In the present paper, the stability of Coalescence Hidden variable Fractal Interpolation Surfaces(CHFIS) is established. The estimates on error in approximation of the data generating function by CHFIS are found when there is a perturbation in independent, dependent and hidden variables. It is proved that any small perturbation in any of the variables of generalized interpolation data results in only small perturbation of CHFIS. Our results are likely to be useful in investigations of texture of surfaces arising from the simulation of surfaces of rocks, sea surfaces, clouds and similar natural objects wherein the generating function depends on more than one variable.

In the present paper, the bounds on fractal dimension of Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) in ℝ3 on a equispaced mesh are found. These bounds determine the conditions on the free parameters for fractal dimension of the constructed CHFIS to become close to 3. The results derived here are tested on a tsunami wave surface by computing the lower and upper bounds of the fractal dimension of its CHFIS simulation.

Riemann–Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order [Formula: see text] of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order [Formula: see text] of a CHFIF defined on any interval [Formula: see text] is also a CHFIF.

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In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x 0 , x N ], the interpolating Cubic Spline SFIF g σ (x) ≡ g σ (x) converge respectively to the data generating function y(x) ≡ y (0) (x) and its derivatives y (j) (x) at the rate of h 2−j+ǫ (0 < ǫ < 1), j = 0, 1, 2, as the norm h of the partition of [x 0 , x N ] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.