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Solving neutral delay differential equations with state-dependent delays
No full textIn this paper we consider a class of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving
rise to subsequent points, called “breaking points”, where the solution derivative is still discontinuous. As a consequence, in a right neighbourhood of each such point we have to face a Cauchy problem where the equation has a discontinuous right-hand side. In this case the existence and the uniqueness of the solution is no longer guaranteed to the right
of such points and hence the solution of the neutral equation may either cease to exist or bifurcate. After illustrating why uniqueness and existence of the solution is no longer guaranteed for general state-dependent problems and showing a possible way to detect
these occurrences automatically, we explain how to generalize/regularize the problem in order to suitably extend the solution beyond the breaking point. This is important, for example, when exploring numerically the presence of possible periodic orbits
Parallel algorithms for initial value problems for nonlinear difference and differential equations
No full textSulla ricerca di soluzioni periodiche di equazioni e disequazioni differenziali ordinarie e con ritardo
No full text
Algoritmi paralleli per problemi iniziali di equazioni differenziali lineari ordinarie e con ritardo
No full textBoundary value problems for systems of functional differential equations
Full text linksummary:Algorithms for finding an approximate solution of boundary value problems for systems of functional ordinary differential equations are studied. Sufficient conditions for consistency and convergence of these methods are given. In the last section, a construction of methods of arbitrary order is presented
On the contractivity and asymptotic stability of systems of delayn differential equations of neutral type
No full textIn this paper we investigate both the contractivity and the asymptotic stability of the solutions of linear systems of delay differential equations of neutral type (NDDEs) of the form y(t) = Ly(t) + M(t)y(t – (t)) + N(t)y(t – (t)). Asymptotic stability properties of numerical methods applied to NDDEs have been recently studied by numerous authors. In particular, most of the obtained results refer to the constant coefficient version of the previous system and are based on algebraic analysis of the associated characteristic polynomials. In this work, instead, we play on the contractivity properties of the solutions and determine sufficient conditions for the asymptotic stability of the zero solution by considering a suitable reformulation of the given system. Furthermore, a class of numerical methods preserving the above-mentioned stability properties is also presented
Recent trends in the numerical solution of retarded functional differential equations
No full textRetarded functional differential equations (RFDEs) form a wide class of evolution equations which share the property that, at any point, the rate of the
solution depends on a discrete or distributed set of values attained by the
solution itself in the past. Thus the initial problem for RFDEs is an infinitedimensional problem, taking its theoretical and numerical analysis beyond
the classical schemes developed for differential equations with no functional
elements. In particular, numerically solving initial problems for RFDEs is a
difficult task that cannot be founded on the mere adaptation of well-known
methods for ordinary, partial or integro-differential equations to the presence
of retarded arguments. Indeed, efficient codes for their numerical integration
need specific approaches designed according to the nature of the equation and
the behaviour of the solution.
By defining the numerical method as a suitable approximation of the solution map of the given equation, we present an original and unifying theory for
the convergence and accuracy analysis of the approximate solution. Two particular approaches, both inspired by Runge–Kutta methods, are described.
Despite being apparently similar, they are intrinsically different. Indeed, in
the presence of specific types of functionals on the right-hand side, only one
of them can have an explicit character, whereas the other gives rise to an
overall procedure which is implicit in any case, even for non-stiff problems.
In the panorama of numerical RFDEs, some critical situations have been
recently investigated in connection to specific classes of equations, such as the
accurate location of discontinuity points, the termination and bifurcation of
the solutions of neutral equations, with state-dependent delays, the regularization of the equation and the generalization of the solution behind possible
termination points, and the treatment of equations stated in the implicit form,
which include singularly perturbed problems and delay differential-algebraic
equations as well. All these issues are tackled in the last three sections.
In this paper we have not considered the important issue of stability, for
which we refer the interested reader to the comprehensive book by Bellen and
Zennaro (2003)
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