Quintic B-spline Method Research Articles

Numerical Simulation of Three Dimensional Chaotic System by A Quintic B-Spline Method

Numerical Simulation of Three Dimensional Chaotic System by A Quintic B-Spline Method

Analytical and Numerical solutions for fourth order Lane–Emden–Fowler equation

This model attracted attention due to its importance as it emerge in the modeling of many physical systems. We present an approximate solution for the fourth-order nonlinear Lane–Emden–Fowler equation using two simple and powerful methods, the first one is the Adomian decomposition method and the second one is the quintic B-spline method (QBSM). We transform the differential equation into an integral equation to overcome the singularity at the origin. Moreover, we discuss the convergence and the uniqueness of the three types of the model using the Adomian decomposition method. Finally, We present tables and graphs to make a comparison between the two methods and the exact solutions to exhibit the accuracy and efficiency of the proposed methods.

Multi-objective Trajectory Planning Method based on the Improved Elitist Non-dominated Sorting Genetic Algorithm

Robot manipulators perform a point-point task under kinematic and dynamic constraints. Due to multi-degree-of-freedom coupling characteristics, it is difficult to find a better desired trajectory. In this paper, a multi-objective trajectory planning approach based on an improved elitist non-dominated sorting genetic algorithm (INSGA-II) is proposed. Trajectory function is planned with a new composite polynomial that by combining of quintic polynomials with cubic Bezier curves. Then, an INSGA-II, by introducing three genetic operators: ranking group selection (RGS), direction-based crossover (DBX) and adaptive precision-controllable mutation (APCM), is developed to optimize travelling time and torque fluctuation. Inverted generational distance, hypervolume and optimizer overhead are selected to evaluate the convergence, diversity and computational effort of algorithms. The optimal solution is determined via fuzzy comprehensive evaluation to obtain the optimal trajectory. Taking a serial-parallel hybrid manipulator as instance, the velocity and acceleration profiles obtained using this composite polynomial are compared with those obtained using a quintic B-spline method. The effectiveness and practicability of the proposed method are verified by simulation results. This research proposes a trajectory optimization method which can offer a better solution with efficiency and stability for a point-to-point task of robot manipulators.

Quintic B-Spline Method for Solving Sharma Tasso Oliver Equation

When analysing the thermal conductivity of magnetic fluids, the traditional Sharma-Tasso-Olver (STO) equation is crucial. The Sharma-Tasso-Olive equation’s approximate solution is the primary goal of this work. The quintic B-spline collocation method is used for solving such nonlinear partial differential equations. The developed plan uses the collocation approach and finite difference method to solve the problem under consideration. The given problem is discretized in both time and space directions. Forward difference formula is used for temporal discretization. Collocation method is used for spatial discretization. Additionally, by using Von Neumann stability analysis, it is demonstrated that the devised scheme is stable and convergent with regard to time. Examining two analytical approaches to show the effectiveness and performance of our approximate solution.

Numerical simulations of Zakharov’s (ZK) non-dimensional equation arising in Langmuir and ion-acoustic waves

The trigonometric quintic B-spline scheme is used in this research paper to research Zakharov’s (ZK) nonlinear dimensional equation’s numerical solution. The ZK model’s solutions explain the relationship between the high-frequency Langmuir and the low-frequency ion-acoustic waves with many applications in optical fiber, coastal engineering, and fluid mechanics of electromagnetic waves, plasma physics, and signal processing. Three recent computational schemes (the expanded [Formula: see text]-expansion method, generalized Kudryashov method, and modified Khater method) have recently been used to investigate this model’s moving wave solution. Many innovative solutions have been established in this paper to determine the original and boundary conditions that allow numerous numerical schemes to be implemented. Here, the trigonometric quintic B-spline method is used to analyze the precision of the collected analytical solutions. To illustrate the precision of the numerical and computational solutions, distinct drawings are depicted.

Numerical approximation for inverse problem of the Ostrovsky-Burgers equation

This article considers a nonlinear inverse problem of the Ostrovsky-Burgers equation by using noisy data. This equation was appeared in modeling internal waves in the ocean or surface waves in a shallow channel with an uneven bottom under the effects of the interfacial friction. Two B-Splines with different levels, the quintic B-spline and septic B-spline, are used to study this problem. For both B-splines, convergence analysis is calculated and results show that an excellent estimation of the unknown functions of the nonlinear inverse problem. Indeed, by comparing the numerical results, we showed that the accuracy and stability of the septic B-spline method is more than those for the quintic B-spline method.

Trigonometric quintic B-spline collocation method for singularly perturbed turning point boundary value problems

A trigonometric quintic B-spline method is proposed for the solution of a class of turning point singularly perturbed boundary value problems (SP-BVPs) whose solution exhibits either twin boundary layers near both endpoints of the interval of consideration or an interior layer near the turning point. To resolve the boundary/interior layer(s) trigonometric quintic B-spline basis functions are used with a piecewise-uniform mesh generated with the help of a transition parameter that separates the layer and regular regions. The proposed method reduces the problem into a system of algebraic equations which can be written in matrix form with the penta-diagonal coefficient matrix. The well-known fast penta-diagonal system solver algorithm is used to solve the system. The method is shown almost fourth-order convergent irrespective of the size of the diffusion parameter ϵ. The theoretical error bounds are verified by taking some relevant test examples computationally.

On the numerical solution of the nonlinear Bratu type equation via quintic B-spline method

In this paper, the nonlinear Bratu type equation is numerically solved via quintic B-spline method. It is shown that the proposed technique has fourth order convergence. The efficiency and applicability of this method is confirmed by considering some examples. The proposed procedure provides better results in comparison to some existing methods.

Quintic B-Spline Method for Singularly Perturbed Fourth-Order Ordinary Differential Equations of Reaction–Diffusion Type

A quintic B-spline method has been proposed in this paper for the numerical solution of fourth order singularly perturbed two-point boundary value problems of reaction–diffusion type. Such types of problems arise in various fields of science and engineering, such as electrical network and vibration problems with large Peclet numbers, Navier–Stokes flows with large Reynolds numbers in the theory of hydrodynamics stability, reaction–diffusion process, quantum mechanics and optimal control theory etc. These types of problems are very important physically and difficult to solve because of boundary layers lies at the end point of domain. This method is directly applied to the problem without reducing the order of differential equation. The convergence analysis of the present method is also given and it has been proved to be of second order. The proposed method is applied on two numerical examples which verify theoretical estimates and numerical results are compared with the existing method. The numerical results are calculated with the help of MATLAB software.

Quintic B-spline method for numerical solution of fourth order singular perturbation boundary value problems

In this communication, we have studied an ecient numerical ap- proach based on uniform mesh for the numerical solutions of fourth order singular perturbation boundary value problems. Such type of problems arises in various elds of science and engineering, such as electrical network and vi- bration problems with large Peclet numbers, Navier-Stokes ows with large Reynolds numbers in the theory of hydrodynamics stability, reaction-diusion process, quantum mechanics and optimal control theory etc. In the present study, a quintic B-spline method has been discussed for the approximate solu- tion of the fourth order singular perturbation boundary value problems. The convergence analysis is also carried out and the method is shown to have con- vergence of second order. The performance of present method is shown through some numerical tests. The numerical results are compared with other existing method available in the literature.

Solution of a Class of Fourth Order Singular Singularly Perturbed Boundary Value Problems by Haar Wavelets Method and Quintic B-Spline Method

In this paper, Haar wavelet method is described for solving a class of fourth order singular singularly perturbed boundary value problems. Its efficiency is tested by solving two examples for which the exact solution is known. Numerical comparisons have been carried out to demonstrate the efficiency and the performance of the proposed method.

Quintic B-spline method for solving second order linear and nonlinear singularly perturbed two-point boundary value problems

In this paper, we have studied a numerical scheme to solve second order singularly perturbed two-point linear and nonlinear boundary value problems. The boundary layer of this type of problems exhibits at one end (left or right) point of the domain due to the presence of perturbation parameter ε. The quintic B-spline method and suitable piecewise uniform Shishkin mesh have been used. Linear and nonlinear second order singularly perturbed boundary value problems have been solved by the present method. The convergence analysis is also provided and the method is shown to have uniform convergence of fourth order. Numerical results have demonstrated the efficiency of the present method.

Quintic B-spline method for time-fractional superdiffusion fourth-order differential equation

The main objective of this paper is to obtain the approximate solution of superdiffusion fourth-order partial differential equations. Quintic B-spline collocation method is employed for fractional differential equations (FPDEs). The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order alpha, 1<alpha <2. The given problem is discretized in both time and space directions. Central difference formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.

Collocation method with quintic b-spline method for solving hirota-satsuma coupled KDV equation

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of KdV (CKdV) equation with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms, are computed. Three invariants of motion are predestined to determine the preservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term.

Solution of a class of fourth order singular singularly perturbed boundary value problems by quintic B-spline method

In this paper, quintic B-spline method is described for solving a class of fourth order singular singularly perturbed boundary value problems. This method is applied directly to the solution of the problems without reducing the order of the problems. The convergence analysis is also given and the method is shown to have uniform convergence of the second order. Numerical results are shown which demonstrate the efficiency of our method.

Quintic B-spline collocation method for fourth order partial integro-differential equations with a weakly singular kernel

Quintic B-spline collocation method for the numerical solution of fourth order partial integro-differential equations is considered. The scheme is based on the second order backward differential formula in the time direction and the quintic B-spline method for the spatial derivative. Some numerical experiments are included to demonstrate the effectiveness of the discrete technique. From the numerical experiments, we can find that the present scheme possesses high accuracy and efficiency, and the numerical results are in good agreement with the exact solutions.

B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations

In this work, we discuss two methods for solving a fourth order parabolic partial differential equation. In Method-I, we decompose the given equation into a system of second order equations and solve them by using cubic B-spline method with redefined basis functions. In Method-II, the equation is solved directly by applying quintic B-spline method with redefined basis functions. Stability of these methods have been discussed. Both methods are unconditionally stable. These methods are tested on four examples. The computed results are compared wherever possible with those already available in literature. We have developed Method-I for fourth order non homogeneous parabolic partial differential equation from which we can obtain displacement and bending moment both simultaneously, while Method-II gives only displacement. The results show that the derived methods are easily implemented and approximate the exact solution very well.