Three different spline-based approaches for solving Bratu and Bratu-type equations are presented. The classical cubic spline collocation method, an adaptive spline collocation on nonuniform partitions, and an optimal collocation method are derived for solving Bratu-type equations. Numerical examples are presented to verify the efficiency and accuracy of these methods when compared to other nume...

A C curve interpolation scheme for monotonic data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The monotone rational cubic spline scheme has a unique representation.

A general setting is given for smooth interpolating splines depending on a parameter such that as this parameter approaches infinity the spline converges to the piecewise linear interpolant. The theory includes the standard exponential spline in tension, a rational spline, and several cubic splines. An algorithm is given for one of the cubics; the parameter for this example controls the spacing...

A cubic spline approximation in C1 to the solution of a general Volterra integral equation of the second kind is constructed. Under certain conditions, convergence of the approximation and its first two derivatives is proved and error bounds are obtained. The question of stability is not examined.

In this paper, a high-order method for solving the Schrodinger equation is introduced. We apply a compact finite difference approximation for discretizing spatial derivatives and we use the C-cubic spline collocation method for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. W...

This paper gives a construction to complete, at extraordinary points, an otherwise bi-cubic spline surface – so that the resulting surface is curvature continuous everywhere. To fill the n-sided gap in the bi-cubic surface, a cap is constructed from n spline patches, each consisting of 2× 2 pieces of polynomial degree bi-5. Particular care is taken to continue the curvature distribution from th...

Defects in articular cartilage can be repaired through osteochondral transplantation (mosaic arthroplasty), where osteochondral plugs from non-weight-bearing areas of the joint are transferred to the defect site. Incongruity between the plug surface and the adjacent cartilage results in increased contact pressures and poorer outcomes. We compare three methods to predict the desired repair surfa...