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The fractals literature develops the now well-known concept of local iterated function systems (using affine maps) with grey-level maps (LIFSM) as an approach to function representation in terms of the associated fixed point of the so-called fractal transform. While originally explored as a method to achieve signal (and 2-D image) compression, more recent work has explored various aspects of signal and image processing using this machinery. In this paper, we develop a similar framework for function representation using circle inversion map systems. Given a circle C with centre õ and radius r, inversion with respect to C transforms the point p to the point p′ such that p and p′ lie on the same radial half-line from õ and d(õ,p)d(õ; p′) = r², where d is Euclidean distance. We demonstrate the results with an example. [ABSTRACT FROM AUTHOR]

The collage method for inverse problems has become well-established in the literature in recent years. Initial work developed a collage theorem, based upon Banach's fixed point theorem, for treating inverse problems for ordinary differential equations (ODEs). Amongst the subsequent work was a generalized collage theorem, based upon the Lax-Milgram representation theorem, useful for treating inverse problems for elliptic partial differential equations (PDEs). Each of these two different approaches can be applied to elliptic PDEs in one space dimension. In this paper, we explore and compare how the two different approaches perform for the estimation of the diffusivity for a steady-state heat equation. [ABSTRACT FROM AUTHOR]

The method of generalized quasilinearization has been extended for forced Duffing equations with integral boundary conditions. Natural lower and upper solutions have been considered. Two sequences that converge uniformly and monotonically to the unique solution of the forced Duffing equation have been obtained. The rate of convergence of these sequences is cubic when the nonlinearity of the iterates is quadratic. [ABSTRACT FROM AUTHOR]

Fractal interpolation functions provide a new insight to the natural deterministic approximation and modelling of complex phenomena. Cardinal natural cubic spline Coalescence Fractal Interpolantion Functions (CFIFs) are constructed through moments. Using tensor product of univariate bases of cardinal natural cubic CFIFs, the present paper proposes the construction of natural bicubic spline CFIFs over a rectangular grid ℛ. Natural bicubic CFIFs are self-affine or non-self-affine in nature depending on the IFS parameters of these univariate natural cubic CFIFs. An upper bound of the difference between the natural bicubic CFIFs and an original function ƒ ∈ C4[ℛ], and their derivatives are deduced. The effects of hidden variables are demonstrated through suitably chosen examples. [ABSTRACT FROM AUTHOR]