Quartic Spline Research Articles

Quartic Non-Polynomial Spline Method for Singularly Perturbed Differential-difference Equation with Two Parameters

Quartic non-polynomial spline method is presented to solve the singularly perturbed differential-difference equation containing two parameters. The considered equation is transformed into an asymptotical equivalent differential equation, and the derivatives are replaced finite difference approximation using the quartic non-polynomial spline method. The convergence analysis of the method has been established. Numerical experimentation is carried out on model examples, and the results are presented both in tables and graphs. Furthermore, the present method gives a more accurate solution than some existing methods reported in the literature.

A New Nonparametric Estimate of the Risk-Neutral Density with Applications to Variance Swaps

Estimates of risk-neutral densities of future asset returns have been commonly used for pricing new financial derivatives, detecting profitable opportunities, and measuring central bank policy impacts. We develop a new nonparametric approach for estimating the risk-neutral density of asset prices and reformulate its estimation into a double-constrained optimization problem. We evaluate our approach using the S&P 500 market option prices from 1996 to 2015. A comprehensive cross-validation study shows that our approach outperforms the existing nonparametric quartic B-spline and cubic spline methods, as well as the parametric method based on the normal inverse Gaussian distribution. As an application, we use the proposed density estimator to price long-term variance swaps, and the model-implied prices match reasonably well with those of the variance future downloaded from the Chicago Board Options Exchange website.

Quartic Non-Polynomial Spline for Solving the Third-Order Dispersive Partial Differential Equation

In the present paper, we introduce a non-polynomial quadratic spline method for solving third-order boundary value problems. Third-order singularly perturbed boundary value problems occur frequently in many areas of applied sciences such as solid mechanics, quantum mechanics, chemical reactor theory, Newtonian fluid mechanics, optimal control, convection-diffusion processes, hydrodynamics, aerodynamics, etc. These problems have various important applications in fluid dynamics. The procedure involves a reduction of a third-order partial differential equation to a first-order ordinary differential equation. Truncation errors are given. The unconditional stability of the method is analysed by the Von-Neumann stability analysis. The developed method is tested with an illustrated example, and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, a graphical comparison between analytical and approximate solutions is also shown for the illustrated example.

Optimal properties for deficient quartic splines of Marsden type

In this work, we obtain an improved error estimate in the interpolation with the Hermite \(C^{2}\)-smooth deficient complete quartic spline that has the distribution of nodes following the Marsden type scheme and investigate the possibilities to compute the derivatives on the knots such that the obtained spline \(S\in C^{1}[a,b]\) has minimal curvature and minimal \(L^{2}\)-norm of \(S^{\prime }\) and \(S^{\prime \prime \prime }\). In each case, the interpolation error estimate is performed in terms of the modulus of continuity.%MCEPASTEBIN%

Numerical simulation of Rayleigh-Bénard convection and three-phase Rayleigh-Taylor instability using a modified MPS method

The main objective of the current work is to enhance consistency and capabilities of Moving Particle Semi-implicit (MPS) method for simulating a wide range of free-surface flows and convection heat transfer. For this purpose, two novel high-order gradient and Laplacian operators are derived from the Taylor series expansion and are applied for the discretization of governing equations. Furthermore, the combination of the explicit Third-order TVD Runge-Kutta scheme and two-step projection algorithm is employed to approximate transient terms in the Navier-stokes and energy equations. To further improve the accuracy and performance of the method, a new kernel function is constructed by a combination of the Gaussian and cosine functions and then implemented for modeling the 1D Sod shock tube problem. Validation and verification of the proposed model are conducted through the simulations of several canonical test cases such as: dam break, rotation of a square patch of fluid, two-phase Rayleigh-Taylor instability, oscillating concentric circular drop and good agreement are achieved. The proposed model is then employed to simulate three-phase Rayleigh-Taylor instability and entropy generation due to natural convection heat transfer (Differentially Heated Cavity and Rayleigh-Bénard convection). The obtained results reveal that, the newly constructed kernel function provides more reliable results in comparison with two frequently used kernel functions namely; quartic spline and Wendland. Furthermore, it is found that, the enhanced MPS model is capable of handling multiphase flow problems with low and high density contrast.

On the minimal derivative oscillation for deficient quartic splines of Marsden’s type

In this note we investigate the possibility to minimize the derivative oscillation for deficient quartic splines of Marsden’s type, motivated by the purpose to reduce the excessive oscillations near midpoints. The error estimate of the obtained quartic spline interpolant is derived considering a comparison with the cases of the smooth quartic spline having continuous second derivative and the quartic spline that has minimal Euclidean norm of the second derivative.

Scattered Data Interpolation Using Quartic Triangular Patch for Shape-Preserving Interpolation and Comparison with Mesh-Free Methods

Scattered data interpolation is important in sciences, engineering, and medical-based problems. Quartic Bézier triangular patches with 15 control points (ordinates) can also be used for scattered data interpolation. However, this method has a weakness; that is, in order to achieve C 1 continuity, the three inner points can only be determined using an optimization method. Thus, we cannot obtain the exact Bézier ordinates, and the quartic scheme is global and not local. Therefore, the quartic Bézier triangular has received less attention. In this work, we use Zhu and Han’s quartic spline with ten control points (ordinates). Since there are only ten control points (as for cubic Bézier triangular cases), all control points can be determined exactly, and the optimization problem can be avoided. This will improve the presentation of the surface, and the process to construct the scattered surface is local. We also apply the proposed scheme for the purpose of positivity-preserving scattered data interpolation. The sufficient conditions for the positivity of the quartic triangular patches are derived on seven ordinates. We obtain nonlinear equations that can be solved using the regula-falsi method. To produce the interpolated surface for scattered data, we employ four stages of an algorithm: (a) triangulate the scattered data using Delaunay triangulation; (b) assign the first derivative at the respective data; (c) form a triangular surface via convex combination from three local schemes with C 1 continuity along all adjacent triangles; and (d) construct the scattered data surface using the proposed quartic spline. Numerical results, including some comparisons with some existing mesh-free schemes, are presented in detail. Overall, the proposed quartic triangular spline scheme gives good results in terms of a higher coefficient of determination (R2) and smaller maximum error (Max Error), requires about 12.5% of the CPU time of the quartic Bézier triangular, and is on par with Shepard triangular-based schemes. Therefore, the proposed scheme is significant for use in visualizing large and irregular scattered data sets. Finally, we tested the proposed positivity-preserving interpolation scheme to visualize coronavirus disease 2019 (COVID-19) cases in Malaysia.

A time-domain method for load identification using moving weighted least square technique

Based on the thought of Green’s kernel function method (GKFM), an improved time-domain load identification method using moving weighted least square technique (MWLST) which can accurately fit dynamic load is proposed. Better than the traditional shape function method using moving least square fitting (SFM_MLSF), the proposed method considers continuity and correlation of dynamic load between two adjacent sampling points, and involves the weighted contribution of sampling points to the fitting point. In numerical examples, Gauss, Cubic and Quartic spline weight functions are utilized in the proposed method to realize the reconstruction of kernel matrix. It is found that the accuracies of load identification are almost same when their optimum supported domain radii are adopted. Furthermore, the numerical results illustrate that the proposed method can identify dynamic load more accurately and smoothly than GKFM and SFM_MLSF significantly by the same regularization method for ill-posedness, and the proposed method has excellent stability and robustness. Additionally, a special technique combining both the whole identification and the truncated-processing identification is proposed to identify external dynamic loads during hoisting process, which solves the oscillation problem caused by using inversing methods directly.

Convergence of Quartic Interpolating Splines

The problem of interpolation by quartic splines according to Marsden’s scheme is considered. It is shown that the calculation of an interpolating spline in terms of the coefficients of expansion of its second derivative in L1-normalized quadratic B-splines yields a system of linear equations for the chosen parameters. The matrix of the system is pentadiagonal and has a column diagonal dominance, which makes it possible to efficiently calculate the required parameters and establish the convergence of the spline interpolation process according to Marsden’s scheme for any function from the class C1 on an arbitrary sequence of grids without any constraints. In Marsden’s scheme, it is assumed that a knot grid is given and the interpolation nodes are chosen strictly in the middle. The established results are transferred to the case of interpolation by quartic splines according to Subbotin’s scheme (the data grid and knot grid are swapped). Here the system of equations for the coefficients of expansion of the third derivative in L∞-normalized B-splines has a diagonal dominance, and the interpolation process converges for any interpolated function from the class C3.

An Efficient Radial Basis Function Meshfree Local Petrov–Galerkin Method for Modeling the Unidirectional Fully Developed Fluid Flow

In hydrodynamics the applications range of incompressible flows is very wide. In this study, a robust, high order modeling approach is introduced, based on the MLPG meshfree method -based radial basis functions (RBF- MLPG) method, for solving the incompressible flow field. In other words a MLPG meshfree method based on an interpolation function is presented to solve the 2‐D unsteady incompressible fully developed fluid flows. This meshfree method is based on the quartic (4^th order) spline. The method is then compared against the Finite Element Method on a test case of unidirectional fully developed incompressible fluid. The performance of this weight function proved that the quartic (4th order) spline gains the highest accuracy, convergence and efficiency. Finally, it can be concluded that the presented method is formidable for simulating fluid dynamics.

Application of Non Polynomial Spline to Solve the Third Order Boundary Value Problems

In this paper we used quartic non-polynomial splines to improve a new numerical method for computing approximations to the solution of third order boundary value problems, and shown the new method gives approximations, which are better than those produced by other collocation, finite-difference, and spline methods. Convergence analysis of the method is discussed through standard procedures. A numerical example is given here to illustrate the efficiency and applicability of the novel method.

Spline-RRT⁎: Coordinated Motion Planning of Dual-Arm Space Robot

Abstract This paper addresses the coordinated motion planning issue for a dual-arm free-floating space robot. Based on the sampling-based planning method, a novel coordinated RRT⁎-based path planning framework is proposed for a dual-arm manipulation. First, an asymptotically optimal sampling-based method, RRT⁎, is employed to generate the initial rough path for each end-effector in the task-space. To avoid self-collision, we present a coordinated strategy which samples from separated inertial spaces when performing RRT⁎ algorithm. Second quartic, splines are used to smooth the generated RRT⁎ path so that the robot can execute smoothly. Physical constraints including the end-effectors’ limit, joint limit and boundary conditions are all controlled within the design of the quartic splines. The effectiveness of the proposed path planning framework is illustrated and demonstrated via a kinematically redundant dual-arm space robot.

An Improved Empirical Mode Decomposition Based on Adaptive Weighted Rational Quartic Spline for Rolling Bearing Fault Diagnosis

As a powerful time-frequency signal analysis technique, empirical mode decomposition (EMD) has been commonly applied in fault diagnosis. However, the cubic spline-curve often causes outstanding over and undershoot problem, which significantly limits the performance of conventional EMD. To address this problem, an improved EMD (I-EMD) based on adaptive weighted rational quartic spline is proposed. Firstly, the original cubic spline interpolation in conventional EMD is replaced with the weighted rational quartic spline interpolation (WRQSI) which has two adjustable shape control parameters. Secondly, a novel parameter selection criterion termed envelope characteristic frequency ratio (ECFR) is designed to guide the construction of the optimal local envelope. And simulation analysis proved that ECFR is not only sensitive to the characteristic information but also robust to the weak noise and sudden impulses. Subsequently, the optimal shape control parameter can be searched by grasshopper optimization algorithm (GOA) using the maximum ECFR as the objective function. Then the sensitive modes selected via weighted Kurtosis index are employed for further Hilbert envelope spectrum analysis. Finally, two case studies on rolling bearing fault diagnosis are constructed to verify the rationality and effectiveness of the I-EMD method. The results show that I-EMD method can evidently solve the over and undershoot problem and restrain the mode mixing phenomenon. Moreover, I-EMD also performs better fault feature extraction ability under the same conditions compared with EMD, VMD and CEEMDAN. So it is expected that I-EMD will serve as a potential improvement for signal processing, fault feature extraction and fault diagnosis.

Positivity preserving interpolation by using rational quartic spline

In this study, a new scheme for positivity preserving interpolation is proposed by using C1 rational quartic spline of (quartic/quadratic) with three parameters. The sufficient condition for the positivity rational quartic interpolant is derived on one parameter meanwhile the other two are free parameters for shape modification. These conditions will guarantee to provide positive interpolating curve everywhere. We tested the proposed positive preserving scheme with four positive data and compared the results with other established schemes. Based on the graphical and numerical results, we found that the proposed scheme is better than existing schemes, since it has extra free parameter to control the positive interpolating curve.

SHAPE PRESERVING RATIONAL QUARTIC FRACTAL FUNCTIONS

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct [Formula: see text]-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the [Formula: see text]-fractal rational quartic spline when the original function is in [Formula: see text]. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the [Formula: see text]-fractal rational quartic spline to [Formula: see text]. The elements of the iterated function system are identified befittingly so that the class of [Formula: see text]-fractal function [Formula: see text] incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ [Formula: see text]. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

Improved error estimate and applications of the complete quartic spline

Abstract In this paper an improved error bound is obtained for the complete quartic spline with deficiency 2, in the less smooth class of continuous functions. In the case of Lipschitzian functions, the obtained estimate improves the constant from Theorem 3, in J. Approx. Theory 58 (1989) 58-67. Some applications of the complete quartic spline in the numerical integration and in the construction of an iterative numerical method for fourth order two-point boundary value problems with pantograph type delay are presented.

Interpolation with C2 quartic macro-elements based on 10-splits

In this paper interpolation methods for the construction of quartic splines on triangulations refined with 10-splits are proposed. After examining the C2 macro-structure on a single triangle in terms of the Bernstein–Bézier representation, three methods that can be applied on general triangulations are developed. The methods make use of Hermite interpolation data prescribed at the vertices and (optionally) on the edges of the triangulation. The first two approaches lead to splines that are C2 continuous on triangles and C1 continuous across the interior edges of the triangulation. The third method gives rise to splines with overall C2 continuity, which is an exceptionally high order of smoothness for splines of degree four, but comes at the cost of solving a global system of linear equations. The derived results are accompanied with a few numerical examples that show an interesting behavior of splines in dependence of interpolated data.

A new non-polynomial spline method for solution of linear and non-linear third order dispersive equations

In this paper, a new three-level implicit method is developed to solve linear and non-linear third order dispersive partial differential equations. The presented method is obtained by using exponential quartic spline to approximate the spatial derivative of third order and finite difference discretization to approximate the first order spatial and temporal derivative. The developed method is tested on four examples and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, the truncation error and stability analysis of the presented method are investigated, and graphical comparison between analytical and approximate solution is also shown for each example.

Point and differential C1 quasi-interpolation on three direction meshes

In this paper we construct and analyse C 1 cubic and quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data, without using minimal determining sets and without defining the approximating splines as linear combinations of compactly supported bivariate spanning functions. In particular, the C 1 cubic splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values without using prescribed derivatives at any point of the domain, in such a way that the C 1 -smoothness conditions are satisfied and approximation order three is guaranteed, for smooth functions. We also propose some numerical tests that confirm the theoretical results. Then, from the above C 1 cubic splines we obtain C 1 quartic splines exact on P 3 , achieving approximation order four. The associated differential quasi-interpolation operator involves the values of the first partial derivatives in its definition.

Quasi-interpolation by C1 quartic splines on type-1 triangulations

In this paper we construct two new families of C 1 quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain. The quasi-interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation, reproduce cubic polynomials and yield approximation order four for smooth functions. We also propose some numerical tests that confirm the theoretical results.