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.

scholarly journals Crank-Nicholson difference scheme for the system of nonlinear parabolic equations observing epidemic models with general nonlinear incidence rate

2021 ◽  
Vol 18 (6) ◽  
pp. 8883-8904
Author(s):  
Allaberen Ashyralyev ◽  
◽  
Evren Hincal ◽  
Bilgen Kaymakamzade ◽  
◽  
...  
Keyword(s):  
Difference Scheme ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Bounded Solution ◽  
Nonlinear Parabolic Equations ◽  
Nonlinear Incidence Rate ◽  
Time Step ◽  
Multidimensional Problems ◽  
Nonlinear Parabolic ◽  
Theoretical Results

<abstract><p>In this work, we study second order Crank-Nicholson difference scheme (DS) for the approximate solution of problem (1). The existence and uniqueness of the theorem on a bounded solution of Crank-Nicholson DS uniformly with respect to time step $ \tau $ is proved. In practice, theoretical results are presented on four systems of nonlinear parabolic equations to explain how it works on one and multidimensional problems. Numerical results are provided.</p></abstract>

Kinetics of oxidation of ammonia on cobalt catalyst I model of the reactor with catalytically active wall

1982 ◽  
Vol 47 (8) ◽  
pp. 2087-2096 ◽  
Author(s):  
Bohumil Bernauer ◽  
Antonín Šimeček ◽  
Jan Vosolsobě
Keyword(s):  
Parabolic Equations ◽  
Cobalt Catalyst ◽  
Nonlinear Parabolic Equations ◽  
Catalytic Reactions ◽  
Temperature Profiles ◽  
Dimensional Model ◽  
Kinetics Of Oxidation ◽  
Nonlinear Parabolic ◽  
Catalytically Active ◽  
Kinetics Of

A two dimensional model of a tabular reactor with the catalytically active wall has been proposed in which several exothermic catalytic reactions take place. The derived dimensionless equations enable evaluation of concentration and temperature profiles on the surface of the active component. The resulting nonlinear parabolic equations have been solved by the method of orthogonal collocations.

scholarly journals Harnack’s inequality for doubly nonlinear equations of slow diffusion type

2021 ◽  
Vol 60 (6) ◽  
Author(s):  
Verena Bögelein ◽  
Andreas Heran ◽  
Leah Schätzler ◽  
Thomas Singer
Keyword(s):  
Vector Field ◽  
Harnack Inequality ◽  
Parabolic Equations ◽  
Full Range ◽  
Nonlinear Parabolic Equations ◽  
Growth Conditions ◽  
Diffusion Type ◽  
Slow Diffusion ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

Sign in / Sign up

Export Citation Format

Share Document