Relevant bibliographies by topics / Lattice Differential-Difference Equation

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Lattice Differential-Difference Equation'

Author: Grafiati
Published: 4 June 2021
Last updated: 3 February 2022

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

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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Lattice Differential-Difference Equation"

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Segal, Joseph. "STANDING WAVES OF SPATIALLY DISCRETE FITZHUGH-NAGUMO EQUATIONS." Master's thesis, University of Central Florida, 2009. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/3511.

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We study a system of spatially discrete FitzHugh-Nagumo equations, which are nonlinear differential-difference equations on an infinite one-dimensional lattice. These equations are used as a model of impulse propagation in nerve cells. We employ McKean's caricature of the cubic as our nonlinearity, which allows us to reduce the nonlinear problem into a linear inhomogeneous problem. We find exact solutions for standing waves, which are steady states of the system. We derive formulas for all 1-pulse solutions. We determine the range of parameter values that allow for the existence of standing waves. We use numerical methods to demonstrate the stability of our solutions and to investigate the relationship between the existence of standing waves and propagation failure of traveling waves.
M.S.
Department of Mathematics
Sciences
Mathematical Science MS
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Books on the topic "Lattice Differential-Difference Equation"

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Kikuchi, Tetsuya. Studies on commuting difference systems arising from solvable lattice models. Sendai, Japan: Tohoku University, 2000.

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Book chapters on the topic "Lattice Differential-Difference Equation"

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Caraballo, Tomás, Francisco Morillas, and José Valero. "Pullback Attractors of Stochastic Lattice Dynamical Systems with a Multiplicative Noise and Non-Lipschitz Nonlinearities." In Differential and Difference Equations with Applications, 341–49. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-7333-6_27.

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Ashrafizaadeh, Mahmud, and A. Ghavaminia. "Development of a Lattice Boltzmann Model for the Solution of Partial Differential Equations, A Performance Comparison Study with that of the Finite Difference Method." In Springer Proceedings in Mathematics & Statistics, 255–65. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-63591-6_24.

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Conference papers on the topic "Lattice Differential-Difference Equation"

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Velivelli, Aditya C., and Kenneth M. Bryden. "An Improved Lattice Boltzmann Method for Steady Fluid Flows." In ASME 2004 International Mechanical Engineering Congress and Exposition. ASMEDC, 2004. http://dx.doi.org/10.1115/imece2004-61900.

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The use of the lattice Boltzmann method in computational fluid dynamics has been steadily increasing. The highly local nature of lattice Boltzmann computations have allowed for easy cache optimization and parallelization. This bestows the lattice Boltzmann method with considerable superiority in computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size. The time step size is limited by the Courant-Friedrichs-Lewy (CFL) condition and local gradients in the solution, the latter limitation being more extreme. This paper describes a novel explicit discretization for the lattice Boltzmann method that can perform simulations with larger time step sizes. The new algorithm is applid to the steady Burger’s equation, uux = μ(uxx + uyy), which is a nonlinear partial differential equation containing both convection and diffusion terms. A comparison between the original lattice Boltzmann method and the new algorithm is performed with regard to time for computation and accuracy.
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Qi, Xuele, and C. Steve Suh. "Thermomechanical Characteristics of Silicon Wafer in Response to Ultrafast Laser Heating." In ASME 2008 International Mechanical Engineering Congress and Exposition. ASMEDC, 2008. http://dx.doi.org/10.1115/imece2008-66329.

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Material processing using ultra-short pulse duration lasers of low fluence produces no plasma and inflicts no damage to the material being treated. The benefit provided by the brief heating time on the order of femtoseconds also enables precision control over the spread of the heat affected zone. In this paper, the interaction between femtosecond lasers and silicon wafer is investigated. A multiscale two dimensional axisymmetric model that governs the transport dynamics in silicon is presented based on the relaxation-time approximation of the Boltzmann equation. Temperature-dependent multi-phonons, free-carrier absorptions, and the recombination and impact ionization processes are considered using a set of balance equations through which the spatial and temporal evolutions of the electron and lattice temperatures along with the electron-hole carrier density are obtained. The mechanical response of the lattice, on the other hand, is described by momentum equations. To solve the model of 17 coupled time-dependent partial differential equations without having to be concerned with non-physical oscillations in the solution, an implicit finite difference scheme on a staggered grid is developed. Unlike the conventional finite difference method in which primary variables are evaluated at grid points, the staggered finite difference scheme allows velocities and first order spatial derivative terms to be calculated at locations midway between two consecutive grid points, and shear stresses to be evaluated at the center of each element. A multi-time-scale approach involving the use of varying time steps ranging from 2.5fsec to 5psec is implemented to successfully obtain time integration results up to 10nsec. The temporal evolution and two dimensional spatial distributions of electron and lattice temperatures are presented in the first few hundreds of picoseconds when thermal equilibrium is reached. It is shown that the calculated material damage threshold based on the carrier density criterion agrees well with the published experimental data, thus validating the feasibility of the model in describing the various thermomechanical responses subject to femtosecond heating of low fluence input. The displacement and velocity distributions explain how the particles initially located at the grid nodes move as time increases and how the mechanical waves are generated by the laser heating. The thermal stress waves induced in a short time scale are proved to be highly dispersive and characteristically of broadband, low amplitude, and extremely high frequency.
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Velivelli, Aditya C., and Kenneth M. Bryden. "Parallel Performance of Lattice Boltzmann and Implicit Finite Difference Approaches to the Approximation of the Two-Dimensional Diffusion Equation." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-41280.

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The reduction of computation times is an important aspect of interactive computational fluid dynamics simulations. The lattice Boltzmann method has proved to be an important technique for the numerical solution of partial differential equations because it has nearly ideal scalability on parallel computers for many applications. Utilizing the two-dimensional diffusion equation, Tt=μ(Txx+Tyy), this paper examines the parallel performance for the lattice Boltzmann method and the alternating direction implicit (ADI) method. In this study for 50 time steps the non-cache optimized parallel lattice Boltzmann method was on average two times faster than the parallel ADI method. The cache optimized parallel lattice Boltzmann was on average seven times faster than the parallel ADI method.
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