On n‐dimensional quadratic B‐splines

Abstract

In this paper, we present new constructions of the n‐dimensional quadratic B‐splines method. The quadratic B‐splines method format is displayed in a single, two, and three‐dimensional format. These constructions are of utmost importance in solving differential equations in their various dimensions, which have applications in many fields of science. Some numerical examples are also presented, through which we can show the accuracy of this construction by giving some comparisons of our results using this new construction and between the results in other works. Some figures also give in this paper for the solutions.

A new structure to n-dimensional trigonometric cubic B-spline functions for solving n-dimensional partial differential equations

“…We are working on a method for solving partial differential equations in two-and three-dimensional cases, as well as other problems, using a trigonometric cubic B-spline collocation method. Our aim is a continuation of what was done previously [16,17]. Frazer et al [18] started the collocation approach in 1937.…”

“…The first test problem [17]. We take the first test problem in the two dimensions in the form u xx (x, y) + u yy (x, y) + u x (x, y) + u y (x, y) -3e 2x+3y x 2 18y 2 -4y -5 Figure 1 The exact results at y = 0.5 and the numerical results…”

“…6 and 7, we display the velocity profile with various Hartmann numbers at [-0.5, 0.5] using a mesh of 50 × 50. The third test problem: [17,[32][33][34][35]. We take the third test problem in dimension two of the form…”

“…The proposed method Quadratic B-spline approach [17] MCBDQM approach [32] Spline-based DQM approach [33] Haar wavelet approach [34] Spectral collocation approach based on Haarwavelets [35] 7.78 E-5 Now, we compare the results of the proposed method to the results of various methods [17,[32][33][34][35], which are shown in Table 5 using mesh 15 × 15 grid points.…”

On n-dimension Extended cubic B-splines collocation algorithms

“…Now, we are continuing to work on developing a different basis B-spline collocation method to find numerical solutions to partial differential equations in two and three dimensions and so on. This work is a continuation of the works in [17,18]. The collocation strategy began by Frazer et al [19] in 1937.…”