scholarly journals Exact Solutions and Continuous Numerical Approximations of Coupled Systems of Diffusion Equations with Delay

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1560
Author(s):  
Elia Reyes ◽  
M. Ángeles Castro ◽  
Antonio Sirvent ◽  
Francisco Rodríguez
Keyword(s):  
Exact Solutions ◽  
Numerical Solutions ◽  
Separation Of Variables ◽  
Coupled Systems ◽  
Diffusion Equations ◽  
Series Solutions ◽  
Numerical Approximations ◽  
Initial Value ◽  
Mixed Problems ◽  
Equations With Delay

In this work, we obtain exact solutions and continuous numerical approximations for mixed problems of coupled systems of diffusion equations with delay. Using the method of separation of variables, and based on an explicit expression for the solution of the separated vector initial-value delay problem, we obtain exact infinite series solutions that can be truncated to provide analytical–numerical solutions with prescribed accuracy in bounded domains. Although usually implicit in particular applications, the method of separation of variables is deeply correlated with symmetry ideas.

Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions

2018 ◽  
Vol 39 (2) ◽  
pp. 607-625 ◽  
Author(s):  
Qiang Du ◽  
Yunzhe Tao ◽  
Xiaochuan Tian ◽  
Jiang Yang
Keyword(s):  
Green's Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Optimal Order ◽  
Numerical Approximations ◽  
Splitting Technique ◽  
Singular Measures ◽  
Nonlocal Models

AbstractNonlocal diffusion equations and their numerical approximations have attracted much attention in the literature as nonlocal modeling becomes popular in various applications. This paper continues the study of robust discretization schemes for the numerical solution of nonlocal models. In particular, we present quadrature-based finite difference approximations of some linear nonlocal diffusion equations in multidimensions. These approximations are able to preserve various nice properties of the nonlocal continuum models such as the maximum principle and they are shown to be asymptotically compatible in the sense that as the nonlocality vanishes, the numerical solutions can give consistent local limits. The approximation errors are proved to be of optimal order in both nonlocal and asymptotically local settings. The numerical schemes involve a unique design of quadrature weights that reflect the multidimensional nature and require technical estimates on nonconventional divided differences for their numerical analysis. We also study numerical approximations of nonlocal Green’s functions associated with nonlocal models. Unlike their local counterparts, nonlocal Green’s functions might become singular measures that are not well defined pointwise. We demonstrate how to combine a splitting technique with the asymptotically compatible schemes to provide effective numerical approximations of these singular measures.

scholarly journals Note on the Numerical Solutions of the General Matrix Convolution Equations by Using the Iterative Methods and Box Convolution Product

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Adem Kılıçman ◽  
Zeyad Al zhour
Keyword(s):  
Exact Solutions ◽  
Iterative Methods ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Convolution Product ◽  
Initial Value ◽  
General Matrix ◽  
Convolution Equations

We define the so-called box convolution product and study their properties in order to present the approximate solutions for the general coupled matrix convolution equations by using iterative methods. Furthermore, we prove that these solutions consistently converge to the exact solutions and independent of the initial value.

scholarly journals On the Long Time Simulation of Reaction-Diffusion Equations with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Dongfang Li ◽  
Chengjian Zhang
Keyword(s):  
Numerical Method ◽  
Numerical Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Open Problems ◽  
Time Simulation ◽  
Long Time ◽  
Continuous Problem ◽  
Equations With Delay

For a consistent numerical method to be practically useful, it is widely accepted that it must preserve the asymptotic stability of the original continuous problem. However, in this study, we show that it may lead to unreliable numerical solutions in long time simulation even if a classical numerical method gives a larger stability region than that of the original continuous problem. Some numerical experiments on the reaction-diffusion equations with delay are presented to confirm our findings. Finally, some open problems on the subject are proposed.

scholarly journals Conditional Lie-Bäcklund Symmetry Reductions and Exact Solutions of a Class of Reaction-Diffusion Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Xinyang Wang ◽  
Junquan Song
Keyword(s):  
Dynamical Systems ◽  
Exact Solutions ◽  
Separation Of Variables ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Financial Mathematics ◽  
Symmetry Reductions ◽  
Finite Dimensional ◽  
Wide Range

The method of conditional Lie-Bäcklund symmetry is applied to solve a class of reaction-diffusion equations ut+uxx+Qxux2+Pxu+Rx=0, which have wide range of applications in physics, engineering, chemistry, biology, and financial mathematics theory. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamical systems. The exact solutions obtained in concrete examples possess the extended forms of the separation of variables.

scholarly journals Accuracy Analysis on Solution of Initial Value Problems of Ordinary Differential Equations for Some Numerical Methods with Different Step Sizes

2021 ◽  
Vol 10 (1) ◽  
pp. 118-133
Author(s):  
Mohammad Asif Arefin ◽  
Biswajit Gain ◽  
Rezaul Karim
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Numerical Solutions ◽  
Initial Value ◽  
Step Size ◽  
Runge Kutta Method ◽  
Efficiency Comparison

In this article, three numerical methods namely Euler’s, Modified Euler, and Runge-Kutta method have been discussed, to solve the initial value problem of ordinary differential equations. The main goal of this research paper is to find out the accurate results of the initial value problem (IVP) of ordinary differential equations (ODE) by applying the proposed methods. To achieve this goal, solutions of some IVPs of ODEs have been done with the different step sizes by using the proposed three methods, and solutions for each step size are analyzed very sharply. To ensure the accuracy of the proposed methods and to determine the accurate results, numerical solutions are compared with the exact solutions. It is observed that numerical solutions are best fitted with exact solutions when the taken step size is very much small. Consequently, all the proposed three methods are quite efficient and accurate for solving the IVPs of ODEs. Error estimation plays a significant role in the establishment of a comparison among the proposed three methods. On the subject of accuracy and efficiency, comparison is successfully implemented among the proposed three methods.

Application of Differential Transformation Method for Numerical Computation of Regularized LongWave Equation

2012 ◽  
Vol 67 (3-4) ◽  
pp. 160-166 ◽  
Author(s):  
Babak Soltanalizadeh ◽  
Ahmet Yildirim
Keyword(s):  
Exact Solutions ◽  
Numerical Computation ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Series Solutions ◽  
Numerical Approximations ◽  
Differential Transformation ◽  
Long Wave ◽  
Rlw Equation

In this article, the differential transformation method (DTM) is utilized for finding the solution of the regularized long wave (RLW) equation. Not only the exact solutions have been achieved by the known forms of the series solutions, but also for the finite terms of series, and the corresponding numerical approximations have been computed

Sign in / Sign up

Export Citation Format

Share Document