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Journal articles on the topic 'Lattice Differential-Difference Equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 3 February 2022

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1

Levi, Decio, and Miguel A. Rodriguez. "ON THE CONSTRUCTION OF PARTIAL DIFFERENCE SCHEMES II: DISCRETE VARIABLES AND SCHWARZIAN LATTICES." Acta Polytechnica 56, no. 3 (June 30, 2016): 236. http://dx.doi.org/10.14311/ap.2016.56.0236.

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In the process of constructing invariant difference schemes which approximate partial differential equations we write down a procedure for discretizing a partial differential equation on an arbitrary lattice. An open problem is the meaning of a lattice which does not satisfy the Clairaut–Schwarz–Young theorem. To analyze it we apply the procedure on a simple example, the potential Burgers equation with two different lattices, an orthogonal lattice which is invariant under the symmetries of the equation and satisfies the commutativity of the partial difference operators and an exponential lattice which is not invariant and does not satisfy the Clairaut–Schwarz–Young theorem. A discussion on the numerical results is presented showing the different behavior of both schemes for two different exact solutions and their numerical approximations.

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2

Gepreel, Khaled A., Taher A. Nofal, and Fawziah M. Alotaibi. "Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/756896.

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We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.

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3

Lu, Jun-Feng. "GDTM-Padé technique for the non-linear differential-difference equation." Thermal Science 17, no. 5 (2013): 1305–10. http://dx.doi.org/10.2298/tsci1305305l.

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This paper focuses on applying the GDTM-Pad? technique to solve the non-linear differential-difference equation. The bell-shaped solitary wave solution of Belov-Chaltikian lattice equation is considered. Comparison between the approximate solutions and the exact ones shows that this technique is an efficient and attractive method for solving the differential-difference equations.

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4

Bekir, Ahmet, Ozkan Guner, Burcu Ayhan, and Adem C. Cevikel. "Exact Solutions for Fractional Differential-Difference Equations by (G'/G)-Expansion Method with Modified Riemann-Liouville Derivative." Advances in Applied Mathematics and Mechanics 8, no. 2 (January 27, 2016): 293–305. http://dx.doi.org/10.4208/aamm.2014.m798.

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AbstractIn this paper, the (G'/G)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.

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5

Mousa, Mohamed M., and Aidarkhan Kaltayev. "Homotopy Perturbation Method for Solving Nonlinear Differential- Difference Equations." Zeitschrift für Naturforschung A 65, no. 6-7 (July 1, 2010): 511–17. http://dx.doi.org/10.1515/zna-2010-6-705.

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In this paper, the homotopy perturbation method (HPM) is extended to obtain analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Kortewegde Vries (mKdV) lattice equation and the discretized nonlinear Schrodinger equation are taken as examples to illustrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons between the results of the presented method and exact solutions are made. The results reveal that the HPM is very effective and convenient for solving such kind of equations.

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6

Liu, Jian-Gen, Xiao-Jun Yang, and Yi-Ying Feng. "Analytical solutions of some integral fractional differential–difference equations." Modern Physics Letters B 34, no. 01 (December 9, 2019): 2050009. http://dx.doi.org/10.1142/s0217984920500098.

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The invariant subspace method (ISM) is a powerful tool for investigating analytical solutions to fractional differential–difference equations (FDDEs). Based on previous work by other people, we apply the ISM to the space-time fractional differential and difference equations, including the cases of the scalar space-time FDDEs and the multi-coupled space-time FDDEs. As a result, we obtain some new analytical solutions to the well-known scalar space-time Lotka–Volterra equation, the space-time fractional generalized Hybrid lattice equation and the space-time fractional Burgers equation as well as two couple space-time FDDEs. Furthermore, some properties of the analytical solutions are illustrated by graphs.

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7

Gepreel, Khaled A., Taher A. Nofal, and Ali A. Al-Thobaiti. "The Modified Rational Jacobi Elliptic Functions Method for Nonlinear Differential Difference Equations." Journal of Applied Mathematics 2012 (2012): 1–30. http://dx.doi.org/10.1155/2012/427479.

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We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

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8

Gepreel, Khaled A., and A. R. Shehata. "Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics." Journal of Applied Mathematics 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/710375.

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We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

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9

Levi, D., O. Ragnisco, and A. B. Shabat. "Construction of higher local (2 + 1) dimensional exponential lattice equations." Canadian Journal of Physics 72, no. 7-8 (July 1, 1994): 439–41. http://dx.doi.org/10.1139/p94-059.

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It is shown that the 2-dimensional Toda equation can be decomposed into two integrable differential-difference equations, thus implying that we can associate to it a triad of linear problems. Using this same trick we construct new 2-dimensional differential-difference equations with exponential interactions.

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10

HON, Y. C., YUFENG ZHANG, and JIANQIN MEI. "EXACT SOLUTIONS FOR DIFFERENTIAL-DIFFERENCE EQUATIONS BY BÄCKLUND TRANSFORMATION OF RICCATI EQUATION." Modern Physics Letters B 24, no. 27 (October 30, 2010): 2713–24. http://dx.doi.org/10.1142/s0217984910025012.

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Based on a Bäcklund transformation of the Riccati equation and its known soliton solutions, we obtain in this paper some exact traveling-wave solutions, including triangle function solutions and hyperbolic function solutions, of a hybrid lattice equation. The proposed method can be easily extended to locate exact solitary wave solutions for other types of differential-difference equations.

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11

He, Ji-Huan, S. K. Elagan, and Guo-Cheng Wu. "Solitary-Solution Formulation for Differential-Difference Equations Using an Ancient Chinese Algorithm." Abstract and Applied Analysis 2012 (2012): 1–6. http://dx.doi.org/10.1155/2012/861438.

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This paper applies an ancient Chinese algorithm to differential-difference equations, and a solitary-solution formulation is obtained. The discrete mKdV lattice equation is used as an example to elucidate the solution procedure.

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12

COMMON, A. K., and M. MUSETTE. "Non-integrable lattice equations supporting kink and soliton solutions." European Journal of Applied Mathematics 12, no. 6 (December 2001): 709–18. http://dx.doi.org/10.1017/s0956792501004648.

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Nonintegrable differential-difference equations are constructed which support two-kink and two-soliton solutions. These equations are related to the discrete Burgers hierarchy and a discrete form of the Korteweg-de Vries equation. In particular, discretisations of equations related to the Fitzhugh-Nagumo-Kolmogorov-Petrovskii-Piskunov, Satsuma-Burgers-Huxley equations are derived. Methods presented here can also be used to derive non-integrable differential-difference equations describing the elastic collision of more than two kinks or solitary waves.

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13

Mousa, Mohamed Medhat, Aidarkan Kaltayev, and Hasan Bulut. "Extension of the Homotopy Perturbation Method for Solving Nonlinear Differential-Difference Equations." Zeitschrift für Naturforschung A 65, no. 12 (December 1, 2010): 1060–64. http://dx.doi.org/10.1515/zna-2010-1207.

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In this paper, we have extended the homotopy perturbation method (HPM) to find approximate analytical solutions for some nonlinear differential-difference equations (NDDEs). The discretized modified Korteweg-de Vries (mKdV) lattice equation and the discretized nonlinear Schr¨odinger equation are taken as examples to demonstrate the validity and the great potential of the HPM in solving such NDDEs. Comparisons are made between the results of the presented method and exact solutions. The obtained results reveal that the HPM is a very effective and convenient tool for solving such kind of equations.

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14

WELLS, J. C., V. E. OBERACKER, M. R. STRAYER, and A. S. UMAR. "SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD." International Journal of Modern Physics C 06, no. 01 (February 1995): 143–67. http://dx.doi.org/10.1142/s0129183195000125.

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We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

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15

Martinsson, Per-Gunnar, and Gregory J. Rodin. "Boundary algebraic equations for lattice problems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2108 (June 2, 2009): 2489–503. http://dx.doi.org/10.1098/rspa.2008.0473.

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Procedures for constructing boundary integral equations equivalent to linear boundary-value problems governed by partial differential equations are well established. In this paper, it is demonstrated how these procedures can be extended to linear boundary-value problems defined on lattices and governed by algebraic (‘difference’) equations. The boundary equations that arise are then themselves algebraic equations. Such ‘boundary algebraic equations’ (BAEs) are derived for fundamental boundary-value problems defined on both perfect lattices and lattices with defects. It is demonstrated that key advantages of representing a continuum boundary-value problem as an equation on the boundary, such as favourable spectral properties and minimal problem size, are preserved in the lattice environment. Certain spectral properties of BAEs are established rigorously, whereas others are supported by numerical experiments.

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16

Zhang, Sheng, Ying Ying Zhou, and Bin Cai. "Kink-Type Solutions of the MKdV Lattice Equation with an Arbitrary Function." Advanced Materials Research 989-994 (July 2014): 1716–19. http://dx.doi.org/10.4028/www.scientific.net/amr.989-994.1716.

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In this paper, the exp-function method is improved for constructing exact solutions of nonlinear differential-difference equations with variable coefficients. To illustrate the validity and advantages of the improved method, the mKdV lattice equation with an arbitrary function is considered. As a result, kink-type solutions are obtained which possess rich spatial structures. It is shown that the improved exp-function method can be applied to some other nonlinear differential-difference equations with variable coefficients.

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17

Polat, Refet. "Finite Difference Solution to the Space-Time Fractional Partial Differential-Difference Toda Lattice Equation." Journal of Mathematical Sciences and Modelling 1, no. 3 (December 30, 2018): 202–5. http://dx.doi.org/10.33187/jmsm.460001.

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18

Fu, Wei, and Frank W. Nijhoff. "On non-autonomous differential-difference AKP, BKP and CKP equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2245 (January 2021): 20200717. http://dx.doi.org/10.1098/rspa.2020.0717.

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Based on the direct linearization framework of the discrete Kadomtsev–Petviashvili-type equations presented in the work of Fu & Nijhoff (Fu W, Nijhoff FW. 2017 Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations. Proc. R. Soc. A 473 , 20160915 ( doi:10.1098/rspa.2016.0915 )), six novel non-autonomous differential-difference equations are established, including three in the AKP class, two in the BKP class and one in the CKP class. In particular, one in the BKP class and the one in the CKP class are both in (2 + 2)-dimensional form. All the six models are integrable in the sense of having the same linear integral equation representations as those of their associated discrete Kadomtsev–Petviashvili-type equations, which guarantees the existence of soliton-type solutions and the multi-dimensional consistency of these new equations from the viewpoint of the direct linearization.

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19

Levi, D., and M. A. Rodriguez. "Symmetry group of partial differential equations and of differential difference equations: the Toda lattice versus the Korteweg-de Vries equation." Journal of Physics A: Mathematical and General 25, no. 15 (August 7, 1992): L975—L979. http://dx.doi.org/10.1088/0305-4470/25/15/013.

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20

GENG, XIANGUO, FANG LI, and BO XUE. "A GENERALIZATION OF TODA LATTICES AND THEIR BI-HAMILTONIAN STRUCTURES." Modern Physics Letters B 26, no. 13 (April 26, 2012): 1250078. http://dx.doi.org/10.1142/s0217984912500789.

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A hierarchy of new nonlinear differential-difference equations associated with fourth-order discrete spectral problem is proposed, in which a typical member is a generalization of the Toda lattice equation. The bi-Hamiltonian structures for this hierarchy are obtained with the help of trace identity.

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21

Meng, Fanwei. "A New Variable-Coefficient Riccati Subequation Method for Solving Nonlinear Lattice Equations." Abstract and Applied Analysis 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/810363.

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We propose a new variable-coefficient Riccati subequation method to establish new exact solutions for nonlinear differential-difference equations. For illustrating the validity of this method, we apply it to the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new and generalized traveling wave solutions including hyperbolic function solutions, trigonometric function solutions, and rational function solutions are obtained.

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22

Yildirim, O., and S. Caglak. "On the Lie symmetries of the boundary value problems for differential and difference sine-Gordon equations." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 102, no. 2 (June 30, 2021): 142–53. http://dx.doi.org/10.31489/2021m2/142-153.

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In general, due to the nature of the Lie group theory, symmetry analysis is applied to single equations rather than boundary value problems. In this paper boundary value problems for the sine-Gordon equations under the group of Lie point symmetries are obtained in both differential and difference forms. The invariance conditions for the boundary value problems and their solutions are obtained. The invariant discretization of the difference problem corresponding to the boundary value problem for sine-Gordon equation is studied. In the differential case an unbounded domain is considered and in the difference case a lattice with points lying in the plane and stretching in all directions with no boundaries is considered.

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23

LI, XINGLI, ZHIPENG LI, XIANGLIN HAN, and SHIQIANG DAI. "JAMMING TRANSITION IN EXTENDED COOPERATIVE DRIVING LATTICE HYDRODYNAMIC MODELS INCLUDING BACKWARD-LOOKING EFFECT ON TRAFFIC FLOW." International Journal of Modern Physics C 19, no. 07 (July 2008): 1113–27. http://dx.doi.org/10.1142/s0129183108012698.

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Two extended cooperative driving lattice hydrodynamic models are proposed by incorporating the intelligent transportation system and the backward-looking effect in traffic flow under certain conditions. They are the lattice versions of the hydrodynamic model of traffic: one (model A) is described by the differential-difference equation where time is a continuous variable and space is a discrete variable, and the other (model B) is the difference-difference equation in which both time and space variables are discrete. In light of the real traffic situations, the appropriate forward and backward optimal velocity functions are selected, respectively. Then the stability conditions for the two models are investigated with the linear stability theory and it is found that the new consideration leads to the improvement of the stability of traffic flow. The modified Korteweg-de Vries equations (the mKdV equation, for short) near the critical point are derived by using the nonlinear perturbation method to show that the traffic jam could be described by the kink-antikink soliton solutions for the mKdV equations. Moreover, the anisotropy of traffic flow is further discussed through examining the negative propagation velocity as the effect of following vehicle is involved.

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24

Liu, Ning Ning. "The Numerical Solution of Richards Equation Using the Lattice Boltzmann Method." Applied Mechanics and Materials 188 (June 2012): 90–95. http://dx.doi.org/10.4028/www.scientific.net/amm.188.90.

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The Richards equation is applied to describe the unsaturated soil moisture movement. The Lattice Boltzmann method is developed to solve this partial differential equation. The accuracy and efficiency of the Lattice Boltzmann method in modeling unsaturated soil moisture movement are compared to the Philip series method as well as Crank-Nicolson finite difference scheme. The results reveal that all three methods provide solutions of comparable accuracy. The computation efficiency, accuracy and simplicity of the Lattice Boltzmann method indicate that it has the capacity to model unsaturated soil moisture movement.

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25

Zhang, Sheng, and Dong Liu. "Multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation via Hirota’s bilinear method." Canadian Journal of Physics 92, no. 3 (March 2014): 184–90. http://dx.doi.org/10.1139/cjp-2013-0341.

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In this paper, Hirota’s bilinear method is extended to construct multisoliton solutions of a (2+1)-dimensional variable-coefficient Toda lattice equation. As a result, new and more general one-soliton, two-soliton, and three-soliton solutions are obtained, from which the uniform formula of the N-soliton solution is derived. It is shown that Hirota’s bilinear method can be used for constructing multisoliton solutions of some other nonlinear differential-difference equations with variable coefficients.

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26

CARUTHERS, J. E., J. S. STEINHOFF, and R. C. ENGELS. "AN OPTIMAL FINITE DIFFERENCE REPRESENTATION FOR A CLASS OF LINEAR PDE'S WITH APPLICATION TO THE HELMHOLTZ EQUATION." Journal of Computational Acoustics 07, no. 04 (December 1999): 245–52. http://dx.doi.org/10.1142/s0218396x99000163.

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A finite difference procedure for linear partial differential equations is shown to be optimal in the sense that the L2 error norm obtained from interpolating an arbitrarily large set of solutions is minimized. The method is valid for any spatial lattice of discretizing points, however, formal 6th order error equivalence is proven for the special cases of the Helmholtz and Laplace equations for a square 2-D lattice.

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27

Li, Zhi-Fang, and Hang-Yu Ruan. "(2+1)-Dimensional Davey-Stewartson II Equation for a Two-Dimensional Nonlinear Monatomic Lattice." Zeitschrift für Naturforschung A 61, no. 1-2 (February 1, 2006): 45–52. http://dx.doi.org/10.1515/zna-2006-1-207.

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A two-dimensional monatomic lattice with nearest-neighbor interaction is investigated by the method of multiple scales combined with a quasidiscreteness approximation. The Davey-Stewartson II equation (DS-II) is obtained from the original two-dimensional (2D) differential-difference system. By solving the DS-II, explicit periodic solutions, soliton solutions and rational function solutions are obtained, and the leading order approximated solutions of the 2D monatomic lattice are constructed by explicit solutions of the DS-II.

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28

Li, Xinfu, and Guang Zhang. "Positive Solutions of a General Discrete Dirichlet Boundary Value Problem." Discrete Dynamics in Nature and Society 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/7456937.

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A steady state equation of the discrete heat diffusion can be obtained by the heat diffusion of particles or the difference method of the elliptic equations. In this paper, the nonexistence, existence, and uniqueness of positive solutions for a general discrete Dirichlet boundary value problem are considered by using the maximum principle, eigenvalue method, sub- and supersolution technique, and monotone method. All obtained results are new and valid on anyn-dimension finite lattice point domain. To the best of our knowledge, they are better than the results of the corresponding partial differential equations. In particular, the methods of proof are different.

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29

Aslan, İsmail. "The discrete (G ′/G )-expansion method applied to the differential-difference Burgers equation and the relativistic Toda lattice system." Numerical Methods for Partial Differential Equations 28, no. 1 (July 15, 2010): 127–37. http://dx.doi.org/10.1002/num.20611.

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30

Polievoda, Yurii, Igor Tverdokhlib, and Valentina Bandura. "MODELLING OF OILY RAW MATERIAL EXCRACTION PROCESS." Vibrations in engineering and technology, no. 3(94) (November 26, 2019): 92–101. http://dx.doi.org/10.37128/2306-8744-2019-3-12.

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This paper concentrates on approaches to mathematical modelling of oily raw material extraction process. «Evolution» of modeling hypothesis based on analysis of differential diffusion equation under the right boundary and initial conditions; on simplified model of Fick's equation and on material balance equation; on mass transfer model in adsorption pore volume; on model for the surface layer by analogy of Van der Waals equation; on Gibbs' model based on the abrupt change of phases due to the intermolecular forces; on the simplest Langmuir equation model; on lattice-based models of Guggenheim, Pryhozhyn, Everett, Ohm, Briukhovetskyi and others. The fact that we need to know a large number of micro - parameters makes these models difficult to use in practice. Under normal extraction conditions, the flow, which comes out of solid phase, collides with the resistance of the diffusion boundary layer, which presents a tangible obstacle affecting the duration and quantity of special-purpose component extraction. As the boundary layer thickness depends on the hydrodynamics of the process, under the influence of the microwave field his obstacle is almost insensible, as the intense movement of liquid reduces its thickness. The main factor acting on the quantity of extracted substance is a pressure difference in capillaries and in the flow of extraction agent and mass transfer coefficient. The effect of pulse microwave input during the extraction transfers the process of diffusion from the external environment to internal, because internal pressure diffusion dominates in this process, but not the convective diffusion with the influence of external agent.

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ABDOU, M. A. "APPROXIMATE SOLUTIONS OF NONLINEAR DIFFERENTIAL DIFFERENCE EQUATIONS." International Journal of Computational Methods 06, no. 04 (December 2009): 569–83. http://dx.doi.org/10.1142/s0219876209002005.

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An extend of He's homotopy perturbation method (HPM) is used for finding a new approximate and exact solutions of nonlinear difference differential equations arising in mathematical physics. To illustrate the effectiveness and the advantage of the proposed method, two models of nonlinear difference equations of special interest in physics are chosen, namely, Ablowitz–Ladik lattice equations and Relativistic Toda lattice difference equations. Comparisons are made between the results of the proposed method and exact solutions. The results show that the HPM is a attracted method in solving the differential difference equations (DDEs). The proposed method will become a much more interesting method for solving nonlinear DDEs in science and engineering.

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32

ŠESTOVIĆ, DRAGAN. "TREE METHOD FOR OPTION PRICING UNDER STOCHASTIC VARIANCE." International Journal of Theoretical and Applied Finance 03, no. 03 (July 2000): 557. http://dx.doi.org/10.1142/s0219024900000565.

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We develop a recombining tree method for pricing of options by using a general two-factor stochastic-variance (SV) diffusion model for asset price dynamics. We show that it is possible to construct a riskless hedge by including additional short-term options in the hedging portfolio. This procedure gives us Partial Differential Equation (PDE) that can be solved by using standard numerical techniques giving us a unique option's price. We show that the option's price does not depend on the long run volatility forecast but only on the parameters of the model, which are related to the volatility of variance. We show one particular transformation of PDE to the Finite Difference Equation (FDE) that leads to the three-dimensional lattice method similar to the standard binomial-tree method. Our tree grows in the price-variance space and similarly to the binomial-tree, the coefficients of the FDE can be interpreted as risk neutral probabilities for jumping between the tree nodes. By investigating an error accumulation in the tree we found the stability criterion and the method that can be applied in order to achieve a stabile procedure. Our option pricing method can be used for European and American options and various payoffs. Since the procedure converges quickly and can be easily implemented, we believe that it could be useful for practitioners. The SV model we used here was shown earlier to be a diffusion limit of various GARCH-type models giving the possibility of using parameters obtained in the discrete-time GARCH framework as an input for our option pricing method.

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33

Xie, Fuding, Zhen Wang, and Min Ji. "Application of Symbolic Computation in Nonlinear Differential-Difference Equations." Discrete Dynamics in Nature and Society 2009 (2009): 1–8. http://dx.doi.org/10.1155/2009/158142.

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A method is proposed to construct closed-form solutions of nonlinear differential-difference equations. For the variety of nonlinearities, this method only deals with such equations which are written in polynomials in function and its derivative. Some closed-form solutions of Hybrid lattice, Discrete mKdV lattice, and modified Volterra lattice are obtained by using the proposed method. The travelling wave solutions of nonlinear differential-difference equations in polynomial in function tanh are included in these solutions. This implies that the proposed method is more powerful than the one introduced by Baldwin et al. The results obtained in this paper show the validity of the proposal.

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34

Igarashi, Yuji, Katsumi Itoh, and Ken Nakanishi. "Toda Lattice Solutions of Differential-Difference Equations for Dissipative Systems." Journal of the Physical Society of Japan 68, no. 3 (March 15, 1999): 791–96. http://dx.doi.org/10.1143/jpsj.68.791.

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35

Gepreel, Khaled A., and A. R. Shehata. "Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations." Applied Mathematics Letters 25, no. 9 (September 2012): 1173–78. http://dx.doi.org/10.1016/j.aml.2012.02.028.

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36

Xenitidis, Pavlos, Frank Nijhoff, and Sarah Lobb. "On the Lagrangian formulation of multidimensionally consistent systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2135 (July 13, 2011): 3295–317. http://dx.doi.org/10.1098/rspa.2011.0124.

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Multidimensional consistency has emerged as a key integrability property for partial difference equations (PΔEs) defined on the ‘space–time’ lattice. It has led, among other major insights, to a classification of scalar affine-linear quadrilateral PΔEs possessing this property, leading to the so-called Adler–Bobenko–Suris (ABS) list. Recently, a new variational principle has been proposed that describes the multidimensional consistency in terms of discrete Lagrangian multi-forms. This description is based on a fundamental and highly non-trivial property of Lagrangians for those integrable lattice equations, namely the fact that on the solutions of the corresponding PΔE the Lagrange forms are closed, i.e. they obey a closure relation . Here, we extend those results to the continuous case: it is known that associated with the integrable PΔEs there exist systems of partial differential equations (PDEs), in fact differential equations with regard to the parameters of the lattice as independent variables, which equally possess the property of multidimensional consistency. In this paper, we establish a universal Lagrange structure for affine-linear quad-lattices alongside a universal Lagrange multi-form structure for the corresponding continuous PDEs, and we show that the Lagrange forms possess the closure property.

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Hon, Benny Y. C., Engui Fan, and Qi Wang. "Homotopy Analysis Method for Ablowitz–Ladik Lattice." Zeitschrift für Naturforschung A 66, no. 10-11 (November 1, 2011): 599–605. http://dx.doi.org/10.5560/zna.2011-0022.

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In this paper, the homotopy analysis method is successfully applied to solve the systems of differential-difference equations. The Ablowitz-Ladik lattice system are chosen to illustrate the method. Comparisons between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective and simple in solving systems of differential-difference equations.

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Wang, Qi. "Application of the Homotopy Analysis Method for Systems of Differential-Difference Equations." Zeitschrift für Naturforschung A 65, no. 10 (October 1, 2010): 811–17. http://dx.doi.org/10.1515/zna-2010-1007.

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The homotopy analysis method is used for solving systems of differential-difference equations. To demonstrate the validity and applicability of the presented technique the Volterra lattice system is taken as example. Analysis results show that the method is very effective and yields very accurate results.

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Berezin, A. A. "Das an Electric Current have an Acoustic Component?" Journal of Energy Conservation 1, no. 2 (March 9, 2019): 1–14. http://dx.doi.org/10.14302/issn.2642-3146.jec-19-2663.

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The quantum model of electric current suggested by Feynman has been enlarged by n difference-differential Hamiltonian equations describing the phonon dynamics in one dimensional crystallyne lattice. The process of interaction between the electron and phonon components in a crystalline lattice of a conductor has been described by 2n parametrically coupled difference-differential Hamiltonian equations. Computer analysis of the system of these coupled equations showed that their solutions represent a form of the quantum recurrence similar to the Fermi-Pasta-Ulam recurrence. The results of the research might reconsider the existing concept of electric current and will be possibly helpful in developing an acoustic «laser».

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Fučík, Radek, and Robert Straka. "Equivalent finite difference and partial differential equations for the lattice Boltzmann method." Computers & Mathematics with Applications 90 (May 2021): 96–103. http://dx.doi.org/10.1016/j.camwa.2021.03.014.

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Zhang, Sheng, and Hong-Qing Zhang. "Exp-Function Method for N-Soliton Solutions of Nonlinear Differential-Difference Equations." Zeitschrift für Naturforschung A 65, no. 11 (November 1, 2010): 924–34. http://dx.doi.org/10.1515/zna-2010-1105.

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In this paper, the exp-function method is generalized to construct N-soliton solutions of nonlinear differential-difference equations. With the aid of symbolic computation, we choose the Toda lattice to illustrate the validity and advantages of the generalized work. As a result, 1-soliton, 2-soliton, and 3-soliton solutions are obtained, from which the uniform formula of N-soliton solutions is derived. It is shown that the exp-function method may provide us with a straightforward and effective mathematical tool for generating N-soliton solutions of nonlinear differential-difference equations in mathematical physics.

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Steinbach, Bernd, and Christian Posthoff. "Boolean differential equations: A common model for classes, lattices, and arbitrary sets of Boolean functions." Facta universitatis - series: Electronics and Energetics 28, no. 1 (2015): 51–76. http://dx.doi.org/10.2298/fuee1501051s.

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The Boolean Differential Calculus (BDC) significantly extends the Boolean Algebra because not only Boolean values 0 and 1, but also changes of Boolean values or Boolean functions can be described. A Boolean Differential Equation (BDe) is a Boolean equation that includes derivative operations of the Boolean Differential Calculus. This paper aims at the classification of BDEs, the characterization of the respective solutions, algorithms to calculate the solution of a BDe, and selected applications. We will show that not only classes and arbitrary sets of Boolean functions but also lattices of Boolean functions can be expressed by Boolean Differential equations. In order to reach this aim, we give a short introduction into the BDC, emphasize the general difference between the solutions of a Boolean equation and a BDE, explain the core algorithms to solve a BDe that is restricted to all vectorial derivatives of f (x) and optionally contains Boolean variables. We explain formulas for transforming other derivative operations to vectorial derivatives in order to solve more general BDEs. New fields of applications for BDEs are simple and generalized lattices of Boolean functions. We describe the construction, simplification and solution. The basic operations of XBOOLE are sufficient to solve BDEs. We demonstrate how a XBooLe-problem program (PRP) of the freely available XBooLe-Monitor quickly solves some BDes.

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Dai, Chao-Qing, and Yue-Yue Wang. "Exact Travelling Wave Solutions of Toda Lattice Equations Obtained via the Exp-Function Method." Zeitschrift für Naturforschung A 63, no. 10-11 (November 1, 2008): 657–62. http://dx.doi.org/10.1515/zna-2008-10-1109.

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We generalize the exp-function method, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, we study two Toda lattices and obtain some new travelling wave solutions by means of the exp-function method. As some special examples, some new exact travelling wave solutions can degenerate into the kink-type solitary wave solutions reported in open literatures.

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Picandet, Vincent, and Noël Challamel. "Nonlocality of one-dimensional bilinear hardening–softening elastoplastic axial lattices." Mathematics and Mechanics of Solids 25, no. 2 (October 17, 2019): 475–97. http://dx.doi.org/10.1177/1081286519881668.

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The static behaviour of an elastoplastic axial lattice is studied in this paper through both discrete and nonlocal continuum analyses. The elastoplastic lattice system is composed of piecewise linear hardening–softening elastoplastic springs connected between each other via nodes, loaded by concentrated tension forces. This inelastic lattice evolution problem is ruled by some difference equations, which are shown to be equivalent to the finite difference formulation of a continuous elastoplastic bar problem under distributed tension load. Exact solutions of this inelastic discrete problem are obtained from the resolution of this piecewise linear difference equations system. Localization of plastic strain in the first elastoplastic spring, connected to the fixed end, is observed in the softening range. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process, based on a rational asymptotic expansion of the associated pseudo-differential operators. The continualized lattice-based model is equivalent to a distributed nonlocal continuous elastoplastic theory coupled to a cohesive elastoplastic model, which is shown to capture efficiently the scale effects of the reference axial lattice. The hardening–softening localization process of the nonlocal elastoplastic continuous model strongly depends on the lattice spacing, which controls the size of the nonlocal length scales. An analogy with the one-dimensional lattice system in bending is also shown. The considered microstructured elastoplastic beam is a Hencky bar-chain connected by elastoplastic rotational springs. It is shown that the length scale calibration of the nonlocal model strongly depends on the degree of the difference equations of each lattice model (namely axial or bending lattice). These preliminary results valid for one-dimensional systems allow possible future developments of new nonlocal elastoplastic models, including two- or even three-dimensional elastoplastic interactions.

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Adler, Mark, Pierre van Moerbeke, and Pol Vanhaecke. "Singularity confinement for a class of m -th order difference equations of combinatorics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1867 (July 17, 2007): 877–922. http://dx.doi.org/10.1098/rsta.2007.2090.

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In a recent publication, it was shown that a large class of integrals over the unitary group U ( n ) satisfy nonlinear, non-autonomous difference equations over n , involving a finite number of steps; special cases are generating functions appearing in questions of the longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the discrete Painlevé property ; roughly speaking, this means that after a finite number of steps the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be finite again (‘ singularity confinement ’). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the Painlevé property for the discrete relations is inherited from the Painlevé property of the (continuous) Toeplitz lattice.

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Foupouagnigni, Mama, and Salifou Mboutngam. "On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices." Axioms 8, no. 2 (April 21, 2019): 47. http://dx.doi.org/10.3390/axioms8020047.

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In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type.

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DAI, CHAO-QING, and JIE-FANG ZHANG. "TRAVELLING WAVE SOLUTIONS TO THE COUPLED DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS." International Journal of Modern Physics B 19, no. 13 (May 20, 2005): 2129–43. http://dx.doi.org/10.1142/s0217979205029778.

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In this paper, we have utilized the extended Jacobian elliptic function approach to construct seven families of new Jacobian elliptic function solutions for the coupled discrete nonlinear Schrödinger equations. When the modulus m → 1 or 0, some of these obtained solutions degenerate to the soliton solutions (the moving bright-bright and dark-dark solitons), the solitonic solutions and the trigonometric function solutions. This integrable model possesses the moving solitons because there is no PN barrier to block their motion in the lattice. We also find that some solutions in differential-difference equations (DDEs) are essentially identical to the continuous cases, while some solutions such as sec-type and tan-type in differential-difference models present different properties.

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Challamel, Noël, Attila Kocsis, and C. M. Wang. "Higher-order gradient elasticity models applied to geometrically nonlinear discrete systems." Theoretical and Applied Mechanics 42, no. 4 (2015): 223–48. http://dx.doi.org/10.2298/tam1504223c.

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The buckling and post-buckling behavior of a nonlinear discrete repetitive system, the discrete elastica, is studied herein. The nonlinearity essentially comes from the geometrical effect, whereas the constitutive law of each component is reduced to linear elasticity. The paper primarily focuses on the relevancy of higher-order continuum approximations of the difference equations, also called continualization of the lattice model. The pseudo-differential operator of the lattice equations are expanded by Taylor series, up to the second or the fourth-order, leading to an equivalent second-order or fourth-order gradient elasticity model. The accuracy of each of these models is compared to the initial lattice model and to some other approximation methods based on a rational expansion of the pseudo-differential operator. It is found, as anticipated, that the higher level of truncation is chosen, the better accuracy is obtained with respect to the lattice solution. This paper also outlines the key role played by the boundary conditions, which also need to be consistently continualized from their discrete expressions. It is concluded that higher-order gradient elasticity models can efficiently capture the scale effects of lattice models.

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Poklonski, N. A., A. O. Bury, N. G. Abrashina-Zhadaeva, and S. A. Vyrko. "Diffusion-drift model of ion migration over interstitial sites of a two-dimensional lattice." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 3 (October 7, 2019): 355–65. http://dx.doi.org/10.29235/1561-2430-2019-55-3-355-365.

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An analytical and numerical modeling of the process of obtaining hydroxyl radicals OH0 and atomic hydrogen H0 from water molecules on a square lattice based on electrical neutralization of ions OH− on an anode and ions H+ on a cathode is conducted. The numerical solution of a system of equations describing a stationary migration of ions H+ and OH− over the interstitial sites of a square lattice located in an external electric field is considered. The ions H+ and OH− in the interstitial sites of a square lattice are generated as a result of dissociation of a water molecule under the action of external electromagnetic radiation and external constant (stationary) electric field. It is assumed that anode and cathode are unlimited ion sinks. The problem is solved using the finite difference approximation for the initial system of differential equations with the construction of an iterative process due to the nonlinearity of the constituent equations. It is shown by using calculation that the dependence of the ion current on a difference of electric potentials between anode and cathode is sublinear.

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ABRAHAM, RALPH H., JOHN B. CORLISS, and JOHN E. DORBAND. "ORDER AND CHAOS IN THE TORAL LOGISTIC LATTICE." International Journal of Bifurcation and Chaos 01, no. 01 (March 1991): 227–34. http://dx.doi.org/10.1142/s0218127491000154.

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Cellular dynamical systems, alias lattice dynamical systems, emerged as a new mathematical structure and modeling strategy in the 1980s. Based, like cellular automata, on finite difference methods for partial differential equations, they provide challenging patterns of spatiotemporal organization, in which chaos and order cooperate in novel ways. Here we present initial findings of our exploration of a two-dimensional logistic lattice with the Massively Parallel Processor (MPP) at NASA's Goddard Space Flight Center, a machine capable of 200 megaflops per second. A video tape illustrating these findings is available.

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