Laplace Partial Differential Equation Research Articles

Induced homomorphism: Kirchhoff’s law in photonics

Abstract When solving, modeling or reasoning about complex problems, it is usually convenient to use the knowledge of a parallel physical system for representing it. This is the case of lumped-circuit abstraction, which can be used for representing mechanical and acoustic systems, thermal and heat-diffusion problems and in general partial differential equations. Integrated photonic platforms hold the prospective to perform signal processing and analog computing inherently, by mapping into hardware specific operations which relies on the wave-nature of their signals, without trusting on logic gates and digital states like electronics. Here, we argue that in absence of a straightforward parallelism a homomorphism can be induced. We introduce a photonic platform capable of mimicking Kirchhoff’s law in photonics and used as node of a finite difference mesh for solving partial differential equation using monochromatic light in the telecommunication wavelength. Our approach experimentally demonstrates an arbitrary set of boundary conditions, generating a one-shot discrete solution of a Laplace partial differential equation, with an accuracy above 95% with respect to commercial solvers. Our photonic engine can provide a route to achieve chip-scale, fast (10 s of ps), and integrable reprogrammable accelerators for the next generation hybrid high-performance computing. Summary A photonic integrated platform which can mimic Kirchhoff’s law in photonics is used for approximately solve partial differential equations noniteratively using light, with high throughput and low-energy levels.

Solutions to inverse moment estimation problems in dimension 2, using best constrained approximation

We study an inverse problem that consists in estimating the first (zero-order) moment of some R2-valued distribution m that is supported within a closed interval S̄⊂R, from partial knowledge of the solution to the Poisson–Laplace partial differential equation with source term equal to the divergence of m on another interval parallel to and located at some distance from S. Such a question coincides with a 2D version of an inverse magnetic “net” moment recovery question that arises in paleomagnetism, for thin rock samples. We formulate and constructively solve a best approximation problem under constraint in L2 and in Sobolev spaces involving the restriction of the Poisson extension of the divergence of m. Numerical results obtained from the described algorithms for the net moment approximation are also furnished.

A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition

A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance. This kind of problem has been studied by different researchers suggesting different techniques to solve the problem of how to design a trajectory of a mobile robot avoiding collisions with dynamic obstacles. One of these algorithms is the artificial potential field (APF), proposed by O. Khatib in 1986, where a set of an artificial potential field is generated to attract the mobile robot to the goal and to repel the obstacles. This is one of the best options to obtain the trajectory of a mobile robot in real-time (RT). However, the main disadvantage is the presence of deadlocks. The mobile robot can be trapped in one of the local minima. In 1988, J.F. Canny suggested an alternative solution using harmonic functions satisfying the Laplace partial differential equation. When this article appeared, it was nearly impossible to apply this algorithm to RT applications. Years later a novel technique called proper generalized decomposition (PGD) appeared to solve partial differential equations, including parameters, the main appeal being that the solution is obtained once in life, including all the possible parameters. Our previous work, published in 2018, was the first approach to study the possibility of applying the PGD to designing a path planning alternative to the algorithms that nowadays exist. The target of this work is to improve our first approach while including dynamic obstacles as extra parameters.

On Anisotropic Optical Flow Inpainting Algorithms

This work describes two anisotropic optical flow inpainting algorithms. The first one recovers the missing flow values using the Absolutely Minimizing Lipschitz Extension partial differential equation (also called infinity Laplacian equation) and the second one uses the Laplace partial differential equation, both defined on a Riemmanian manifold. The Riemannian manifold is defined by endowing the plane domain with an appropriate metric depending on the reference video frame. A detailed analysis of both approaches is provided and their results are compared on three different applications: flow densification, occlusion inpainting and large hole inpainting.

Error estimation of bilinear Galerkin finite element method for 2D thermal problems

This study demonstrates a two-dimensional steady state heat conduction Laplace partial differential equation solution using the bilinear Galerkin finite element method. Heat transfer analysis is of vital importance in many engineering applications and devising computationally inexpensive numerical methods while maintaining accuracy is one of the primary concerns. The method uses structured mesh grid over a two-dimensional rectangular domain and solved using a stiffness matrix for the bilinear elements, calculated using the proposed modified numerical scheme. Several numerical experiments are conducted by controlling the number of nodes and changing element sizes of the presented scheme, and comparison made between analytical solution and software generated solution.

Total Head Evaluation using Exact and Finite Element Solutions of Laplace Equation for Seepage of Water under Sheet Pile

In this study, the strong and weak forms of Darcy flow equation has been derived. The final obtained equation is Laplace partial differential equation. For total head evaluation, Laplace equation is solved using exact solution depending on Fourier series and numerical solution depending on finite element method. A computer program has been developed in MATLAB program to solve the Laplace equation using both exact and finite element solutions and the seepage of water under sheet pile is taken as a case study. Different number of points and nodes has been selected for the two methods. It is shown that both exact and finite element solutions have a good match in distribution for the study area especially when the Fourier points are increased.

Three-dimensional solutions of the magnetohydrostatic equations for rigidly rotating magnetospheres in cylindrical coordinates

We present new analytical three-dimensional solutions of the magnetohydrostatic equations, which are applicable to the co-rotating frame of reference outside a rigidly rotating cylindrical body, and have potential applications to planetary magnetospheres and stellar coronae. We consider the case with centrifugal force only, and use a transformation method in which the governing equation for the “pseudo-potential” (from which the magnetic field can be calculated) becomes the Laplace partial differential equation. The new solutions extend the set of previously found solutions to those of a “fractional multipole” nature, and offer wider possibilities for modelling than before. We consider some special cases, and present example solutions.

An Unsteady Two-Dimensional Complex Variable Boundary Element Method

The Complex Variable Boundary Element Method (CVBEM) procedure is extended to modeling applications of the two-dimensional linear diffusion partial differential equation (PDE) on a rectangular domain. The methodology in this work is suitable for modeling diffusion problems with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The underpinning of the modeling approach is to decompose the global initial-boundary value problem into a steady-state component and a transient component. The steady-state component is governed by the Laplace PDE and is modeled using the Complex Variable Boundary Element Method. The transient component is governed by the linear diffusion PDE and is modeled by a linear combination of basis functions that are the products of a two-dimensional Fourier sine series and an exponential function. The global approximation function is the sum of the approximate solutions from the two components. The boundary conditions of the steady-state problem are specified to match the boundary conditions from the global problem so that the CVBEM approximation function satisfies the global boundary conditions. Consequently, the boundary conditions of the transient problem are specified to be continuously zero. The initial condition of the transient component is specified as the difference between the initial condition of the global initial-boundary value problem and the CVBEM approximation of the steady-state solution. Therefore, when the approximate solutions from the two components are summed, the resulting global approximation function approximately satisfies the global initial condition. In this work, it will be demonstrated that the coupled global approximation function satisfies the governing diffusion PDE. Lastly, a procedure for developing streamlines at arbitrary model time is discussed.

A Conceptual Numerical Model of the Wave Equation Using the Complex Variable Boundary Element Method

In this work, a conceptual numerical solution of the two-dimensional wave partial differential equation (PDE) is developed by coupling the Complex Variable Boundary Element Method (CVBEM) and a generalized Fourier series. The technique described in this work is suitable for modeling initial-boundary value problems governed by the wave equation on a rectangular domain with Dirichlet boundary conditions and an initial condition that is equal on the boundary to the boundary conditions. The new numerical scheme is based on the standard approach of decomposing the global initial-boundary value problem into a steady-state component and a time-dependent component. The steady-state component is governed by the Laplace PDE and is modeled with the CVBEM. The time-dependent component is governed by the wave PDE and is modeled using a generalized Fourier series. The approximate global solution is the sum of the CVBEM and generalized Fourier series approximations. The boundary conditions of the steady-state component are specified as the boundary conditions from the global BVP. The boundary conditions of the time-dependent component are specified to be identically zero. The initial condition of the time-dependent component is calculated as the difference between the global initial condition and the CVBEM approximation of the steady-state solution. Additionally, the generalized Fourier series approximation of the time-dependent component is fitted so as to approximately satisfy the derivative of the initial condition. It is shown that the strong formulation of the wave PDE is satisfied by the superposed approximate solutions of the time-dependent and steady-state components.

A new iso-parametric machining algorithm for free-form surface

In this paper, a new iso-parametric tool path generation technique has been presented for machining truncated free-form surfaces by means of re-parameterisation. Three methods of re-parameterisation have been discussed, namely, the Coons method, partial differential equation method and the newly developed ‘Boundary Interpolation method.’ Tool paths have been generated based on the iso-parametric curves produced by re-parameterisation. The complete mathematical formulations have been given and subsequently their applications have been studied through some typical cases. Case studies show that the Coons and Laplace partial differential equation have some drawbacks in generating tool paths when the surface boundaries are irregular but the new Boundary Interpolation algorithm gives smooth and efficient iso-parametric tool paths by eliminating the problems like overcrossing (in case of Coons method) and uneven distribution of iso-parametrics (in case of Laplace partial differential equation method).

New automated Markov–Gibbs random field based framework for myocardial wall viability quantification on agent enhanced cardiac magnetic resonance images

A novel automated framework for detecting and quantifying viability from agent enhanced cardiac magnetic resonance images is proposed. The framework identifies the pathological tissues based on a joint Markov-Gibbs random field (MGRF) model that accounts for the 1st-order visual appearance of the myocardial wall (in terms of the pixel-wise intensities) and the 2nd-order spatial interactions between pixels. The pathological tissue is quantified based on two metrics: the percentage area in each segment with respect to the total area of the segment, and the trans-wall extent of the pathological tissue. This transmural extent is estimated using point-to-point correspondences based on a Laplace partial differential equation. Transmural extent was validated using a simulated phantom. We tested the proposed framework on 14 datasets (168 images) and validated against manual expert delineation of the pathological tissue by two observers. Mean Dice similarity coefficients (DSC) of 0.90 and 0.88 were obtained for the observers, approaching the ideal value, 1. The Bland-Altman statistic of infarct volumes estimated by manual versus the MGRF estimation revealed little bias difference, and most values fell within the 95% confidence interval, suggesting very good agreement. Using the DSC measure we documented statistically significant superior segmentation performance for our MGRF method versus established intensity-based methods (greater DSC, and smaller standard deviation). Our Laplace method showed good operating characteristics across the full range of extent of transmural infarct, outperforming conventional methods. Phantom validation and experiments on patient data confirmed the robustness and accuracy of the proposed framework.

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Borner-Durr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.

A general construction of fractal interpolation functions on grids of n

We generalise the notion of fractal interpolation functions (FIFs) to allow data sets of the form where I=[0,1]n. We introduce recurrent iterated function systems whose attractors G are graphs of continuous functions f:I→, which interpolate the data. We show that the proposed constructions generalise the previously existed ones on . We also present some relations between FIFs and the Laplace partial differential equation with Dirichlet boundary conditions. Finally, the fractal dimensions of a class of FIFs are derived and some methods for the construction of functions of class Cp using recurrent iterated function systems are presented.

Contact angles

Young [Philos. Trans. (1805)] derived an equation for the contact angle between a liquid–gas interface and a solid boundary, but doubts have been raised about its validity. This issue is reexamined on the basis of a new integral equation for the interface [J. B. Keller and G. J. Merchant, J. Stat. Phys. 63, 1039 (1991)]. The equation is solved asymptotically by the method of matched asymptotic expansions for small values of the range of intermolecular forces divided by a typical macroscopic length. The leading term in the outer expansion satisfies the Young–Laplace partial differential equation for the interface. The leading term in the boundary-layer expansion satisfies a simplified integral equation. Matching the solutions of these two equations shows that the slope angle at the solid boundary, of the leading term in the outer expansion, is indeed given by the Young equation. Numerical solutions of the boundary-layer integral equation are presented to show how the interface varies near the solid boundary.

Hybrid computer aided design of thick electrostatic electron lenses

The synthesis of space charge free electron lenses can be approached by first assuming solutions for Laplace's partial differential equation in terms of cylindrical harmonics. The hybrid computer can quickly generate these assumed solutions and their gradient by working with the differential equations obtained from product separation of variables. The gradient is used in the integration for the trajectory to determine if the lens is useful. If so, either equipotential surfaces or the potentials along a desired surface are generated with the computer to guide the physical realization of the lens. Certain cases where space charge cannot be neglected are solved by truncation of rectilinear electron flow. Poisson's equation is solved in the computer to serve as a Cauchy boundary condition to Laplace's equation outside the electron beam. The hybrid computing technique and the experimental verification of some of the results in cathode-ray tube electron lenses are presented.

Membrane Apparatus for Analogic Experiments

This paper describes a membrane apparatus which has been used successfully in obtaining analogic soap-film solutions to boundary-value field problems in which either the Poisson or the Laplace partial-differential equation must be satisfied.