Quintic B-spline Research Articles

Highly efficient approach to numerical solutions of two different forms of the modified Kawahara equation via contribution of two effective methods

The numerical solutions of the two different forms of the modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different form of the KdV equation have been investigated. To improve the numerical solutions, two efficient methods have been used together. Firstly, Crank–Nicolson discretization algorithm for time integration is used and then fifth-order quintic B-spline based differential quadrature method for space integration is used. To observe the performance of the present algorithm bell-shaped soliton solution and travelling wave solutions are surveyed. The error norms L2 and L∞ are obtained quite less than earlier papers. The invariants and relative changes of invariants are added to sympathize with superior present results.

A Self-Adaptive Control for Phase-Controlled Electromagnetic Contactor Using Weighted Moving Average Filter

This article proposes a self-adaptive control strategy for controlling the holding current of a novel phase-controlled contactor. The closing moment of contact can be captured by calculating the drop slope of current and the pulsewidth modulation is then applied to decrease the holding current. The weighted moving average filter is employed to filter the current and voltage signals. A novel approach using a quintic B-spline curve-fitting algorithm is proposed to determine the number of sampling values and their weights. Numerous theoretical analyses of electromagnetic force and magnetic flux density distributions are developed. A cosimulation coupling magnetic field, machinery, and circuit are carried out with the Maxwell and circuit editor software. A prototype is manufactured and the experimental results validate the effectiveness of the proposed self-adaptive control strategy for minimizing the holding current.

Quintic B-spline collocation method to solve [formula omitted]-dimensional stochastic Itô-Volterra integral equations

In this paper, the n-dimensional stochastic Itô-Volterra integral equation is numerically solved via quintic B-spline collocation method. To reach this aim, the quintic B-spline interpolation, Gauss–Legendre quadrature formula and Itô approximation are presented. By using the quintic B-spline collocation method, n-dimensional stochastic Itô-Volterra integral equation can be reduced to a linear or nonlinear system of algebraic equations. Also, convergence analysis of the proposed method is discussed. Finally, it is used in some numerical examples to illustrate the effectiveness of the proposed method.

A Sixth-Order Numerical Method Based on Shishkin Mesh for Singularly Perturbed Boundary Value Problems

Recently, Lodhi and Mishra (J Comput Appl Math 319:170–187, 2017) presented the standard B-spline method based on quintic B-spline basis functions to solve a type of singularly perturbed boundary value problems (SPBVP). We note that their method provides only fourth-order convergence approximation to the solution of such problem. In this paper, we present a novel optimal B-spline technique, based on same quintic B-spline basis function as used in Lodhi and Mishra (2017), for solving linear and nonlinear SPBVP. The advantage of the suggested method over the method in Lodhi and Mishra (2017) is that our method has sixth-order rate of convergence. To obtain higher order of convergence, a high-order perturbation of the SPBVP is generated. The method is tested for its efficiency by applying it on five test problems.

A novel B-spline method to analyze convection-diffusion-reaction problems in anisotropic inhomogeneous medium

We present a B-spline based semi-analytical technique for solving 2D convection-diffusion-reaction equations. The main feature of the presented technique is to separate the satisfaction of the conditions on the boundary and the elliptic partial differential equation inside. To be more precise, we transform the original equation into the problem with homogeneous boundary conditions and seek the approximate solution as a sum of the modified B-spline tensor products which satisfy the homogeneous boundary conditions of the problem. The cubic and quintic B-spline are used in the framework of the method. The coefficients of linear combination are determined to satisfy the governing equation. Eight numerical examples have been studied to demonstrate the high effectivity of the presented technique in solving 2D convection-diffusion-reaction problems in single and multi-connected domains.

Global G3 continuity toolpath smoothing for a 5-DoF machining robot with parallel kinematics

• A global G 3 continuity toolpath smoothing method is proposed for 5-DoF parallel machining robots. • Splines are constructed for SLSGs and to guarantee the G 3 continuity between the fitted curves and the adjacent toolpaths. • Smoothing for test toolpaths is conducted to show the validity of the proposed method in motion smoothness. Toolpath smoothing is an important approach to improve robots’ operational stability and machining quality. Nowadays, the corner rounding smoothing and curve fitting smoothing algorithms are usually adopted to process the linear toolpath segments to improve its continuity. But the high order continuity between the fitted curve and its adjacent curves is difficult to be guaranteed. For parallel machining robots (PMRs), the tangential, curvature and curvature derivative discontinuities at the junction may lead to the self-excited vibration of mechanical structure, consequently the machining efficiency and quality are decreased. Under this consideration, a global G 3 continuity toolpath smoothing method for five degrees of freedom (5-DoF) PMRs is proposed. The linear segments toolpath generated by the Computer-Aided Manufacturing (CAM) system is first divided into long linear segments (LLSs) and short linear segments groups (SLSGs) through breakpoint searching. At the junction point, a B-spline transition curve is inserted to blend adjacent toolpaths. For the SLSG, the quintic B-spline is adopted to fit the discrete data points, constraint equations about the derivatives at the start and end points are established to achieve G 3 continuity with the adjacent transition curves. Based on the proposed method, the smoothing for two test toolpaths is carried out, and experiments on a 5-DoF PMR are conducted to show the validity of the method in motion smoothness.

B-Spline Method of Lines for Simulation of Contaminant Transport in Groundwater

In this study, we propose a new numerical method, which can be effectively applied to the advection-dispersion equation, based on B-spline functions and method of lines approach. In the proposed approach, spatial derivatives are calculated using quintic B-spline functions. Thanks to the method of lines approach, the partial differential equation governing the contaminant transport in groundwater is converted into time-dependent ordinary differential equations. After this transformation, the time-integration of this system is realized by using an adaptive Runge–Kutta formula. In order to test the accuracy of the proposed method, four numerical examples were solved and the obtained results compared with various analytical and numerical solutions given in the literature. It is proven that the proposed method is faster and more reliable than other methods referenced herein and is a good alternative for simulation of contaminant transport problems as a result of these comparisons.

Numerical treatment of fourth-order singular boundary value problems using new quintic B-spline approximation technique

Numerical treatment of fourth-order singular boundary value problems using new quintic B-spline approximation technique

Nonlinear waves propagation and stability analysis for planar waves at far field using quintic B-spline collocation method

A study of the dynamics of nonlinear waves at far field has been presented for real gases using quintic B-spline collocation method. To examine the dynamics at far field, “fast variable” in the governing system of equations has been introduced. Using asymptotic expansion of the flow variables in new fast variables, an evolution equation has been obtained for the behavior of waves at far from source. The evolution equation is modified Burger’s equation, which is known to have exact solution for planar flow. However, the solution for the corresponding cylindrical and spherical symmetric flow for the equation is unknown. The obtained numerical results have been compared with the exact solution for the planar flow. The error norms of numerically computed values are found very small when compared to corresponding exact solution. A Von Neumann stability analysis of the scheme is presented. The scheme is found unconditionally stable. It shows that the method is quite efficient to capture the dynamics of flow at far field. Results corresponding to other two geometries have been presented at different time. Further, a study on the effect of the real medium on the flow is presented. Schematic shows that with the increase in non-ideal property of the medium, the formation of shock becomes faster.

Numerical simulation of Fisher's type equation via a collocation technique based on re-defined quintic B-splines

PurposeA collocation technique based on re-defined quintic B-splines over Crank-Nicolson is presented to solve the Fisher's type equation. We take three cases of aforesaid equation. The stability analysis and rate of convergence are also done.Design/methodology/approachThe quintic B-splines are re-defined which are used for space integration. Taylor series expansion is applied for linearization of the nonlinear terms. The discretization of the problem gives up linear system of equations. A Gaussian elimination method is used to solve these systems.FindingsThree examples are taken for analysis. The analysis gives guarantee that the present method provides much better results than previously presented methods in literature. The stability analysis and rate of convergence show that the method is unconditionally stable and quadratic convergent for Fisher's type equation. Moreover, the present method is simple and easy to implement, so it may be considered as an alternative method to solve PDEs.Originality/valueThis work is the original work of authors which is neither published nor submitted anywhere else for publication.

On complex wave structures related to the nonlinear long–short wave interaction system: Analytical and numerical techniques

This article presents a survey on the exact and numerical solutions of the nonlinear long–short wave interaction system. The system performs an optical domain, which does not alter during multiplication according to a ticklish equipoise between nonlinear and linear influences in elastic surrounding (the medium that can alter the figure due to the existence of a deforming strength and comes back to its original shape in the absence of this force). The wave in this medium is obtained by vibrations that are the outcomes of the acoustic power. The modified auxiliary equation and the quintic B-spline approaches are investigated in our model to obtain a bundle of solutions to discuss new physical behaviors for this model. Moreover, the stability property is discussed for the analytical solutions via the properties related to the Hamiltonian system to show the range of the ability of solutions to be used in the applications of the model. These novel properties are explained by different types of figures. Finally, the convergence and the absolute error between the obtained solutions are discussed in a table.

An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods

The numerical solutions of the coupled Korteweg-de Vries equation is investigated via combination of two efficient methods that includes Crank–Nicolson scheme for time integration and quintic B-spline based differential quadrature method for space integration. The differential quadrature method has the advantage of using less number of grid points. And the advantage of the Crank–Nicolson scheme is the prevention of long and tedious algebraic computations for time integration. Those advantages come together and produce better results. To display the accuracy and efficiency of the present hybrid method three well-known test problems, namely single soliton, interaction of two soliton and birth of solitons are solved and the error norms $$L_{2}$$ and $$ L _{\infty }$$ are computed and compared with earlier works. Present hybrid method obtained superior results than earlier works by using the same parameters and less number of grid points. This situation is shown by comparison of the earlier works. At the same time, two lowest invariants and numerical and analytical values of the amplitudes of the solitons during the simulations are computed and tabulated. Besides those, relative changes of invariants are computed. Properties of solitons observed clearly at the all of the test problems and figures of the all of the simulations are given.

Dynamic Trajectory Planning for a Three Degrees-of-Freedom Cable-Driven Parallel Robot Using Quintic B-Splines

Abstract This paper presents a new trajectory planning method based on the improved quintic B-splines curves for a three degrees-of-freedom (3-DOF) cable-driven parallel robot (CDPR). First, the conditions of positive cables’ tension are expressed in terms of the position and acceleration constraints of the end-effector. Then, an improved B-spline curve is introduced, which is employed for generating a pick-and-place path by interpolating a set of given via-points. Meanwhile, by expressing the position and acceleration of the end-effector in terms of the first and second derivatives of the improved B-spline, the cable tension constraints are described in the form of B-spline parameters. According to the properties of the defined pick-and-place path, the proposed motion profile is dominated by two factors: the time taken for the end-effector to pass through all the via-points and the ratio between the nodes of B-spline. The two factors are determined through multi-objective optimization based on the efficiency coefficient method. Finally, experimental results on a 3-DOF CDPR show that the improved B-spline exhibits overall superior behavior in terms of velocity, acceleration, and cables force compared with the traditional B-spline. The validity of the proposed trajectory planning method is proved through the experiments.

Analytical, semi-analytical, and numerical solutions for the Cahn\u2013Allen equation

This paper studies the analytical, semi-analytical, and numerical solutions of the Cahn–Allen equation, which plays a vital role in describing the structure of the dynamics for phase separation in Fe–Cr–X (X=Mo,Cu) ternary alloys. The modified Khater method, the Adomian decomposition method, and the quintic B-spline scheme are implemented on our suggested model to get distinct kinds of solutions. These solutions describe the dynamics of the phase separation in iron alloys and are also used in solidification and nucleation problems. The applications of this model arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The comparison between the obtained solutions is represented by using figures and tables to explain the value of the error between exact and numerical solutions. All solutions are verified by using Mathematica software.

POLYNOMIAL AND RATIONAL WAVE SOLUTIONS OF KUDRYASHOV-SINELSHCHIKOV EQUATION AND NUMERICAL SIMULATIONS FOR ITS DYNAMIC MOTIONS

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions are investigated. Conservation flows of the dynamic motion are obtained utilizing multiplier approach. Using the unified method, a collection of exact solitary and soliton solutions of Kudryashov-Sinelshchikov equation is presented. Collocation finite element method based on quintic B-spline functions is implemented to the equation to evidence the accuracy of the proposed method by test problems. Stability analysis of the numerical scheme is studied by employing von Neumann theory. The obtained analytical and numerical results are in good agreement.

Numerical Approximation of the Combined KdV-mKdV Equation via the Quintic B-Spline Differential Quadrature Method

In this paper, quintic B-spline differential quadrature method (QBDQM) has been used to obtain the numerical approximation of the combined Korteweg-de Vries and modified Korteweg-de Vries equation (combined KdV-mKdV). The efficiency and effectiveness of the proposed method has been tested by computing the maximum error norm L_\infinity and discrete root mean square error L_2 . The newly found numerical approximations have been compared to available numerical approximations and this comparison has shown that the proposed method is an efficient one for solving

Trajectory planning of a redundant planar manipulator based on joint classification and particle swarm optimization algorithm

This paper presents a general method for the trajectory planning of the redundant planar manipulator. The mathematical relation between joint space and Cartesian space in a two-dimensional space is first derived. The joint classification is employed to obtain the solutions of corresponding joints. It divides joints into class I redundant joints, class II redundant joints, and nonredundant joints. The new application of knot points in the quintic B-spline curve is introduced to generate inverse solutions of class I redundant joints. Examples show that the number and distribution of knot points have a large effect on their solutions. Moreover, the particle swarm optimization algorithm is extended to generate solutions of class II redundant joints. It also optimizes the initial trajectories of joints and end-effector. Finally, solutions of nonredundant joints can be generated by the derived relation between the joint space and Cartesian space. The proposed methodology is confirmed by a case study. Under the same conditions, results show that the solution obtained by the extended method is not only better than that obtained by the particle swarm optimization algorithm but also closer to the global optimal solution.

A High-Order Accurate Numerical Algorithm for the Numerical Solution of Burgers’ Equation

In this study, a high accurate numerical solution of the Burgers’ equation is obtained. For this, the collocation method based on quintic B-spline functions for space discretization and the fourth-order single step method for time discretization are used. In order to see the efficiency of the algorithm, a test problem with an analytical solution is discussed and compared with the numerical solution obtained.

Quintic B-Spline Technique for Numerical Treatment of Third Order Singular Perturbed Delay Differential Equation

In this paper, a class of third order singularly perturbed delay differential equation with large delay is considered for numerical treatment. The considered equation has discontinuous convection-diffusion coefficient and source term. A quintic trigonometric B-spline collocation technique is used for numerical simulation of the considered singularly perturbed delay differential equation by dividing the domain into the uniform mesh. Further, uniform convergence of the solution is discussed by using the concept of Hall error estimation and the method is found to be of first-order convergent. The existence of the solution is also established. Computation work is carried out to validate the theoretical results.

A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options

This paper is concerned with the design of a high order numerical approach based on a uniform mesh for efficient numerical solution of time-fractional Black-Scholes equation, governing European options. The time-fractional derivative is defined in the Caputo sense. A collocation method based on quintic B-spline basis functions is used for space discretization and time-stepping is done using a backward Euler method. The stability and convergence of the method are analyzed. The method is shown to be unconditionally stable and fourth order accurate in space and (2−β) order accurate in time, where β is the order of the time fractional derivative (0<β<1). Two numerical examples with the known exact solutions are considered to validate theoretical results and demonstrate the accuracy of the method. Moreover, this scheme is used to price three different European options governed by a time-fractional Black-Scholes model; (i) European double barrier knock-out call option (ii) European call option and (iii) European put option. The effect of the order of fractional derivative on the option price is studied.