Stabilization and uncertainty analysis of a time‐fractional reaction diffusion equation cascaded with a time‐fractional hyperbolic partial differential equation

2021 ◽  
Author(s):  
Juan Chen ◽  
Bo Zhuang ◽  
Aleksei Tepljakov ◽  
Eduard Petlenkov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Diffusion Equation ◽  
Uncertainty Analysis ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Hyperbolic Partial Differential Equation ◽  
Partial Differential ◽  
Fractional Reaction

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.

scholarly journals Numerical Analysis WSGD Scheme for One- and Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation with Collocation Method via Fractional B-Spline

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Mohammad Ramezani
Keyword(s):  
Diffusion Equation ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
B Spline ◽  
Computational Work ◽  
Distributed Order ◽  
Fractional Reaction

AbstractThe main propose of this paper is presenting an efficient numerical scheme to solve WSGD scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation. The proposed method is based on fractional B-spline basics in collocation method which involve Caputo-type fractional derivatives for $$0 < \alpha < 1$$ 0 < α < 1 . The most significant privilege of proposed method is efficient and quite accurate and it requires relatively less computational work. The solution of consideration problem is transmute to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. The finally, several numerical WSGD Scheme for one- and two-dimensional distributed order fractional reaction–diffusion equation.

scholarly journals Pemodelan Tsunami Sederhana dengan Menggunakan Persamaan Differensial Parsial

2018 ◽  
Vol 8 (1) ◽  
pp. 26
Author(s):  
Indriati Retno Palupi ◽  
Wiji Raharjo ◽  
Eko Wibowo ◽  
Hafiz Hamdalah
Keyword(s):  
Differential Equation ◽  
Fluid Dynamics ◽  
Partial Differential Equation ◽  
Wave Propagation ◽  
Taylor Expansion ◽  
Tsunami Wave ◽  
Time Function ◽  
Hyperbolic Partial Differential Equation ◽  
Partial Differential ◽  
Space And Time

One way to solve fluid dynamics problem is using partial differential equation. By using Taylor expansion, fluid dynamics can be applied simply. For the example is tsunami wave. It is include to hyperbolic partial differential equation, tsunami wave propagation can describe in space and time function by using Euler FTCS (Forward Time Central Space) formula.

scholarly journals A Numerical Method for Time-Fractional Reaction-Diffusion and Integro Reaction-Diffusion Equation Based on Quasi-Wavelet

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Sachin Kumar ◽  
Jinde Cao ◽  
Xiaodi Li
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Research Work ◽  
Reaction Diffusion ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Fractional Reaction

In this research work, we focused on finding the numerical solution of time-fractional reaction-diffusion and another class of integro-differential equation known as the integro reaction-diffusion equation. For this, we developed a numerical scheme with the help of quasi-wavelets. The fractional term in the time direction is approximated by using the Crank–Nicolson scheme. The spatial term and the integral term present in integro reaction-diffusion are discretized and approximated with the help of quasi-wavelets. We study this model with Dirichlet boundary conditions. The discretization of these initial and boundary conditions is done with a different approach by the quasi-wavelet-based numerical method. The validity of this proposed method is tested by taking some numerical examples having an exact analytical solution. The accuracy of this method can be seen by error tables which we have drawn between the exact solution and the approximate solution. The effectiveness and validity can be seen by the graphs of the exact and numerical solutions. We conclude that this method has the desired accuracy and has a distinctive local property.

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