This article considers a nonlinear inverse problem of the Ostrovsky–Burgers equation by using noisy data. Two B-splines with different levels, the Quintic B-spline and septic B-spline, are used to study this problem. For both B-splines, the stability and convergence analysis are calculated, and results show that an excellent estimation of the unknown functions of the nonlinear inverse problem.

In this article, a Collocation algorithm based on Quintic B-splines is proposed for the numerical solution of the non-linear generalized regularized long wave (GRLW) equation. Also, to analyse the linear stability of the numerical scheme, the von-Neumann technique is used. The numerical approach is discussed on three test examples consisting of a single solitary wave, the collision of two solitary waves and the growth of an undular bore. The accuracy of the method is demonstrated by calculating the error in the L2 and L¥ norms and the conservative quantities I1, I2 and I3. The findings are compared with those of previously reported in the literature. Finally, the motion of solitary waves is graphically plotted according to different parameters.

A hybrid method utilizing the Collocation technique with B-splines and Lie-Trotter splitting algorithm applied for 3 model problems which include a single solitary wave,  two solitary wave interaction, and a Maxwellian initial condition is designed for getting the approximate solutions for the generalized equal width (GEW) equation. Initially, the considered problem has been split into 2 sub-equations as linear $U_t=\hat{A}(U)$ and nonlinear $U_t=\hat{B}(U)$ in the  terms of time.  After, numerical schemes have been constructed for these sub-equations utilizing the finite element method (FEM) together with Quintic B-splines. Lie-Trotter splitting technique $\hat{A}o\hat{B}$ has been  used to generate approximate solutions of the main equation. The stability analysis of acquired numerical schemes has been examined by the Von Neumann method. Also, the error norms $L_2$ and $L_\infty$ with mass, energy, and momentum conservation constants $I_1$, $I_2$, and $I_3$, respectively are calculated to illustrate how perfect solutions this new algorithm applied to the problem generates and the ones produced are compared with those in the literature. These new results exhibit that the algorithm presented in this paper is more accurate and successful, and easily applicable to other non-linear partial differential equations (PDEs) as the present equation.

A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the Quintic spline functions (QSF). In our approach, we , rst formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.

In this paper, we study the class of Quintic (α,β)-metrics. We show that every weakly Landsberg 5-th root (α,β)-metrics has vanishing S-curvature. Using it, we prove that a Quintic (α,β)-metric is a weakly Landsberg metric if and only if it is a Berwald metric. Then, we show that a Quintic (α,β)-metric satisfies Ξ = 0 if and only if S = 0.

The non-local hyperbolic partial differential equations have many applications in sciences and engineering. A Collocation finite element approach based on exponential cubic B-spline and Quintic B-spline are presented for the numerical solution of the wave equation subject to nonlocal boundary condition. Von Neumann stability analysis is used to analyze the proposed methods. The efficiency, accuracy and stability of the methods are assessed by applying it to the test problem. The results are found to be in good agreement with known solutions and with existing Collocation schemes in literature.

This paper presents a C2 Pythagorean-Hodograph (PH) spline curve interpolator for high speed contouring applications. With the knot vector and control points given, the C2 PH Quintic spline curve is a "good" interpolant to the nodal points of the cubic B-spline curve, with the same knot vector and control points. To generate the C2 PH Quintic spline curves, a uniform knot sequence is employed.The S-curve motion planning architecture, with variable feedrate for a planar C2 PH Quintic spline curve, is also developed. In particular, C1 cubic feed acceleration/deceleration is imposed on the first and last PH Quintic spline segments. Several closed C2 PH Quintic spline curve contouring tasks, along with a simple position loop controller, were conducted to verify the effectiveness of the proposed interpolation algorithm. The experimental results were analyzed and discussed. It is found that the proposed CNC interpolator is not only feasible for machining the complicated parametric curves represented in the C2 PH Quintic spline form, but also yields a satisfactory contouring performance for variable feedrate control.

In this paper, Pythagorean-hodograph (PH) quantic spline curves constructed by several connected PH curves with C2 continuity are used to designing the tool path in machining operation. To generate the tool path commands, the acceleration/deceleration phase of motion and the middle regions of the tool path are devised by time-dependent and constant velocity interpolation algorithms, respectively. In the aforementioned interpolation algorithm, the exponential functions are employed to modify the feedrate profile and subsequently using the Pattern Search algorithm (PSA), the proposed feedrate profile is optimized to maintain the maximum jerk within allowable physical limits. Finally, high speed CNC machining for two cases of open and closed C2 PH quantic spline curves along with the commonly used feed forward/feedback controller is simulated in Simulink Matlab software and the contouring error is studied and analyzed.