Shooting methods for state-dependent impulsive boundary value problems, with applications

Auzinger, Winfried; Burkotová, Jana; Rachůnková, Irena; Wenin, Victor

doi:10.1016/j.apnum.2018.02.006

Cited by 9 publications

(3 citation statements)

References 5 publications

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“…x(t) ∈ int K, ẋ(s+) = ẋ(s) + I(x(s), ẋ(s)), if x(s) ∈ ∂K, (2) where K ⊂ R n is some bounded subset which will be specified later, and I : K × R n → R n is an impulse map describing the impact law.…”

Section: Introduction and Notationmentioning

confidence: 99%

Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

Gabor

Tomeček

2022

J. Fixed Point Theory Appl.

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Dirichlet problem in an n-dimensional billiard space is investigated. In particular, the system of ODEs $$\ddot{x}(t) = f(t,x(t))$$ x ¨ ( t ) = f ( t , x ( t ) ) together with Dirichlet boundary conditions $$x(0) = A$$ x ( 0 ) = A , $$x(T) = B$$ x ( T ) = B in an n-dimensional interval K with elastic impact on the boundary of K is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulating it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used.

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Section: Introduction and Notationmentioning

confidence: 99%

Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

Gabor

Tomeček

2022

J. Fixed Point Theory Appl.

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“…Numerical methods for boundary value problems are mainly divided into two categories: direct methods and shooting methods [1]. The shooting strategies are important for using IVP algorithms to solve BVPs, but the computational mechanisms involve additional costs to find out high order initial conditions [2][3][4]. A wellknown direct method for BVPs is the finite difference method widely used in the literature [5][6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

A new implicit-explicit local differential method for boundary value problems

Tunc¹,

Sarı²

2021

Turk J Math

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In this study, an effective numerical method based on Taylor expansions is presented for boundary value problems. This method is arbitrary directional and called as implicit-explicit local differential transform method (IELDTM). With the completion of this study, a reliable numerical method is derived by optimizing the required degrees of freedom. It is shown that the order refinement procedure of the IELDTM does not affect the degrees of freedom. A priori error analysis of the current method is constructed and order conditions are presented in a detailed analysis. The theoretical order expectations are verified for nonlinear BVPs. Stability of the IELDTM is investigated by following the analysis of approximation matrices. To illustrate efficiency of the method, qualitative and quantitative results are presented for various challenging BVPs. It is tested that the current method is reliable and accurate for a broad range of problems even for strongly nonlinear BVPs. The produced results have revealed that the IELDTM is more accurate than the existing ones in literature.

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“…In response to this paper, two papers dealing with one-dimensional case were written. In [2] the authors gave the numerical treatment to the problem. In [9] the author transformed a one dimensional billiard problem to the problem without impulses and proved multiplicity results by the use of the Schauder Fixed Point Theorem.…”

Section: Introduction and Notationmentioning

confidence: 99%

On the Dirichlet problem in billiard spaces

Gabor

2016

Journal of Mathematical Analysis and Applications

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The constrained Dirichlet boundary value problemẍ = f (t, x), x(0) = x(T ), is studied in billiard spaces, where impacts occur in boundary points. Therefore we develop the research on impulsive Dirichlet problems with state-dependent impulses. Inspiring simple examples lead to an approach enabling to obtain both the existence and multiplicity results in one dimensional billiards. Several observations concerning the multidimensional case are also given.

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Shooting methods for state-dependent impulsive boundary value problems, with applications

Cited by 9 publications

References 5 publications

Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

A new implicit-explicit local differential method for boundary value problems

On the Dirichlet problem in billiard spaces

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