Related papers: Cauchy Problem and Green's Functions for First Ord…
The paper considers the Cauchy problem for the system of partial differential equations of fractional order $D_t^{\mathcal{B}} {U}(t,x) + \mathbb{A}(D) {U} (t,x)=H(t,x) $. Here $U$ and $H$ are vector-functions, the $m\times m$ matrix of…
The well-known Green's function method has been recently generalized to nonlinear second order differential equations. In this paper we study possibilities of exact Green's function solutions of nonlinear differential equations of higher…
We develop a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which cover many examples of derived critical loci for gauge-theoretic quadratic action functionals in Lorentzian signature. We define…
We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The…
We use the contracting mapping principle for proving that under some mild restrictions the Cauchy problem for quasilinear systems of functional differential equations with retarded arguments has the unique solution. As a consequence from…
We set-up and solve the Cauchy problem for Schr\"odinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of…
In this paper, we study the existence of positive solutions for nonlinear fractional differential equations with a singular weight. We derive Green's function and corresponding integral operator and then examine the compactness of the…
We construct an expression for the Green function of a differential operator satisfying nonlocal, homogeneous boundary conditions starting from the fundamental solution of the differential operator. This also provides the solution to the…
The initial value problem is well-defined on a class of spacetimes broader than the globally hyperbolic geometries for which existence and uniqueness theorems are traditionally proved. Simple examples are the time-nonorientable spacetimes…
In the present paper, we consider a Cauchy problem for a linear second order in time abstract differential equation with pure delay. In the absence of delay, this problem, known as the harmonic oscillator, has a two-dimensional eigenspace…
In this work we study Cauchy problem for a high-order differential equation $\frac{\partial u(y,x)}{\partial y}+P(\frac{\partial}{\partial x})u(y,x)=\gamma\frac{\partial}{\partial x}(u^2(y,x))+F(y,x)$. We prove that the problem is…
For the Cauchy problem for an operator differential equation of the form $y'(z) = Ay(z)$, where $A$ is a closed linear operator on a Banach space over the completion of an algebraic closure of the field of $p$-adic numbers, a criterion of…
In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Green's function. In particular, we show that if the nonlinear term possesses a special…
The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…
We show the short time existence and uniqueness of solutions to the Cauchy problem for fully nonlinear systems of arbitrary even order on closed manifolds which are strongly parabolic at the initial values. The proof uses a linearization…
It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the…
The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown…
In this paper we define and construct advanced and retarded Green operators for the wave operator on spacetimes with low regularity. In order to do so we require that the spacetime satisfies the condition of generalised hyperbolicity which…
The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order {\alpha}\in(0,2) is investigated. The principal part of the operator has kernel…
We compute a Green's function giving rise to the solution of the Cauchy problem for the source-free Maxwell's equations on a causal domain $\mathcal{D}$ contained in a geodesically normal domain of the Lorentzian manifold $AdS^5 \times…