Quintic Spline Solutions of Fourth Order Boundary-Value Problems

We consider a non-polynomial cubic spline to develop the classes of methods for the numerical solution of singularly perturbed two-point boundary value problems. The proposed methods are second and fourth order accurate and applicable to problems both in singular and non-singular cases. Numerical results are given to illustrate the efficiency of our methods and compared with the methods given by different authors.

We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using seventh degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline Method is formulated as an approximation solution. We apply the presented method to solve an example of seventh-order boundary value problem which the resul...

This paper presents a novel approach for numerical solution of a class of fourth order time fractional partial differential equations (PDE's). The finite difference formulation has been used for temporal discretization, whereas, the space discretization is achieved by means of non polynomial quintic spline method. The proposed algorithm is proved to be stable and convergent. In order to corroborate this work, some test problems have been considered and the computational outco...

This work presents a new interpolation tool, namely, cubic $q$-spline. Our new analogue generalizes a well known classical cubic spline. This analogue, based on the Jackson $q$-derivative, replaces an interpolating piecewise cubic polynomial function by $q$-polynomials of degree three at most. The parameter $q$ provides a solution flexibility.

We describe some new univariate spline quasi-interpolants on uniform partitions of bounded intervals. Then we give some applications to numerical analysis: integration, differentiation and approximation of zeros.

We provide explicit expressions for quadrature rules on the space of $C^1$ quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires the minimal number of nodes. For each of $n$ subintervals, generically, only two nodes are required which reduces the evaluation cost by $2/3$ when compared to the classic...

We present a local interpolation method in four dimensions utilising cubic splines. An extension of the three-dimensional tricubic method, the interpolated function has C$^1$ continuity and its partial derivatives are analytically accessible. The specific example of application of this work to a time-varying three-dimensional magnetic field is given, but this method would work equally well for a time-independent four-dimensional field. Implementations of both of these methods...

In this work, we present some new integration formulas for any order of accuracy as an application of the B-spline relations obtained in [1]. The resulting rules are defined as a perturbation of the trapezoidal integration method. We prove the order of approximation and extend the results to several dimensions. Finally, some numerical experiments are performed in order to check the theoretical results.

Traditional CNC technology mostly uses the method of increasing the degree of interpolation polynomial when constructing $C^2$ continuous NURBS curves, but this often leads to the appearance of Runge phenomenon in interpolation curves. Alternatively,the method of adding boundary conditions at the endpoints can often make it difficult to control the error range of the interpolation curve. This article presents a $C^2$ continuous cubic B-spline curve interpolation method,which ...

In this paper, we consider a boundary value problem (BVP) for a fourth order nonlinear functional integro-differential equation. We establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for the total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.