Abstract
Abstract Purpose In the present paper, we describe a survey of B-spline techniques which have been used for numerical solutions of mathematical problems recently. Methods Here, we discussed the definition of B-splines of various degrees by two different approaches to generate the recurrence relation to drive the formulation of B-splines. Results Cubic B-spline applied on two test equations and absolute errors in interpolation are compared with cubic and quintic splines. Some remarks have been included. Conclusions Numerical results are tabulated in tables; these tables show that the results obtained by cubic B-spline are considerable and accurate with respect to the cubic spline and more or less similar to the quintic spline.
Highlights
A simple graph G 1⁄4 ðV; EÞ together with an assignment of f1; À 1g to its edges is called a signed graph, and will be denoted by ðG; RÞ, where R is the set of negative edges
The spectrum of a signed graph is the eigenvalues of its adjacency matrix
In a signed graph ðG; RÞ by resigning at a vertex v 2 VðGÞ, we mean multiplying the signs of all the edges incident to v by À 1
Summary
A simple graph G 1⁄4 ðV; EÞ together with an assignment of f1; À 1g to its edges is called a signed graph, and will be denoted by ðG; RÞ, where R is the set of negative edges. V in a signed graph by u ¿ v, u $ þv, or u ¿ Àv, we mean there is no edge, positive edge, or negative edge between them, respectively. The adjacency matrix, As of the signed graph ðG; RÞ on the vertex set V 1⁄4 fv; v2; . À 1; 0; vi $ Àvj; vi ¿ vj: Note that any symmetric ð0; Æ 1Þ-matrix A with zero entries on the diagonal can be considered as a signature of the graph with adjacency matrix |A|. The spectrum of a signed graph is the eigenvalues of its adjacency matrix. In a signed graph ðG; RÞ by resigning at a vertex v 2 VðGÞ, we mean multiplying the signs of all the edges incident to v by À 1.
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