scholarly journals Stable novel and accurate solitary wave solutions of an integrable equation: Qiao model

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 742-752
Author(s):  
Dexu Zhao ◽  
Dianchen Lu ◽  
Samir A. Salama ◽  
Mostafa M. A. Khater
Keyword(s):  
Hamiltonian Structure ◽  
Numerical Solutions ◽  
Necessary Conditions ◽  
Approximate Solutions ◽  
Integrable Equation ◽  
Integrable Hierarchy ◽  
Lax Representation ◽  
Solitary Wave Solutions ◽  
Mathematical Formula ◽  
Physical Behavior

Abstract This article investigates the dynamical and physical behavior of the second positive member in a new, utterly integrable hierarchy. This investigation depends on constructing novel analytical and approximate solutions to the Qiao model. The model’s name is after the researcher who derived the mathematical formula of it in 2007. This model possesses a Lax representation and bi-Hamiltonian structure. This study employs the unified and variational iteration (VI) method to obtain analytical and numerical solutions to the considered model. The obtained analytical solutions are used to calculate the necessary conditions for applying the suggested numerical method that makes checking the obtained solutions’ accuracy a valuable option. The obtained solutions are sketched in different techniques to explain more physical and dynamics details of the Qiao model and show the matching between obtained solutions.

A stochastic operational matrix method for numerical solutions of multi-dimensional stochastic Itô–Volterra integral equations

2020 ◽  
Vol 28 (3) ◽  
pp. 209-216
Author(s):  
S. Singh ◽  
S. Saha Ray
Keyword(s):  
Integral Equations ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Approximate Solutions ◽  
Volterra Integral Equations ◽  
Numerical Examples ◽  
Operational Matrix Method ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Block Pulse Function

AbstractIn this article, hybrid Legendre block-pulse functions are implemented in determining the approximate solutions for multi-dimensional stochastic Itô–Volterra integral equations. The block-pulse function and the proposed scheme are used for deriving a methodology to obtain the stochastic operational matrix. Error and convergence analysis of the scheme is discussed. A brief discussion including numerical examples has been provided to justify the efficiency of the mentioned method.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

A generalized integrable hierarchy related to the relativistic Toda lattice: Hamiltonian structure, Darboux transformation, soliton solution and conservation law

2021 ◽  
Author(s):  
Zhiguo Xu
Keyword(s):  
Darboux Transformation ◽  
Hamiltonian Structure ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Integrable Hierarchy ◽  
Integrable Lattice ◽  
Pass Through ◽  
Matrix Spectral Problem ◽  
Relativistic Toda Lattice ◽  
Toda Lattice Hierarchy

Starting from a more generalized discrete [Formula: see text] matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the [Formula: see text]-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete [Formula: see text]-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

scholarly journals Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

2003 ◽  
Vol 2003 (2) ◽  
pp. 87-114 ◽  
Author(s):  
J. R. Fernández ◽  
M. Sofonea
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Mechanical Damage ◽  
Parabolic Type ◽  
Approximate Solutions ◽  
Damage Function ◽  
Optimal Order ◽  
Order Error ◽  
Fully Discrete ◽  
Solution Regularity

We consider the quasistatic Signorini′s contact problem with damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

scholarly journals Numerical Solutions of Coupled Systems of Fractional Order Partial Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Yongjin Li ◽  
Kamal Shah
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Coupled Systems ◽  
Test Problems ◽  
Operational Matrices ◽  
Partial Differential ◽  
Established Technique

We develop a numerical method by using operational matrices of fractional order integrations and differentiations to obtain approximate solutions to a class of coupled systems of fractional order partial differential equations (FPDEs). We use shifted Legendre polynomials in two variables. With the help of the aforesaid matrices, we convert the system under consideration to a system of easily solvable algebraic equation of Sylvester type. During this process, we need no discretization of the data. We also provide error analysis and some test problems to demonstrate the established technique.

The Residue Harmonic Balance Method for Duffing-Van Der Pol Oscillator with Fractional Derivative

2013 ◽  
Vol 774-776 ◽  
pp. 103-106
Author(s):  
Xin Xue ◽  
Lian Zhong Li ◽  
Dan Sun
Keyword(s):  
Fractional Derivative ◽  
Harmonic Balance ◽  
Numerical Solutions ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Solution Procedure ◽  
Balance Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Fractional Orders

Duffing-van der Pol oscillator with fractional derivative was constructed in this paper. The solution procedure was proposed with the residue harmonic balance method. The effect of different fractional orders on resonance responses of the system in steady state were analyzed for an example without parameters. The approximate solutions were contrasted with numerical solutions. The results show that the residue harmonic balance method to Duffing-van der Pol differential equation with fractional derivative is very valid.

scholarly journals Application of Input–State of the System Transformation for Linearization of Selected Electrical Circuits

2016 ◽  
Vol 67 (3) ◽  
pp. 199-205 ◽  
Author(s):  
Andrzej Zawadzki ◽  
Sebastian Różowicz
Keyword(s):  
Nonlinear System ◽  
Feedback Loop ◽  
Numerical Solutions ◽  
Necessary Conditions ◽  
Electric Circuit ◽  
Input State ◽  
System Transformation ◽  
State Equations ◽  
Local Diffeomorphism ◽  
Linearized System

Abstract The paper presents a transformation of nonlinear electric circuit into linear one through changing coordinates (local diffeomorphism) with the use of closed feedback loop. The necessary conditions that must be fulfilled by nonlinear system to enable carrying out linearizing procedures are presented. Numerical solutions of state equations for the nonlinear system and equivalent linearized system are included.

scholarly journals Exact and Numerical Solitary Wave Structures to the Variant Boussinesq System

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1473 ◽  
Author(s):  
Abdulghani Alharbi ◽  
Mohammed B. Almatrafi
Keyword(s):  
Solitary Wave ◽  
Evolution Equations ◽  
Numerical Solutions ◽  
Nonlinear Evolution ◽  
Numerical Technique ◽  
Method Of Lines ◽  
Nonlinear Evolution Equations ◽  
Boussinesq System ◽  
Solitary Wave Solutions ◽  
The Method Of Lines

Solutions such as symmetric, periodic, and solitary wave solutions play a significant role in the field of partial differential equations (PDEs), and they can be utilized to explain several phenomena in physics and engineering. Therefore, constructing such solutions is significantly essential. This article concentrates on employing the improved exp(−ϕ(η))-expansion approach and the method of lines on the variant Boussinesq system to establish its exact and numerical solutions. Novel solutions based on the solitary wave structures are obtained. We present a comprehensible comparison between the accomplished exact and numerical results to testify the accuracy of the used numerical technique. Some 3D and 2D diagrams are sketched for some solutions. We also investigate the L2 error and the CPU time of the used numerical method. The used mathematical tools can be comfortably invoked to handle more nonlinear evolution equations.

Optimal stopping with discount and observation costs

2000 ◽  
Vol 37 (01) ◽  
pp. 64-72 ◽  
Author(s):  
Robert Kühne ◽  
Ludger Rüschendorf
Keyword(s):  
Differential Equations ◽  
Optimal Stopping ◽  
Point Processes ◽  
Numerical Solutions ◽  
Domain Of Attraction ◽  
Approximate Solutions ◽  
Poisson Approximation ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Optimal Stopping Times

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

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