The wave equation ?tt?c2?xu(x,t)=e?tf(x,t) in the cone {(x,t):?x??t,x?Rd,t?R+} is shown to have a unique solution if u and its partial derivatives x are L2(e?t) on cone, can be explicit given Fourier series of orthogonal polynomials cone. This provides particular for boundary value problems non-homogeneous which combined with homogeneous obtain full solution.

In this paper a 1-D model for the evolution of inelastic particles in a thermal bath is studied. We derive an equation for the evolution of the probability density of the particles in phase space. The analysis of the homogeneous equation is the main objective of this paper. The existence and uniqueness of this equation is addressed. Finally, we prove that the asymptotic behavior is governed by ...

Abstract. This paper investigates the exit-time for a broad class of symmetric finite-range jump processes via the corresponding master equation, a nonlocal diffusion equation suitably constrained. In direct analogy to the classical diffusion equation with a homogeneous Dirichlet boundary condition, the nonlocal diffusion equation is augmented with a homogeneous volume constraint. The volume-co...

We prove the generalized Hyers--Ulam stability  of $n$-th order linear differential equation of the form $$y^{(n)}+p_{1}(x)y^{(n-1)}+ cdots+p_{n-1}(x)y^{prime}+p_{n}(x)y=f(x),$$ with condition that there exists a non--zero solution of corresponding homogeneous equation. Our main results extend and improve the corresponding results obtained by many authors.

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.