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This paper discusses the sufficient conditions for positivity preserving odd degree-n Said-Ball boundary curves defined on a rectangular grid.We derive a sufficient condition on boundary curves of rectangular Said-Ball patches where the lower bound ordinates are adjusted independently.To construct the boundary curves for each rectangular patch, the Said-Ball polynomial solution of fourth order PDE will be considered where its coefficients can be calculated using edge Said-Ball ordinates which fulfill the positivity preserving conditions.Graphical examples are presented using well-known test functions.

In this paper, the sufficient condition for the positive preservation of boundary curves for each edges of bicubic rectangular Said-Ball patches will be discussed.With the use of polynomial solution of fourth order linear PDEs, these curves are defined on rectangular grid for the purpose of enhancing the positivity preservation of the interpolating surface. Through an independent adjustment of the lower boundary of edge Said-Ball ordinates, a sufficient condition on boundary curves for individual edge of bicubic rectangular Said-Ball patches is derived. By employing two well recognized test functions, finding reveals that the approach proposed is appropriate with regard to positive preservation of boundary curves and that there is improvement in the positive preservation of the whole interpolating surfaces.

This paper proposes the sufficient conditions for positivity preserving cubic boundary curves defined on rectangular grid using polynomial solution of fourth order linear PDEs in order to improve the positivity preserving of the interpolating surface. We derive a sufficient condition on boundary curves for each each of bicubic rectangular Bezier patches where the lower bounds of edge Bezier ordinates are adjusted independently.By using two well-known test functions, our result shows that the proposed method is well performed in terms of preserving the positivity of boundary curves and improves the positivity preserving of overall interpolating surfaces.

In this article, the collocation method based on Said-Ball polynomials have been used to solve the singular nonlinear ordinary differential equations of various orders numerically. An operational matrix forms of these ordinary differential equations are obtained from Said-Ball polynomial with variated relations of solution and different derivatives. The presented method reduces the given problem to a system of nonlinear algebraic equations, which removed the singularity of ordinary differential equations. Resulting system is solved using Newton's iteration method to get the coefficients of Said-Ball polynomials. We obtained approximate solutions of the problem under study. Numerical results have been obtained and compared with exact and other works. The presented method gives impressive solutions, that show the accuracy and reliability of the proposed method .