wave packet and an auxiliary scalar field. The periodic solutions may take the forms of breathers (pulsating modes) and line solitons. The localized solutions can include propagating lumps and rogue waves. These nonsingular solutions are obtained by appropriate choice of parameters in the Hirota expansion. Doubly periodic solutions are also computed with elliptic and theta functions. In sharp contrast with the local Mel′nikov equation, the auxiliary scalar field in the present set of solutions can attain complex values. Through a coordinate transformation, the governing equation can reduce to the Schrödinger–Boussinesq system.

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31

El-Nabulsi, Rami Ahmad. "Modified field equations from a complexified nonlocal metric." Canadian Journal of Physics 97, no. 8 (August 2019): 816–27. http://dx.doi.org/10.1139/cjp-2018-0168.

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We argue that it is possible to obtain higher-derivative Einstein’s field equations by means of an extended complexified backward–forward nonlocal extension of the space–time metric, which depends on space–time vectors. Our approach generalizes the notion of the covariant derivative along tangent vectors of a given manifold, and accordingly many of the differential geometrical operators and symbols used in general relativity. Equations of motion are derived and a nonlocal complexified general relativity theory is formulated. A number of illustrations are proposed and discussed accordingly.

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32

Hobiny, Aatef, and Ibrahim Abbas. "The Effect of a Nonlocal Thermoelastic Model on a Thermoelastic Material under Fractional Time Derivatives." Fractal and Fractional 6, no. 11 (November 2, 2022): 639. http://dx.doi.org/10.3390/fractalfract6110639.

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This article develops a novel nonlocal theory of generalized thermoelastic material based on fractional time derivatives and Eringen’s nonlocal thermoelasticity. An ultra-short pulse laser heats the surface of the medium’s surrounding plane. Using the Laplace transform method, the basic equations and their accompanying boundary conditions were numerically solved. The distribution of thermal stress, temperature and displacement are physical variables for which the eigenvalues approach was employed to generate the analytical solution. Visual representations were used to examine the influence of the nonlocal parameters and fractional time derivative parameters on the wave propagation distributions of the physical fields for materials. The consideration of the nonlocal thermoelasticity theory (nonlocal elasticity and heat conduction) with fractional time derivatives may lead us to conclude that the variations in physical quantities are considerably impacted.

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33

Shi, Yue, Chen Wang, Weiao Yang, and Xiangpeng Xin. "Exact solutions of the high-dimensional extended generalized Broer-Kaup equations with nonlocal symmetry." Physica Scripta 99, no. 10 (September 23, 2024): 105048. http://dx.doi.org/10.1088/1402-4896/ad7994.

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Abstract To analyze the generalized Broer-Kaup (GBK) system, we have employed the methodology suggested by Lou et al, which involves constructing high-dimensional equations from conservation laws associated with low-dimensional partial differential equations. For the first time, we have derived the (2+1)-dimensional integrable GBK equations by leveraging the conservation law of the (1+1)-dimensional GBK equations. Through the imposition of constraints within the (2+1)-dimensional GBK equations, we have obtained a novel (1+1)-dimensional GBK equations. Subsequently, we pioneered the use of a nonlocal approach to analyze this new equations. Specifically, we constructed a closed system incorporating nonlocal symmetries. This was achieved by introducing the Lax pairs associated with potential functions and GBK equations, utilizing conservation laws. By applying the nonlocal symmetries to this closed system, we were able to deduce the generating element. The exact solution of the equation is achieved by combining finite symmetry transformations with a symmetry reduction technique that involves approximations. The dynamic behavior of the equations is studied by means of figures of the exact solutions.

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34

Yuan, Yueding, and Zhiming Guo. "Global Asymptotic Stability in a Class of Reaction-Diffusion Equations with Time Delay." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/378172.

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We study a very general class of delayed reaction-diffusion equations in which the reaction term can be nonmonotone and spatially nonlocal. By using a fluctuation method, combined with the careful analysis of the corresponding characteristic equations, we obtain some sufficient conditions for the global asymptotic stability of the trivial solution and the positive steady state to the equations subject to the Neumann boundary condition.

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35

Zhang, Xiaoju, Kai Zheng, Yao Lu, and Huanhuan Ma. "Global existence and long-time behavior of solutions for fully nonlocal Boussinesq equations." Electronic Research Archive 31, no. 9 (2023): 5406–24. http://dx.doi.org/10.3934/era.2023274.

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<abstract><p>In this paper, we study initial boundary value problems for the following fully nonlocal Boussinesq equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ _0^{C}D_{t}^{\beta}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma}{_0^{C}D_{t}^{\beta}}u = -(-\Delta)^{\sigma}f(u) $\end{document} </tex-math></disp-formula></p> <p>with spectral fractional Laplacian operators and Caputo fractional derivatives. To our knowledge, there are few results on fully nonlocal Boussinesq equations. The main difficulty is that each term of this equation has nonlocal effect. First, we obtain explicit expressions and some rigorous estimates of the Green operators for the corresponding linear equation. Further, we get global existence and some decay estimates of weak solutions. Second, we establish new chain and Leibnitz rules concerning $ (-\Delta)^{\sigma} $. Based on these results and small initial conditions, we obtain global existence and long-time behavior of weak solutions under different dimensions $ N $ by Banach fixed point theorem.</p></abstract>

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36

Liu, Jinshan, Huanhe Dong, Yong Fang, and Yong Zhang. "The Soliton Solutions for Nonlocal Multi-Component Higher-Order Gerdjikov–Ivanov Equation via Riemann–Hilbert Problem." Fractal and Fractional 8, no. 3 (March 19, 2024): 177. http://dx.doi.org/10.3390/fractalfract8030177.

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The Lax pairs of the higher-order Gerdjikov–Ivanov (HOGI) equation are extended to the multi-component formula. Then, we first derive four different types of nonlocal group reductions to this new system. To construct the solution of these four nonlocal equations, we utilize the Riemann–Hilbert method. Compared to the local HOGI equation, the solutions of nonlocal equations not only depend on the local spatial and time variables, but also the nonlocal variables. To exhibit the dynamic behavior, we consider the reverse-spacetime multi-component HOGI equation and its Riemann–Hilbert problem. When the Riemann–Hilbert problem is regular, the integral form solution can be given. Conversely, the exact solutions can be obtained explicitly. Finally, as concrete examples, the periodic solutions of the two-component nonlocal HOGI equation are given, which is different from the local equation.

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37

Avalishvili, Gia, and Mariam Avalishvili. "On nonclassical problems for first-order evolution equations." gmj 18, no. 3 (July 14, 2011): 441–63. http://dx.doi.org/10.1515/gmj.2011.0028.

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Abstract The present paper deals with nonclassical initial-boundary value problems for parabolic equations and systems and their generalizations in abstract spaces. Nonclassical problems with nonlocal initial conditions for an abstract first-order evolution equation with time-dependent operator are considered, the existence and uniqueness results are proved and the algorithm of approximation of nonlocal problems by a sequence of classical problems is constructed. Applications of the obtained general results to initial-boundary value problems for parabolic equations and systems are considered.

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Carrillo, José A., Alina Chertock, and Yanghong Huang. "A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure." Communications in Computational Physics 17, no. 1 (November 28, 2014): 233–58. http://dx.doi.org/10.4208/cicp.160214.010814a.

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AbstractWe propose a positivity preserving entropy decreasing finite volume scheme for nonlinear nonlocal equations with a gradient flow structure. These properties allow for accurate computations of stationary states and long-time asymptotics demonstrated by suitably chosen test cases in which these features of the scheme are essential. The proposed scheme is able to cope with non-smooth stationary states, different time scales including metastability, as well as concentrations and self-similar behavior induced by singular nonlocal kernels. We use the scheme to explore properties of these equations beyond their present theoretical knowledge.

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39

Kulagin, Anton E., Alexander V. Shapovalov, and Andrey Y. Trifonov. "Semiclassical Spectral Series Localized on a Curve for the Gross–Pitaevskii Equation with a Nonlocal Interaction." Symmetry 13, no. 7 (July 17, 2021): 1289. http://dx.doi.org/10.3390/sym13071289.

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We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension.

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Nowinski, J. L. "On the Wave Propagation in Elastic Multilayer Periodic Media With Nonlocal Interactions." Journal of Applied Mechanics 57, no. 4 (December 1, 1990): 937–40. http://dx.doi.org/10.1115/1.2897664.

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After a brief review of the main concepts of the nonlocal theory of elasticity, the equations of the nonlocal elastic moduli are derived, and the constitutive equations of the nonlocal medium established. Propagation of a longitudinal time-periodic wave normal to the laminae of the layered medium is then analyzed, and the equation of the wave dispersion determined. The dispersion originates from two sources: the configuration (discreteness) of the structure, and the nonlocal constitution of the material of the laminae. A numerical example accompanied by a graph illustrates the dependence of the effective wave velocity on the wave number in the entire Brillouin zone. It is found that for very short waves the wave velocity decreases to about 64 percent of its conventional value established for waves of long wavelength.

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Shen, Wenxian, and Zhongwei Shen. "Transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity." Discrete & Continuous Dynamical Systems - A 37, no. 2 (2017): 1013–37. http://dx.doi.org/10.3934/dcds.2017042.

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Yang, Bo, and Yong Chen. "Several reverse-time integrable nonlocal nonlinear equations: Rogue-wave solutions." Chaos: An Interdisciplinary Journal of Nonlinear Science 28, no. 5 (May 2018): 053104. http://dx.doi.org/10.1063/1.5019754.

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Pukal’s’kyi, І. D., and B. О. Yashan. "Nonlocal Multipoint (In Time) Problem for Parabolic Equations with Degeneration." Journal of Mathematical Sciences 243, no. 1 (October 11, 2019): 34–44. http://dx.doi.org/10.1007/s10958-019-04523-3.

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Pinigina, N. R. "Time-Nonlocal Boundary Value Problem for Degenerate Sobolev Type Equations." Journal of Mathematical Sciences 211, no. 6 (November 9, 2015): 811–23. http://dx.doi.org/10.1007/s10958-015-2636-6.

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Stratis, Ioannis G., and Athanasios N. Yannacopoulos. "Electromagnetic fields in linear and nonlinear chiral media: a time-domain analysis." Abstract and Applied Analysis 2004, no. 6 (2004): 471–86. http://dx.doi.org/10.1155/s1085337504306287.

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We present several recent and novel results on the formulation and the analysis of the equations governing the evolution of electromagnetic fields in chiral media in the time domain. In particular, we present results concerning the well-posedness and the solvability of the problem for linear, time-dependent, and nonlocal media, andresults concerning the validity of the local approximation of the nonlocal medium (optical response approximation). The paper concludes with the study of a class of nonlinear chiral media exhibiting Kerr-like nonlinearities, for which the existence of bright and dark solitary waves is shown.

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Ashurov, R. R., and Yu E. Fayziev. "On the nonlocal problems in time for subdiffusion equations with the Riemann-Liouville derivatives." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 106, no. 2 (June 30, 2022): 18–37. http://dx.doi.org/10.31489/2022m2/18-37.

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Initial boundary value problems with a time-nonlocal condition for a subdiffusion equation with the Riemann-Liouville time-fractional derivatives are considered. The elliptical part of the equation is the Laplace operator, defined in an arbitrary N−dimensional domain Ω with a sufficiently smooth boundary ∂Ω. The existence and uniqueness of the solution to the considered problems are proved. Inverse problems are studied for determining the right-hand side of the equation and a function in a time-nonlocal condition. The main research tool is the Fourier method, so the obtained results can be extended to subdiffusion equations with a more general elliptic operator.

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HAMBER, H. W., and R. M. WILLIAMS. "NONLOCAL EFFECTIVE FIELD EQUATIONS FOR QUANTUM COSMOLOGY." Modern Physics Letters A 21, no. 09 (March 21, 2006): 735–42. http://dx.doi.org/10.1142/s0217732306019979.

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The possibility that the strength of gravitational interactions might slowly increase with distance, is explored by formulating a set of effective field equations, which incorporate the gravitational, vacuum-polarization induced, running of Newton's constant G. The resulting long distance (or large time) behavior depends on only one adjustable parameter ξ, and the implications for the Robertson–Walker universe are calculated, predicting an accelerated power-law expansion at later times t ~ ξ ~ 1/H.

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Guha, Partha, and A. Ghose-Choudhury. "Nonlocal transformations of the generalized Liénard type equations and dissipative Ermakov-Milne-Pinney systems." International Journal of Geometric Methods in Modern Physics 16, no. 07 (July 2019): 1950107. http://dx.doi.org/10.1142/s021988781950107x.

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We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Liénard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov–Lewis invariant for a time-dependent oscillator to (coupled) Liénard and Liénard type equations. We also study the linearization problem for the coupled Liénard equation using nonlocal transformations and derive coupled dissipative Ermakov-Milne-Pinney equation. As an offshoot of this nonlocal transformation method when the standard Liénard equation, [Formula: see text], is mapped to that of the linear harmonic oscillator equation, we obtain a relation between the functions [Formula: see text] and [Formula: see text] which is exactly similar to the condition derived in the context of isochronicity of the Liénard equation.

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Zhang, Wei, Jiang Yang, Jiwei Zhang, and Qiang Du. "Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain." Communications in Computational Physics 21, no. 1 (December 5, 2016): 16–39. http://dx.doi.org/10.4208/cicp.oa-2016-0033.

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AbstractThis paper is concerned with numerical approximations of a nonlocal heat equation define on an infinite domain. Two classes of artificial boundary conditions (ABCs) are designed, namely, nonlocal analog Dirichlet-to-Neumann-type ABCs (global in time) and high-order Padé approximate ABCs (local in time). These ABCs reformulate the original problem into an initial-boundary-value (IBV) problem on a bounded domain. For the global ABCs, we adopt a fast evolution to enhance computational efficiency and reduce memory storage. High order fully discrete schemes, both second-order in time and space, are given to discretize two reduced problems. Extensive numerical experiments are carried out to show the accuracy and efficiency of the proposed methods.

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Akter, Sharmin, M. D. Hossain, M. F. Uddin, and M. G. Hafez. "Collisional Solitons Described by Two-Sided Beta Time Fractional Korteweg-de Vries Equations in Fluid-Filled Elastic Tubes." Advances in Mathematical Physics 2023 (July 18, 2023): 1–12. http://dx.doi.org/10.1155/2023/9594339.

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This article deals with the basic features of collisional radial displacements in a prestressed thin elastic tube filled having inviscid fluid with the presence of nonlocal operator. By implementing the extended Poincare–Lighthill–Kuo method and a variational approach, the new two-sided beta time fractional Korteweg-de-Vries (BTF-KdV) equations are derived based on the concept of beta fractional derivative (BFD). Additionally, the BTF-KdV equations are suggested to observe the effect of related parameters on the local and nonlocal coherent head-on collision phenomena for the considered system. It is observed that the proposed equations along with their new solutions not only applicable with the presence of locality but also nonlocality to study the resonance wave phenomena in fluid-filled elastic tube. The outcomes reveal that the BFD and other physical parameters related to tube and fluid have a significant impact on the propagation of pressure wave structures.

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