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.

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.

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.

Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
İhsan Çelikkaya
Keyword(s):  
Solitary Wave ◽  
Boundary Conditions ◽  
Numerical Solutions ◽  
Test Problems ◽  
Von Neumann ◽  
Nonhomogeneous Boundary ◽  
Whitham Equations ◽  
Fourier Series Method ◽  
Nonhomogeneous Boundary Conditions ◽  
The Stability

Abstract In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L 2 and L ∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical scheme.

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.

Approximation of 3D convection diffusion equation using DQM based on modified cubic trigonometric B-splines

2020 ◽  
pp. 1-10
Author(s):  
Mohammad Tamsir ◽  
Neeraj Dhiman ◽  
F.S. Gill ◽  
Robin
Keyword(s):  
Diffusion Equation ◽  
Numerical Solutions ◽  
Basis Functions ◽  
B Spline ◽  
Convection Diffusion ◽  
B Splines ◽  
Convection Diffusion Equation ◽  
First Order ◽  
The Stability ◽  
Derivatives Of

This paper presents an approximate solution of 3D convection diffusion equation (CDE) using DQM based on modified cubic trigonometric B-spline (CTB) basis functions. The DQM based on CTB basis functions are used to integrate the derivatives of space variables which transformed the CDE into the system of first order ODEs. The resultant system of ODEs is solved using SSPRK (5,4) method. The solutions are approximated numerically and also presented graphically. The accuracy and efficiency of the method is validated by comparing the solutions with existing numerical solutions. The stability analysis of the method is also carried out.

scholarly journals Constructions of the soliton solutions to the good Boussinesq equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mohammed Bakheet Almatrafi ◽  
Abdulghani Ragaa Alharbi ◽  
Cemil Tunç
Keyword(s):  
Traveling Wave ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Solution Stability ◽  
Exact Results ◽  
The Stability ◽  
Stability And Accuracy

Abstract The principal objective of the present paper is to manifest the exact traveling wave and numerical solutions of the good Boussinesq (GB) equation by employing He’s semiinverse process and moving mesh approaches. We present the achieved exact results in the form of hyperbolic trigonometric functions. We test the stability of the exact results. We discretize the GB equation using the finite-difference method. We also investigate the accuracy and stability of the used numerical scheme. We sketch some 2D and 3D surfaces for some recorded results. We theoretically and graphically report numerical comparisons with exact traveling wave solutions. We measure the $L_{2}$ L 2  error to show the accuracy of the used numerical technique. We can conclude that the novel techniques deliver improved solution stability and accuracy. They are reliable and effective in extracting some new soliton solutions for some nonlinear partial differential equations (NLPDEs).

scholarly journals Numerical approach for solving time fractional diffusion equation

2017 ◽  
Vol 7 (3) ◽  
pp. 281-287
Author(s):  
Dilara Altan Koç ◽  
Mustafa Gülsu
Keyword(s):  
Stability Analysis ◽  
Numerical Solutions ◽  
Numerical Approach ◽  
Fractional Diffusion ◽  
Fractional Partial Differential Equations ◽  
Von Neumann ◽  
Liouville Derivative ◽  
Time Fractional Diffusion Equation ◽  
The Stability ◽  
Space Method

In this article one of the fractional partial differential equations was solved by finite difference scheme  based on five point and three point central space method with discretization in time. We use between the Caputo and the Riemann-Liouville derivative definition and the Grünwald-Letnikov operator for the fractional calculus. The stability analysis of this scheme is examined by using von-Neumann method. A comparison between exact solutions and numerical solutions is made. Some figures and tables are included.

Analytical properties and numerical solutions of the derivative nonlinear Schrödinger equation

1988 ◽  
Vol 40 (3) ◽  
pp. 585-602 ◽  
Author(s):  
Silvina Ponce Dawson ◽  
Constantino Ferro Fontán
Keyword(s):  
Numerical Solutions ◽  
Temporal Evolution ◽  
Soliton Solutions ◽  
Nonlinear Schrödinger ◽  
Analytical Properties ◽  
Derivative Nonlinear Schrödinger Equation ◽  
The Stability ◽  
A Charge ◽  
The Relationship ◽  
Dnls Equation

From the analysis of the symmetries of the derivative nonlinear Schrödinger (DNLS) equation, we obtain a new constant of motion, which may be formally considered as a charge and which is related to the helicity of the physical System. From comparison of these symmetries and those of the soliton solutions, we draw conclusions about the number of constraints that must be imposed and the way a Liapunov functional must be constructed in order to study the solitons' stability. We also examine the relationship between the stability with respect to form and the symmetries that are broken by the soliton solutions. We complete the analysis with some numerical simulations: we solve the DNLS equation taking a slightly perturbed soliton as an initial condition and study its temporal evolution, finding that, as expected, they are stable with respect to form.

scholarly journals Análisis de Von Neumann para el métodoLocal Discontinuous Galerkin en 1D

2019 ◽  
Vol 37 (2) ◽  
pp. 199-217
Author(s):  
Paul Castillo ◽  
Sergio Gómez
Keyword(s):  
Discontinuous Galerkin ◽  
Numerical Experiments ◽  
High Order ◽  
Stability Conditions ◽  
Von Neumann ◽  
Theoretical Tool ◽  
Von Neumann Analysis ◽  
Time Marching ◽  
The Stability ◽  
Theoretical Results

Using the von Neumann analysis as a theoretical tool, an analysisof the stability conditions of some explicit time marching schemes, in com-bination with the spatial discretizationLocal Discontinuous Galerkin(LDG)and high order approximations, is presented. The stabilityconstant, CFL(Courant-Friedrichs-Lewy), is studied as a function of theLDG parametersand the approximation degree. A series of numerical experiments is carriedout to validate the theoretical results.

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.

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