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Journal articles on the topic 'Nonlocal equation'

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Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Lukashchuk, Stanislav Yu. "Approximate Nonlocal Symmetries for a Perturbed Schrödinger Equation with a Weak Infinite Power-Law Memory." AppliedMath 2, no. 4 (October 17, 2022): 585–608. http://dx.doi.org/10.3390/appliedmath2040034.

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A nonlocally perturbed linear Schrödinger equation with a small parameter was derived under the assumption of low-level fractionality by using one of the known general nonlocal wave equations with an infinite power-law memory. The problem of finding approximate symmetries for the equation is studied here. It has been shown that the perturbed Schrödinger equation inherits all symmetries of the classical linear equation. It has also been proven that approximate symmetries corresponding to Galilean transformations and projective transformations of the unperturbed equation are nonlocal. In addition, a special class of nonlinear, nonlocally perturbed Schrödinger equations that admits an approximate nonlocal extension of the Galilei group is derived. An example of constructing an approximately invariant solution for the linear equation using approximate scaling symmetry is presented.

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2

Peng, Linyu. "Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations." Symmetry 11, no. 7 (July 5, 2019): 884. http://dx.doi.org/10.3390/sym11070884.

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In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.

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3

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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4

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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5

Xu, Hai Jing, and Song Lin Zhao. "Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions." Symmetry 13, no. 1 (December 24, 2020): 23. http://dx.doi.org/10.3390/sym13010023.

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In this paper, local and nonlocal reductions of two nonisospectral Ablowitz-Kaup-Newell-Segur equations, the third order nonisospectral AKNS equation and the negative order nonisospectral AKNS equation, are studied. By imposing constraint conditions on the double Wronskian solutions of the aforesaid nonisospectral AKNS equations, various solutions for the local and nonlocal nonisospectral modified Korteweg-de Vries equation and local and nonlocal nonisospectral sine-Gordon equation are derived, including soliton solutions and Jordan block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis.

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6

Delgado, M., A. Suárez, and I. B. M. Duarte. "Nonlocal problems arising from the birth-jump processes." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 149, no. 2 (December 27, 2018): 447–69. http://dx.doi.org/10.1017/prm.2018.34.

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In this paper, we prove the existence and uniqueness of a positive solution for a nonlocal logistic equation arising from the birth-jump processes. For this, we establish a sub-super solution method for nonlocal elliptic equations, we perform a study of the eigenvalue problems associated with these equations and we apply these results to the nonlocal logistic equation.

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7

Feng, Wei, Song-Lin Zhao, and Ying-Ying Sun. "Double Casoratian solutions to the nonlocal semi-discrete modified Korteweg-de Vries equation." International Journal of Modern Physics B 34, no. 05 (February 3, 2020): 2050021. http://dx.doi.org/10.1142/s0217979220500216.

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Two nonlocal versions of the semi-discrete modified Korteweg-de Vries equation are derived by different nonlocal reductions from a coupled equation set in the Ablowitz–Ladik hierarchy. Different kinds of exact solutions in terms of double Casoratians to the reduced equations are obtained by imposing constraint conditions on the double Casorati determinant solutions of the coupled equation set. Dynamics of the soliton solutions for the real and complex nonlocal semi-discrete modified Korteweg-de Vries equations are analyzed and illustrated by asymptotic analysis.

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8

Gaygusuzoglu, Guler, Metin Aydogdu, and Ufuk Gul. "Nonlinear Wave Modulation in Nanorods Using Nonlocal Elasticity Theory." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 7-8 (December 19, 2018): 709–19. http://dx.doi.org/10.1515/ijnsns-2017-0225.

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AbstractIn this study, nonlinear wave modulation in nanorods is examined on the basis of nonlocal elasticity theory. Eringen's nonlocal elasticity theory is employed to derive nonlinear equations for the motion of nanorods. The analysis of the modulation of axial waves in nonlocal elastic media is performed, and the reductive perturbation method is used for the solution of the nonlinear equations. The propagation of weakly nonlinear and strongly dispersive waves is investigated, and the nonlinear Schrödinger (NLS) equation is acquired as an evolution equation. For the purpose of a numerical investigation of the nonlocal impacts on the NLS equation, it has been investigated whether envelope solitary wave solutions exist by utilizing the physical and geometric features of the carbon nanotubes. Amplitude dependent wave frequencies, phase and group velocities have been obtained and they have compared for the linear local, the linear nonlocal, the nonlinear local and the nonlinear nonlocal cases.

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9

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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10

Wang, Xiaodong, Jianping Wu, Yazi Wang, and Can Chen. "Extended Tanh-Function Method and Its Applications in Nonlocal Complex mKdV Equations." Mathematics 10, no. 18 (September 7, 2022): 3250. http://dx.doi.org/10.3390/math10183250.

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In order to construct the multiple traveling wave solutions of the nonlocal modified Korteweg de Vires (mKdV) equation, the modified tanh-function approach for local soliton equations is extended to a nonlocal complex mKdV equation. The central idea of this method is to use the solution of the Riccati equation to replace the tanh function in the tanh function (THF) method. As an application, we investigate a new traveling wave solution for the nonlocal complex mKdV equation of Ablowitz and Musslimani. Moreover, some exciting diagrams show the underlying dynamics of some given solutions.

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11

Shen, Xiao Long, Yong Xin Luo, Lai Xi Zhang, and Hua Long. "Natural Frequency Computation Method of Nonlocal Elastic Beam." Advanced Materials Research 156-157 (October 2010): 1582–85. http://dx.doi.org/10.4028/www.scientific.net/amr.156-157.1582.

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After adopting the constitutive equations of the nonlocal elastic media in the form of Eringen, and making use of the Laplace transformation, the vibration governing equation of nonlocal elastic beam in the Kelvin media are established. Unlike classical elastic models, the stress of a point in a nonlocal model is obtained as a weighted average of the field over the spatial domain, determined by a kernel function based on distance measures. The motion equation of nonlocal elastic beam is an integral differential equation, rather than the differential equation obtained with a classical local model. Solutions for natural frequencies and modes are obtained. Numerical examples demonstrate the efficiency of the proposed method for the beam with simple boundary conditions.

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12

Sapagovas, Mifodijus. "Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition." Nonlinear Analysis: Modelling and Control 7, no. 1 (June 5, 2002): 93–104. http://dx.doi.org/10.15388/na.2002.7.1.15204.

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Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

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13

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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14

Su, Yuan-Hang, Wan-Tong Li, and Fei-Ying Yang. "Asymptotic behaviors for nonlocal diffusion equations about the dispersal spread." Analysis and Applications 18, no. 04 (February 14, 2020): 585–614. http://dx.doi.org/10.1142/s0219530519500222.

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This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator [Formula: see text] and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalized principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter [Formula: see text]. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter [Formula: see text] is in a different range. In particular, for the case [Formula: see text], we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition.

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15

Liu, Wei. "High-order rogue waves of the Benjamin–Ono equation and the nonlocal nonlinear Schrödinger equation." Modern Physics Letters B 31, no. 29 (October 17, 2017): 1750269. http://dx.doi.org/10.1142/s0217984917502694.

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High-order rogue wave solutions of the Benjamin–Ono equation and the nonlocal nonlinear Schrödinger equation are derived by employing the bilinear method, which are expressed by simple polynomials. Typical dynamics of these high-order rogue waves are studied by analytical and graphical ways. For the Benjamin–Ono equation, there are two types of rogue waves, namely, bright rogue waves and dark rogue waves. In particular, the fundamental rogue wave pattern is different from the usual fundamental rogue wave patterns in other soliton equations. For the nonlocal nonlinear Schrödinger equation, the exact explicit rogue wave solutions up to the second order are presented. Typical rogue wave patterns such as Peregrine-type, triple and fundamental rogue waves are put forward. These high-order rogue wave patterns have not been shown before in the nonlocal Schrödinger equation.

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16

TANG, GANG, and BENKUN MA. "SCALING OF THE NONLOCAL GROWTH EQUATIONS WITH SPATIALLY AND TEMPORALLY CORRELATED NOISE." International Journal of Modern Physics B 15, no. 16 (June 30, 2001): 2275–83. http://dx.doi.org/10.1142/s0217979201004824.

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The Flory-type approach proposed by Hentschel and Family [Phys. Rev. Lett.66, 1982 (1991)] is generalized to analyze the scaling behavior of the nonlocal surface growth equations with long-range spatially and temporally correlated noise. The scaling exponents in both the weak- and strong-coupling regions are obtained. The growth equations studied include the nonlocal Kardar–Parisi–Zhang, nonlocal Sun–Guo–Grant, and nonlocal Lai–Das Sarma–Villain equation.

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17

Li, Meng, Yi Zhan, and Lidan Zhang. "Nonlocal Variational Model for Saliency Detection." Mathematical Problems in Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/518747.

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We present a nonlocal variational model for saliency detection from still images, from which various features for visual attention can be detected by minimizing the energy functional. The associated Euler-Lagrange equation is a nonlocalp-Laplacian type diffusion equation with two reaction terms, and it is a nonlinear diffusion. The main advantage of our method is that it provides flexible and intuitive control over the detecting procedure by the temporal evolution of the Euler-Lagrange equation. Experimental results on various images show that our model can better make background details diminish eventually while luxuriant subtle details in foreground are preserved very well.

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18

Putra, Gusrian, Hanifah Septaningtiyas, Elsa Nabila, and Lisa Arianti Br Tarigan. "Analisis Kestabilan Solusi Soliton pada Persamaan Schrodinger Nonlinier Diskrit Nonlokal." Indonesian Journal of Applied Mathematics 2, no. 1 (April 15, 2022): 17. http://dx.doi.org/10.35472/indojam.v2i1.730.

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In this paper, the Nonlocal Discrete Nonlinear Schrodinger (DNLS) equation that interpolates the Nonlocal Ablowitz-Ladik DNLS and the Nonlocal Cubic DNLS equations and its stability are studied in detail. The solution of the Nonlocal SNLD equation is a soliton wave in the form of a Gaussian ansatz obtained using the method of Variational Approximation (VA). The stability of the solution is also analyzed using the VA. These semi-analytical results are then compared to numerical results. The soliton and its stability obtained via VA is concluded to be having a fairly good conformity with numerical results.

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19

Keimer, Alexander, Manish Singh, and Tanya Veeravalli. "Existence and uniqueness results for a class of nonlocal conservation laws by means of a Lax–Hopf-type solution formula." Journal of Hyperbolic Differential Equations 17, no. 04 (December 2020): 677–705. http://dx.doi.org/10.1142/s0219891620500204.

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We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax–Hopf-type solution formula for the corresponding Hamilton–Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax–Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax–Oleinik-type formula for the nonlocal conservation law.

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20

Das, Ashok, and Z. Popowicz. "Supersymmetric nonlocal gas equation." Journal of Mathematical Physics 46, no. 8 (August 2005): 082702. http://dx.doi.org/10.1063/1.1993547.

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21

Gómez-Aguilar, J. F., R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, M. Benavides-Cruz, and C. Calderón-Ramón. "Nonlocal electrical diffusion equation." International Journal of Modern Physics C 27, no. 01 (January 2016): 1650007. http://dx.doi.org/10.1142/s0129183116500078.

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In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is [Formula: see text] and for the time domain is [Formula: see text]. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.

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22

Shakhmurov, Veli. "Regularity properties of nonlocal fractional differential equations and applications." Georgian Mathematical Journal 29, no. 2 (February 5, 2022): 275–84. http://dx.doi.org/10.1515/gmj-2021-2128.

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Abstract The regularity properties of nonlocal fractional elliptic and parabolic equations in vector-valued Besov spaces are studied. The uniform B p , q s B_{p,q}^{s} -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. In particular, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, the maximal regularity properties of the nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional parabolic equations are studied.

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23

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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24

Gilev, A. V. "A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH A DOMINANT MIXED DERIVATIVE." Vestnik of Samara University. Natural Science Series 26, no. 4 (August 17, 2021): 25–35. http://dx.doi.org/10.18287/2541-7525-2020-26-4-25-35.

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In this article, we consider the Goursat problem with nonlocal integral conditions for a hyperbolic equation with a dominant mixed derivative. Research methods of solvability of classical boundary value problems for partial differential equations cannot be applied without serious modifications. The choice of a research method of solvability of a nonlocal problem depends on the form of the integral condition. In the process of developing methods that are effective for nonlocal problems, integral conditions of various types were identified [1]. The solvability of the nonlocal Goursat problem with integral conditions of the first kind for a general equation with dominant mixed derivative of the second order was investigated in [2]. In our problem, the integral conditions are nonlocal conditions of the second kind, therefore, to investigate the solvability of the problem, we propose another method, which consists in reducing the stated nonlocal problem to the classical Goursat problem, but for a loaded equation. In this article, we obtain conditions that guarantee the existence of a unique solution of the problem. The main instrument of the proof is the a priori estimates obtained in the paper.

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25

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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26

Belakroum, Kh, A. Ashyralyev, and A. Guezane-Lakoud. "A note on the nonlocal boundary value problem for a third order partial differential equation." Filomat 32, no. 3 (2018): 801–8. http://dx.doi.org/10.2298/fil1803801b.

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The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space with a self-adjoint positive definite operator is considered. Applying operator approach, the theorem on stability for solution of this nonlocal boundary value problem is established. In applications, the stability estimates for the solution of three nonlocal boundary value problems for third order partial differential equations are obtained.

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27

Xie, Dexuan, and Hans W. Volkmer. "A Modified Nonlocal Continuum Electrostatic Model for Protein in Water and Its Analytical Solutions for Ionic Born Models." Communications in Computational Physics 13, no. 1 (January 2013): 174–94. http://dx.doi.org/10.4208/cicp.170811.211011s.

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AbstractA nonlocal continuum electrostatic model, defined as integro-differential equations, can significantly improve the classic Poisson dielectric model, but is too costly to be applied to large protein simulations. To sharply reduce the model’s complexity, a modified nonlocal continuum electrostatic model is presented in this paper for a protein immersed in water solvent, and then transformed equivalently as a system of partial differential equations. By using this new differential equation system, analytical solutions are derived for three different nonlocal ionic Born models, where a monoatomic ion is treated as a dielectric continuum ball with point charge either in the center or uniformly distributed on the surface of the ball. These solutions are analytically verified to satisfy the original integro-differential equations, thereby, validating the new differential equation system.

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28

Ren, Yan-Ming, and Hai Qing. "Bending and Buckling Analysis of Functionally Graded Euler–Bernoulli Beam Using Stress-Driven Nonlocal Integral Model with Bi-Helmholtz Kernel." International Journal of Applied Mechanics 13, no. 04 (May 2021): 2150041. http://dx.doi.org/10.1142/s1758825121500411.

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Static bending and elastic buckling of Euler–Bernoulli beam made of functionally graded (FG) materials along thickness direction is studied theoretically using stress-driven integral model with bi-Helmholtz kernel, where the relation between nonlocal stress and strain is expressed as Fredholm type integral equation of the first kind. The differential governing equation and corresponding boundary conditions are derived with the principle of minimum potential energy. Several nominal variables are introduced to simplify differential governing equation, integral constitutive equation and boundary conditions. Laplace transform technique is applied directly to solve integro-differential equations, and the nominal bending deflection and moment are expressed with six unknown constants. The explicit expression for nominal deflection for static bending and nonlinear characteristic equation for the bucking load can be determined with two constitutive constraints and four boundary conditions. The results from this study are validated with those from the existing literature when two nonlocal parameters have same value. The influence of nonlocal parameters on the bending deflection and buckling loads for Euler–Bernoulli beam is investigated numerically. A consistent toughening effect is obtained for stress-driven nonlocal integral model with bi-Helmholtz kernel.

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29

BARCI, D. G., L. E. OXMAN, and M. ROCCA. "CANONICAL QUANTIZATION OF NONLOCAL FIELD EQUATIONS." International Journal of Modern Physics A 11, no. 12 (May 10, 1996): 2111–26. http://dx.doi.org/10.1142/s0217751x96001061.

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We consistently quantize a class of relativistic nonlocal field equations characterized by a nonlocal kinetic term in the Lagrangian. We solve the classical nonlocal equations of motion for a scalar field and evaluate the on-shell Hamiltonian. The quantization is realized by imposing Heisenberg’s equation, which leads to the commutator algebra obeyed by the Fourier components of the field. We show that the field operator carries, in general, a reducible representation of the Poincaré group. We also consider the Gupta-Bleuler quantization of a nonlocal gauge theory and analyze the propagators and the physical modes of the gauge field.

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30

Li, Fang, Jin Liang, Tzon-Tzer Lu, and Huan Zhu. "A Nonlocal Cauchy Problem for Fractional Integrodifferential Equations." Journal of Applied Mathematics 2012 (2012): 1–18. http://dx.doi.org/10.1155/2012/901942.

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This paper is concerned with a nonlocal Cauchy problem for fractional integrodifferential equations in a separable Banach spaceX. We establish an existence theorem for mild solutions to the nonlocal Cauchy problem, by virtue of measure of noncompactness and the fixed point theorem for condensing maps. As an application, the existence of the mild solution to a nonlocal Cauchy problem for a concrete integrodifferential equation is obtained.

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31

Shapovalov, Alexander V., and Anton E. Kulagin. "Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media." Mathematics 9, no. 23 (November 23, 2021): 2995. http://dx.doi.org/10.3390/math9232995.

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A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

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32

Yuan, Cui-Lian, and Xiao-Yong Wen. "Discrete nonlocal N-fold Darboux transformation and soliton solutions in a reverse space-time nonlocal nonlinear self-dual network equation." Modern Physics Letters B 35, no. 19 (June 15, 2021): 2150314. http://dx.doi.org/10.1142/s0217984921503140.

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In this paper, we construct a discrete nonlocal integrable lattice hierarchy related to a reverse space-time nonlocal nonlinear self-dual network equation which may have the potential applications in designing nonlocal electrical circuits and understanding the propagation of electrical signals. By means of nonlocal version of [Formula: see text]-fold Darboux transformation (DT) technique, discrete multi-soliton solutions in determinant form are constructed for the reverse space-time nonlocal nonlinear self-dual network equation. Through the asymptotic and graphic analysis, unstable soliton structures of one-, two- and three-soliton solutions are discussed graphically. We observe that the single components in this nonlocal equation display instability while the combined potential terms with nonlocal [Formula: see text]-symmetry show stable soliton structures. It is shown that these nonlocal solutions are clearly different from those of its corresponding local equation. The results given in this paper may explain the soliton propagation in electrical signals.

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33

Ashyralyev, Allaberen, and Ozgur Yildirim. "A Note on the Second Order of Accuracy Stable Difference Schemes for the Nonlocal Boundary Value Hyperbolic Problem." Abstract and Applied Analysis 2012 (2012): 1–29. http://dx.doi.org/10.1155/2012/846582.

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The second order of accuracy absolutely stable difference schemes are presented for the nonlocal boundary value hyperbolic problem for the differential equations in a Hilbert spaceHwith the self-adjoint positive definite operatorA. The stability estimates for the solutions of these difference schemes are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions and multidimensional hyperbolic equation with Dirichlet conditions are considered. The stability estimates for the solutions of these difference schemes for the nonlocal boundary value hyperbolic problem are established. Finally, a numerical method proposed and numerical experiments, analysis of the errors, and related execution times are presented in order to verify theoretical statements.

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Sun, Hong-Qian, and Zuo-Nong Zhu. "Darboux Transformation and Soliton Solution of the Nonlocal Generalized Sasa–Satsuma Equation." Mathematics 11, no. 4 (February 8, 2023): 865. http://dx.doi.org/10.3390/math11040865.

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This paper aims to seek soliton solutions for the nonlocal generalized Sasa–Satsuma (gSS) equation by constructing the Darboux transformation (DT). We obtain soliton solutions for the nonlocal gSS equation, including double-periodic wave, breather-like, KM-breather solution, dark-soliton, W-shaped soliton, M-shaped soliton, W-shaped periodic wave, M-shaped periodic wave, double-peak dark-breather, double-peak bright-breather, and M-shaped double-peak breather solutions. Furthermore, interaction of these solitons, as well as their dynamical properties and asymptotic analysis, are analyzed. It will be shown that soliton solutions of the nonlocal gSS equation can be reduced into those of the nonlocal Sasa–Satsuma equation. However, several of these properties for the nonlocal Sasa–Satsuma equation are not found in the literature.

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Fernández Bonder, Julián, Antonella Ritorto, and Ariel Martin Salort. "A class of shape optimization problems for some nonlocal operators." Advances in Calculus of Variations 11, no. 4 (October 1, 2018): 373–86. http://dx.doi.org/10.1515/acv-2016-0065.

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AbstractIn this work we study a family of shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

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LACHOWICZ, MIROSŁAW, and DARIUSZ WRZOSEK. "NONLOCAL BILINEAR EQUATIONS: EQUILIBRIUM SOLUTIONS AND DIFFUSIVE LIMIT." Mathematical Models and Methods in Applied Sciences 11, no. 08 (November 2001): 1393–409. http://dx.doi.org/10.1142/s0218202501001380.

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This paper deals with the qualitative analysis of a class of bilinear systems of equations describing the dynamics of individuals undergoing kinetic (stochastic) interactions. A corresponding evolution problem is formulated in terms of integro-differential (nonlocal) system of equations. A general existence theory is provided. Under the assumption of periodic boundary conditions and the interaction rates expressed in terms of convolution operators two classes of equilibrium solutions are distinguished. The first class contains only constant functions and the second one contains some nonconstant functions. In the scalar case (one equation) under suitable scaling, related to the shrinking of interaction range of each individual, the limit to the corresponding "macroscopic" equation is studied. The limiting equation turns out to be the (nonlinear) porous medium equation.

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37

Zhang, Wen-Xin, and Yaqing Liu. "Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV) equations." AIMS Mathematics 6, no. 10 (2021): 11046–75. http://dx.doi.org/10.3934/math.2021641.

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<abstract><p>In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.</p></abstract>

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Márquez, Almudena P., Elena Recio, and María L. Gandarias. "Lie Symmetries and Conservation Laws for the Viscous Cahn-Hilliard Equation." Symmetry 14, no. 5 (April 22, 2022): 861. http://dx.doi.org/10.3390/sym14050861.

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In this paper, we study a viscous Cahn–Hilliard equation from the point of view of Lie symmetries in partial differential equations. The analysis of this equation is motivated by its applications since it serves as a model for many problems in physical chemistry, developmental biology, and population movement. Firstly, a classification of the Lie symmetries admitted by the equation is presented. In addition, the symmetry transformation groups are calculated. Afterwards, the partial differential equation is transformed into ordinary differential equations through symmetry reductions. Secondly, all low-order local conservation laws are obtained by using the multiplier method. Furthermore, we use these conservation laws to determine their associated potential systems and we use them to investigate nonlocal symmetries and nonlocal conservation laws. Finally, we apply the multi-reduction method to reduce the equation and find a soliton solution.

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39

Avalishvili, Gia, and Mariam Avalishvili. "Nonclassical Problem for Ultraparabolic Equation in Abstract Spaces." Journal of Function Spaces 2016 (2016): 1–15. http://dx.doi.org/10.1155/2016/5687920.

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Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.

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Parasidis, Ioannis, Efthimios Providas, and Vassilios Dafopoulos. "Dafopoulos Loaded differential and Fredholm integro-differential equations with nonlocal integral boundary conditions." Applied Mathematics and Control Sciences, no. 3 (October 9, 2018): 50–68. http://dx.doi.org/10.15593/2499-9873/2018.3.04.

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A direct method for the exact solution of loaded differential or Fredholm Integro-Differential Equations (IDEs) with nonlocal integral boundary conditions is proposed. We consider the abstract operator equations of the form Bu = Au − gΨ(u) = f with abstract nonlocal boundary conditions. In this paper we investigate the correctness of the equation Bu = f and its exact solution in closed form.

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Kirane, Mokhtar, Batirkhan Kh Turmetov, and Berikbol T. Torebek. "A nonlocal fractional Helmholtz equation." Fractional Differential Calculus, no. 2 (2017): 225–34. http://dx.doi.org/10.7153/fdc-2017-07-08.

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42

Chicone, C., and B. Mashhoon. "Nonlocal gravity: Modified Poisson's equation." Journal of Mathematical Physics 53, no. 4 (April 2012): 042501. http://dx.doi.org/10.1063/1.3702449.

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Bidégaray, B. "On a nonlocal Zakharov equation." Nonlinear Analysis: Theory, Methods & Applications 25, no. 3 (August 1995): 247–78. http://dx.doi.org/10.1016/0362-546x(94)00136-6.

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Ignat, Liviu I., and Julio D. Rossi. "A nonlocal convection–diffusion equation." Journal of Functional Analysis 251, no. 2 (October 2007): 399–437. http://dx.doi.org/10.1016/j.jfa.2007.07.013.

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Ablowitz, Mark J., Xu-Dan Luo, Ziad H. Musslimani, and Yi Zhu. "Integrable nonlocal derivative nonlinear Schrödinger equations." Inverse Problems 38, no. 6 (April 19, 2022): 065003. http://dx.doi.org/10.1088/1361-6420/ac5f75.

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Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation.

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D’Elia, Marta, Qiang Du, Max Gunzburger, and Richard Lehoucq. "Nonlocal Convection-Diffusion Problems on Bounded Domains and Finite-Range Jump Processes." Computational Methods in Applied Mathematics 17, no. 4 (October 1, 2017): 707–22. http://dx.doi.org/10.1515/cmam-2017-0029.

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AbstractA nonlocal convection-diffusion model is introduced for the master equation of Markov jump processes in bounded domains. With minimal assumptions on the model parameters, the nonlocal steady and unsteady state master equations are shown to be well-posed in a weak sense. Then the nonlocal operator is shown to be the generator of finite-range nonsymmetric jump processes and, when certain conditions on the model parameters hold, the generators of finite and infinite activity Lévy and Lévy-type jump processes are shown to be special instances of the nonlocal operator.

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ABLOWITZ, M. J., and T. S. HAUT. "SPECTRAL FORMULATION OF THE TWO FLUID EULER EQUATIONS WITH A FREE INTERFACE AND LONG WAVE REDUCTIONS." Analysis and Applications 06, no. 04 (October 2008): 323–48. http://dx.doi.org/10.1142/s0219530508001213.

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A nonlocal spectral formulation of classic water waves is derived, and its connection to the classic water wave equations and the Dirichlet–Neumann operator is explored. The nonlocal spectral formulation is also extended to a two-fluid system with a free interface, from which long wave asymptotic reductions are obtained. Of particular interest is an asymptotically distinguished (2 + 1)-dimensional generalization of the intermediate long wave equation, which includes the Kadomtsev–Petviashvili equation and the Benjamin–Ono equation as limiting cases. Lump-type solutions to this (2 + 1)-dimensional ILW equation are obtained, and the speed versus amplitude relationship is shown to be linear in the shallow, intermediate, and deep water regimes.

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Ngoc, Tran, and Nguyen Tuan. "Existence of mild solutions to semilinear fractional evolution equation using Krasnoselskii fixed point theorem." Filomat 36, no. 4 (2022): 1099–112. http://dx.doi.org/10.2298/fil2204099n.

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This paper is devoted to study the existence and stability of mild solutions for semilinear fractional evolution equations with a nonlocal final condition. The analysis is based on analytic semigroup theory, Krasnoselskii fixed point theorem, and a special probability density function. An application to a time fractional diffusion equation with nonlocal final condition is also given.

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HERNÁNDEZ HEREDERO, RAFAEL, and ENRIQUE G. REYES. "NONLOCAL SYMMETRIES, COMPACTON EQUATIONS, AND INTEGRABILITY." International Journal of Geometric Methods in Modern Physics 10, no. 09 (August 30, 2013): 1350046. http://dx.doi.org/10.1142/s0219887813500461.

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We review the theory of nonlocal symmetries of nonlinear partial differential equations and, as examples, we present infinite-dimensional Lie algebras of nonlocal symmetries of the Fokas–Qiao and Kaup–Kupershmidt equations. Then, we consider nonlocal symmetries of a family which contains the Korteweg–de Vries (KdV) and (a subclass of) the Rosenau–Hyman compacton-bearing K(m, n) equations. We find that the only member of the family which possesses nonlocal symmetries (of a kind specified in Sec. 3 below) is precisely the KdV equation. We take this fact as an indication that the K(m, n) equations are not integrable in general, and we use the formal symmetry approach of Shabat to check this claim: we prove that the only integrable cases of the full K(m, n) family are the KdV and modified KdV equations.

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BO, LIJUN, KEHUA SHI, and YONGJIN WANG. "ON A NONLOCAL STOCHASTIC KURAMOTO–SIVASHINSKY EQUATION WITH JUMPS." Stochastics and Dynamics 07, no. 04 (December 2007): 439–57. http://dx.doi.org/10.1142/s0219493707002104.

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In this paper, we study a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by compensated Poisson random measures and show the existence and uniqueness of the weak solution to the equation. Furthermore, we prove that an invariant measure of the equation indeed exists under some appropriate assumptions.

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