scholarly journals Analytic Approximation of Solutions of Parabolic Partial Differential Equations with Variable Coefficients

2017 ◽  
Vol 2017 ◽  
pp. 1-5 ◽  
Author(s):  
Vladislav V. Kravchenko ◽  
Josafath A. Otero ◽  
Sergii M. Torba
Keyword(s):  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Cauchy Data ◽  
Variable Coefficients ◽  
Dirichlet Boundary Conditions ◽  
Reaction Diffusion Equation ◽  
Analytic Approximation ◽  
Transmutation Operator ◽  
Initial Boundary Value

A complete family of solutions for the one-dimensional reaction-diffusion equation, uxx(x,t)-q(x)u(x,t)=ut(x,t), with a coefficient q depending on x is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper, the Dirichlet boundary conditions are considered; however, the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.

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.

scholarly journals Identifying an Unknown Coefficient in the Reaction-Diffusion Equation Using He’s VIM

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
F. Parzlivand ◽  
A. M. Shahrezaee
Keyword(s):  
Unknown Parameter ◽  
Reaction Diffusion ◽  
Variational Iteration Method ◽  
Main Property ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equation ◽  
Unknown Coefficient ◽  
Initial Boundary Value ◽  
Inverse Heat Problem

An inverse heat problem of finding an unknown parameter p(t) in the parabolic initial-boundary value problem is solved with variational iteration method (VIM). For solving the discussed inverse problem, at first we transform it into a nonlinear direct problem and then use the proposed method. Also an error analysis is presented for the method and prior and posterior error bounds of the approximate solution are estimated. The main property of the method is in its flexibility and ability to solve nonlinear equation accurately and conveniently. Some examples are given to illustrate the effectiveness and convenience of the method.

scholarly journals Maximum Principles for Discrete and Semidiscrete Reaction-Diffusion Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Petr Stehlík ◽  
Jonáš Volek
Keyword(s):  
Maximum Principle ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Maximum Principles ◽  
Time Step ◽  
Reaction Diffusion Model ◽  
Continuous Reaction ◽  
Initial Boundary Value

We study reaction-diffusion equations with a general reaction functionfon one-dimensional lattices with continuous or discrete timeux′  (or  Δtux)=k(ux-1-2ux+ux+1)+f(ux),x∈Z. We prove weak and strong maximum and minimum principles for corresponding initial-boundary value problems. Whereas the maximum principles in the semidiscrete case (continuous time) exhibit similar features to those of fully continuous reaction-diffusion model, in the discrete case the weak maximum principle holds for a smaller class of functions and the strong maximum principle is valid in a weaker sense. We describe in detail how the validity of maximum principles depends on the nonlinearity and the time step. We illustrate our results on the Nagumo equation with the bistable nonlinearity.

scholarly journals A Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Alexander Gladkov ◽  
Alexandr Nikitin
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Local Existence ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Initial Boundary Value

We consider initial boundary value problem for a reaction-diffusion system with nonlinear and nonlocal boundary conditions and nonnegative initial data. We prove local existence, uniqueness, and nonuniqueness of solutions.

GRID APPROXIMATION OF SINGULARLY PERTURBED PARABOLIC REACTION‐DIFFUSION EQUATIONS WITH PIECEWISE SMOOTH INITIAL‐BOUNDARY CONDITIONS

2007 ◽  
Vol 12 (2) ◽  
pp. 235-254 ◽  
Author(s):  
Grigorii Shishkin
Keyword(s):  
Boundary Conditions ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Rectangular Domain ◽  
Piecewise Smooth ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Lateral Part ◽  
Boundary And Interior Layers

A Dirichlet problem is considered for a singularly perturbed parabolic reaction–diffusion equation with piecewise smooth initial‐boundary conditions on a rectangular domain. The higher‐order derivative in the equation is multiplied by a parameter ϵ 2; ɛ ϵ (0, 1]. For small values of ϵ, a boundary and an interior layer arises, respectively, in a neighbourhood of the lateral part of the boundary and in a neighbourhood of the characteristic of the reduced equation passing through the point of nonsmoothness of the initial function. Using the method of special grids condensing either in a neighbourhood of the boundary layer or in neighbourhoods of the boundary and interior layers, special ϵ‐uniformly convergent difference schemes are constructed and investigated. It is shown that the convergence rate of the schemes crucially depends on the type of nonsmoothness in the initial–boundary conditions.

On a singular initial-boundary-value problem for a reaction-diffusion equation arising from a simple model of isothermal chemical autocatalysis

1992 ◽  
Vol 437 (1901) ◽  
pp. 657-671 ◽  
Keyword(s):  
Simple Model ◽  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Rate Constant ◽  
Boundary Value ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reaction Diffusion Equation ◽  
Initial Boundary Value

In this paper we examine the evolution that occurs when a localized input of an autocatalyst B is introduced into an expanse of a reactant A. The reaction is autocatalytic of order p,so A -> B at rate k [A] [B] p with rate constant k . We examine the case when 0 < p < 1, with p>/ 1 having been examined by Needham & Merkin (Phil. Trans. R. Soc. Lond. A 337, 261—274 (1991)). In particular, we show that the fully reacted state is not achieved (as t-> oo) via the propagation of a travelling wavefront (as for p>/ 1) but is approached uniformly in space as t-00.

scholarly journals The dirichlet boundary conditions and related natural boundary conditions in strengthened sobolev spaces for discretized parabolic problems

2000 ◽  
Vol 4 (4) ◽  
pp. 269-281
Author(s):  
E. D'Yakonov
Keyword(s):  
Boundary Conditions ◽  
Sobolev Spaces ◽  
Dirichlet Boundary ◽  
Initial Boundary ◽  
Dirichlet Boundary Conditions ◽  
Parabolic Problems ◽  
Natural Boundary ◽  
Optimal Perturbation ◽  
Natural Boundary Conditions ◽  
Initial Boundary Value

Correctness of initial boundary value problems and their discretizations are analyzed under unusual second-order boundary conditions, which can be considered as natural boundary conditions in strengthened Sobolev spaces and as improvements (in some cases) of the classical Dirichlet boundary conditions. Special attention is paid to optimal perturbation estimates for new variants of the penalty method with respect to the Dirichlet conditions.

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