scholarly journals An ancient Chinese algorithm for two-point boundary problems and its application to the Michaelis-Menten kinetics

2021 ◽  
Vol 1 (4) ◽  
pp. 172-176
Author(s):  
Ji-Huan He ◽  
◽  
Shuai-Jia Kou ◽  
Hamid M. Sedighi ◽  
◽  
...  
Keyword(s):  
Infinite Series ◽  
Solution Process ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Initial Condition ◽  
Point Boundary ◽  
Boundary Problems ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation ◽  
Taylor Series Method

<abstract><p>Taylor series method is simple, and an infinite series converges to the exact solution for initial condition problems. For the two-point boundary problems, the infinite series has to be truncated to incorporate the boundary conditions, making it restrictively applicable. Here is recommended an ancient Chinese algorithm called as <italic>Ying Buzu Shu</italic>, and a nonlinear reaction diffusion equation with a Michaelis-Menten potential is used as an example to show the solution process.</p></abstract>

scholarly journals Some New Traveling Wave Solutions of the Nonlinear Reaction Diffusion Equation by Using the Improved (G′/G)-Expansion Method

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah
Keyword(s):  
Diffusion Equation ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Expansion Method ◽  
Reaction Diffusion Equation ◽  
Wave Solutions ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation ◽  
Constant Coefficients

We construct new exact traveling wave solutions involving free parameters of the nonlinear reaction diffusion equation by using the improved (G′/G)-expansion method. The second-order linear ordinary differential equation with constant coefficients is used in this method. The obtained solutions are presented by the hyperbolic and the trigonometric functions. The solutions become in special functional form when the parameters take particular values. It is important to reveal that our solutions are in good agreement with the existing results.

1-Soliton Solution of the Nonlinear Reaction-Diffusion Equation

2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Anjan Biswas
Keyword(s):  
Solitary Wave ◽  
Diffusion Equation ◽  
Soliton Solution ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation

In this paper, the 1-soliton solution of the nonlinear reaction-diffusion equation is obtained. The solitary wave ansatz is employed to obtain the solution.

Stability and Bifurcations in a Model of Phase Transitions With Order Parameter

1997 ◽  
Author(s):  
Janusz Sikora ◽  
Joseph P. Cusumano ◽  
William A. Jester
Keyword(s):  
Phase Transitions ◽  
Phase Plane ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Governing Equations ◽  
Nonlinear Reaction ◽  
Nonlinear Reaction Diffusion Equation ◽  
Spatially Periodic ◽  
Stability And Bifurcations ◽  
Special Case

Abstract A one-dimensional model of phase transitions with convex strain energy is investigated within the limits of nonlinear bar theory. The model is a special case of a coupled field theory that has been developed by Fried and Gurtin to study the nucleation and propagation of phase boundaries. The system of governing equations studied here consists of a wave equation coupled to a nonlinear reaction-diffusion equation. Using phase plane methods, the equilibria of the system have been constructed in order to obtain the macroscopic response of the bar and the bifurcation diagram. It is demonstrated that a large number of coexisting spatially periodic, inhomogeneous solutions can occur, with the number of these solutions being inversely proportional to the diffusion coefficient in the reaction-diffusion subsystem. A stability analysis of the equilibria is presented.

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