41,922 research outputs found
New developments in Functional and Fractional Differential Equations and in Lie Symmetry
Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis
Superconvergence for finite element approximation of a convection-diffusion equation using graded meshes
In this paper we analyse the approximation of a model convection-diffusion equation by standard bilinear finite elements using the graded meshes introduced in Durán & Lombardi (2006, Finite element approximation of convection-diffusion problems using graded meshes. Appl. Numer. Math., 56, 1314-1325). Our main goal is to prove superconvergence results of the type known for standard elliptic problems, namely, that the difference between the finite element solution and the Lagrange interpolation of the exact solution, in the ε-weighted H 1-norm, is of higher order than the error itself. The constant in our estimate depends only weakly on the singular perturbation parameter. As a consequence of the superconvergence result we obtain optimal order error estimates in the L 2-norm. Also we show how to obtain a higher order approximation by a local postprocessing of the computed solution.Fil: Durán, Rodrigo Gonzalo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Lombardi, Ariel Luis. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Prieto, Mariana Ines. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin
Methodology to Estimating Aquatic Dispersion of Effluents from Accidental and Routine Releases
This paper presents a methodology to analysis of dispersion of radioactive materials in an aquatic environment, specifically for estuaries, based on the Regulatory Guide 1.113. The objective is to present an adaptation of methodology for computational user, that it is possible by means of the use of numerical approximations techniques.
The methodology to be present consist in a numerical approximation of the Navier-Stokes Equation applied in a finite medium with known transport mechanisms, such as Coriolis Effect, floor drag, diffusion, salinity, temperature difference and adhesion per water molecule. The basis of methodology is substantiated in a transport diffusive-convection equation, which has similarity with the Partial Differential Burgues’ Equation for one dimension and with the Kardar-Parisi-Zhang Equation for multidimensional cases
Matrix-equation-based strategies for convection-diffusion equations
We are interested in the numerical solution of nonsymmetric linear systems
arising from the discretization of convection-diffusion partial differential
equations with separable coefficients and dominant convection. Preconditioners
based on the matrix equation formulation of the problem are proposed, which
naturally approximate the original discretized problem. For certain types of
convection coefficients, we show that the explicit solution of the matrix
equation can effectively replace the linear system solution. Numerical
experiments with data stemming from two and three dimensional problems are
reported, illustrating the potential of the proposed methodology
VAGO method for the solution of elliptic second-order boundary value problems
Mathematical physics problems are often formulated using differential
oprators of vector analysis - invariant operators of first order, namely,
divergence, gradient and rotor operators. In approximate solution of such
problems it is natural to employ similar operator formulations for grid
problems, too. The VAGO (Vector Analysis Grid Operators) method is based on
such a methodology. In this paper the vector analysis difference operators are
constructed using the Delaunay triangulation and the Voronoi diagrams. Further
the VAGO method is used to solve approximately boundary value problems for the
general elliptic equation of second order. In the convection-diffusion-reaction
equation the diffusion coefficient is a symmetric tensor of second order
Numerical solving unsteady space-fractional problems with the square root of an elliptic operator
An unsteady problem is considered for a space-fractional equation in a
bounded domain. A first-order evolutionary equation involves the square root of
an elliptic operator of second order. Finite element approximation in space is
employed. To construct approximation in time, regularized two-level schemes are
used. The numerical implementation is based on solving the equation with the
square root of the elliptic operator using an auxiliary Cauchy problem for a
pseudo-parabolic equation. The scheme of the second-order accuracy in time is
based on a regularization of the three-level explicit Adams scheme. More
general problems for the equation with convective terms are considered, too.
The results of numerical experiments are presented for a model two-dimensional
problem.Comment: 21 pages, 7 figures. arXiv admin note: substantial text overlap with
arXiv:1412.570
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
A RBF partition of unity collocation method based on finite difference for initial-boundary value problems
Meshfree radial basis function (RBF) methods are popular tools used to
numerically solve partial differential equations (PDEs). They take advantage of
being flexible with respect to geometry, easy to implement in higher
dimensions, and can also provide high order convergence. Since one of the main
disadvantages of global RBF-based methods is generally the computational cost
associated with the solution of large linear systems, in this paper we focus on
a localizing RBF partition of unity method (RBF-PUM) based on a finite
difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation
method, which can successfully be applied to solve time-dependent PDEs. This
approach allows to significantly decrease ill-conditioning of traditional
RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix
system, reducing the computational effort but maintaining at the same time a
high level of accuracy. Numerical experiments show performances of our
collocation scheme on two benchmark problems, involving unsteady
convection-diffusion and pseudo-parabolic equations
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