A diierential equation of the form Dp(x)Du(x) = f(x; u(x)) is solved by collocation, using a class of spline functions of order 3. A restriction of each spline function to any subinterval of the partition is piecewisely in the kernel of the operator DDp(x)D. A coeecient p(x) may be singular. An algorithm is described by means of which a B-spline basis could be eeciently constructed. Some numeri...

In [4] the first named author discussed the explicit solutions of the cubic spline interpolation problems. We are now concerned with quintic spline functions. Let 5B[0, n] denote the class of quintic spline functions S(x) defined in the interval [0, n] and having the points 0, 1, • • • , n — 1 as knots. This means that the restriction of S(x) to the interval (*>, J > + 1 ) (P = 0, • • • , n — \...

We introduce a modeling constructor for micro-structures and porous geometry via curve-trivariate, surface-trivariate and trivariate-trivariate function (symbolic) compositions. By using 1-, 2and 3-manifold based tiles and paving them multiple times inside the domain of a 3-manifold deforming trivariate function, smooth, precise and watertight, yet general, porous/micro-structure geometry might...

A Class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional Bratu's problem are presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Associated boundary f...

Based on analysis of basic cubic spline interpolation, the clamped cubic spline interpolation is generalized in this paper. The methods are presented on the condition that the first derivative and second derivative of arbitrary node are given. The Clamped spline and Curvature-adjusted cubic spline are also generalized. The methods are presented on the condition that the first derivatives of arb...

In this article, a fourth order quintic spline method has been developed to obtain numerical solutions for second order boundary value problems with Dirichlet boundary conditions. The developments of the quintic spline method and convergence analysis were presented. Three test problems have been considered for comparison purposes. The numerical results showed that the quintic spline method is m...

In the context of radial basis function interpolation, the construction of native spaces and the techniques for proving error bounds deserve some further clari cation and improvement. This can be described by applying the general theory to the special case of cubic splines. It shows the prevailing gaps in the general theory and yields a simple approach to local error bounds for cubic spline int...