一类常微分方程的非多项式样条函数逼近解

本文提出一种新的数值方法来获得三阶带障碍边值问题的常微分方程的数值解,这类方法的基函数包括三角函数、指数函数、多项式函数。本文将给出一数值例子说明这种方法优于其他差分法、配置法、多项式样条函数法。     

Sextic spline method for the solution of a system of obstacle problems

We use sextic spline function to develop numerical method for the solution of system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the approximate solutions obtained by the present method are better than those produced by other collocation, finite difference and spline methods. A numerical example is given to illustrate practical usefulness of our method.     

Numerical solutions for system of second order boundary value problems

We investigate some numerical methods for computing approximate solutions of a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that cubic spline method gives approximations which are better than that computed by higher order spline and finite difference techniques.     

Modified Numerov method for solving system of second-order boundary-value problems

We introduce and discuss a new numerical method for solving system of second order boundary value problems, where the solution is required to satisfy some extra continuity conditions on the subintervals in addition to the usual boundary conditions. We show that the present method gives approximations which are better than that produced by other collocation, finite difference and spline methods. Numerical example is presented to illustrate the applicability of the new method.     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

Quintic nonpolynomial spline solutions for fourth order two-point boundary value problem

In this paper, we develop quintic nonpolynomial spline methods for the numerical solution of fourth order two-point boundary value problems. Using this spline function a few consistency relations are derived for computing approximations to the solution of the problem. The present approach gives better approximations and generalizes all the existing polynomial spline methods up to order four. This approach has less computational cost. Convergence analysis of these methods is discussed. Two numerical examples are included to illustrate the practical usefulness of our methods.     

Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary-Value Problems

In this paper, we use parametric quintic splines to derive some consistency relations which are then used to develop a numerical method for computing the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Numerical evidence is presented to show the applicability and superiority of the new method over other collocation, finite difference, and spline methods.     

Family of numerical methods based on non-polynomial splines for solution of contact problems

In this paper a new technique based on quartic non-polynomial spline functions connecting spline functions values at mid knots and their corresponding values of the fourth-order derivatives is developed. This approach leads to a family of numerical methods for computing approximations to the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is shown that the present family of methods gives better approximations. Existing second and fourth-order finite-difference and spline functions based methods developed at mid knots become special cases of the new approach. Numerical examples are given to illustrate applicability and efficiency of the new methods.     

B‐spline spectral method for constrained fractional optimal control problems

In this paper, we present a direct B‐spline spectral collocation method to approximate the solutions of fractional optimal control problems with inequality constraints. We use the location of the maximum of B‐spline functions as collocation points, which leads to sparse and nonsingular matrix B whose entries are the values of B‐spline functions at the collocation points. In this method, both the control and Caputo fractional derivative of the state are approximated by B‐spline functions. The fractional integral of these functions is computed by the Cox‐de Boor recursion formula. The convergence of the method is investigated. Several numerical examples are considered to indicate the efficiency of the method.     

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors.