Class Of Partial Differential Equations Research Articles

NonLocal Boundary Value Problems of a Stochastic Variational Inequality Modeling an Elasto-Plastic Oscillator Excited by a Filtered Noise

In the literature, failure risk analysis on most elasto-perfectly-plastic oscillators is essentially focused on those excited by white noise, which is rather restrictive from the modeling perspective. Our present article is motivated by the study of the probability distribution of the solution of a stochastic variational inequality modeling an elasto-plastic oscillator excited by a filtered noise. We introduce a class of partial differential equations (PDEs) with nonlocal Dirichlet conditions and we establish the unique existence of solutions of these PDEs by extending the method developed in [A. Bensoussan and J. Turi, Applied and Numerical Partial Differential Equations, Comput. Methods Appl. Sci. 15, Springer, New York, 2009, pp. 9--23]. A major mathematical challenge here is to carry out the analysis of boundary value problems for elliptic equations in dimension two rather than that in dimension one.

Acceleration and Parallelization of ZENO/Walk-on-Spheres

This paper describes our on-going work to accelerate ZENO, a software tool based on Monte Carlo methods (MCMs), used for computing material properties at nanoscale. ZENO employs three main algorithms: (1) Walk on Spheres (WoS), (2) interior sampling, and (3) surface sampling. We have accelerated the first two algorithms. For the sake of brevity, the paper will discuss our work on the first one only as it is the most commonly used and the acceleration techniques were similar in both cases.WoS is a Brownian motion MCM for solving a class of partial differential equations (PDEs). It provides a stochastic solution to a PDE by estimating the probability that a random walk, which started at infinity, will hit the surface of the material under consideration. WoS is highly effective when the problem's geometry is additive, as this greatly reduces the number of walk steps needed to achieve accurate results. The walks start on the surface of an enclosing sphere and can make much larger jumps than in a direct simulation of Brownian motion. Our current implementation represents the molecular structure of nanomaterials as a union of possibly overlapping spheres. The core processing bottleneck in WoS is a Computational Geometry one, as the algorithm repeatedly determines the distance from query point to the material surface in each step of the random walk.In this paper, we present results from benchmarking spatial data structures, including several open-source implementations of k-D trees, for accelerating WoS algorithmically. The paper also presents results from our multicore and cluster parallel implementation to show that it exhibits linear strong scaling with the number of cores and compute nodes; this implementation delivers up to 4 orders of magnitude speedup compared to the original FORTRAN code when run on 8 nodes (each with dual 6-core Intel Xeon CPUs) with 24 threads per node.

Construction of Global Solutions of Partial Differential Equations with Deviating Arguments in the Time Variable

A class of partial differential equations with deviating arguments in the time variable is investigated. We establish conditions under which it is possible to construct global solutions of these equations and describe the structure of these solutions and the algorithm used for their construction. We also consider some special cases and examples and prove the theorems substantiating the proposed method.

Spectral analysis and structure preserving preconditioners for fractional diffusion equations

Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grunwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.

A Sparse Reformulation of the Green's Function Formalism Allows Efficient Simulations of Morphological Neuron Models.

We prove that when a class of partial differential equations, generalized from the cable equation, is defined on tree graphs and the inputs are restricted to a spatially discrete, well chosen set of points, the Green's function (GF) formalism can be rewritten to scale as O(n) with the number n of inputs locations, contrary to the previously reported O(n(2)) scaling. We show that the linear scaling can be combined with an expansion of the remaining kernels as sums of exponentials to allow efficient simulations of equations from the aforementioned class. We furthermore validate this simulation paradigm on models of nerve cells and explore its relation with more traditional finite difference approaches. Situations in which a gain in computational performance is expected are discussed.

On strongly decreasing solutions of cyclic systems of second-order nonlinear differential equations

The n-dimensional cyclic system of second-order nonlinear differential equationsis analysed in the framework of regular variation. Under the assumption that αi and βi are positive constants such that α1 … αn > β1 … βn and pi and qi are regularly varying functions, it is shown that the situation in which the system possesses decreasing regularly varying solutions of negative indices can be completely characterized, and moreover that the asymptotic behaviour of such solutions is governed by a unique formula describing their order of decay precisely. Examples are presented to demonstrate that the main results for the system can be applied effectively to some classes of partial differential equations with radial symmetry to provide new accurate information about the existence and the asymptotic behaviour of their radial positive strongly decreasing solutions.

The automatic solution of partial differential equations using a global spectral method

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank 2, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree (nx,ny) is computed in O((nxny)3/2) operations. Partial differential operators of splitting rank ≥3 are solved via a linear system involving a block-banded matrix in O(min⁡(nx3ny,nxny3)) operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in Matlab and is publicly available as part of Chebfun. It can resolve solutions requiring over a million degrees of freedom in under 60 seconds. An experimental implementation in the Julia language can currently perform the same solve in 10 seconds.

Efficient gluing of numerical continuation and a multiple solution method for elliptic PDEs

Numerical continuation calculations for ordinary differential equations (ODEs) are, by now, an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences. Although several excellent standard software packages are available for ODEs, there are - for good reasons - no standard numerical continuation toolboxes available for partial differential equations (PDEs), which cover a broad range of different classes of PDEs automatically. A natural ach to this problem is to look for efficient gluing computation approaches, with independent components developed by researchers in numerical analysis, dynamical systems, scientific computing and mathematical modeling. In this paper, we shall study several elliptic PDEs (Lane–Emden–Fowler, Lane–Emden–Fowler with microscopic force, Caginalp) via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the PDEs. In particular, we solve the initialization problem of numerical continuation for PDEs via a minimax algorithm to find multiple unstable solution. Furthermore, for the Caginalp system, we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the FEM MatLab pdetoolbox. Even though the approach works efficiently due to the high-level programming language and without developing any new algorithms, we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic PDEs we study, in particular with respect to symmetry-breaking. In particular, we show for a modified Lane–Emden–Fowler equation with an asymmetric microscopic force, how a fully connected bifurcation diagram splits up into C-shaped isolas on which localized pattern deformation appears towards two different regimes. We conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient, time-saving, gluing computation for dynamical systems analysis of PDEs in the near future.

Parallel adaptive wavelet collocation method for PDEs

A parallel adaptive wavelet collocation method for solving a large class of Partial Differential Equations is presented. The parallelization is achieved by developing an asynchronous parallel wavelet transform, which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution. The data are stored using tree-like structure with tree roots starting at a priori defined level of resolution. Both static and dynamic domain partitioning approaches are developed. For the dynamic domain partitioning, trees are considered to be the minimum quanta of data to be migrated between the processes. This allows fully automated and efficient handling of non-simply connected partitioning of a computational domain. Dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process. The parallel efficiency of the approach is discussed based on parallel adaptive wavelet-based Coherent Vortex Simulations of homogeneous turbulence with linear forcing at effective non-adaptive resolutions up to 20483 using as many as 2048 CPU cores.

Phase dynamics of periodic waves leading to the Kadomtsev–Petviashvili equation in 3+1 dimensions

The Kadomstev–Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper, it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservation of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the case of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocusing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to N >3 is also discussed.

Gradient and Laplacian-Based Hyperspectral Anisotropic Diffusion

To improve accuracy and efficiency of object detection and classification with hyperspectral imagery (HSI), we propose a novel smoothing algorithm by coupling of a Laplacian-based reaction term to a classical divergence-based anisotropic diffusion partial differential equation (PDE). In addition, an adaptive parameter is introduced to regularize this nonlinear reaction-diffusion PDE by explicitly integrating the interband correlations with the noise level of each band in HSI. It is also well-known that the interband correlations can be implicitly embedded into the diffusion coefficient of the divergence-based PDE, to allow a selective smoothing that reduces the local homogeneous area variability while preventing smoothing across edges. Therefore, the interband correlations in HSI are exploited in the proposed method in both explicit and implicit ways. As a result, our algorithm is more effective at controlling the behavior of the diffusion evolution when compared to previous multi/hyperspectral diffusion algorithms. The simulations based on both synthetic data and real hyperspectral remote sensing data show that our algorithm can improve the hyperspectral data quality in terms of both visual inspection and image quality indices.

μ-Pseudo Almost Periodic and Automorphic Solutions in the α-Norm for Some Partial Functional Differential Equations

In this work, we prove the existence and uniqueness of μ-pseudo almost periodic and μ-pseudo almost automorphic solutions in the α −norm for some classes of partial differential equations in Banach spaces. For illustration, we study some models arising in physical systems.

Symmetries of second-order PDEs and conformal Killing vectors

We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first derivatives we show that the Lie point symmetries are given by the conformal algebra of the metric modulo a constraint involving the linear part of the PDE. Important elements in this class are the Klein-Gordon equation and the Laplace equation. We apply the general results and determine the Lie point symmetries of these equations in various general classes of Riemannian spaces. Finally we study the type II hidden symmetries of the wave equation in a Riemannian space with a Lorenzian metric.

From Godunov to a unified hybridized discontinuous Galerkin framework for partial differential equations

By revisiting the basic Godunov approach for system of linear hyperbolic Partial Differential Equations (PDEs) we show that it is hybridizable. As such, it is a natural recipe for us to constructively and systematically establish a unified hybridized discontinuous Galerkin (HDG) framework for a large class of PDEs including those of Friedrichs' type. The unification is fourfold. First, it provides a single constructive procedure to devise HDG schemes for elliptic, parabolic, hyperbolic, and mixed-type PDEs. The key that we exploit is the fact that, for many PDEs, irrespective of their type, the first order form is a hyperbolic system. Second, it reveals the nature of the trace unknowns as the upwind states. Third, it provides a parameter-free HDG framework, and hence eliminating the “usual complaint” that HDG is a parameter-dependent method. Fourth, it allows us to rediscover most existing HDG methods and furthermore discover new ones.We apply the proposed unified framework to three different PDEs: the convection–diffusion–reaction equation, the Maxwell equation in frequency domain, and the Stokes equation. The purpose is to present a step-by-step construction of various HDG methods, including the most economic ones with least trace unknowns, by exploiting the particular structure of the underlying PDEs. The well-posedness of the resulting HDG schemes, i.e. the existence and uniqueness of the HDG solutions, is proved. The well-posedness result is also extended and proved for abstract Friedrichs' systems. We also discuss variants of the proposed unified framework and extend them to the popular Lax–Friedrichs flux and to nonlinear PDEs. Numerical results for transport equation, convection–diffusion equation, compressible Euler equation, and shallow water equation are presented to support the unification framework.

A NEW DEFY FOR ITERATION METHODS

The work presents an adaptation of iteration method for solving a class of thirst order partial nonlinear differential equation with mixed derivatives.The class of partial differential equations present here is not solvable with neither the method of Green function, the most usual iteration methods for instance variational iteration method, homotopy perturbation method and Adomian decomposition method, nor integral transform for instance Laplace,Sumudu, Fourier and Mellin transform. We presented the stability and convergence of the used method for solving this class of nonlinear chaotic equations.Using the proposed method, we obtained exact solutions to this kind of equations.

Linearisable nonlinear partial differential equations in multidimensions

We show that the complexification of the independent variables of a given partial differential equation (PDE) provides a straightforward approach for both unifying certain large classes of PDEs, as well as generating new classes of linearisable PDEs in multidimensions. For example, by complexifying the Burgers equation, which is a linearisable PDE in 1 + 1, i.e., an evolution equation in one spatial dimension, it is possible to construct a linearisable PDE in N + 1, i.e., an evolution PDE in N > 1 spatial dimensions. Among these equations in 2 + 1 are certain linearisable PDEs which have been recently introduced in the literature.

Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities

We consider \(n\)-dimensional cyclic systems of second order differential equations \[(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},\] \[\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}\] under the assumption that the positive constants \(\alpha_i\) and \(\beta_i\) satisfy \(\alpha_1{\ldots}\alpha_n \gt \beta_1{\ldots}\beta_n\) and \(p_i(t)\) and \(q_i(t)\) are regularly varying functions, and analyze positive strongly increasing solutions of system (\(\ast\)) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (\(\ast\)) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (\(\ast\)) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions.

Boundary Integrability of Multiple Stokes--Dirac Structures

A distributed port-Hamiltonian (DPH) system is a framework of boundary controls based on energy for partial differential equations (PDEs). A DPH system is defined in terms of a Stokes--Dirac structure that implies boundary integrability in the sense of Stokes' theorem. Because of the boundary integrability, the energy flows of the systems distributed on a domain can be transformed into energy flows across the boundary of the domain. In general, there might exist boundary nonintegrable energy flows in PDEs. Such energy flows can be modeled by an extended representation of DPH systems, called a distributed energy flow. This paper shows that DPH systems with distributed energy flows can be transformed into standard DPH systems without distributed energy flows by adding extra DPH systems. As a result, we can apply boundary controls and boundary interconnections based on energy to a larger class of PDEs. We call this transformation the boundary completion of DPH systems. This paper derives the necessary and sufficient conditions for such boundary completions.

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations areO(N2M),O(NM2), andO(NM(M+N)) compared withO(MN) for the classical partial differential equations with finite difference methods, whereM,Nare the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.

Impacts of noise on a class of partial differential equations

This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen–Cahn equation does not have finite time singularities and the unique solution exists globally.