Standard books on the theory of differential equations deal with scalar equations and systems of equations in normal or Cauchy-Kovalevskaya form, i.e., systems that are solvable with respect to the highest-order derivative. Traditionally, one considers systems where a distinguished independent variable t exists such that the system can be written in evolution form, u t = φ(t, x, u, u x), where ...

using the mean-value theorem for integrals we tried to solved the nonlinear integral equations of hammerstein type . the mean approach is to obtain an initial guess with unknown coefficients for unknown function y(x). the procedure of this method is so fast and don't need high cpu and complicated programming. the advantages of this method are that we can applied for those integral equation...

2 Linear Equations 14 2.1 Matrix Exponentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Non-Autonomous Linear Equations . . . . . . . . . . . . . . . . . . . . . 19 2.4 Inhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Stability and Boundedness . . . . . . ....

With the ultimate goal of iteratively solving piecewise smooth (PS) systems, we consider the solution of piecewise linear (PL) equations. As shown in [Gri13] PL models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of PS functions with a second order error in the distance to a given reference point. The resulting PL functions are obtained quite...

The normal equations constructed by a Toeplitz matrix are studied, in order to nd a suitable preconditioner related to the discrete sine transform. New results are given about the structure of the product of two Toeplitz matrices, which allow the CGN method to achieve a superlinear rate of convergence. This preconditioner outperforms the circulant one for the iterative solution of Toeplitz leas...

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term dynamics from undesirably detailed microscale dynamics. I aim to explore normal forms of stochastic differential equations when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic...

Parabolic Monge-Ampère equations are divided into contact inequivalent classes, for each of which a normal form is given. Two of these are new in the C category.

| We consider di erential equations A(x) _ x = g(x), where A is an n n-matrix of C 1 functions and g is C 1 . We investigate the above di erential equation about singular points x 0 that are standard in the sense of Rabier. In particular, the null space of A(x 0 ) is of dimension 1. We show that there is a C 1 -di eomorphism that transforms the above equation about x 0 into x 1 _ x 1 = 1, _ x 2...