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We propose a class of numerical methods for solving nonlinear random differential equations with piecewise constant argument, called gPCRK methods as they combine generalised polynomial chaos with Runge-Kutta methods. An error analysis is presented involving the error arising from a finite-dimensional noise assumption, the projection error, the aliasing error and the discretisation error. A numerical example is given to illustrate the effectiveness of this approach.

By applying the theory of exponential dichotomies and contraction mapping, we establish some existence and uniqueness results for weighted pseudo almost periodic solutions of some differential equations with piecewise constant arguments. For this purpose, we also describe some basic properties of weighted pseudo almost periodic sequences.

Using the method of exponential dichotomies, we establish a new existence and uniqueness theorem for almost automorphic solutions of differential equations with piecewise constant argument of the form

$$\begin{eqnarray*}{x}^{\prime } (t)= A(t)x(t)+ B(t)x(\lfloor t\rfloor )+ f(t), \quad t\in \mathbb{R} ,\end{eqnarray*}$$

where

$\lfloor \cdot \rfloor $ denotes the greatest integer function, and

$A(t), B(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q\times q} $,

$f(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q} $ are all almost automorphic.

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Application of gPCRK Methods to Nonlinear Random Differential Equations with Piecewise Constant Argument

WEIGHTED PSEUDO ALMOST PERIODIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

ALMOST AUTOMORPHIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT

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3 results

Application of gPCRK Methods to Nonlinear Random Differential Equations with Piecewise Constant Argument

WEIGHTED PSEUDO ALMOST PERIODIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

ALMOST AUTOMORPHIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT