The Modified Localized Method of Approximated Particular Solutions for Linear and Nonlinear Convection-Diffusion-Reaction PDEs

2020 ◽  
Vol 12 (5) ◽  
pp. 1113-1136
Author(s):  
global sci
Keyword(s):  
Diffusion Reaction ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Particular Solutions ◽  
Linear And Nonlinear

scholarly journals Hermite Method of Approximate Particular Solutions for Solving Time-Dependent Convection-Diffusion-Reaction Problems

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 188
Author(s):  
Jen-Yi Chang ◽  
Ru-Yun Chen ◽  
Chia-Cheng Tsai
Keyword(s):  
Shape Parameter ◽  
Source Term ◽  
Time Dependent ◽  
Diffusion Reaction ◽  
System Matrix ◽  
Time Step ◽  
Convection Diffusion ◽  
Particular Solutions ◽  
Particular Solution ◽  
Reaction Problem

This article describes the development of the Hermite method of approximate particular solutions (MAPS) to solve time-dependent convection-diffusion-reaction problems. Using the Crank-Nicholson or the Adams-Moulton method, the time-dependent convection-diffusion-reaction problem is converted into time-independent convection-diffusion-reaction problems for consequent time steps. At each time step, the source term of the time-independent convection-diffusion-reaction problem is approximated by the multiquadric (MQ) particular solution of the biharmonic operator. This is inspired by the Hermite radial basis function collocation method (RBFCM) and traditional MAPS. Therefore, the resultant system matrix is symmetric. Comparisons are made for the solutions of the traditional/Hermite MAPS and RBFCM. The results demonstrate that the Hermite MAPS is the most accurate and stable one for the shape parameter. Finally, the proposed method is applied for solving a nonlinear time-dependent convection-diffusion-reaction problem.

Optimization analysis of the inverse coefficient problem for the nonlinear convection-diffusion-reaction equation

2018 ◽  
Vol 26 (6) ◽  
pp. 821-833 ◽  
Author(s):  
Roman V. Brizitskii ◽  
Zhanna Y. Saritskaya
Keyword(s):  
Boundary Value ◽  
Diffusion Reaction ◽  
Stability Estimates ◽  
Coefficient Problem ◽  
Reaction Equation ◽  
Nonlinear Convection ◽  
Inverse Coefficient Problem ◽  
Convection Diffusion ◽  
Reaction Coefficient ◽  
Inverse Coefficient

AbstractThe inverse coefficient problem for the nonlinear convection-diffusion-reaction equation is considered. A velocity vector and a mass-transfer coefficient are considered as the unknown coefficients and are recovered with the help of the additional information about the boundary value problem’s solution. The inverse coefficient problem is reduced to a two-parameter problem of multiplicative control, the solvability of which is proved in a general form. For a cubic reaction coefficient the local stability estimates of the control problem’s solutions are obtained regarding to a rather small perturbation of either the cost functional or the specified functions of the boundary value problem.

scholarly journals Implicit-Explicit Methods for a Convection-Diffusion-Reaction Model of the Propagation of Forest Fires

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1034 ◽  
Author(s):  
Raimund Bürger ◽  
Elvis Gavilán ◽  
Daniel Inzunza ◽  
Pep Mulet ◽  
Luis Miguel Villada
Keyword(s):  
Forest Fires ◽  
Control Strategies ◽  
Reaction Diffusion ◽  
Reaction Model ◽  
Diffusion Reaction ◽  
Nonlinear Convection ◽  
Splitting Technique ◽  
Convection Diffusion ◽  
Fuel Density ◽  
Diffusion Convection

Numerical techniques for approximate solution of a system of reaction-diffusion-convection partial differential equations modeling the evolution of temperature and fuel density in a wildfire are proposed. These schemes combine linearly implicit-explicit Runge–Kutta (IMEX-RK) methods and Strang-type splitting technique to adequately handle the non-linear parabolic term and the stiffness in the reactive part. Weighted essentially non-oscillatory (WENO) reconstructions are applied to the discretization of the nonlinear convection term. Examples are focused on the applicative problem of determining the width of a firebreak to prevent the propagation of forest fires. Results illustrate that the model and numerical scheme provide an effective tool for defining that width and the parameters for control strategies of wildland fires.

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