Related papers: On Solving the Cauchy Problem with Propagators

A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper sub-diffusive cases are…

We present methods for constructing Taylor series surrogate models for covariance preconditioned high dimensional mappings that depend implicitly on the solution of a system of nonlinear equations, e.g., the solution of a partial…

In this paper, we investigate the Cauchy problem for a higher order shallow water type equation \begin{eqnarray*} u_{t}-u_{txx}+\partial_{x}^{2j+1}u-\partial_{x}^{2j+3}u+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0, \end{eqnarray*} where $x\in…

In this paper we consider a reduction of a non-homogeneous linear system of first order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant…

This paper studies the Cauchy problem for variable coefficient weakly hyperbolic first order systems of partial differential operators. The hyperbolicity assumption is that for each $t, x$ the principal symbol is hyperbolic. No hypothesis…

We introduce a discrete delayed exponential depending on sequence of matrices. This discrete matrix gives a representation of a solution to the Cauchy problem for a discrete linear system with pure delay with sequence of matrices. We…

We consider solutions to the Cauchy problem for the incompressible Euler equations on the 3-dimensional torus which are continuous or H\"older continuous for any exponent $\theta<\frac{1}{16}$. Using the techniques introduced in \cite{DS12}…

The paper investigates the initial problem for a linear system of ordinary differential equations with the fractional differentiation operator Dzhrbashyan -- Nersesyan with constant coefficients. The existence and uniqueness theorems of the…

We solve the Cauchy problem of the Ward equation with both continuous and discrete scattering data.

An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder…

This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest orders, $\alpha$ and $\alpha_m$, satisfy the conditions $1<\alpha\le 2$ and…

We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a solution to a stochastic problem. Namely, we construct diffusion processes which allow us to obtain a probabilistic…

To every Darboux integrable system there is an associated Lie group $G$ which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial…

In this article the unique solution of the Cauchy problem is founded by the Riemann method. Some relations for given here confluent hypergeometric functions of two and three variables are used.

The main result establishes the existence of a solution in a generalized sense for a nonlinear Dirichlet problem driven by a competing operator and exhibiting a convection term composed with an intrinsic operator. A finite dimensional…

The Taylor expansion of wave fields with respect to shape parameters has a wide range of applications in wave scattering problems, including inverse scattering, optimal design, and uncertainty quantification. However, deriving the high…

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…

Solution operators for the equation $\bar \partial u=f$ are constructed on general product domains in $\mathbb{C}^n$. When the factors are one-dimensional, the operator is a simple integral operator: it involves specific derivatives of $f$…

To study the existence and uniqueness of solutions to Cauchy-type problems for fractional q-difference equations with the bi-ordinal Hilfer fractional q-derivative which is an extension of the Hilfer fractional q-derivative. An approach is…

We solve the Cauchy problem for the Korteweg-de Vries equation with steplike quasi-periodic, finite-gap initial conditions under the assumption that the perturbations have a given number of derivatives with finite moments.