scholarly journals New Results of the Time-Space Fractional Derivatives of Kortewege-De Vries Equations via Novel Analytic Method

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2296
Author(s):  
Mariam Sultana ◽  
Uroosa Arshad ◽  
Md. Nur Alam ◽  
Omar Bazighifan ◽  
Sameh Askar ◽  
...  
Keyword(s):  
Numerical Solutions ◽  
Kdv Equation ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Analytic Method ◽  
The Novel ◽  
Time Space ◽  
Essential Function ◽  
Derivatives Of ◽  
Taylor’S Series

Symmetry performs an essential function in finding the correct techniques for solutions to time space fractional differential equations (TSFDEs). In this article, we present the Novel Analytic Method (NAM) for approximate solutions of the linear and non-linear KdV equation for TSFDs. To enunciate the non-integer derivative for the aforementioned equation, the Caputo operator is manipulated. Furthermore, the formula implemented is a numerical way that is postulated from Taylor’s series, which confirms an analytical answer in the form of a convergent series. For delineation of the efficiency and functionality of the method in question, four applications are exemplified along with graphical interpretation and numerical solutions to finitely illustrate the behavior of the solution to this equation. Moreover, the 3D graphs of some of these numerical examples are plotted with specific values. Observing the effectiveness of this process, we can easily decide that this process can be implemented to other TSFDEs applied in the mathematical modeling of a real-world aspect.

Richardson Extrapolation: An Info-Gap Analysis of Numerical Uncertainty

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
Yakov Ben-Haim ◽  
François Hemez
Keyword(s):  
Solid Mechanics ◽  
Gap Analysis ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Richardson Extrapolation ◽  
Numerical Application ◽  
Main Method ◽  
Time Space ◽  
Computational Modeling And Simulation ◽  
Central Tool

Abstract Computational modeling and simulation is a central tool in science and engineering, directed at solving partial differential equations for which analytical solutions are unavailable. The continuous equations are generally discretized in time, space, energy, etc., to obtain approximate solutions using a numerical method. The aspiration is for the numerical solutions to asymptotically converge to the exact-but-unknown solution as the discretization size approaches zero. A generally applicable procedure to assure convergence is unavailable. The Richardson extrapolation is the main method for dealing with this challenge, but its assumptions introduce uncertainty to the resulting approximation. We use info-gap decision theory to model and manage its main uncertainty, namely, in the rate of convergence of numerical solutions. The theory is illustrated with a numerical application to Hertz contact in solid mechanics.

scholarly journals Spectrally accurate approximate solutions and convergence analysis of fractional Burgers’ equation

2020 ◽  
Vol 9 (3) ◽  
pp. 633-644
Author(s):  
A. K. Mittal
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
Numerical Technique ◽  
Time Derivative ◽  
Pseudospectral Method ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Time And Space ◽  
Time Space ◽  
Coupled Burgers Equation

Abstract In this paper, a new numerical technique implements on the time-space pseudospectral method to approximate the numerical solutions of nonlinear time- and space-fractional coupled Burgers’ equation. This technique is based on orthogonal Chebyshev polynomial function and discretizes using Chebyshev–Gauss–Lobbato (CGL) points. Caputo–Riemann–Liouville fractional derivative formula is used to illustrate the fractional derivatives matrix at CGL points. Using the derivatives matrices, the given problem is reduced to a system of nonlinear algebraic equations. These equations can be solved using Newton–Raphson method. Two model examples of time- and space-fractional coupled Burgers’ equation are tested for a set of fractional space and time derivative order. The figures and tables show the significant features, effectiveness, and good accuracy of the proposed method.

scholarly journals Numerical Solutions of a Quadratic Integral Equations by Using Variational Iteration and Homotopy Perturbation Methods

2017 ◽  
Vol 9 (2) ◽  
pp. 134
Author(s):  
Hind Al-badrani ◽  
F. A. Hendi ◽  
Wafa Shammakh
Keyword(s):  
Integral Equations ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Variational Iteration ◽  
Homotopy Perturbation ◽  
Quadratic Integral ◽  
Quadratic Integral Equations

In this paper, the approximate solutions for  quadratic integral equations (QIEs) are given by the variational iteration method(VIM) and homotopy perturbation method (HPM). These methods produce the solutions in terms of convergent series without needing to restrictive assumptions, to illustrate the ability and credibility of the methods, we deal with some examples that show simplicity and effectiveness.

scholarly journals Solving Famous Nonlinear Coupled Equations with Parameters Derivative by Homotopy Analysis Method

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Sohrab Effati ◽  
Hassan Saberi Nik ◽  
Reza Buzhabadi
Keyword(s):  
Homotopy Analysis Method ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Reaction Diffusion Equations ◽  
Analysis Method ◽  
Coupled Equations ◽  
Homotopy Analysis ◽  
Special Case

The homotopy analysis method (HAM) is employed to obtain symbolic approximate solutions for nonlinear coupled equations with parameters derivative. These nonlinear coupled equations with parameters derivative contain many important mathematical physics equations and reaction diffusion equations. By choosing different values of the parameters in general formal numerical solutions, as a result, a very rapidly convergent series solution is obtained. The efficiency and accuracy of the method are verified by using two famous examples: coupled Burgers and mKdV equations. The obtained results show that the homotopy perturbation method is a special case of homotopy analysis method.

On the Soliton Solution and Jacobi Doubly Periodic Solution of the Fractional Coupled Schrödinger–KdV Equation by a Novel Approach

2015 ◽  
Vol 16 (2) ◽  
pp. 79-95 ◽  
Author(s):  
S. Saha Ray
Keyword(s):  
Present Method ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
New Novel ◽  
Novel Approach ◽  
Adomian Decomposition ◽  
Novel Method ◽  
Doubly Periodic Solution

AbstractIn this paper, fractional coupled Schrödinger–Korteweg–de Vries equation (or Sch–KdV) equation with appropriate initial values has been solved by using a new novel method. The fractional derivatives are described in the Caputo sense. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. Basically, the present method originated from generalized Taylor’s formula [1]. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of fractional coupled Schrödinger–KdV equation. Numerical solutions are presented graphically to show the reliability and efficiency of the method. The method does not need linearization, weak nonlinearity assumptions or perturbation theory. The convergence of the method as applied to Sch–KdV is illustrated numerically as well as derived analytically. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM).

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.

Tumor Necrosis Factor-α Production-Enhancing Properties of Novel Adamantylalkylthio Derivatives of Some Heterocyclic Compounds

2005 ◽  
Vol 60 (4) ◽  
pp. 471-475 ◽  
Author(s):  
Barbara Orzeszko ◽  
Tomasz Świtaj ◽  
Anna B. Jakubowska-Mućka ◽  
Witold Lasek ◽  
Andrzej Orzeszko ◽  
...  
Keyword(s):  
Tumor Necrosis Factor ◽  
Heterocyclic Compounds ◽  
Tumor Necrosis ◽  
The Novel ◽  
Necrosis Factor Alpha ◽  
Tnf Α ◽  
Factor Alpha ◽  
Factor Α ◽  
Derivatives Of ◽  
Necrosis Factor

Certain adamantylated heterocycles were previously shown to enhance the secretion of tumor necrosis factor alpha (TNF-α) by murine melanoma cells that have been transduced with the gene for human TNF-α and constitutively expressed this cytokine. The stimulatory potency of those compounds depended, among other factors, on the structure of the linker between the adamantyl residue and the heterocyclic core. In the present study, a series of (1-adamantyl)alkylsulfanyl derivatives of heterocyclic compounds was prepared by alkylation of the corresponding thioheterocyles. Of the novel adamantylalkylthio compounds tested in the aforementioned cell line, 2-(2-adamantan-1-ylethylsulfanyl)- 4-methyl-pyrimidine was found to be the most active

Transcritical two-layer flow over topography

1987 ◽  
Vol 178 ◽  
pp. 31-52 ◽  
Author(s):  
W. K. Melville ◽  
Karl R. Helfrich
Keyword(s):  
Steady Flow ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Single Layer ◽  
Cubic Nonlinearity ◽  
Arbitrary Length ◽  
Weakly Nonlinear ◽  
Flow Over Topography ◽  
Layer Flow

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

Thio derivatives of β-diketones and their metal chelates. XIX. Metal chelates of two new fluorinated Monothio-β-diketones and their mass spectra

1975 ◽  
Vol 28 (6) ◽  
pp. 1249 ◽  
Author(s):  
SE Livingstone ◽  
N Saha
Keyword(s):  
Mass Spectra ◽  
Metal Chelates ◽  
The Novel ◽  
Copper Chelates ◽  
Derivatives Of

The nickel(II), palladium(II), platinum(II), copper(II), zinc(II) and rhodium(III) chelates of the new fluorinated monothio-β-diketones RC(SH)=CHCOCF3 (R = Pri, Bui) and the iron(III), ruthenium(III) and cobalt(III) chelates of PriC(SH)=CHCOCF3 have been prepared. The mass spectra of all but the two copper chelates have been obtained. The novel features of the spectra are the occurrence of the ions M-(R-H), M-R, and M-LH and the loss of H2S from the ions M-LH and M-L.

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