General finite difference approximation to the Cauchy problem for nonlinear parabolic differential-functional equations
Convergence results for unbounded solutions of first order non-linear differential-functional equations
We consider the Cauchy problem in an unbounded region for equations of the type either or . We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.
A note on a Herzog’s and Lemmert’s paper
We give an alternative view of the results published in the Herzog’s and Lemmert’s paper “On maximal and minimal solutions for , ”, Comment. Math. XL (2000), 93-102. One can observe that these results can be obtained by classical (elementary) methods, instead of Tarski’s fixed point theorems in partially ordered spaces.
Stability of finite difference schemes for certain problems in biology
We consider a generalized 1-D von Foerster equation. We present two discretization methods for the initial value problem and study stability of finite difference schemes on regular meshes.
On the Cauchy problem for hyperbolic functional-differential equations
We consider the Cauchy problem for a nonlocal wave equation in one dimension. We study the existence of solutions by means of bicharacteristics. The existence and uniqueness is obtained in topology. The existence theorem is proved in a subset generated by certain continuity conditions for the derivatives.
Existence of solutions to generalized von Foerster equations with functional dependence
We prove the existence of solutions to a differential-functional system which describes a wide class of multi-component populations dependent on their past time and state densities and on their total size. Using two different types of the Hale operator, we incorporate in this model classical von Foerster-type equations as well as delays (past time dependence) and integrals (e.g. influence of a group of species).
Existence of solutions with exponential growth for nonlinear differential-functional parabolic equations
We consider the Cauchy problem for nonlinear parabolic equations with functional dependence. We prove Schauder-type existence results for unbounded solutions. We also prove existence of maximal solutions for a wide class of differential functional equations.
The Rothe method for the McKendrick-von Foerster equation
We present the Rothe method for the McKendrick-von Foerster equation with initial and boundary conditions. This method is well known as an abstract Euler scheme in extensive literature, e.g. K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel, Dordrecht, 1982. Various Banach spaces are exploited, the most popular being the space of bounded and continuous functions. We prove the boundedness of approximate solutions and stability of the Rothe method in and...
Thermal ablation modeling via the bioheat equation and its numerical treatment
The phenomenon of thermal ablation is described by Pennes' bioheat equation. This model is based on Newton's law of cooling. Many approximate methods have been considered because of the importance of this issue. We propose an implicit numerical scheme which has better stability properties than other approaches.
Quasi-linearisation methods for a nonlinear heat equation with functional dependence.
Iterative methods for generalized von Foerster equations with functional dependence.
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