Book ChapterDOI
Numerical Methods for Evolutionary Equations with Delay and Software Package PDDE
V. G. Pimenov,Andrey Lozhnikov +1 more
- pp 437-444
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TLDR
A survey of the author's results on the grid-based numerical algorithms for solving the evolutionary equations parabolic and hyperbolic with the effect of heredity on a time variable.Abstract:
The paper gives a survey of the author's results on the grid-based numerical algorithms for solving the evolutionary equations parabolic and hyperbolic with the effect of heredity on a time variable. From uniform positions we construct analogs of schemes with weights for the one-dimensional heat conduction equation with delay of general form, analog of a method of variable directions for the equation of parabolic type with time delay and two spatial variables, analog of the scheme with weights for the equation of hyperbolic type with delay. For the one-dimensional heat conduction equation and the wave equation we obtained conditions on the weight coefficients that ensure stability on the prehistory of the initial function. Numerical algorithms are implemented in the form of software package Partial Delay Differential Equations PDDE toolbox.read more
Citations
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Journal ArticleDOI
First order partial differential equations with time delay and retardation of a state variable
TL;DR: A finite difference scheme is constructed for the numerical solution of a first order partial differential equation with a time delay and retardation of a state variable to model the dynamics of structured cell populations when age and maturity level are taken into account.
Book ChapterDOI
Numerical Methods for a Class of Fractional Advection-Diffusion Models with Functional Delay
V. G. Pimenov,Ahmed S. Hendy +1 more
TL;DR: The algorithm is a fractional analogue of the pure implicit numerical method in which the model is reduced on each time step to the solution of linear algebraic system and the order of convergence is obtained.
Difference scheme for multidimensional transfer equation with time delay
TL;DR: In this paper, a finite difference scheme for a first order multidimensional partial differential equation including a time delay is presented. But the authors focus on the first order partial differential equations and do not consider the time delay.
Journal ArticleDOI
A difference scheme for multidimensional transfer equations with time delay
TL;DR: A finite difference scheme for a first order multidimensional partial differential equation including a time delay used to model different time lapse phenomena, e.g. birds migration, proliferation of viruses or bacteria and transfer of nuclear particles is developed.
Proceedings Article
A Parallel Algorithm for Solving the Advection Equation with a Retarded Argument
TL;DR: A parallel implementation of a difference scheme for the advection equation with time delay on a hybrid architecture computation system that has the second order in space and the first order in time and is unconditionally stable is described.
References
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Book
Theory and Applications of Partial Functional Differential Equations
TL;DR: In this paper, the existence and compactness of solution semiflows of linear systems are investigated. But the authors focus on the nonhomogeneous systems and do not consider the linearized stability of non-homogeneous solutions.
Book
The Theory of Difference Schemes
TL;DR: The theory of difference schemes was introduced in this paper, where homogeneous difference schemes and different schemes for elliptic and time-dependent equations with constant coefficients were studied. And the theory of differenceschemes symbols was discussed.
MonographDOI
Numerical Methods for Delay Differential Equations
Alfredo Bellen,Marino Zennaro +1 more
TL;DR: A review of DDE methods can be found in this paper, where the standard approach via continuous runge-kutta methods for ODEs is described and a stability analysis of runge kutta method for DDE is presented.
Journal ArticleDOI
Finite Difference Approximations for a Class of Semilinear Volterra Evolution Problems
TL;DR: In this article, the authors deal with the numerical integration of some initial and initial-boundary value problems in partial differential equations where the derivatives may depend on the history of the solution by the inclusion of certain "memory" terms.