Singly diagonally implicit runge-kutta method for time-dependent reaction-diffusion equation

2010 ◽  
Vol 27 (6) ◽  
pp. 1423-1441 ◽  
Author(s):  
Wenyuan Liao ◽  
Yulian Yan
Keyword(s):  
Diffusion Equation ◽  
Reaction Diffusion ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta

Modeling on Falling Velocity of Sodiumtetraborate Aqueous Solution Drops before the Gelation of PVA-TiO2 Suspensions by the Runge-Kutta Method in Matlab 6.5

2008 ◽  
Vol 368-372 ◽  
pp. 1683-1685
Author(s):  
Cheng Long Yu ◽  
Xiu Feng Wang ◽  
Jun Xin Zhou ◽  
Hong Tao Jiang ◽  
Yan Wang
Keyword(s):  
Differential Equation ◽  
Aqueous Solution ◽  
Ordinary Differential Equation ◽  
Elevation Angle ◽  
Time Dependent ◽  
Quadratic Term ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Air Resistance ◽  
Runge Kutta

Numerical modeling on falling of sodiumtetraborate aqueous solution drops as the initiator before the gelation of PVA-TiO2 suspensions was conducted. Effect of time and elevation angle of the PVA-TiO2 suspensions on the falling velocity of the sodiumtetraborate aqueous solution drops was analyzed. An ordinary differential equation was given. Integration of the ordinary differential equation was fulfilled using the fourth-order Runge-Kutta method in Matlab 6.5. From the model, a two-order nonlinear effect of time on the velocity of the drops during falling is determined and the quadratic term -3.408t2 serves as the time dependent air resistance. The component of the falling velocity along the suspensions increases with the increasing of the elevation angle. However, for the component vertical to the suspensions, with elevation angle increasing, it decreases.

scholarly journals Gradient Optimal Control of the Bilinear Reaction–Diffusion Equation

2021 ◽  
Author(s):  
El Hassan Zerrik ◽  
Abderrahman Ait Aadi
Keyword(s):  
Optimal Control ◽  
Diffusion Equation ◽  
Minimization Problem ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Energy Term ◽  
Bounded Controls ◽  
Special Cases

In this chapter, we study a problem of gradient optimal control for a bilinear reaction–diffusion equation evolving in a spatial domain Ω⊂Rn using distributed and bounded controls. Then, we minimize a functional constituted of the deviation between the desired gradient and the reached one and the energy term. We prove the existence of an optimal control solution of the minimization problem. Then this control is characterized as solution to an optimality system. Moreover, we discuss two special cases of controls: the ones are time dependent, and the others are space dependent. A numerical approach is given and successfully illustrated by simulations.

New exact solutions of the reaction–diffusion equation with variable coefficients via the mathematical computation

2018 ◽  
Vol 11 (04) ◽  
pp. 1850051 ◽  
Author(s):  
Jin Hyuk Choi ◽  
Hyunsoo Kim
Keyword(s):  
Diffusion Equation ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Painlevé Test ◽  
New Exact Solutions ◽  
Mathematical Computation

In this paper, we construct new exact solutions of the reaction–diffusion equation with time dependent variable coefficients by employing the mathematical computation via the Painlevé test. We describe the behaviors and their interactions of the obtained solutions under certain constraints and various variable coefficients.

Molecular Motion through a Fluctuating Bottleneck

1994 ◽  
Vol 366 ◽  
Author(s):  
N. Eizenberg ◽  
J. Klafter
Keyword(s):  
Steady State ◽  
Diffusion Equation ◽  
Molecular Motion ◽  
Reaction Diffusion ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Langevin Equations ◽  
Molecular Pathways

ABSTRACTMolecular motion in a series of cavities dominated by time dependent bottlenecks is studied as a model for molecular pathways in biomolecules. The problem is formulated by coupled rate and Langevin equations and is shown to be equivalent to n-dimensional reaction-diffusion equation where n is the number of cavities visited by the molecules. Results are presented for two cavities and a comparison is made between steady state and non steady state results.

scholarly journals Lie Group Method of the Diffusion Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jian-Qiang Sun ◽  
Rong-Fang Huang ◽  
Xiao-Yan Gu ◽  
Ling Yu
Keyword(s):  
Differential Equations ◽  
Diffusion Equation ◽  
Ordinary Differential Equations ◽  
Lie Group ◽  
Diffusion Equations ◽  
Group Method ◽  
Kutta Method ◽  
Lie Group Method ◽  
Runge Kutta Method ◽  
Runge Kutta

The diffusion equation is discretized in spacial direction and transformed into the ordinary differential equations. The ordinary differential equations are solved by Lie group method and the explicit Runge-Kutta method. Numerical results showed that Lie group method is more stable than the corresponding explicit Runge-Kutta method.

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