On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equation

2017 ◽  
Vol 175 (3) ◽  
pp. 652-682
Author(s):  
Enrique Fernández-Cara ◽  
Dany Nina-Huamán ◽  
Miguel R. Nuñez-Chávez ◽  
Franciane B. Vieira
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Numerical Control ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
One Dimensional ◽  
Nonlinear Parabolic

Uniqueness of the null solution to a nonlinear partial differential equation satisfied by the explosion probability of a branching diffusion

2016 ◽  
Vol 53 (3) ◽  
pp. 938-945 ◽  
Author(s):  
K. Bruce Erickson
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Null Solution ◽  
Nonlinear Parabolic ◽  
Branching Diffusion ◽  
Creation Rate

AbstractThe explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

scholarly journals A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Wenyuan Liao
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Time Integration ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Rosenbrock Method ◽  
Ode System ◽  
Nonlinear Parabolic

The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.

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