scholarly journals Age-structured delayed SIPCV epidemic model of HPV and cervical cancer cells dynamics I. Numerical method

BIOMATH ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 2110027
Author(s):  
Vitalii Akimenko ◽  
Fajar Adi-Kusumo
Keyword(s):  
Numerical Method ◽  
Cancer Cells ◽  
Epidemic Model ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Second Order ◽  
Phase Portraits ◽  
Order Of Accuracy ◽  
Age Structured

The numerical method for simulation dynamics of nonlinear epidemic model of age-structured sub-populations of susceptible, infectious, precancerous and cancer cells and unstructured population of human papilloma virus (HPV) is developed (SIPCV model). Cell population dynamics is described by the initial-boundary value problem for the delayed semi-linear hyperbolic equations with age- and time-dependent coefficients and HPV dynamics is described by the initial problem for nonlinear delayed ODE. The model considers two time-delay parameters: the time between viral entry into a target susceptible cell and the production of new virus particles, and duration of the first stage of delayed immune response to HPV population growing. Using the method of characteristics and method of steps we obtain the exact solution of the SIPCV epidemic model in the form of explicit recurrent formulae. The numerical method designed for this solution and used the trapezoidal rule for integrals in recurrent formulae has a second order of accuracy. Numerical experiments with vanished mesh spacing illustrate the second order of accuracy of numerical solution with respect to the benchmark solution and show the dynamical regimes of cell-HPV population with the different phase portraits.

Bounded solutions of second order of accuracy difference schemes for semilinear fractional schrödinger equations

2020 ◽  
Vol 23 (6) ◽  
pp. 1723-1761
Author(s):  
Allaberen Ashyralyev ◽  
Betul Hicdurmaz
Keyword(s):  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Second Order ◽  
Difference Schemes ◽  
Bounded Solutions ◽  
Differential Problem ◽  
Order Of Accuracy ◽  
Fractional Schrödinger Equations

AbstractThe present paper deals with initial value problem (IVP) for semilinear fractional Schrödinger integro-differential equation$$\begin{array}{} \displaystyle i\!\frac{du}{dt}+Au = \int\limits_{0}^{t}f\left( s,D_{s}^{\alpha }u(s)\right) ds,\, \, \, 0 \lt t \lt T,\, u\left( 0\right) = 0 \end{array} $$in a Hilbert space H with a self-adjoint positive definite (SAPD) operator A. Stable difference schemes (DSs) have significant interest in investigations of fractional partial differential equations. The main theorem concerns the existence and uniqueness of the uniformly bounded solutions (UBSs) with respect to step time of second order of accuracy DSs for this semilinear fractional Schrödinger differential problem. In practice, existence and uniqueness theorems for a UBS of the one-dimensional initial boundary value problem (BVP) with nonlocal condition and multi-dimensional problem with local condition on the boundary are proved. Numerical results and explanatory illustrations are presented to show the validation of the theoretical results.

Numerical analysis and simulation of delay parabolic partial differential equation involving a small parameter

2019 ◽  
Vol 37 (1) ◽  
pp. 289-312 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Hybrid Scheme ◽  
Content Type ◽  
Convection Diffusion ◽  
Initial Boundary Value

Purpose The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem. Design/methodology/approach The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh. Findings The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature. Originality/value A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates.

scholarly journals Numerical Solotion of a Problem of Fluid Filtration in a Viscoelastic Porous Medium

2020 ◽  
pp. 72-76
Author(s):  
R.A. Virts ◽  
A.A. Papin ◽  
W.A. Weigant
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Order Equation ◽  
Second Order ◽  
Initial System ◽  
Effective Pressure ◽  
Initial Boundary Value

The paper considers a model for filtering a viscous incompressible fluid in a deformable porous medium. The filtration process can be described by a system consisting of mass conservation equations for liquid and solid phases, Darcy's law, rheological relation for a porous medium, and the law of conservation of balance of forces. This paper assumes that the poroelastic medium has both viscous and elastic properties. In the one-dimensional case, the transition to Lagrange variables allows us to reduce the initial system of governing equations to a system of two equations for effective pressure and porosity, respectively. The aim of the work is a numerical study of the emerging initial-boundary value problem. Paragraph 1 gives the statement of the problem and a brief review of the literature on works close to this topic. In paragraph 2, the initial system of equations is transformed, as a result of which a second-order equation for effective pressure and the first-order equation for porosity arise. Paragraph 3 proposes an algorithm to solve the initial-boundary value problem numerically. A difference scheme for the heat equation with the righthand side and a Runge–Kutta second-order approximation scheme are used for numerical implementation.

scholarly journals Some features of numerical diagnostics of instantaneous blow-up of the solution by the example of solving the equation of slow diffusion

2021 ◽  
pp. 77-86
Author(s):  
И.В. Пригорный ◽  
А.А. Панин ◽  
Д.В. Лукьяненко
Keyword(s):  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Ill Posed ◽  
The Difference ◽  
Initial Boundary Value ◽  
Richardson Extrapolation Method ◽  
Posteriori Estimation

В работе демонстрируется, как метод апостериорной оценки порядка точности разностной схемы по Ричардсону позволяет сделать вывод о некорректности постановки (в смысле отсутствия решения) решаемой численно начально-краевой задачи для уравнения в частных производных. Это актуально в ситуации, когда аналитическое доказательство некорректности постановки ещё не получено или принципиально невозможно. The paper demonstrates how the method of a posteriori estimation of the order of accuracy for the difference scheme according to the Richardson extrapolation method allows one to conclude that the formulation of the numerically solved initial-boundary value problem for a partial differential equation is ill-posed (in the sense of the absence of a solution). This is important in a situation when the ill-posedness of the formulation is not analytically proved yet or cannot be proved in principle.

scholarly journals FINITE ELEMENT APPROXIMATION OF A TIME-FRACTIONAL DIFFUSION PROBLEM FOR A DOMAIN WITH A RE-ENTRANT CORNER

2017 ◽  
Vol 59 (1) ◽  
pp. 61-82 ◽  
Author(s):  
KIM NGAN LE ◽  
WILLIAM MCLEAN ◽  
BISHNU LAMICHHANE
Keyword(s):  
Piecewise Linear ◽  
Finite Element Approximation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Second Order ◽  
Element Approximation ◽  
Second Order Convergence ◽  
Initial Boundary Value

An initial-boundary value problem for a time-fractional diffusion equation is discretized in space, using continuous piecewise-linear finite elements on a domain with a re-entrant corner. Known error bounds for the case of a convex domain break down, because the associated Poisson equation is no longer $H^{2}$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation.

scholarly journals Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Ülkü Dinlemez ◽  
Esra Aktaş
Keyword(s):  
Blow Up ◽  
Hyperbolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Energy ◽  
Nonlinear Hyperbolic Equations ◽  
Damping Term ◽  
Linear Damping ◽  
Initial Boundary Value ◽  
Nonlinear String

We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time.

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