Order Elliptic Partial Differential Equations Research Articles

Atmospheric Ekman flows of the Northern Hemisphere with latitude and height in ellipsoidal coordinates

In this paper, we study the atmospheric Ekman flows of the Northern Hemisphere related to latitude and height in ellipsoidal coordinates. Firstly, by using a suitable transformation, we convert the atmospheric Ekman equation into a second order elliptic partial differential equation. Further, we prove it is equivalent to a second order homogeneous ordinary differential equation with variable coefficients. Finally, we construct the explicit solution and give the laws of motion of the atmospheric Ekman flows.

SOME FUNCTION SPACES AND THEIR APPLICATIONS TO ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space or a Stummel class. We also show that the logarithm of a non-negative weak solution to a second order elliptic partial differential equation with potential in a generalized Morrey space or a Stummel class, under some assumptions, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuation property. An example of an elliptic partial differential equation with potential in a Morrey space or a Stummel class which does not satisfy the strong unique continuation is presented.

A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations

Within the framework of parameter dependent Partial Differential Equations (PDEs), we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n n -width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n n -width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by numerical experiments, where we compare the proposed approach with classical Principal Orthogonal Decomposition (POD)-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Penalty parameter and dual-wind discontinuous Galerkin approximation methods for elliptic second order PDEs

This article analyzes the effect of the penalty parameter used in symmetric dual-wind discontinuous Galerkin (DWDG) methods for approximating second order elliptic partial differential equations (PDE). DWDG methods follow from the DG differential calculus framework that defines discrete differential operators used to replace the continuous differential operators when discretizing a PDE. We establish the convergence of the DWDG approximation to a continuous Galerkin approximation as the penalty parameter tends towards infinity. We also test the influence of the regularity of the solution for elliptic second-order PDEs with regards to the relationship between the penalty parameter and the error for the DWDG approximation. Numerical experiments are provided to validate the theoretical results and to investigate the relationship between the penalty parameter and the L^2-error. For more information see https://ejde.math.txstate.edu/conf-proc/26/l1/abstr.html

A synthetic analysis of a recently published book that presents a NUMERICAL method of solving 2nd order Elliptic Partial Differential Equations leading towards ANALYTICAL solutions

A synthetic analysis of a recently published book that presents a NUMERICAL method of solving 2nd order Elliptic Partial Differential Equations leading towards ANALYTICAL solutions

Grid-free Monte Carlo for PDEs with spatially varying coefficients

Partial differential equations (PDEs) with spatially varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in nature, since they must first discretize the problem---leading to spatial aliasing, and global meshing/sampling that is costly and error-prone. We describe a method that approximates neither the domain geometry, the problem data, nor the solution space, providing the exact solution (in expectation) even for problems with extremely detailed geometry and intricate coefficients. Our main contribution is to extend the walk on spheres (WoS) algorithm from constant- to variable-coefficient problems, by drawing on techniques from volumetric rendering. In particular, an approach inspired by null-scattering yields unbiased Monte Carlo estimators for a large class of 2nd order elliptic PDEs, which share many attractive features with Monte Carlo rendering: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations.

The Neumann problem for a type of fully nonlinear complex equations

In this paper we study the Neumann problem for a type of fully nonlinear second order elliptic partial differential equations on domains in Cn without any curvature assumptions on the domain.

Error and stability estimates of a least-squares variational kernel-based method for second order elliptic PDEs

We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is self-adjoint or positive definite as it should be in the Rayleigh-Ritz setting. However, the new scheme leads to a symmetric and positive definite algebraic system of equations. Moreover, the resulting method does not rely on certain subspaces satisfying the boundary conditions. The trial space for discretization is provided via standard kernels that reproduce the Sobolev spaces as their native spaces. The error analysis of the method is given, but it is partly subjected to an inverse inequality on the boundary which is still an open problem. The condition number of the final linear system is approximated in terms of the smoothness of the kernel and the discretization quality. Finally, the results of some computational experiments support the theoretical error bounds.

Constrained semilinear elliptic systems on $\mathbb R^N$

The existence of solutions $u$ in $H^1(\mathbb R^N,\mathbb R^M)\cap H^2_{loc}(\mathbb R^N,\mathbb R^M)$ of a coupled semilinear system of second order elliptic partial differential equations on $\mathbb R^N$ of the form \[ \mathcal{P}[u] = f(x,u,\partial u), \quad x\in \mathbb R^N, \] under pointwise constraints is considered. The problem is studied via the constructed suitable topological invariant, the so-called constrained topological degree, which allows to get the existence of solutions of abstract problems considered as $L^2$-realizations of the approximating sequence of systems obtained by the truncation of the initial system to bounded subdomains. The key step of the proof consists in showing the relative $H^1$-compactness of the sequence of solutions to the truncated systems by the use of the so-called tail estimates. The constructions rely on the semigroup approach combined with topological methods, as well as invariance/viability techniques.

High-frequency conductivity at Larmor-frequency in human brain using moving local window multilayer perceptron neural network.

Magnetic resonance electrical properties tomography (MREPT) aims to visualize the internal high-frequency conductivity distribution at Larmor frequency using the B1 transceive phase data. From the magnetic field perturbation by the electrical field associated with the radiofrequency (RF) magnetic field, the high-frequency conductivity and permittivity distributions inside the human brain have been reconstructed based on the Maxwell’s equation. Starting from the Maxwell’s equation, the complex permittivity can be described as a second order elliptic partial differential equation. The established reconstruction algorithms have focused on simplifying and/or regularizing the elliptic partial differential equation to reduce the noise artifact. Using the nonlinear relationship between the Maxwell’s equation, measured magnetic field, and conductivity distribution, we design a deep learning model to visualize the high-frequency conductivity in the brain, directly derived from measured magnetic flux density. The designed moving local window multi-layer perceptron (MLW-MLP) neural network by sliding local window consisting of neighboring voxels around each voxel predicts the high-frequency conductivity distribution in each local window. The designed MLW-MLP uses a family of multiple groups, consisting of the gradients and Laplacian of measured B1 phase data, as the input layer in a local window. The output layer of MLW-MLP returns the conductivity values in each local window. By taking a non-local mean filtering approach in the local window, we reconstruct a noise suppressed conductivity image while maintaining spatial resolution. To verify the proposed method, we used B1 phase datasets acquired from eight human subjects (five subjects for training procedure and three subjects for predicting the conductivity in the brain).

Two-step MPS-MFS ghost point method for solving partial differential equations

The fictitious boundary surrounding the domain is required in the implementation of the method of fundamental solutions (MFS). A similar approach of taking a portion of the centres of radial basis functions (RBFs) outside the domain in the context of the method of particular solutions (MPS) is proposed to further improve the performance of a two-step MPS-MFS method. Since the shape parameter of RBFs is problem dependent, several known procedures on how to determine a good shape parameter are introduced for solving various types of problems. The proposed method is not only highly accurate but also fairly stable due to the use of the particular solutions and fundamental solutions. To demonstrate the effectiveness of the proposed method, five numerical examples in highly complicate and irregular domains, which include second and fourth order elliptic partial differential equations in 2D and 3D, are presented.

Second order elliptic partial differential equations driven by Lévy white noise

This paper deals with linear stochastic partial differential equations with variable coefficients driven by Lévy white noise. First, an existence theorem for integral transforms of Lévy white noise is derived and the existence of generalized and mild solutions of second order elliptic partial differential equations is proved. Further, the generalized electric Schrödinger operator for different potential functions V is discussed.

Second order splitting of a class of fourth order PDEs with point constraints

We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments.

Global bifurcation of positive solutions from zero in nonlinearizable elliptic problems with indefinite weight

This paper is devoted to the study of global bifurcation of solutions from zero of some nonlinearizable eigenvalue problems for second order elliptic partial differential equations with indefinite weight. The existence of global continua of solutions emanating from bifurcation intervals surrounding the trivial solutions corresponding to the principal eigenvalues of the linear problem obtained by setting the nonlinear term equal to zero and contained in the classes of positive and negative functions is shown.

Effect of algorithm parameters in development of spiral tool path for machining of 2.5D star-shaped pockets

2.5D pocket milling is extensively used in aerospace, shipyard, automobile, dies and moulds industries. In machining of 2.5D pockets, directional parallel tool-path and contour parallel tool-path are widely used. However, these tool paths significantly limit the machining efficiency. In the present work, an attempt has been made to generate a spiral tool path for machining of 2.5D star-shaped pocket for improving machining efficiency. The successful generation of spiral tool path depends on various algorithm parameters such as mesh size, permissible error and number of degree-steps. The effect of these parameters on spiral tool path generation is discussed and the best values are reported. The spiral tool path is developed using second order elliptic partial differential equation (PDE) and it is free from sharp corners inside the pocket region. Further, the implementation of proposed method is presented on complex non-star-shaped polygon, pocket bounded by free-form curve and pocket with island.

Novel numerical method based on cubic B-splines for a class of nonlinear generalized telegraph equations in irregular domains

In this work, a cubic B-spline method based on finite difference and meshless approaches for solving 2D generalized telegraph equations in irregular single and multi-connected domains is presented. The three-layer Crank-Nicolson time-stepping scheme is applied for temporal derivatives and systems of second order elliptic partial differential equations (EPDEs) of general type with mixed derivatives and variable coefficients are obtained. The Dirichlet and Robin boundary conditions are considered. The approximate solution on each time layer is sought as series over basis functions which are taken in the form of the tensor products of cubic B-splines. The coefficients of the series are chosen to satisfy the governing EPDEs in the solution domain. The stability and accuracy of the proposed scheme is demonstrated by several examples.

Multiscale mortar expanded mixed discretization of nonlinear elliptic problems

We consider the discretization of nonlinear second order elliptic partial differential equations by multiscale mortar expanded mixed method. This is a domain decomposition method in which the model problem is restricted to the small pieces by dividing the computational domain into the non-overlapping subdomains. An unknown (Lagrange multiplier) is introduced on the interfaces which serve as pressure Dirichlet boundary condition for local subdomain problems. A finite element space is defined on interface to approximate the pressure boundary such that the normal fluxes match weakly on interface.We demonstrated solvability of discretization, and established a priori L2-error estimates for both vector and scalar approximations. We proved the optimal order convergence rates by an appropriate choice of mortar space and polynomial degree of approximation. The uniqueness of discrete problem is shown for sufficiently small values of mesh size. An error estimate for the mortar pressure is derived via linear interface formulation with pressure dependent coefficient. We also present the analysis of the linear second order elliptic problem and prove the similar results. The computational experiments are presented to validate theory.

A boundary element and radial basis function approximation method for a second order elliptic partial differential equation with general variable coefficients

A numerical method based on boundary integral equation and radial basis function approximation is presented for solving boundary value problems governed by a second‐order elliptic partial differential equation with variable coefficients. The equation arises in the analysis of steady‐state anisotropic heat or mass diffusion in nonhomogeneous media with properties that vary according to general smoothly varying functions of space. The method requires only the boundary of the solution domain to be discretized into elements. To check the validity and accuracy of the numerical solution, some specific problems with known solutions are solved.

Spinning Q-balls in Five-Dimensional Spacetime

We investigate q-balls in the d=5 dimensional Minkowski spacetime background with axially symmetric ansatz. In this case, q-balls have non-zero angular momentum for spinning problems. We construct spinning q-balls in five dimensions and solve the solutions of second order elliptic partial differential equations both using analytical and numerical approach since no closed form of the equations following by appropriate boundary conditions. This work only considers the case with equal angular momentum rather than two different angular momenta. Passing the solutions, we discuss the global charge and their properties compared to Coleman q-balls. The solutions exist in some range of frequencies. The main features of q-balls are still recovered in this construction.

On bivectors and jay-vectors

A combination $\mathbf{a}+\mathrm{i}\mathbf{b}$ where ${\mathrm i}^2=-1$ and $\mathbf{a}, \, \mathbf{b}$ are real vectors is called a bivector. Gibbs developed a theory of bivectors, in which he associated an ellipse with each bivector. He obtained results relating pairs of conjugate semi-diameters and in particular considered the implications of the scalar product of two bivectors being zero. This paper is an attempt to develop a similar formulation for hyperbolas by the use of jay-vectors - a jay-vector is a linear combination $\mathbf{a}+\mathrm{j}\mathbf{b}$ of real vectors $\mathbf{a}$ and $\mathbf{b}$, where ${\mathrm j}^2=+1$ but ${\mathrm j}$ is not a real number, so ${\mathrm j}\neq\pm1$. The implications of the vanishing of the scalar product of two jay-vectors is also considered. We show how to generate a triple of conjugate semi-diameters of an ellipsoid from any orthonormal triad. We also see how to generate in a similar manner a triple of conjugate semi-diameters of a hyperboloid and its conjugate hyperboloid. The role of complex rotations (complex orthogonal matrices) is discussed briefly. Application is made to second order elliptic and hyperbolic partial differential equations. Keywords: Split complex numbers, Hyperbolic numbers, Coquaternions, Conjugate semi-diameters, Hyperboloids and ellipsoids, Complex rotations, PDEs MSC (2010) 35J05, 35L10, 74J05