On the difference scheme for a nonlinear diffusion-reaction-type problem

1993 ◽  
Vol 33 (1) ◽  
pp. 12-22 ◽  
Author(s):  
R. Čiegis ◽  
M. Meilūnas
Keyword(s):  
Difference Scheme ◽  
Nonlinear Diffusion ◽  
Diffusion Reaction ◽  
Type Problem ◽  
Reaction Type ◽  
The Difference

scholarly journals An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

Mathematics ◽  
2022 ◽  
Vol 10 (1) ◽  
pp. 131
Author(s):  
Mikhail K. Kolev ◽  
Miglena N. Koleva ◽  
Lubin G. Vulkov
Keyword(s):  
Difference Scheme ◽  
Nonlinear Parabolic Equations ◽  
Nonlinear Ordinary Differential Equation ◽  
Diffusion Reaction ◽  
Migration And Invasion ◽  
Reaction Type ◽  
Truncation Errors ◽  
Cancer Migration ◽  
Nonlinear Parabolic ◽  
Unconditional Positivity

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

scholarly journals A structural approach to control of transitional waves of nonlinear diffusion-reaction systems: Theory and possible applications to bio-medicine

2002 ◽  
Vol 7 (1) ◽  
pp. 27-40 ◽  
Author(s):  
Victor Kardashov ◽  
Shmuel Einav
Keyword(s):  
Nonlinear Diffusion ◽  
Structural Parameters ◽  
Deterministic Chaos ◽  
Reaction Diffusion ◽  
Diffusion Reaction ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Novel Approach ◽  
Dynamics Models ◽  
And Control

This paper has considered a novel approach to structural recognition and control of nonlinear reaction-diffusion systems (systems with density dependent diffusion). The main consistence of the approach is interactive variation of the nonlinear diffusion and sources structural parameters that allows to implement a qualitative control and recognition of transitional system conditions (transients). The method of inverse solutions construction allows formulating the new analytic conditions of compactness and periodicity of the transients that is also available for nonintegrated systems. On the other hand, using of energy conservations laws, allows transfer to nonlinear dynamics models that gives the possiblity to apply the modern deterministic chaos theory (particularly the Feigenboum's universal constants and scenario of chaotic transitions).

Accuracy of the difference scheme of solving the eigenvalue problem for the Laplacian

2011 ◽  
Vol 47 (5) ◽  
pp. 783-790 ◽  
Author(s):  
N. V. Maiko ◽  
V. G. Prikazchikov ◽  
V. L. Ryabichev
Keyword(s):  
Eigenvalue Problem ◽  
Difference Scheme ◽  
The Difference

scholarly journals On the properties of numerical solutions of dynamical systems obtained using the midpoint method

2019 ◽  
Vol 27 (3) ◽  
pp. 242-262 ◽  
Author(s):  
Vladimir P. Gerdt ◽  
Mikhail D. Malykh ◽  
Leonid A. Sevastianov ◽  
Yu Ying
Keyword(s):  
Difference Scheme ◽  
Coupled Oscillators ◽  
Numerical Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integrals Of Motion ◽  
Midpoint Method ◽  
Nonlinear Algebraic Equations ◽  
The Difference

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

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