Related papers: Initial value problems for quad equations
In order to solve an initial value problem by the variational iteration method, a sequence of functions is produced which converges to the solution under some suitable conditions. In the nonlinear case, after a few iterations the terms of…
Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects…
We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These…
We study the initial value problem for Lovelock and Horndeski theories of gravity. We show that the equations of motion of these theories can be written in a form that, at weak coupling, is strongly hyperbolic and therefore admits a…
A new iterative technique is presented for solving of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations. Proposed iterative scheme does not require any discretization,…
We show that the so called Robinson-Trautman-Maxwell equations do not constitute a well posed initial value problem. That is, the dependence of the solution on the initial data is not continuous in any norm built out from the initial data…
The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced…
This paper introduces, up to the author's knowledge, for the first time the generalized initial value problem. In this problem, given an ordinary differential equation defined in some set, the initial conditions are mapped to a subset of…
We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…
We present a stable and convergent method for solving initial value problems based on the use of differentiation matrices obtained by Lagrange interpolation. This implicit multistep-like method is easy-to-use and performs pretty well in the…
Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix…
Integrable difference equations commonly have more low-order conservation laws than occur for nonintegrable difference equations of similar complexity. We use this empirical observation to sift a large class of difference equations, in…
We point out that the initial-value (Cauchy) problem for self-interacting vector fields presents the same well-posedness issues as for first-order derivative self-interacting scalar fields (often referred to as $k$-essence). For the latter,…
This paper investigates the initial value problem for a system of one-dimensional fourth-order dispersive partial differential-integral equations with nonlinearity involving derivatives up to second order. Examples of the system arise in…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
In recent work, we used pseudo-differential theory to establish conditions that the initial-boundary value problem for second order systems of wave equations be strongly well-posed in a generalized sense. The applications included the…
In this paper, we study the well-posedness of boundary value problems for a special class of degenerate elliptic equations coming from geometry. Such problems is intimately tied to rigidity problem arising in infinitesimal isometric…
We study a class of initial boundary value problems of hyperbolic type. A new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result.
We consider a characteristic initial value problem for a class of symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces. We prove existence of solutions on a future neighborhood of the…
In this paper, we are concerned with the first initial boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C^2 estimates is our main part. Based on…