Related papers: Analytic solutions for the Burgers equation with s…
We derive equations for the source terms appearing in structure function equations for the fourth and sixth order under the assumption of homogeneity and isotropy. The source terms can be divided into two classes, namely those stemming from…
Approximation theorems, analogous to known results for linear elliptic equations, are obtained for solutions of the heat equation. Via the Cole-Hopf transformation, this gives rise to approximation theorems for a nonlinear parabolic…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
We study the inviscid Burgers equation in the presence of spatially periodic potential force. We prove that for foliated initial value problem there are always solutions developing shocks in a finite time. We give an application of this…
The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and…
A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra…
Gradual argumentation is a field of symbolic AI which is attracting attention for its ability to support transparent and contestable AI systems. It is considered a useful tool in domains such as decision-making, recommendation, debate…
Using the fractional derivative, considered in the Caputo sense, we study an analytical technique associated with the variational iteration method for the fractional generalized $\alpha$-time Burgers' equation with $\alpha>0$ and obtain…
We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation…
In this article we investigate the spatial Sobolev regularity of mild solutions to stochastic Burgers equations with additive trace class noise. Our findings are based on a combination of suitable bootstrap-type arguments and a detailed…
A new analytical operator method is discussed which solves linear ordinary differential equations with regular singularities. Solutions are obtained in analytic series form and also in Mellin-Barnes-type contour integral form. Exact series…
Gathering together some existing results, we show that the solutions to the one-dimensional Burgers equation converge for long times towards the stationary solutions to the steady Burgers equation, whose Fourier spectrum is not integrable.…
The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper we investigate for the…
We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…
In this paper, we present a solution to the problem of the analytic classification of germs of plane curves with several irreducible components. Our algebraic approach follows precursive ideas of Oscar Zariski and as a subproduct allow us…
A method to calculate the adjoint solution for a large class of partial differential equations is discussed. It differs from the known continuous and discrete adjoint, including automatic differentiation. Thus, it represents an alternative,…
This paper is an introduction to the theory of 1d stochastic Burgers equation under periodic boundary conditions and with a stochastic force, sufficiently smooth in the space variable. We prove the classical results on the existence and…
We consider multidimensional stochastic Burgers equation on the torus $\mathbb{T}^d$ and the whole space $\Rd$. In both cases we show that for positive viscosity $\nu>0$ there exists a unique strong global solution in $L^p$ for $p>d$. In…
I analyse a generalised Burger's equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the…
An approach is proposed to obtain some exact explicit solutions in terms of the Weierstrass' elliptic function $\wp$ to a generalized Benjamin-Bona-Mahony (BBM) equation. Conditions for periodic and solitary wave like solutions can be…