In this paper we present numerical methods – finite differences and finite elements – for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is a phenomenon connected with the interactions within the complex and non-homogeneous background. In order to consider physical initial-value conditions we use...

We consider the class of polynomial differential equation x&= , 2(,)(,)(,)nnmnmPxyPxyPxy++++2(,)(,)(,)nnmnmyQxyQxyQxy++&=++. For where and are homogeneous polynomials of degree i. Inside this class of polynomial differential equation we consider a subclass of Darboux integrable systems. Moreover, under additional conditions we proved such Darboux integrable systems can have at most 1 limit cycle.

The usual way to model heat conduction is to solve a partial differential equation (PDE) with boundary conditions (BCs), i.e., a boundary value problem (BVP). For the case of steady-state heat conduction in a homogeneous isotropic solid with thermal conductivity independent of temperature and with no heat source inside the solid, temperature T (x, y, z) in the solid verifies Laplace equation [5] :

The goal of this paper is to clarify for which starting points the state processes of a stochastic partial differential equation with an affine realization are time-homogeneous. We will illustrate our results by means of the HJMM equation from mathematical finance.

We study the wave equation on a bounded Lipschitz set, characterizing all homogeneous boundary conditions for which this partial differential equation generates a contraction semigroup in the energy space L(Ω). The proof uses boundary triplet techniques. MSC 2010 — 35F15, 35L05, 93C20

In this paper we develop a dressing method for constructing and solving some classes of matrix quasi-linear Partial Differential Equations (PDEs) in arbitrary dimensions. This method is based on a homogeneous integral equation with a nontrivial kernel, which allows one to reduce the nonlinear PDEs to systems of non-differential (algebraic or transcendental) equations for the unknown fields. In ...

The Perona-Malik equation is an ill-posed forward-backward parabolic equation with some application in image processing. In this paper, we study the Perona-Malik type equation on a ball in an arbitrary dimension n and show that there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for smooth nonconstant radially symmetric initial data. Our approach is to ...

In this paper, a Laplace Homotopy perturbation method is employed for solving onedimensional non-homogeneous partial differential equation with a variable coefficient. This method is a combination of the Laplace transform and Homotopy Perturbation Method (LHPM). LHPM presents an accurate methodology to solve non-homogeneous partial differential equation with variable coefficient. The aim of usi...

In a previous article [1] we have shown how one can employ Artificial Neu-ral Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigen-value problems for differential and integrodifferential operators, using ANNs. We start by considering the Schrödinger equation for the Morse potential that has an analyt...

the numerical methods are of great importance for approximating the solutions of nonlinear ordinary or partial differential equations, especially when the nonlinear differential equation under consideration faces difficulties in obtaining its exact solution. in this latter case, we usually resort to one of the efficient numerical methods. in this paper, the chebyshev collocation method is sugge...