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MQSI is a Fortran 2003 subroutine for constructing monotone quintic spline interpolants to univariate monotone data. Using sharp theoretical monotonicity constraints, first and second derivative estimates at data provided by a quadratic facet model are refined to produce a univariate C2 monotone interpolant. Algorithm and implementation details, complexity and sensitivity analyses, usage information, a brief performance study, and comparisons with other spline approaches are included. [ABSTRACT FROM AUTHOR]

Mathematical model to account for non-homogeneity of plate material is designed, keeping in mind all the physical aspects, and analyzed by applying quintic spline technique for the first time. This method has been applied earlier for other geometry of plates which shows its utility. Accuracy and versatility of the technique are established by comparing with the well-known existing results. Effect of quadratic thickness variation, an exponential variation of non-homogeneity in the radial direction, and variation in density; for the three different outer edge conditions namely clamped, simply supported and free have been computed using MATLAB for the first three modes of vibration. For all the three edge conditions, normalized transverse displacements for a specific plate have been presented which shows the shiftness of nodal radii with the effect of taperness. [ABSTRACT FROM AUTHOR]

Abstract Approximate curve is constructed using quadratic, quintic, and cubic splines and examination between these splines. The point of this construction is to predict sediment yield index (SYI) corresponding to curve number. This strategy is outlined with a contextual analysis of Manot watershed of Narmada Basin, India. The relation among calculated SYI and observed SYI esteems is associated with a coefficient of determination (R 2) of 0.36 and 0.48 for the corresponding quadratic and quintic splines, while the cubic spline showed R 2 of 0.87 (Meshram et al. Arab J Geosci 10:155–168, 2017b; Appl Water Sci 7:1773–1779, 2017c). Numerical results seemed to indicate that the cubic spline method is more accurate than the quadratic/quintic spline method.

In this paper, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems of the reaction–diffusion equation type. The convergence analysis is discussed and the method is shown to have fourth order convergence. To validate the applicability of the method, two model examples have been solved for different values of the perturbation parameter and mesh sizes. The numerical results have been tabulated and also presented in graphs. It can be observed from the results that the present method approximates the exact solution very well.

In this paper, we tackle the numerical treatment of a fourth-order fractional diffusion-wave problem. By using parametric quintic spline in the spatial dimension and an approximation of Caputo derivatives at half-points, we propose a numerical scheme and rigorously prove its solvability, convergence and stability in maximum norm. It is shown that the theoretical convergence order improves those of earlier work. To confirm, simulation is carried out to demonstrate the numerical efficiency of the proposed scheme as well as the better performance over other methods. [ABSTRACT FROM AUTHOR]

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.

In this paper, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems of the reaction–diffusion equation type. The convergence analysis is discussed and the method is shown to have fourth order convergence. To validate the applicability of the method, two model examples have been solved for different values of the perturbation parameter and mesh sizes. The numerical results have been tabulated and also presented in graphs. It can be observed from the results that the present method approximates the exact solution very well. [ABSTRACT FROM AUTHOR]

In this paper, we study a new method for approximating a function and its derivatives from the given integral values over the grid intervals and two given first order derivatives at boundary grid nodes by using four times continuously differentiable quintic integro-differential spline. The method mainly includes two steps: constructing three extra values and then solving a linear system. It is easy to implement. We analyze the approximation properties of the spline. We prove that the spline and its first four derivatives are able to approximate the original function and its first four derivatives respectively with high accuracy. To examine the theoretical results, we perform some numerical tests and also compare the method with some other methods. Numerical results show that the new method is very effective and practical. [ABSTRACT FROM AUTHOR]

The spline collocation method is a competent and highly effective mathematical tool for constructing the approximate solutions of boundary value problems arising in science, engineering and mathematical physics. In this paper, a quintic polynomial spline collocation method is employed for a class of fractional boundary value problems (FBVPs). The FBVPs are expressed in terms of Caputo’s fractional derivative in this approach. The consistency relations are derived in order to compute the approximate solutions of FBVPs. Finally, numerical results are given, which demonstrate the effectiveness of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, to overcome the innate drawbacks of some old methods, we present a new quintic spline method for integro interpolation. The method is free of any exact end conditions, and it can reconstruct a function and its first order to fifth order derivatives with high accuracy by only using the given integral values of the original function. The approximation properties of the obtained integro quintic spline are well studied and examined. The theoretical analysis and the numerical tests show that the new method is very effective for integro interpolation. [ABSTRACT FROM AUTHOR]

In this paper we shall develop a numerical scheme for a fractional sub-diffusion problem using parametric quintic spline. The solvability, convergence and stability of the scheme will be established and it is shown that the convergence order is higher than some earlier work done. We also present some numerical examples to illustrate the efficiency of the numerical scheme as well as to compare with other methods. [ABSTRACT FROM AUTHOR]

In this paper, we study an effective quintic polynomial spline method for numerical solution, and first order to fifth order numerical derivatives of the analytic solution at the knots for a class of sixth order two-point boundary value problems. Our new method is based on a quintic spline interpolation problem. It is easy to implement and is able to provide sixth order accurate numerical results at the knots. Numerical tests show that our method is very practical and effective. [ABSTRACT FROM AUTHOR]

In this paper, we study two new spline methods for the Birkhoff interpolation problem proposed in Costabile and Longo (Calcolo 47:49-63, ). Our methods are based on cubic spline and quintic spline respectively. The new methods are easy to be implemented. They are effective not only in approximating the original function but also in approximating its derivatives. The respective remainders and error bounds are given. Numerical tests are performed. Numerical results confirm the theoretical analysis and show that our methods are very practical. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

In this paper, a new difference scheme based on C¹-quintic splines is derived for the numerical solution of the stiff delay differential equations. Convergence results shows that the methods have a convergence of order five. Moreover, the stability analysis properties of these methods have been studied. Finally, numerical results illustrating the behavior of the methods when faced with some difficult problems are presented. [ABSTRACT FROM AUTHOR]

In this article, we consider various boundary conditions for interpolation of quintic splines of defect 1 on a uniform mesh. We obtain an asymptotic representation of the approximation error for the spline for different boundary conditions. Boundary conditions that are optimal by approximation accuracy are found. [ABSTRACT FROM AUTHOR]

Abstract: Methods of order two, four and six based on Exponential spline functions consisting of a polynomial part of degree three and an exponential part is developed to find approximation of linear and nonlinear fourth order two point boundary value problems. It is shown that the free parameter k of the exponential part can be used to raise the order of accuracy of the new scheme. Convergence analysis of these methods is shown along with numerical examples each for linear and nonlinear are included to illustrate the practical usefulness of our methods. [Copyright &y& Elsevier]

Wang in [1] solved an open problem described by Farm in [2], he constructed a C2 quintic spline interpolation scheme on a refined triangulation. In this paper, we would like to improve Wang's study by enforcing some additional smoothness conditions across the interior edges of Wang's refinement of triangulation Δ. The results imply that our improvement interpolation scheme is more effective than other counterparts. [ABSTRACT FROM AUTHOR]

We develop a class of methods for the numerical solution of singularly perturbed two-point boundary-value problems. Using non-polynomial quintic spline in grid points we can obtain the fourth-order method only. But we develop the quintic spline in off-step points to raise the order of accuracy. Based on such spline, the purposed methods are fourth, sixth, and eighth-order accurate. These methods are applicable to problems both in singular and non-singular cases. The convergence analysis of the fourth-order method is stablished. Numerical results are given to illustrate the efficiency of our methods. [ABSTRACT FROM AUTHOR]

In the present paper we describe a survey on recent spline techniques for solving boundary value problems in ordinary differential equations. Here we discuss the summary of the articles between 2000 and 2007 based on cubic, quintic, and sextic splines. Comparisons of methods with our own critical comments as remarks have been included. [ABSTRACT FROM AUTHOR]

We develop a numerical method for computing smooth approximations to the solution of a system of second-order boundary value problems associated with obstacle, unilateral and contact problems based on uniform mesh quintic splines. It is shown that this method gives better approximations than those produced by other collocation, finite-difference and spline methods. A numerical example is given to illustrate the applicability of the new method. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a synthesis method of designing flexible cam profiles by using smoothing spline curves. Our approach simultaneously considers displacement, velocity and acceleration rather than displacement alone in the design process. The inherent characteristic of the smoothing spline ensures the smoothness of cam profiles designed by the proposed method. The essential difference between the proposed method and other existing approaches is that our method allows designers to refine the cam motion by local modifications while still satisfying profile smoothness and discrete constraints. Particularly, the method can control not only the level but also the shape of the velocity and acceleration curves for better kinematical and dynamic properties. Examples show that the obtained cam profiles provide solutions for some problems in automotive engine valve trains. [Copyright &y& Elsevier]

Abstract: There are few techniques available to numerically solve sixth-order boundary value problems with two point boundary conditions. In this paper, quintic spline method is developed for the numerical solutions of linear special case sixth-order boundary value problems. Two examples are considered for the numerical illustration of the method developed. The method is also compared with those developed by EL-Gamel et al. [Mohamed EL-Gamel, John R. Cannon, Ahmed I. Zayed, Sinc–Galerkin method for solving sixth-order BVPs, Math. Comput. 73 (247) (2003)] and Wazwaz [A. Wazwaz, The numerical solution of sixth-order boundary value problems by the modified decomposition method, Appl. Math. Comput. 118 (2001) 311–325]. The comparison shows that the quintic spline method is more efficient and effective tool and yields better results. [Copyright &y& Elsevier]

Abstract: In this paper, we apply Adomian decomposition method (shortly, ADM) to develop a fast and accurate algorithm of a special second-order ordinary initial value problems. The ADM does not require discretization and consequently of massive computations. This paper is particularly concerned with the ADM and the results obtained are compared with previously known results using the Quintic C 2-spline integration methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented. [Copyright &y& Elsevier]

A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the question of convergence of this approach is completely guaranteed here, because a limited number of term included in the series can describe a sufficient exact solution. Comparisons with the solutions of the quintic spline, and finite difference are presented. [ABSTRACT FROM AUTHOR]

Abstract: Quintic spline procedure for the solution of a fourth-order boundary value problem is developed, spline relations are derived, boundary condition formula of orders O(h 6) and O(h 8) are formulated. The convergence of prescribed method is proved, the matrix associated with the system of linear equations that arises is not required to be monotone, the only requirement is that f(x) be positive. Numerical illustration are given. [Copyright &y& Elsevier]

Abstract: This survey paper contains a large amount of material and indeed can serve as an introduction to some of the ideas and methods for the solution of ordinary and partial differential equations starting from Schoenberg’s work [Quart. Appl. Math. 4 (1946) 345–369]. The parametric spline function which depends on a parameter ω >0, is reduces to the ordinary cubic or quintic spline for ω =0. A note on parametric spline function approximation, which is special case of this work has been published in [Comp. Math. Applics. 29 (1995) 67–73]. This article deals with the odd-order parametric spline relations. [Copyright &y& Elsevier]

Abstract: Singularly perturbed self-adjoint boundary-value problems are considered in this article. A difference scheme based on quintic spline is proposed. This scheme is applied to the subproblems obtained from the given problem by dividing the whole domain into nonoverlapping subdomains. The proposed scheme is of fourth-order convergent and more suitable for parallel computers. Stability and convergence of the method are discussed. Numerical examples are provided to show the efficiency and accuracy. [Copyright &y& Elsevier]

This paper presents a new motion planning primitive to be used for the iterative steering of vision-based autonomous vehicles. This primitive is a parameterized quintic spline, denoted as η-spline, that allows interpolating an arbitrary sequence of points with overall second-order geometric (G2-) continuity. Issues such as completeness, minimality, regularity, symmetry, and flexibility of these G2-splines are addressed in the exposition. The development of the new primitive is tightly connected to the inversion control of nonholonomic car-like vehicles. The paper also exposes a supervisory strategy for iterative steering that integrates feedback vision data processing with the feedforward inversion control [ABSTRACT FROM PUBLISHER]