Collocation and finite difference-collocation methods for the solution of nonlinear Klein-Gordon equation

Two numerical techniques based on the finite difference and collocation methods are presented for the solution of nonlinear Klein-Gordon equation. The operational matrix of derivative for the cubic B-spline scaling functions is presented and is utilized to reduce the solution of nonlinear Klein-Gordon equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new techniques.     

Numerical solution of nonlinear system of second-order boundary value problems using cubic B-spline scaling functions

A numerical technique is presented for the solution of nonlinear system of second-order boundary value problems. This method uses the cubic B-spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B-spline scaling function. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.     

Numerical solution of fourth-order integro-differential equations using Chebyshev cardinal functions

A numerical technique is presented for the solution of fourth-order integro-differential equations. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results.     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

Chebyshev expansions for wave functions

Chebyshev series expansion of solutions of linear differential equations which occur in atomic scattering problems is discussed. We apply this technique to obtain both the regular and the irregular radial Coulomb wave functions. The Chebyshev expansion technique is extended to evaluate linearly independent solutions for the modified Coulomb potential. It is further shown that relativistic Coulomb wave functions may also be evaluated using Chebyshev expansion techniques.An advantage of this technique is that wave functions and their derivatives can be represented to a very high accuracy in terms of only a small number of Chebyshev expansion coefficients over a wide range of values of the independent variable. Moreover, in certain cases it is possible to evaluate matrix elements involving functions so represented by using properties of Chebyshev polynomials and thus avoiding numerical integration altogether.     

求解对流扩散方程的混合三次B-样条配点法

通过对三次B-样条和三次三角B-样条基函数引入权因子[ω],给出了对流扩散方程的混合三次B-样条配点法。对对流扩散方程空间离散采用混合三次B-样条配点法和时间离散采用向前有限差分,引入参数[θ],建立差分格式。对差分格式的稳定性进行分析,得到稳定性条件。数值实验表明所构造方法的有效性,并且适当调整权因子[ω]和参数[θ]的值,可提高计算的精度。     

Fast B-spline transforms for continuous image representation andinterpolation

Efficient algorithms for the continuous representation of a discrete signal in terms of B-splines (direct B-spline transform) and for interpolative signal reconstruction (indirect B-spline transform) with an expansion factor m are described. Expressions for the z-transforms of the sampled B-spline functions are determined and a convolution property of these kernels is established. It is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering. Fast computational algorithms based on the recursive implementations of these filters are proposed. A B-spline interpolator can also be characterized in terms of its transfer function and its global impulse response (cardinal spline of order n). The case of the cubic spline is treated in greater detail. The present approach is compared with previous methods that are reexamined from a critical point of view. It is concluded that B-spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner     

B-Spline Collocation Methods For Numerical Solutions Of The Rlw Equation

The numerical solution of the RLW equation is obtained by using a splitting up technique and both quadratic and cubic B-splines. Both quadratic and cubic B-spline collocation methods are applied to the resulting equation. Solutions without splitting the RLW equation are also obtained with the method of the cubic collocation method. Results are substantiated by studying propagation of a solitary wave and undular bore development. Comparison is made with results of the proposed schemes.     

二次均匀B样条曲线的扩展

为便于对均匀B样条曲线进行形状修改,利用二次均匀B样条基函数所需满足的条件,扩展二次均匀B样条基函数,构造出三次多项式调配函数．基于给出的调配函数,建立1种带形状参数的分段多项式曲线．调整形状参数可使三次多项式曲线在二次均匀B样条曲线两侧摆动．最后给出实例,构造出带局部调节参数G^1的连续曲线．该方法可以通过调整参数扩大二次均匀B样条曲线的调整范围．     

Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic conservation laws with the singular source by spectral Chebyshev collocation methods. The regularization is obtained by convolution with a high-order compactly supported Dirac-delta approximation whose overall accuracy is controlled by the number of vanishing moments, degree of smoothness and length of the support (scaling parameter). An optimal scaling parameter that leads to a high-order accurate representation of the singular source at smooth parts and full convergence order away from the singularities in the spectral solution is derived. The accuracy of the regularization and the spectral solution is assessed by solving an advection and Burgers equation with smooth initial data. Numerical results illustrate the enhanced accuracy of the spectral method through the proposed regularization.     

A statistical approach for economization of the polynomial functions

Due to having the minimax property, Chebyshev polynomials are used today to economize the arbitrary polynomial functions. In this work, we present a statistical approach to show that, contrary to current thought, the Chebyshev polynomials of the first kind are not appropriate for economizing these polynomials if one uses this statistical approach. In this way, a numerical results section is also given to clearly prove our claim.     

Numerical method using cubic B-spline for the heat and wave equation

In the present paper, the cubic B-splines method is considered for solving one-dimensional heat and wave equations. A typical finite difference approach had been used to discretize the time derivative while the cubic B-spline is applied as an interpolation function in the space dimension. The accuracy of the method for both equations is discussed. The efficiency of the method is illustrated by some test problems. The numerical results are found to be in good agreement with the exact solution.     

周期B样条基函数系数的并行算法

在现有周期B样条插值方法中,需要用迭代算法确定B样条基函数系数。针对现有方法的不足,建立B样条基函数系数的并行算法。首先构造周期区域的正交B样条基,得出正交B样条基函数系数的并行算法;进一步利用正交B样条基函数系数与B样条基函数系数的关系,得出B样条基函数系数的并行算法;最后推导二阶、三阶、四阶周期插值B样条基函数系数及插值点函数值的显式算式。实验证明了该方法在实现B样条基函数系数快速并行算法的同时保持了B样条基函数简单的函数关系。     

Computation of stationary points of distance functions

This paper presents an algorithm for computation of the stationary points of the squared distance functions between two point sets. One point set consists of a single space point, a rational B-spline curve, or a rational B-spline surface. The problem is reformulated in terms of solution of n polynomial equations with n variables expressed in the tensor product Bernstein basis. The solution method is based on subdivision relying on the convex hull property of the n-dimensional Bernstein basis and minimization techniques. We also cover classification of the stationary points of these distance functions, and include a method for tracing curves of stationary points in case the solution set is not zerodimensional. The distance computation problem is shown to be equivalent to the geometrically intuitive problem of computing collinear normal points. Finally, examples illustrate the applicability of the method     

A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems

Recently, Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891] proposed a numerical technique based on cubic B-spline for solving a Bratu-type problem. This method provides a second-order convergent approximation to the solution of the problem. In this paper, we develop a high-order numerical method based on quartic B-spline collocation approach for the Bratu-type and Lane–Emden problems. The error analysis of the quartic B-spline interpolation is carried out. Some numerical examples are provided to demonstrate the efficiency and applicability of the method and to verify its rate of convergence. The numerical results are compared with exact solutions and a numerical method based on cubic B-spline approach. Comparison reveals that our method produces more accurate results than the method proposed by Caglar et al. [B-spline method for solving Bratu's problem, Int. J. Comput. Math. 87(8) (2010), pp. 1885–1891].     

Computation of functions of Hamiltonian and skew-symmetric matrices

In this paper we consider numerical methods for computing functions of matrices being Hamiltonian and skew-symmetric. Analytic functions of this kind of matrices (i.e., exponential and rational functions) appear in the numerical solutions of ortho-symplectic matrix differential systems when geometric integrators are involved. The main idea underlying the presented techniques is to exploit the special block structure of a Hamiltonian and skew-symmetric matrix to gain a cheaper computation of the functions. First, we will consider an approach based on the numerical solution of structured linear systems and then another one based on the Schur decomposition of the matrix. Splitting techniques are also considered in order to reduce the computational cost. Several numerical tests and comparison examples are shown.     

Numerical approximation of inhomogeneous time fractional Burgers–Huxley equation with B-spline functions and Caputo derivative

A prototype model used to explain the relationship between mechanisms of reaction, convection effects, and transportation of diffusion is the generalized Burgers–Huxley equation. This study presents numerical solution of non-linear inhomogeneous time fractional Burgers–Huxley equation using cubic B-spline collocation method. For this purpose, Caputo derivative is used for the temporal derivative which is discretized by L1 formula and spatial derivative is interpolated with the help of B-spline basis functions, so the dependent variable is continuous throughout the solution range. The validity of the proposed scheme is examined by solving four test problems with different initial-boundary conditions. The algorithm for the execution of scheme is also presented. The effect of non-integer parameter \(\alpha \) and time on dependent variable is studied. Moreover, convergence and stability of the proposed scheme is analyzed, and proved that scheme is unconditionally stable. The accuracy is checked by error norms. Based on obtained results we can say that the proposed scheme is a good addition to the existing schemes for such real-life problems.

     

B样条函数在模糊系统中的应用

模糊系统的设计可看成是一类函数逼近问题, 从而可以利用数值逼近的方法来设计模糊系统. 本文将B样条函数引入到模糊系统的设计中, 构造了两类多输入单输出的B样条模糊系统, 并证明了它们均能逼近函数及其导函数. 仿真结果表明, 将两类B样条模糊系统应用到模糊系统建模和模糊控制器设计是可行的, 且在大多数情形下, 第1类B样条模糊系统的性能优于本文提到的其他模糊系统.     

Topology optimization using B-spline finite elements

Topology optimization algorithms using traditional elements often do not yield well-defined smooth boundaries. The computed optimal material distributions have problems such as ??checkerboard?? pattern formation unless special techniques, such as filtering, are used to suppress them. Even when the contours of a continuous density function are defined as the boundary, the solution can still have shape irregularities. The ability of B-spline elements to mitigate these problems are studied here by using these elements to both represent the density function as well as to perform structural analysis. B-spline elements can represent the density function and the displacement field as tangent and curvature continuous functions. Therefore, stresses and strains computed using these elements is continuous between elements. Furthermore, fewer quadratic and cubic B-spline elements are needed to obtain acceptable solutions. Results obtained by B-spline elements are compared with traditional elements using compliance as objective function augmented by a density smoothing scheme that eliminates mesh dependence of the solutions while promoting smoother shapes.     

Coefficient formula and matrix of nonuniform B-spline functions

The paper derives a coefficient formula of nonuniform B-spline functions of arbitrary degree from the Coxde Boor recursive algorithm. An efficient numerical algorithm for the coefficient matrix of nonuniform B-spline functions is also presented that is based on this formula. The results in the paper are useful for the evaluation and conversion of NURBS curves and surfaces in CAD/CAM systems. They will promote the application of product data-exchange standards in industry.