189,586 research outputs found
A Recipe for State Dependent Distributed Delay Differential Equations
We use the McKendrick equation with variable ageing rate and randomly
distributed maturation time to derive a state dependent distributed delay
differential equation. We show that the resulting delay differential equation
preserves non-negativity of initial conditions and we characterise local
stability of equilibria. By specifying the distribution of maturation age, we
recover state dependent discrete, uniform and gamma distributed delay
differential equations. We show how to reduce the uniform case to a system of
state dependent discrete delay equations and the gamma distributed case to a
system of ordinary differential equations. To illustrate the benefits of these
reductions, we convert previously published transit compartment models into
equivalent distributed delay differential equations.Comment: 28 page
Algebraic entropy for differential-delay equations
We extend the definition of algebraic entropy to a class of
differential-delay equations. The vanishing of the entropy, as a structural
property of an equation, signals its integrability. We suggest a simple way to
produce differential-delay equations with vanishing entropy from known
integrable differential-difference equations
Mixed stochastic delay differential equations
We consider a stochastic delay differential equation driven by a Holder
continuous process and a Wiener process. Under fairly general assumptions on
its coefficients, we prove that this equation is uniquely solvable. We also
give sufficient conditions for finiteness of its moments and establish a limit
theorem
Dissipativity of the delay semigroup
Under mild conditions a delay semigroup can be transformed into a
(generalized) contraction semigroup by modifying the inner product on the
(Hilbert) state space into an equivalent inner product. Applications to
stability of differential equations with delay and stochastic differential
equations with delay are given as examples
Delay differential equations
Tato bakalářská práce se zabývá problematikou diferenciálních rovnic se zpožděním, které na rozdíl od obyčejných diferenciálních rovnic, obsahují v argumentu neznámé funkce funkci tzv. zpoždění a díky tomu mohou přesněji popisovat některé reálné systémy, jenž se snažíme převést do matematického modelu. V praxi to jsou systémy, v nichž se vyskytují například časové prodlevy potřebné k reakci systému na změnu stavu.\\Přítomnost zpoždění je na druhou stranu komplikací při řešení těchto rovnic a příčinou mnoha odlišností od obyčejných rovnic, z nichž ty hlavní jsou v této práci popsané. Rovněž je ukázán princip použití diferenciálních rovnic se zpožděním při modelování růstu populací.Bachelor thesis focuses on the issue of differential equations with delay, which, unlike ordinary differential equations, contain in the unknown function argument the function of the so-called delay. Therefore, these are capable of a more exact description of certain real systems we want to convert into mathematic models. Practically, these are those systems where time delays, necessary for the reaction of the system to the change of status, occur. The presence of this delay, however, also complicates solution of such equations and sets further differences in comparison with ordinary equations. The crucial differences are described in this thesis. Also the principle is shown for the use of delay-differential equations in population growth models.
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