B-Spline Collocation Method for Solving Singularly Perturbed Boundary Value Problems

We use fifth order B-spline functions to construct the numerical method for solving singularly perturbed boundary value problems. We use B-spline collocation method, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problems. The numerical results are found in good agreement with exact solutions.

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.

An efficient cubic trigonometric B-spline collocation scheme for the time-fractional telegraph equation

In this paper,a proficient numerical technique for the time-fractional telegraph equation(TFTE)is proposed.The chief aim of this paper is to utilize a relatively new type of B-spline called the cubic trigonometric B-spline for the proposed scheme.This technique is based on finite difference formulation for the Caputo time-fractional derivative and cubic trigonometric B-splines based technique for the derivatives in space.A stability analysis of the scheme is presented to confirm that the errors do not amplify.A convergence analysis is also presented.Computational experiments are carried out in addition to verify the theoretical analysis.Numerical results are contrasted with a few present techniques and it is concluded that the presented scheme is progressively right and more compelling.

Wavelet Collocation Method for Solving Elliptic Singularly Perturbed Problem

Wavelet collocation method is used to solve an elliptic singularly perturbed problem with two parameters. The B-spline function is used as a single mother wavelet, which leads to a tri-diagonal linear system. The accuracy of the proposed method is demonstrated by test problem and the result shows the reliability and efficiency of the method.

Collocation Method for Solving the Generalized KdV Equation

In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including;single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, L2 and L∞ and invariants I1, I2 and I3 have been calculated. Our numerical results are compared with some of those available in the literature.

Numerical Solution of the Seventh Order Boundary Value Problems Using B-Spline Method

We develop a numerical method for solving the boundary value problem of The Linear Seventh Ordinary Boundary Value Problem by using the seventh-degree B-Spline function. Formulation is based on particular terms of order of seventh order boundary value problem. We obtain Septic B-Spline formulation and the Collocation B-spline method is formulated as an approximation solution. We apply the presented method to solve an example of seventh order boundary value problem in which the result shows that there is an agreement between approximate solutions and exact solutions. Resulting in low absolute errors shows that the presented numerical method is effective for solving high order boundary value problems. Finally, a general conclusion has been included.

Trigonometric tension B-spline collocation approximations for time fractional Burgers’equation

This manuscript’s aim is to form and examine the numerical simulation of Caputo-time fractional nonlinear Burgers’equation via collocation approach with trigonometric tension B-splines as base functions.First,L 1 discretization formula is utilized for the time fractional derivative and after linearizing the nonlinear term,the trigonometric tension B-spline interpolants are utilized to get a system of simultaneous linear equations that are solved via Gauss elimination method.Thus,numerical approximation at the desired time level is obtained.It is demonstrated via von-Neumann approach that proposed scheme produces stable solutions.The results of six different test examples having their analytical solutions are compared with the results in the literature to validate the accuracy and efficiency of the scheme.

Numerical treatment of Benjamin-Bona-Mahony-Burgers equation with fourth-order improvised B-spline collocation method

In this work,the Benjamin-Bona-Mahony-Burgers(BBMB)equation is solved using an improvised cubic B-spline collocation technique.This equation describes the propagation of small amplitude waves in a non-linear dispersive medium,in the modeling of unidirectional planar waves.Due to the higher smoothness and sparse nature of matrices corresponding to splines,cubic B-splines are chosen as the basis function in the collocation method.But,the optimal accuracy and order of convergence cannot be achieved using the standard B-spline collocation method.So to overcome this,improvised cubic B-splines are formed by making posteriori corrections to cubic B-spline interpolant and its higher-order derivatives.The Crank-Nicolson scheme is used to discretize the temporal domain along with the quasilinearization process to deal with the nonlinear terms.The spatial domain discretization is carried out using the improvised cubic B-spline collocation method(ICSCM).The stability analysis of the technique is performed using the von-Neumann scheme.Several test problems are solved numerically and obtained results are compared with the results available in the literature.The aim of the paper is to show that such improvised techniques which were earlier used to solve ODEs,can be applied to solve the BBMB equation also,with excellent accuracy in results.

Septic B-Spline Solution of Fifth-Order Boundary Value Problems

A numerical method based on septic B-spline function is presented for the solution of linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization technique to reduce the nonlinear problems to linear problems and use B-spline collocation method, which leads to a seven nonzero bands linear system. Illustrative example is included to demonstrate the validity and applicability of the proposed techniques.

Numerical study of self-adjoint singularly perturbed two-point boundary value problems using collocation method with error estimation

A numerical treatment for self-adjoint singularly perturbed second-order two-point boundary value problems using trigonometric quintic B-splines is presented,which depend on different engineering applications.The method is found to have a truncation error of O(h 6)and converges to the exact solution at O(h 4).The numerical examples show that our method is very effective and the maximum absolute error is acceptable.

A numerical technique based on collocation method for solving modified Kawahara equation

In this article,a numerical solution of the modified Kawahara equation is presented by septic B-spline collocation method.Applying the von-Neumann stability analysis,the present method is shown to be unconditionally stable.L 2 and L∞error norms and conserved quantities are given at selected times.The accuracy of the proposed method is checked by test problems including motion of the single solitary wave,interaction of solitary waves and evolution of solitons.

五次B样条配置法求解广义KdV方程

采用有限元方法进行空间离散,构造了求一维非线性广义KdV方程孤立波解的五次B样条配置法.数值计算了p=1,p=2和p=3时该方程的孤立波解,从结果来看,它们满足该方程的守恒律.

变系数波动方程的五次样条配置法

在时间方向使用差分格式,在空间方向以五次样条插值函数作为基函数,对带有变系数的波动方程进行数值求解研究,提出了一种五次样条配置法。从理论上分析了该数值算法的截断误差,并以数值例子验证了该算法的实际可行性和数值精度。

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular kernel.Cubic trigonometric B-spline(CTBS)functions are used for interpolation in both methods.The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations.The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values.An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method.An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method.The methods are tested with three nonhomogeneous problems for their validation.Stability,computational efficiency and numerical convergence of the methods are analyzed.Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made.Convection and diffusion dominant cases are discussed in terms of Peclet number.The results are also compared with cubic B-spline collocation method.

Multi-objective optimal trajectory planning for manipulators based on CMOSPBO

Feasible,smooth,and time-jerk optimal trajectory is essential for manipulators utilized in manufacturing process.A novel technique to generate trajectories in the joint space for robotic manipulators based on quintic B-spline and constrained multi-objective student psychology based optimization(CMOSPBO)is proposed in this paper.In order to obtain the optimal trajectories,two objective functions including the total travelling time and the integral of the squared jerk along the whole trajectories are considered.The whole trajectories are interpolated by quintic B-spline and then optimized by CMOSPBO,while taking into account kinematic constraints of velocity,acceleration,and jerk.CMOSPBO mainly includes improved student psychology based optimization,archive management,and an adaptiveε-constraint handling method.Lévyflights and differential mutation are adopted to enhance the global exploration capacity of the improved SPBO.Theεvalue is varied with iterations and feasible solutions to prevent the premature convergence of CMOSPBO.Solution density estimation corresponding to the solution distribution in decision space and objective space is proposed to increase the diversity of solutions.The experimental results show that CMOSPBO outperforms than SQP,and NSGA-II in terms of the motion efficiency and jerk.The comparison results demonstrate the effectiveness of the proposed method to generate time-jerk optimal and jerk-continuous trajectories for manipulators.