Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method

2018 ◽  
Vol 32 (29) ◽  
pp. 1850365 ◽  
Author(s):  
Asıf Yokuş
Keyword(s):  
Difference Method ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Expansion Method ◽  
Von Neumann ◽  
Forward Difference ◽  
De Vries ◽  
Conformable Derivatives ◽  
Von Neumann Stability Analysis

In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G[Formula: see text])-expansion method and the finite forward difference method. We first obtain the exact wave solutions of the nonlinear time-fractional KdV equation. In addition, we used the finite-forward difference method to obtain numerical solutions in this equations. When these solutions are obtained, the indexed forms of both Caputo and conformable derivatives are used. By using indexing technique, it is shown that the numerical results of the nonlinear time-fractional KdV equation approaches the exact solution. The two- and three-dimensional surfaces of the obtained analytical solutions are plotted. The von Neumann stability analysis of the used numerical scheme with the studied equation is carried out. The L2and L[Formula: see text] error norms are computed. The exact solutions and numerical approximations are compared by supporting with graphical plots and tables.

On the exact and numerical solutions to the FitzHugh–Nagumo equation

2020 ◽  
Vol 34 (17) ◽  
pp. 2050149 ◽  
Author(s):  
Asıf Yokus
Keyword(s):  
Difference Method ◽  
Numerical Solutions ◽  
Transformation Method ◽  
Von Neumann ◽  
Wave Solutions ◽  
Forward Difference ◽  
Package Program ◽  
Von Neumann Stability ◽  
The Stability ◽  
Von Neumann Stability Analysis

In this paper, with the help of a computer package program, the auto-Bäcklund transformation method (aBTM) and the finite forward difference method are used for obtaining the wave solutions and the numeric and exact approximations to the FitzHugh–Nagumo (F-N) equation, respectively. We successfully obtain some wave solutions to this equation by using aBTM. We then employ the finite difference method (FDM) in approximating the exact and numerical solutions to this equation by taking one of the obtained wave solutions into consideration. We also present the comparison between exact and numeric approximations and support the comparison with a graphic plot. Moreover, the Fourier von-Neumann stability analysis is used in checking the stability of the numeric scheme. We also present the [Formula: see text] and [Formula: see text] error norms of the solutions to this equation.

scholarly journals Regarding the numerical solutions of the Sharma-Tasso-Olver equation

2018 ◽  
Vol 22 ◽  
pp. 01036 ◽  
Author(s):  
Tukur Abdulkadir Sulaiman ◽  
Asif Yokus ◽  
Nesrin Gulluoglu ◽  
Haci Mehmet Baskonus
Keyword(s):  
Difference Method ◽  
Numerical Solutions ◽  
Von Neumann ◽  
Wolfram Mathematica ◽  
Forward Difference ◽  
Von Neumann Stability ◽  
Matlab Package ◽  
The Stability ◽  
Von Neumann Stability Analysis ◽  
Singular Soliton

With aid of the Wolfram Mathematica package, this study investigates the solutions of a nonlinear model with strong nonlinear- ity, namely; the Sharma-Tasso-Olver equation. We use the improved Bernoulli sub-equation function method in acquiring the analytical so- lution to this equation, we successfully obtain one-singular soliton so- lution with exponential function structure. Through the obtained ana- lytical solution, the finite forward difference method is used in approx- imating the exact and numerical solutions to this equation. We check the stability of the finite forward difference method with this equation using the Fourier-Von Neumann stability analysis. We find the L2 and L∞ norm error to the numerical approximation. We present the in- teresting 3D and 2D figures of the obtained singular soliton solution. We also plot the graphics of the numerical error, exact and numeri- cal approximations data obtained in this study by using the MATLAB package.

scholarly journals On the exact and numerical solutions to a nonlinear model arising in mathematical biology

2018 ◽  
Vol 22 ◽  
pp. 01061 ◽  
Author(s):  
Asif Yokus ◽  
Tukur Abdulkadir Sulaiman ◽  
Haci Mehmet Baskonus ◽  
Sibel Pasali Atmaca
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Hyperbolic Function ◽  
Von Neumann ◽  
Convection Diffusion Equation ◽  
Wolfram Mathematica ◽  
Forward Difference ◽  
Von Neumann Analysis ◽  
The Stability

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ(η))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L2 and L∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics

2020 ◽  
Vol 34 (29) ◽  
pp. 2050282
Author(s):  
Asıf Yokuş ◽  
Doğan Kaya
Keyword(s):  
Traveling Wave ◽  
Numerical Solutions ◽  
Traveling Wave Solutions ◽  
Transformation Method ◽  
Mkdv Equation ◽  
Traveling Wave Solution ◽  
Von Neumann ◽  
Wave Solutions ◽  
De Vries ◽  
The Stability

The traveling wave solutions of the combined Korteweg de Vries-modified Korteweg de Vries (cKdV-mKdV) equation and a complexly coupled KdV (CcKdV) equation are obtained by using the auto-Bäcklund Transformation Method (aBTM). To numerically approximate the exact solutions, the Finite Difference Method (FDM) is used. In addition, these exact traveling wave solutions and numerical solutions are compared by illustrating the tables and figures. Via the Fourier–von Neumann stability analysis, the stability of the FDM with the cKdV–mKdV equation is analyzed. The [Formula: see text] and [Formula: see text] norm errors are given for the numerical solutions. The 2D and 3D figures of the obtained solutions to these equations are plotted.

On the integrability properties of variable coefficient Korteweg – de Vries equations

1996 ◽  
Vol 74 (9-10) ◽  
pp. 676-684 ◽  
Author(s):  
F. Güngör ◽  
M. Sanielevici ◽  
P. Winternitz
Keyword(s):  
Differential Equations ◽  
Variable Coefficient ◽  
Kdv Equation ◽  
Three Dimensional ◽  
Symmetry Algebra ◽  
Time Symmetry ◽  
Kdv Equations ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

All variable coefficient Korteweg – de Vries (KdV) equations with three-dimensional Lie point symmetry groups are investigated. For such an equation to have the Painlevé property, its coefficients must satisfy seven independent partial differential equations. All of them are satisfied only for equations equivalent to the KdV equation itself. However, most of them are satisfied in all cases. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time. Symmetry reduction is used to obtain particular solutions. The reduced ordinary differential equations are classified.

scholarly journals A Numerical Solution for Hirota-Satsuma Coupled KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
M. S. Ismail ◽  
H. A. Ashi
Keyword(s):  
Vries Equation ◽  
Kdv Equation ◽  
Approximation Technique ◽  
Conserved Quantities ◽  
Test Functions ◽  
Order Accuracy ◽  
Implicit Midpoint Rule ◽  
Von Neumann ◽  
Von Neumann Stability ◽  
Von Neumann Stability Analysis

A Petrov-Galerkin method and product approximation technique are used to solve numerically the Hirota-Satsuma coupled Korteweg-de Vries equation, using cubicB-splines as test functions and a linearB-spline as trial functions. The implicit midpoint rule is used to advance the solution in time. Newton’s method is used to solve the block nonlinear pentadiagonal system we have obtained. The resulting schemes are of second order accuracy in both directions, space and time. The von Neumann stability analysis of the schemes shows that the two schemes are unconditionally stable. The single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitons, three solitons, and birth of solitons is also discussed.

scholarly journals The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Ozlem Ersoy ◽  
Idris Dag
Keyword(s):  
Numerical Experiments ◽  
Numerical Solutions ◽  
Vries Equation ◽  
Kdv Equation ◽  
Algebraic Equations ◽  
Thomas Algorithm ◽  
B Spline ◽  
Suggested Algorithm ◽  
De Vries

The exponential cubic B-spline algorithm is presented to find the numerical solutions of the Korteweg-de Vries (KdV) equation. The problem is reduced to a system of algebraic equations, which is solved by using a variant of Thomas algorithm. Numerical experiments are carried out to demonstrate the efficiency of the suggested algorithm.

scholarly journals Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation

2019 ◽  
Vol 4 (1) ◽  
pp. 35-42 ◽  
Author(s):  
Asıf Yokuş ◽  
Sema Gülbahar
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Von Neumann ◽  
Non Linear ◽  
The Finite Difference Method ◽  
Dym Equation ◽  
Harry Dym Equation ◽  
Linearization Techniques

AbstractIn this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2 and L∞ are computed. Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis.

Self-Similar Crack Expansion Method for Three-Dimensional Crack Analysis

1997 ◽  
Vol 64 (4) ◽  
pp. 729-737 ◽  
Author(s):  
Yonglin Xu ◽  
B. Moran ◽  
T. Belytschko
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Numerical Solutions ◽  
Crack Surface ◽  
Three Dimensional ◽  
Expansion Method ◽  
Infinite Medium ◽  
Boundary Integral ◽  
Intensity Factors ◽  
Self Similar

The self-similar crack expansion method is developed to calculate stress intensity factors for three-dimensional cracks in an infinite medium or semi-infinite medium by the boundary integral element technique. With this method, the stress intensity factors at crack tips are determined by calculating the crack-opening displacements over the crack surface, and the crack expansion rate, which is related to the crack energy release rate, is estimated by using a technique based on self-similar (virtual) crack extension. For elements on the crack surface, regular integrals and singular integrals are evaluated based on closed-form expressions, which improves the accuracy. Examples show that this method yields very accurate results for stress intensity factors of penny-shaped cracks and elliptical cracks in the full space, with errors of less than one percent as compared with exact solutions. The stress intensity factors of subsurface cracks are in good agreement with other numerical solutions.

scholarly journals Fractional System of Korteweg-De Vries Equations via Elzaki Transform

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 673
Author(s):  
Wenfeng He ◽  
Nana Chen ◽  
Ioannis Dassios ◽  
Nehad Ali Shah ◽  
Jae Dong Chung
Keyword(s):  
Fractional Order ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Hybrid Technique ◽  
Transform Method ◽  
De Vries ◽  
In Series ◽  
Analytical Result ◽  
Approximate Result ◽  
Good Agreement

In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable.

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