Numerical Computations with Hausdorff Continuous Functions

Abstract

Hausdorff continuous (H-continuous) functions appear naturally in many areas of mathematics such as Approximation Theory [11], Real Analysis [1], [8], Interval Analysis, [2], etc. From numerical point of view it is significant that the solutions of large classes of nonlinear partial differential equations can be assimilated through H-continuous functions [7]. In particular, discontinuous viscosity solutions are better represented through Hausdorff continuous functions [6]. Hence the need to develop numerical procedures for computations with H-continuous functions. It was shown recently, that the operations addition and multiplication by scalars of the usual continuous functions on $\Omega\subseteq\mathbb{R}^n$ can be extended to H-continuous functions in such a way that the set ℍ(Ω) of all Hausdorff continuous functions is a linear space [4]. In fact ℍ(Ω) is the largest linear space involving interval functions. Furthermore, multiplication can also be extended [5], so that ℍ(Ω) is a commutative algebra. Approximation of ℍ(Ω) by a subspace were discussed in [3]. In the present paper we consider numerical computations with H-continuous functions using ultra-arithmetical approach [9], namely, by constructing a functoid of H-continuous functions. For simplicity we consider $\Omega\subseteq\mathbb{R}$. In the next section we recall the definition of the algebraic operations on ℍ(Ω). The concept of functoid is defined in Section 3. In Section 4 we construct a functoid comprising a finite dimensional subspace of ℍ(Ω) with a Fourier base extended by a set of H-continuous functions. Application of the functoid to the numerical solution of the wave equation is discussed in Section 5.