Related papers: Formal exact operator solutions to nonlinear diffe…
We prove a general local existence theorem for nonlinear systems of Cauchy-Riemann operator of any order in one complex variable with initial values at a given point, which is a counterpart of local existence of ODE.
We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic $\sigma\pi$-ODEs (and, more in general, the real…
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…
Ordinary differential equations of the second order with one constant delay are considered in this paper. An analytical representation of the solution is obtained using the method of steps.
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
Integration of nonlinear partial differential equations with the help of the non-commutative integration over octonions is studied. An apparatus permitting to take into account symmetry properties of PDOs is developed. For this purpose…
Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
We derive the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. The time-variable coefficient is recovered by solving the inverse abstract…
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 shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff…
A general method for solving nonlinear ill-posed problems is developed. The method consists of solving a Cauchy problem with a regularized operator and proving that the solution of this problem tends, as time grows, to a solution of the…
We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable…
Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expensive to compute. In this paper, we derive analytic…
In this paper we study a Cauchy problem for the nonlinear damped wave equations for a general positive operator with discrete spectrum. We derive the exponential in time decay of solutions to the linear problem with decay rate depending on…
The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special…
We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The…
This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the…
In this article we discuss the solvability of some class of fully nonlinear equations, and equations with p-Laplacian in more general conditions by using a new approach given in [1] for studying the nonlinear continuous operator. Moreover…