Mostrando recursos 1 - 20 de 662

  1. Runge-Kutta convolution quadrature and FEM-BEM coupling for the time-dependent linear Schrödinger equation

    Melenk, Jens Markus; Rieder, Alexander
    We propose a numerical scheme to solve the time-dependent linear Schr\"odinger equation. The discretization is carried out by combining a Runge-Kutta time stepping scheme with a finite element discretization in space. Since the Schr\"odinger equation is posed on the whole space $\mathbb{R}^d$, we combine the interior finite element discretization with a convolution quadrature based boundary element discretization. In this paper, we analyze the resulting fully discrete scheme in terms of stability and convergence rate. Numerical experiments confirm the theoretical findings.

  2. Mathematical aspects of variational boundary integral equations for time dependent wave propagation

    Joly, Patrick; Rodríguez, Jerónimo
    In this work, we provide a review of recent results on the mathematical analysis of space-time variational bilinear forms associated to transient boundary integral operators for the wave equation. Most of the results will be proven directly in the time domain and compared to similar results (most of them obtained in the Laplace domain) that can be found in the literature.

  3. A new and improved analysis of the time domain boundary integral operators for the acoustic wave equation

    Hassell, Matthew E.; Qiu, Tianyu; Sánchez-Vizuet, Tonatiuh; Sayas, Francisco-Javier
    We present a novel analysis of the boundary integral operators associated to the wave equation. The analysis is done entirely in the time-domain by employing tools from abstract evolution equations in Hilbert spaces and semi-group theory. We prove a single general theorem from which well-posedness and regularity of the solutions for several boundary integral formulations can be deduced as specific cases. By careful choices of continuous and discrete spaces, we are able to provide a concise analysis for various direct and indirect formulations, both for their Galerkin in space semi-discretizations and at the continuous level. Some of the results here...

  4. Adaptive time domain boundary element methods with engineering applications

    Gimperlein, Heiko; Maischak, Matthias; Stephan, Ernst P.
    Time domain Galerkin boundary elements provide an efficient tool for numerical solution of boundary value problems for the homogeneous wave equation. We review recent advances in their a~posteriori error analysis and the resulting adaptive mesh refinement procedures, as well as basic algorithmic aspects of these methods. Numerical results for adaptive mesh refinements are discussed in two and three dimensions, as are benchmark problems in a half-space related to the transient emission of traffic noise.

  5. Numerical approximation of first kind Volterra convolution integral equations with discontinuous kernels

    Davies, Penny J.; Duncan, Dugald B.
    The cubic ``convolution spline'' method for first kind Volterra convolution integral equations was introduced in P.J. Davies and D.B. Duncan, $\mathit {Convolution\ spline\ approximations\ of\ Volterra\ integral\ equations}$, Journal of Integral Equations and Applications \textbf {26} (2014), 369--410. Here, we analyze its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function and a new error bound obtained from a particular...

  6. Comparison between numerical methods applied to the damped wave equation

    Aimi, A.; Diligenti, M.; Guardasoni, C.
    For the numerical solution of Dirichlet-Neumann problems related to 1D damped wave propagation, from a numerical point of view, we compare the so-called energetic approach, considered here separately for boundary and finite element methods with classical finite difference schemes, both explicit and implicit. The analysis reveals the superiority of energetic approximations with respect to unconditional stability and accuracy with respect to any choice of discretization parameters.

  7. Recent progress in time domain boundary integral equations

    Domínguez, Víctor; Salles, Nicolas; Sayas, Francisco-Javier
    Introduction to this special issue on recent progress in time domain boundary integral equations.

  8. Volume Index for Volume 28


  9. Essential norm of a Volterra-type integral operator from Hardy spaces to some analytic function spaces

    Zhou, Jizhen; Zhu, Xiangling
    In this paper, we obtain some estimates of essential norm of the Volterra-type integral operator $T_g$, where \[ T_gf(z)=\int ^z_0f(\zeta )g'(\zeta )\,d\zeta , \] from Hardy spaces to the BMOA space, Besov spaces, Berg\-man spaces and Bloch-type spaces.

  10. Existence of solutions and controllability for impulsive fractional order damped systems

    Liu, Zhenhai; Li, Xuemei
    In this paper, we are concerned with the controllability of linear and nonlinear Caputo impulsive fractional order damped systems in Banach spaces. Our main purpose is to establish some necessary and sufficient conditions for controllability for this kind of impulsive control system by using Mittag-Leffler matrix functions and the Schauder fixed point theorem.

  11. Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs

    Goodrich, Christopher S.
    We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation \[ y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )\] \[\qquad \qquad \qquad \quad +\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\, ds,\] where certain asymptotic growth properties are imposed on the functions $f$, $H_1$ and $H_2$. Moreover, the functionals $\varphi _1$ and $\varphi _2$ are realizable as Stieltjes integrals with signed measures, which means that the nonlocal elements in the Hammerstein equation are possibly of a very general, sign-changing form. We focus here on the case where the kernel $(t,s)\mapsto G(t,s)$ is allowed to change sign...

  12. Approximation of solutions to a delay equation with a random forcing term and non local conditions

    Chaudhary, Renu; Pandey, Dwijendra N.
    The existence and approximation of a solution to a delay equation with a random forcing term and non local conditions is studied by using a stochastic version of the well-known Banach fixed point theorem and semigroup theory. Moreover, the convergence of Faedo-Galerkin approximations of the solution is shown. An example is given which illustrates the results.

  13. On some regular fractional Sturm-Liouville problems with generalized Dirichlet conditions

    Bensidhoum, Fatima-Zahra; Dib, Hacen
    The present work deals with some spectral properties of the problem ¶ \medskip $(\mathcal{P} )$ $\Bigg \{$\vbox {$D^{\alpha }_{b,-}(p(x)D^{\alpha }_{a,+}y)(x)+\lambda q(x)\,y(x)=0$,\quad $a\lt x\lt b$, ¶ \vspace {-2pt} \qquad \quad $\displaystyle \lim _{\stackrel {x\rightarrow a}{>}}(x-a)^{1-\alpha }y(x)=0=y(b)$,} \smallskip ¶ \noindent where $p,q \in C([a,b])$, $p(x)>0$, $q(x)>0$, for all $x \in [a,b]$ and ${1}/{2} \lt \alpha \lt 1$. $D^{\alpha }_{b,-}$ and $D^{\alpha }_{a,+}$ are the right- and left-sided Riemann-Liouville fractional derivatives of order $\alpha \in (0,1)$, respectively. $\lambda $ is a scalar parameter. ¶ First, we prove, using the spectral theory of linear compact operators, that this problem has an infinite sequence of real eigenvalues and the...

  14. Application of measure of noncompactness to Volterra equations of convolution type

    Alvarez, Edgardo; Lizama, Carlos
    Sufficient conditions for the existence of at least one solution of a nonlinear integral equation with a general kernel are established. The existence result is proved in $C([0,T],E)$, where $E$ denotes an arbitrary Banach space. We use the Darbo-Sadovskii fixed point theorem and techniques of measure of noncompactness. We extend and generalize results obtained by other authors in the context of fractional differential equations. One example illustrates the theoretical results.

  15. Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains

    Domínguez, Víctor; Lyon, Mark; Turc, Catalin
    We present a comparison among the performance of solvers based on Nystr\"om discretizations of several well-posed boundary integral equation formulations of Helmholtz transmission problems in two-dimensional Lipschitz domains. Specifically, we focus on the following four classes of boundary integral formulations of Helmholtz transmission problems: (1)~the classical first kind integral equations for transmission problems~\cite {costabel-stephan}, (2)~the classical second kind integral equations for transmission problems~\cite {KressRoach}, (3)~the single integral equation formulations~\cite {KleinmanMartin}, and (4)~certain direct counterparts of recently introduced generalized combined source integral equations \cite {turc2, turc3}. The former two formulations were the only formulations whose well-posedness in Lipschitz domains was rigorously...

  16. $C^\sigma ,\alpha $ estimates for concave, non-local parabolic equations with critical drift

    Lara, Héctor Chang; Dávila, Gonzalo
    Given a concave integro-differential operator $I$, we study regularity for solutions of fully nonlinear, nonlocal, parabolic equations of the form $u_t-Iu=0$. The kernels are assumed to be smooth but non necessarily symmetric, which accounts for a critical non-local drift. We prove a $C^{\sigma +\alpha }$ estimate in the spatial variable and $C^{1,\alpha }$ estimates in time assuming time regularity for the boundary data. The estimates are uniform in the order of the operator $I$, hence allowing us to extend the classical Evans-Krylov result for concave parabolic equations.

  17. Boundary integral operator for the fractional Laplacian on the boundary of a bounded smooth domain

    Chang, Tongkeun
    We introduce the boundary integral operator induced from the fractional Laplace equation on the boundary of a bounded smooth domain. For~$\frac 12\lt \alpha \lt 1$, we show the bijectivity of the boundary integral operator~$S_{2\alpha }:L^p(\partial \Omega )\to H^{2\alpha -1}_p(\partial \Omega )$ for $1 \lt p \lt \infty $. As an application, we demonstrate the existence of the solution of the Dirichlet boundary value problem of the fractional Laplace equation.

  18. Numerical methods for systems of nonlinear integro-parabolic equations of Volterra type

    Boglaev, Igor
    This paper deals with the numerical solution of systems of nonlinear integro-parabolic problems of Volterra type. The numerical approach is based on the method of upper and lower solutions. A monotone iterative method is constructed. Existence and uniqueness of a solution to the nonlinear difference scheme are established. An analysis of convergence rates of the monotone iterative method is given. Construction of initial upper and lower solutions is discussed. Numerical experiments are presented.

  19. A collocation method solving integral equation models for image restoration

    Liu, Yuzhen; Shen, Lixin; Xu, Yuesheng; Yang, Hongqi
    We propose a collocation method for solving integral equations which model image restoration from out-of-focus images. Restoration of images from out-of-focus images can be formulated as an integral equation of the first kind, which is an ill-posed problem. We employ the Tikhonov regularization to treat the ill-posedness and obtain results of a well-posed second kind integral equation whose integral operator is the square of the original operator. The present of the square of the integral operator requires high computational cost to solve the equation. To overcome this difficulty, we convert the resulting second kind integral equation into an equivalent system...

  20. Approximate solution of Urysohn integral equations with non-smooth kernels

    Kulkarni, Rekha P.; Nidhin, T.J.
    Consider a nonlinear operator equation $ x - K ( x )=f$, where $K$ is a Urysohn integral operator with a kernel of the type of Green's function and defined on $L^\infty [0, 1]$. For $ r \geq 0$, we choose the approximating space to be a space of discontinuous piecewise polynomials of degree $\leq r$ with respect to a quasi-uniform partition of $[0, 1]$ and consider an interpolatory projection at $r+1$ Gauss points. Previous authors have proved that the orders of convergence in the collocation and the iterated collocation methods are $ r+1$ and $r + 2 + \min...

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