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1
Content available remote On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces
Metwali Mohamed M. A.
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
86--94
EN
The existence of a.e. monotonic solutions for functional quadratic Hammerstein integral equations with the perturbation term is discussed in Orlicz spaces. We utilize the strategy of measure of noncompactness related to the Darbo fixed point principle. As an application, we discuss the presence of solution of the initial value problem with nonlocal conditions.
2
Content available remote Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative
Benchohra Mouffak, Bouriah Soufyane, Nieto Juan J.
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
437--450
EN
In this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.
3
Content available A verified method for solving piecewise smooth initial value problems
Auer E., Kiel S., Rauh A.
International Journal of Applied Mathematics and Computer Science
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2013
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Vol. 23, no. 4
731--747
EN
In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.
4
Content available remote Interval multistep predictor-corrector methods of Adams type for solving the initial value problem for ordinary differential equations
Jankowska M. A.
Vibrations in Physical Systems
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2010
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Vol. 24
185--193
EN
In the paper we propose the interval multistep predictor-corrector methods of Adams type for solving the initial value problem (IVP) for ordinary differential equations (ODEs). These methods are based on the explicit interval methods of Adams-Bashforth type and the implicit interval methods of Adams-Moulton type. The interval methods considered belong to a class of algorithms that allow to obtain the guaranteed result, i.e. the interval solution that contain the exact solution of the problem.
PL
W pracy zaproponowane zostały przedziałowe metody wielokrokowe predyktor-korektor typu Adamsa rozwiązywania zagadnienia początkowego dla równań różniczkowych zwyczajnych. Metody te oparte są na jawnych przedziałowych metodach typu Adamsa-Bashfortha oraz niejawnych przedziałowych metodach typu Adamsa-Moultona. Metody przedziałowe należą do klasy algorytmów, które pozwalają otrzymać rozwiązanie danego problemu w postaci przedziału-rozwiązania, który zawiera w sobie rozwiązanie dokładne.
5
Content available remote Comparison of the interval multistep methods of Adams type on some dynamical systems
Jankowska M. A.
Vibrations in Physical Systems
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2010
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Vol. 24
173--184
EN
In the paper we compare the explicit and implicit interval multistep methods of Adams type on some dynamical systems. The methods considered can be used for solving the initial value problem (IVP) for ordinary differential equations (ODEs). As a results we obtain the interval solution that include the exact solution of the IVP. The interval methods are examined on efficiency and numerical precision of the results.
PL
W pracy porównane zostały jawne i niejawne przedziałowe metody typu Adamsa na przykładzie wybranych układów dynamicznych. Rozważane metody mogą być wykorzystane do rozwiązywania zagadnienia początkowego dla równań różniczkowych zwyczajnych. W wyniku zastosowania wspomnianych metod otrzymujemy przedział rozwiązanie, które zawiera w sobie rozwiązanie dokładne danego zagadnienia początkowego. Metody przedziałowe zostały zbadane ze względu na efektywność ich działania oraz dokładność otrzymanego rozwiązania.
6
Content available remote Approximate controllability for systems described by right invertible operators
Hoang V. T.
Control and Cybernetics
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2008
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Vol. 37, no 1
39-50
EN
In this paper, we deal with the approximate controllability for linear systems described by right invertible operators in an infinite dimensional Banach space.
7
Content available remote Existence and uniqueness of solution for initial value problem of first order differential equation involving generalized Lebesgue-Bochner spaces Lp(I,(Xv, II.II))
Lahrech S., Ouahab A., Benbrik A., Mbarki A.
Demonstratio Mathematica
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2007
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Vol. 40, nr 2
303-315
EN
We consider a bitopological vector space (X, v, II.II), where (X, v) is a topological vector space, and II.II is a norm defined on X. This paper deals with the existence and uniqueness of solution for initial value problem of first differential equation: (P)( ˙ x(t) = f(t), t is an element of]alpha, beta[ x(alpha) = x1, where the vector valued function f:]alpha,beta[-› X is assumed to be not necessarily in the classical Lebesgue-Bochner space L1(]alpha,beta[, (X, II.II). Here, by the solution of problem (P), we mean a vector valued function x acting from ]alpha,beta[ into X satisfying the conditions: 1) x is absolutely continuous with respect to the norm II.II; 2) x is almost everywhere differentiable on ]alpha,beta[ with respect to the topology v; 3) ˙ x = f(t) almost everywhere on ]alpha,beta[; 4) x(alpha) = x1. For this, we introduce a special class of integrable functions called generalized Lebesgue- Bochner space denoted L1(]alpha,beta[, (Xv, II.II)) containing (in general, strictly containing, [see the example given at the end of the paper]) the classical Lebesgue-Bochner space L1(]alpha,beta[, (X, II.II). Thus, under some conditions on the pair of topologies (v, II.II) , we prove that if f is an element of L1(]alpha,beta[, (Xv,II.II)), then the initial value problem (P) has an unique solution in the above mentioned sense. Finally, we give an example to illustrate the result given in this paper.
8
Content available remote The controllability of the system described by right invertible operators with constrained controls
Thi H.
Demonstratio Mathematica
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2006
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Vol. 39, nr 3
599-608
EN
This paper is to deal with the controllability of the linear system described by right invertible operators with constrained controls in Banach space.
9
Content available remote Twierdzenie o istnieniu i jednoznaczności rozwiązania zagadnienia początkowego dla nieliniowych równań termodyfuzji w przestrzeni trójwymiarowej
Szymaniec A.
Biuletyn Wojskowej Akademii Technicznej
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2003
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Vol. 52, nr 7
101-109
PL
W pracy udowodniono twierdzenie o istnieniu i jednoznaczności lokalnego w czasie rozwiązania zagadnienia Cauchy'ego dla nieliniowego hiperboliczno-parabolicznego układu równań różniczkowych cząstkowych opisujących trójwymiarowy ośrodek termodyfuzyjny.
EN
Theorem of the existence and uniqueness local (in time) solution the Cauchy problem for nonlinear hyperbolic-parabolic partial differential equations describing the process of the thermodiffusion in 3D space is proved.
10
Content available remote The initial-boundary value problem for the first order degenerated hyperbolic system
Lavrenyuk S., Zaręba L.
Demonstratio Mathematica
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2000
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Vol. 33, nr 1
75-82
EN
In this paper we consider the linear hyperbolic system of the first order with degeneracy at x --0 and x -> l. For such system we assume that initial data arę unbounded on the interval (O, l). Some conditions of the uniqueness, existence and stability of solution for the initial-boundary value problem are obtained.
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