scholarly journals Numerical solutions to Giovannini’s QCD parton branching processes

2020 ◽  
Vol 240 ◽  
pp. 07009 ◽  
Author(s):  
Z. Ong ◽  
P. Agarwal ◽  
H.W. Ang ◽  
E. Wang ◽  
A.H. Chan ◽  
...  
Keyword(s):  
Charged Particle ◽  
Numerical Solution ◽  
Multiplicity Distribution ◽  
Numerical Solutions ◽  
Branching Processes ◽  
Approximate Solutions ◽  
Pair Creation ◽  
Gluon Vertex ◽  
200 Gev ◽  
Gluon Bremsstrahlung

In 1979, Giovannini [1] formulated multiparticle production as a manifestation of four QCD processes: A: g → gg (gluon bremsstrahlung), A-: q → qg (quark bremsstrahlung), B: g → qq (quark pair creation) and C: g → ggg (four-gluon vertex). At present, only approximate solutions exist, such as the Generalized Multiplicity Distribution (GMD) which excludes pro- cesses B and C [2, 3]. These approximate solutions have been very successful in describing charged particle multiplicities at lower energies, at least until the appearance of a shoulder-like structure at √s ≈ 200 GeV and 900 GeV first reported by the UA5 collaboration [4], also known as “KNO-scaling violation”. Here, we extend the work of Sakai [5] and attempt a numerical solution that incorporates all four QCD processes, and see if the addition of processes B and C can explain the shoulder-like structure.

A new finite difference method for computing approximate solutions of boundary value problems including transition conditions

2021 ◽  
Vol 102 (2) ◽  
pp. 54-61
Author(s):  
S. Çavuşoğlu ◽  
◽  
O.Sh. Mukhtarov ◽  
◽  
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Difference Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Approximate Solutions

This article is aimed at computing numerical solutions of new type of boundary value problems (BVPs) for two-linked ordinary differential equations. The problem studied here differs from the classical BVPs such that it contains additional conditions at the point of interaction, so-called transition conditions. Naturally, such type of problems is much more complicated to solve than classical problems. It is not clear how to apply the classical numerical methods to such type of boundary value transition problems (BVTPs). Based on the finite difference method (FDM) we have developed a new numerical algorithm for computing numerical solution of BVTPs for two-linked ordinary differential equations. To demonstrate the reliability and efficiency of the presented algorithm we obtained numerical solution of one BVTP and the results are compared with the corresponding exact solution. The maximum absolute errors (MAEs) are presented in a table.

scholarly journals Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

2021 ◽  
Author(s):  
Wedad Albalawi ◽  
Alvaro H. Salas ◽  
S. A. El-Tantawy
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Classical Mechanics ◽  
Elliptic Functions ◽  
Approximate Solutions ◽  
Electronic Circuits ◽  
Chebyshev Collocation ◽  
Pendulum Equation ◽  
First Time

Abstract In this work, novel semi-analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation on the pivot vertically for arbitrary angles are obtained for the first time. The semi-analytical solution is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical analysis, the Chebyshev collocation numerical method is introduced for analyzing tthe forced damped parametric driven pendulum equation. Moreover, the semi-analytical solution and Chebyshev collocation numerical solution are compared with the Runge-Kutta (RK) numerical solution. Also, the maximum distance error to the obtained approximate solutions is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

scholarly journals NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION USING RANDOMLY GENERATED FINITE GRIDS AND TWO-DIMENSIONAL FRACTIONAL-ORDER LEGENDRE FUNCTION

2021 ◽  
Author(s):  
Sanaullah Mastoi
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Polar Coordinates ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Uniform Grids

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

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.

scholarly journals A fractional order epidemic model for the simulation of outbreaks of Ebola

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Weiqiu Pan ◽  
Tianzeng Li ◽  
Safdar Ali
Keyword(s):  
Numerical Solution ◽  
Root Mean Square ◽  
Fractional Order ◽  
Relative Error ◽  
Numerical Solutions ◽  
Real Data ◽  
Mean Square ◽  
Order Model ◽  
The Real ◽  
Fractional Orders

AbstractThe Ebola outbreak in 2014 caused many infections and deaths. Some literature works have proposed some models to study Ebola virus, such as SIR, SIS, SEIR, etc. It is proved that the fractional order model can describe epidemic dynamics better than the integer order model. In this paper, we propose a fractional order Ebola system and analyze the nonnegative solution, the basic reproduction number $R_{0}$ R 0 , and the stabilities of equilibrium points for the system firstly. In many studies, the numerical solutions of some models cannot fit very well with the real data. Thus, to show the dynamics of the Ebola epidemic, the Gorenflo–Mainardi–Moretti–Paradisi scheme (GMMP) is taken to get the numerical solution of the SEIR fractional order Ebola system and the modified grid approximation method (MGAM) is used to acquire the parameters of the SEIR fractional order Ebola system. We consider that the GMMP method may lead to absurd numerical solutions, so its stability and convergence are given. Then, the new fractional orders, parameters, and the root-mean-square relative error $g(U^{*})=0.4146$ g ( U ∗ ) = 0.4146 are obtained. With the new fractional orders and parameters, the numerical solution of the SEIR fractional order Ebola system is closer to the real data than those models in other literature works. Meanwhile, we find that most of the fractional order Ebola systems have the same order. Hence, the fractional order Ebola system with different orders using the Caputo derivatives is also studied. We also adopt the MGAM algorithm to obtain the new orders, parameters, and the root-mean-square relative error which is $g(U^{*})=0.2744$ g ( U ∗ ) = 0.2744 . With the new parameters and orders, the fractional order Ebola systems with different orders fit very well with the real data.

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 Unconditional Positive Stable Numerical Solution of Partial Integrodifferential Option Pricing Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
M. Fakharany ◽  
R. Company ◽  
L. Jódar
Keyword(s):  
Option Pricing ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Stable Process ◽  
Integration Formula ◽  
Pricing Models ◽  
Integral Term ◽  
Pricing Problems ◽  
Stability And Consistency ◽  
Partial Integrodifferential Equation

This paper is concerned with the numerical solution of partial integrodifferential equation for option pricing models under a tempered stable process known as CGMY model. A double discretization finite difference scheme is used for the treatment of the unbounded nonlocal integral term. We also introduce in the scheme the Patankar-trick to guarantee unconditional nonnegative numerical solutions. Integration formula of open type is used in order to improve the accuracy of the approximation of the integral part. Stability and consistency are also studied. Illustrative examples are included.

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 .̂

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