Dirichlet boundary value problem for differential equation with <i>ϕ</i>-Laplacian and state-dependent impulses

Tomeček, Jan

doi:10.1515/ms-2016-0283

Cited by 11 publications

(3 citation statements)

References 33 publications

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“…Let assumption (5) be satisfied. Assume that A, B ∈ int K, where K is given in (7). Then there exist infinitely many solutions of the impulsive two-point boundary value problem (6).…”

Section: Resultsmentioning

confidence: 99%

“…Among several interesting questions concerning the constraint system (2) there is one we would like to examine in the paper, the two-point boundary value problem with the boundary condition x(0) = A, x(T ) = B, where A, B ∈ K (comp., e.g., [6,7,8], where the two-point boundary value problem has been examined for time-and state-dependent impulses). It is very natural when we want to hit a ball to a hole.…”

Section: Introduction and Notationmentioning

confidence: 99%

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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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“…Let assumption (5) be satisfied. Assume that A, B ∈ int K, where K is given in (7). Then there exist infinitely many solutions of the impulsive two-point boundary value problem (6).…”

Section: Resultsmentioning

confidence: 99%

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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“…For successive applications of numerical methods to solve periodic boundary value problems (PBVP), theorems are needed regarding periodic behaviour of the solutions. Some of these theorems were proven [10], [11]. The existing theorems allow us to use Fourier series coupled with the finite element method (FEM), finite difference method (FDM) or boundary element method (BEM), widely used in PBVP for differential equations.…”

Section: Introductionmentioning

confidence: 99%

Boundary Element Method Analysis of Boundary Value Problems With Periodic Boundary Conditions

Gnitko

Karaiev

Choudhary

et al. 2021

WIT Transactions on Engineering Sciences

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This paper explores non-axisymmetric boundary value problems for the Laplace equation. Neumann's, Dirichlet's and mixed boundary conditions are involved, supposing their periodic behaviour. Boundary value problems arise as auxiliary issues in many practical applications. Among them there are problems related to numerical simulation of vibrations of fluid-filled elastic shells of revolution, coupled vibrations of elastic circular plates resting on a sloshing liquid, crack propagation in elastic mediums, and more. The common feature in these problems is the necessity to obtain the numerical solution of the Laplace equation under different boundary conditions. As these problems are auxiliary, it is necessary to obtain their numerical solutions with high accuracy. The most effective method to solve these problems is the boundary elements method (BEM). Here a new variant of BEM is proposed for the axisymmetric calculation domain with given periodic functions for boundary conditions. The shape of the calculation domain allows us to reduce surface integral equations to one-dimensional ones. In doing so, we must evaluate elliptic-like inner integrals with high accuracy, to elaborate the method of calculation of the outer integrals with logarithmic, Cauchy or Hadamard finite part singularities. An efficient method for evaluating elliptic-like integrals was developed using a special series for integrands, and the quadrature equations were obtained for highprecision calculation of outer integrals. The method developed can be used to determine free vibration modes and frequencies for elastic fluid-filled shells of revolution.

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