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In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional…
This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed…
In this paper, exponential Runge-Kutta methods of collocation type (ERKC) which were originally proposed in (Appl Numer Math 53:323-339, 2005) are extended to semilinear parabolic problems with time-dependent delay. Two classes of the ERKC…
In this work, we present a modification of explicit Runge-Kutta temporal integration schemes that guarantees the preservation of any locally-defined quasiconvex set of bounds for the solution. These schemes operate on the basis of a…
In this paper we characterize the behavior of solutions to systems of boundary integral equations associated with Laplace transmission problems in composite media consisting of regions with polygonal boundaries. In particular we consider…
This paper uses the Modified Projection Method to examine the errors in solving the boundary integral equation from Laplace equation. The analysis uses weighted norms, and parallel algorithms help solve the independent linear systems. By…
The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in…
The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the…
Boundary integral methods are attractive for solving homogeneous linear constant coefficient elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or…
This paper proposes a parallel in time (called also time parareal) method to solve Volterra integral equations of the second kind. The parallel in time approach follows the same spirit as the domain decomposition that consists of breaking…
We propose a reduced basis method to solve time-dependent partial differential equations based on the Laplace transform. Unlike traditional approaches, we start by applying said transform to the evolution problem, yielding a…
There is a growing interest in investigating numerical approximations of the water wave equation in recent years, whereas the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In this…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
In this work we study a multi-step scheme on time-space grids proposed by W. Zhao et al. [28] for solving backward stochastic differential equations, where Lagrange interpolating polynomials are used to approximate the time-integrands with…
When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor…
We present a simple systematic construction and analysis of solutions of the two-dimensional parabolic wave equation that exhibit far-field localisation near a given algebraic plane curve. Our solutions are complex contour integral…
A method is presented for parallelizing the computation of solutions to discrete-time, linear-quadratic, finite-horizon optimal control problems, which we will refer to as LQR problems. This class of problem arises frequently in robotic…
Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle…
We introduce a method for the fast numerical approximation of linear, second-order parabolic partial differential equations (PDEs for short) with time-independent coefficients based on model order reduction techniques and the Laplace…
The recently developed quadrature by expansion (QBX) technique accurately evaluates the layer potentials with singular, weakly or nearly singular, or even hyper singular kernels in the integral equation reformulations of partial…