scholarly journals Comparative Study of Some Numerical Methods for the Burgers–Huxley Equation

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1333 ◽  
Author(s):  
Appanah Appadu ◽  
Bilge İnan ◽  
Yusuf Tijani
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Fully Implicit ◽  
Huxley Equation ◽  
Relative Errors ◽  
First Time

In this paper, we construct four numerical methods to solve the Burgers–Huxley equation with specified initial and boundary conditions. The four methods are two novel versions of nonstandard finite difference schemes (NSFD1 and NSFD2), explicit exponential finite difference method (EEFDM) and fully implicit exponential finite difference method (FIEFDM). These two classes of numerical methods are popular in the mathematical biology community and it is the first time that such a comparison is made between nonstandard and exponential finite difference schemes. Moreover, the use of both nonstandard and exponential finite difference schemes are very new for the Burgers–Huxley equations. We considered eleven different combination for the parameters controlling diffusion, advection and reaction, which give rise to four different regimes. We obtained stability region or condition for positivity. The performances of the four methods are analysed by computing absolute errors, relative errors, L 1 and L ∞ errors and CPU time.

scholarly journals A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems

2021 ◽  
Vol 8 (1) ◽  
pp. 1-11
Author(s):  
Abdul Abner Lugo Jiménez ◽  
Guelvis Enrique Mata Díaz ◽  
Bladismir Ruiz
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Stationary Regime ◽  
Heat Transfer Fluid ◽  
Mimetic Finite Difference Method ◽  
Linear Differential

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

Characteristics of finite difference methods for dispersive shallow water equations

2016 ◽  
Vol 31 (3) ◽  
Author(s):  
Zinaida I. Fedotova ◽  
Gayaz S. Khakimzyanov
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Finite Difference Methods ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hydrodynamic Equations ◽  
Difference Methods

AbstractThe paper contains a description of the most important properties of numerical methods for solving nonlinear dispersive hydrodynamic equations and their distinctions from similar properties of finite difference schemes approximating classic dispersion-free shallow water equations.

scholarly journals Analytical and Numerical Methods for the CMKdV-II Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Ömer Akin ◽  
Ersin Özuğurlu
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Iterative Method ◽  
Soliton Solution ◽  
Bilinear Form ◽  
Difference Method ◽  
Soliton Solutions ◽  
Hirota’S Bilinear Form ◽  
De Vries

Hirota's bilinear form for the Complex Modified Korteweg-de Vries-II equation (CMKdV-II) is derived. We obtain one- and two-soliton solutions analytically for the CMKdV-II. One-soliton solution of the CMKdV-II equation is obtained by using finite difference method by implementing an iterative method.

scholarly journals Computational Analysis of the Stability of 2D Heat Equation on Elliptical Domain Using Finite Difference Method

2020 ◽  
pp. 8-19
Author(s):  
Mehwish Naz Rajput ◽  
Asif Ali Shaikh ◽  
Shakeel Ahmed Kamboh
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Heat Equation ◽  
Stability Condition ◽  
Difference Method ◽  
Stable Solution ◽  
Finite Difference Schemes ◽  
Step Size ◽  
Stability Range ◽  
The Stability

Aims: The aim and objective of the study to derive and analyze the stability of the finite difference schemes in relation to the irregularity of domain. Study Design: First of all, an elliptical domain has been constructed with the governing two dimensional (2D) heat equation that is discretized using the Finite Difference Method (FDM). Then the stability condition has been defined and the numerical solution by writing MATLAB codes has been obtained with the stable values of time domain. Place and Duration of Study: The work has been jointly conducted at the MUET, Jamshoro and QUEST, Nawabshah Pakistan from January 2019 to December 2019.  Methodology: The stability condition over an elliptical domain with the non-uniform step size depending upon the boundary tracing function is derived by using Von Neumann method. Results: From the results it was revealed that stability region for the small number of mesh points remains larger and gets smaller as the number of mesh nodes is increased. Moreover, the ranges for the time steps are defined for varied spatial step sizes that help to find the stable solution. Conclusion: The corresponding stability range for number of nodes N=10, 20, 30, 40, 50, and 60 was found respectively. Within this range the solution remains smooth as time increases. The results of this study attempt to provide the stable solution of partial differential equations on irregular domains.

scholarly journals New schemes for a two-dimensional inverse problem with temperature overspecification

2001 ◽  
Vol 7 (3) ◽  
pp. 283-297 ◽  
Author(s):  
Mehdi Dehghan
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Central Processor ◽  
Two Dimensional ◽  
Alternating Direction Implicit ◽  
Fully Implicit ◽  
Implicit Schemes ◽  
Alternating Direction

Two different finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the (3,3) alternating direction implicit (ADI) finite difference scheme and the (3,9) alternating direction implicit formula. These schemes are unconditionally stable. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett [17]. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit schemes for two-dimensional diffusion with temperature overspecification. The alternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their unconditional stability is significant. The results of numerical experiments are presented, and accuracy and the Central Processor (CPU) times needed for each of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.

scholarly journals Iterative Methods for Obtaining Energy-Minimizing Parametric Snakes with Applications to Medical Imaging

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Alexandru Ioan Mitrea ◽  
Radu Badea ◽  
Delia Mitrea ◽  
Sergiu Nedevschi ◽  
Paulina Mitrea ◽  
...  
Keyword(s):  
Medical Imaging ◽  
Numerical Methods ◽  
Finite Difference ◽  
Iterative Method ◽  
Iterative Methods ◽  
Approximation Error ◽  
Deformable Models ◽  
Difference Schemes ◽  
Finite Difference Schemes

After a brief survey on the parametric deformable models, we develop an iterative method based on the finite difference schemes in order to obtain energy-minimizing snakes. We estimate the approximation error, the residue, and the truncature error related to the corresponding algorithm, then we discuss its convergence, consistency, and stability. Some aspects regarding the prosthetic sugical methods that implement the above numerical methods are also pointed out.

scholarly journals Numerical Approximation of Generalized Burger’s-Fisher and Generalized Burger’s-Huxley Equation by Compact Finite Difference Method

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Ravneet Kaur ◽  
Shallu ◽  
Sachin Kumar ◽  
V. K. Kukreja
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Time Integration ◽  
Strong Stability ◽  
Coefficient Matrix ◽  
Computational Effort ◽  
Compact Finite Difference ◽  
Compact Finite Difference Method ◽  
Huxley Equation

In this work, computational analysis of generalized Burger’s-Fisher and generalized Burger’s-Huxley equation is carried out using the sixth-order compact finite difference method. This technique deals with the nonstandard discretization of the spatial derivatives and optimized time integration using the strong stability-preserving Runge-Kutta method. This scheme inculcates four stages and third-order accuracy in the time domain. The stability analysis is discussed using eigenvalues of the coefficient matrix. Several examples are discussed for their approximate solution, and comparisons are made to show the efficiency and accuracy of CFDM6 with the results available in the literature. It is found that the present method is easy to implement with less computational effort and is highly accurate also.

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