scholarly journals Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

2013 ◽  
Vol 26 (3) ◽  
pp. 315-317 ◽  
Author(s):  
E. Mamontov
Keyword(s):  
Cauchy Problem ◽  
Analytical Solution ◽  
Diffusion Equation ◽  
Source Term ◽  
Exact Analytical Solution ◽  
Reaction Diffusion ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Linear Reaction ◽  
The Cauchy Problem

Emergence of Waves in a Nonlinear Convection-Reaction-Diffusion Equation

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Shoshana Kamin ◽  
Philip Rosenau
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Diffusion Equation ◽  
Travelling Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Nonlinear Convection ◽  
The Cauchy Problem ◽  
The Right

AbstractIn this work we prove that for some class of initial data the solution of the Cauchy problemuu(0; x) = uapproaches the travelling solution, spreading either to the right or to the left, or two travelling waves moving in opposite directions.

On the Hybrid of Fourier Transform and Adomian Decomposition Method for the Solution of Nonlinear Cauchy Problems of the Reaction-Diffusion Equation

2012 ◽  
Vol 67 (6-7) ◽  
pp. 355-362 ◽  
Author(s):  
Salman Nourazar ◽  
Akbar Nazari-Golshan ◽  
Ahmet Yıldırım ◽  
Maryam Nourazar
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Fourier Transform ◽  
Diffusion Equation ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Adomian Decomposition ◽  
The Cauchy Problem

The physical science importance of the Cauchy problem of the reaction-diffusion equation appears in the modelling of a wide variety of nonlinear systems in physics, chemistry, ecology, biology, and engineering. A hybrid of Fourier transform and Adomian decomposition method (FTADM) is developed for solving the nonlinear non-homogeneous partial differential equations of the Cauchy problem of reaction-diffusion. The results of the FTADM and the ADM are compared with the exact solution. The comparison reveals that for the same components of the recursive sequences, the errors associated with the FTADM are much lesser than those of the ADM. We show that as time increases the results of the FTADM approaches 1 with only six recursive terms. This is in agreement with the physical property of the density-dependent nonlinear diffusion of the Cauchy problem which is also in agreement with the exact solution. The monotonic and very rapid convergence of the results of the FTADM towards the exact solution is shown to be much faster than that of the ADM

scholarly journals About the Structure of Attractors for a Nonlocal Chafee-Infante Problem

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 353
Author(s):  
Rubén Caballero ◽  
Alexandre N. Carvalho ◽  
Pedro Marín-Rubio ◽  
José Valero
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Fixed Points ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Stationary Points ◽  
The Cauchy Problem ◽  
The Stability ◽  
Nonlocal Reaction ◽  
Multivalued Semiflow

In this paper, we study the structure of the global attractor for the multivalued semiflow generated by a nonlocal reaction-diffusion equation in which we cannot guarantee the uniqueness of the Cauchy problem. First, we analyse the existence and properties of stationary points, showing that the problem undergoes the same cascade of bifurcations as in the Chafee-Infante equation. Second, we study the stability of the fixed points and establish that the semiflow is a dynamic gradient. We prove that the attractor consists of the stationary points and their heteroclinic connections and analyse some of the possible connections.

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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