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In this work, the numerical approximation of Convection-Reaction-Diffusion equation is investigated using the method based on tension spline function and finite difference approximation. For nonlinear term, nonstandard finite difference method by nonlocal approximation is utilized. We describe the mathematical formulation procedure in detail and also analyze the stability and convergence of the method. Numerical results are provided to justify the good performance of the proposed scheme. Keywords: Convection-Reaction-Diffusion equation, Tension spline, Nonstandard finite difference, Implicit scheme, Convergence analysis, MSC: 65M06, 12, 99

The numerical solution for a class of time fractional diffusion equation via tension spline is studied. Time fractional derivative is considered in the Caputo sense. The numerical method is constructed by means of the Crank-Nicolson method and is proven to be conditionally stable. Convergence analysis is also discussed by using the Fourier series method. Numerical evidences are given to prove the efficiency of the proposed method.

The present paper studies adaptive refinement on multi-patch domains in isogeometric analysis. In particular, we investigate the gluing construction for adaptively refined spline spaces to obtain discretizations that are C 0 smooth across interfaces. We will see that this is closely related to the concept of boundary compatibility of an adaptive spline construction. Given a spline basis (or, more generally, a generating system if linear independence is not guaranteed) on a d -dimensional box domain, there are two possibilities for constructing the spline basis on the domain boundary. Firstly, one can simply restrict the basis functions to the boundary. Secondly, one may restrict the underlying mesh to the boundary and construct the spline basis on the resulting mesh. The two constructions do not necessarily produce the same set of functions. If they do, then the spline bases are said to be compatible. We study this property for hierarchical (HB-) and truncated hierarchical B-splines (THB-splines) and identify sufficient conditions. These conditions are weaker for THB- than for HB-splines. Finally we demonstrate the importance of boundary compatibility for geometric modeling and for adaptive refinement in isogeometric analysis, in particular when considering multi-patch domains.

A new spline based generalized non-linear filter.Adaptive spline function is used as basis function.Provides improved classification accuracy.Successfully applied in dynamic system identification. A new nonlinear filter, which employs an adaptive spline function as the basis function is designed in this paper. The input signal to this filter is used to generate suitable parameters to update the control points in a spline function. The update rule for updating the control points have been derived and a mean square analysis has been carried out. The output of the spline functions are suitably combined together to obtain the filter response. This filter is called the generalized spline nonlinear adaptive filter (GSNAF). The proposed GSNAF is similar to a functional link artificial neural network (FLANN), considering a functional expansion using spline basis functions. GSNAF has been shown to offer improved accuracy in benchmark classification scenarios and provide enhanced modeling accuracy in single input single output as well as in multiple input multiple output dynamic system identification cases.

Wiener systems consist of a linear dynamic block in cascade with static nonlinearity. One of the challenging issues in the identification of a process noise disturbed Wiener system is that the influence of noise is difficult to eliminate. For Wiener systems with process noise, traditional algorithms will result in biased estimates. To solve this problem, a novel recursive Bayesian algorithm based on variable knot spline approximation is proposed in this paper. First, a spline function is taken to approximate the inverse function of the nonlinear part, which can achieve excellent extrapolation and eliminate oscillatory behaviors. A knot selection method is then presented to achieve accurate estimates. Furthermore, a knot variation strategy to improve the accuracy of the spline approximation is described. Finally, the proposed algorithm is validated through a numerical simulation.

The best L 1 approximation of the Heaviside function and the best l 1 approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L 1 spline fits which are the best L 1 approximations in an appropriate spline space obtained by the union of L 1 interpolation splines. We prove here the existence of L 1 spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes.

In this paper, we present a new technique to get superconvergence phenomenon of spline quasi-interpolants at the knots of the partition. This method gives rise to good approximation not only at these knots but also on the whole domain of definition. Moreover, we give an application to numerical integration. Numerical results are given to illustrate the theoretical ones.

We develop a numerical method based on parametric adaptive quintic spline functions for solving the nonlinear Schrodinger (NLS) equation. The truncation error is theoretically analyzed. Based on the von Neumann method and the linearization technique, stability analysis of the method is studied and the method is shown to be unconditionally stable. Two invariants of motion related to mass and momentum are calculated to determine the conservation properties of the problem. Finally, some numerical tests are presented to illustrate the method’s efficiency.

A novel numerical technique is considered for the solution of a multi-term time-fractional diffusion equation. The orthogonal spline collocation method is used for in space, and a finite difference...

In this paper an exponentially fitted spline method is presented for solving singularly perturbed convection delay problems with boundary layer at left (or right) end of the domain. The error analysis of the scheme is investigated. It is shown that the proposed scheme provides second order accuracy, independent of the perturbation parameter. Numerical results are presented to illustrate the efficiency of the method.

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. A high order numerical scheme for fractional sub-diffusion problems is proposed.The solvability, stability and convergence are established rigorously.Theoretical spatial convergence order is improved.Simulation performance is better than previous methods.

In this paper we define Tchebycheffian spline spaces over planar T-meshes and we address the problem of determining their dimension. We extend to the Tchebycheffian spline context the homological approach previously used to characterize polynomial spline spaces over T-meshes, and we exploit this characterization in the study of the dimension. In particular, we give combinatorial lower and upper bounds for the dimension, and we show that these bounds coincide if the dimensions of the underlying extended Tchebycheff section spaces are large enough with respect to the smoothness, under some mild conditions on the T-mesh. Finally, we provide simple examples of Tchebycheffian spline spaces over T-meshes with unstable dimension, which means that their dimension depends on the exact geometry of the T-mesh. These results are extensions of those known in the literature for polynomial spline spaces over T-meshes. We define Tchebycheffian spline spaces over planar T-meshes and we address the problem of determining their dimension.We give combinatorial lower and upper bounds for the dimension.We show that these bounds coincide under certain conditions on the T-mesh and/or the Tchebycheffian spline space.We provide simple examples of Tchebycheffian spline spaces over T-meshes with unstable dimension.These results are extensions of known results in the literature for polynomial spline spaces over T-meshes.

In this paper, two classes of methods are developed for the solution of two space dimensional wave equations with a nonlinear source term. We have used non-polynomial cubic spline function approximations in both space directions. The methods involve some parameters, by suitable choices of the parameters, a new high accuracy three time level scheme of order O(h4 + k4 + ź2 + ź2h2 + ź2k2) has been obtained. Stability analysis of the methods have been carried out. The results of some test problems are included to demonstrate the practical usefulness of the proposed methods. The numerical results for the solution of two dimensional sine-Gordon equation are compared with those already available in literature.

The Holland-Batt spline is well known when it comes to plotting spiral separation performance. The spline, which consists of a linear curve and power curve, has been successfully used to fit test work data that is presented as cumulative recovery of a valuable mineral versus the cumulative mass yield to valuable mineral concentrate. The benefit of this curve fitting process is that it produces a mathematical expression that is essential for simple mass flow modelling calculations. Although the mathematics is simple, the fitting process can be quite cumbersome. This work enhances the Holland-Batt spline with a few adjustments to improve the fit accuracy and ease the fitting process through (1) smoothing the transition zone between the linear and power law curves, (2) applying the principles to both low and high density particles, (3) use Visual Basic user-defined functions to simplify test work sheets in Excel and (4) use Excel Solver to automate the curve fitting process. These steps are applied to an example test work data set to clearly demonstrate the approach. The enhanced method is easy and simple to apply to spiral concentrator mass flow modelling.

We propose a p -version two level method for the spline solutions of the Navier-Stokes equations in stream function formulation. The new two level method can significantly accelerate the corresponding one level method, especially when the degrees of the spline spaces are greater than 5 in two dimensional case. The convergence analysis as well as the error estimation is given, which is asymptotic optimal with respect to the degrees of the spline spaces. Several numerical examples are presented to validate the theoretical analysis and show the efficiency of the proposed two level spline method.

A method of real-time reconstruction of the useful signal and its lower derivatives on the basis of a recurrent smoothing spline is presented. A calculation technique for a spline with the number of measurements at each segment greater than the number of nodes is given, and the spline coefficients are found by the variational approach.

The C 1 $C^{1}$ spline spaces with degree d ≥ 5 $d\geq 5$ over given triangulations are implemented in the framework of multi-variate spline theory. Based on this approach, two-level methods are proposed by using various order spline spaces for the steady state Navier–Stokes equations in the stream function formulation. The proposed method can be reduced to solving a linear equation in the high-order spline space and the nonlinear equations in the low-order spline space. The convergence analysis is given based on the Newton iteration. Besides, the matrix forms of the two-level scheme are also presented. We finally tabulate the numerical results to validate and show the efficiency of the proposed two-level spline methods.

A method of modeling derivatives in a real-time non-stationary process is proposed. This concept is based on approximation of variation smoothing splines. A recurrent estimation formula for parameters of a spline where the number of measurements of every segment is higher than the number of node is given. Dependence between accuracy of estimations of derivatives and values of the spline parameters is studied. Optimal values of parameters are found.

In this article, we develop numerical method by constructing ninth degree spline function using extended cubic spline Bickley’s method to find the approximate solution of seventh order linear boundary value problems at different step lengths. The approximate solution is compared with the solution obtained by eighth degree splines and exact solution. It has been observed that the approximate solution is an excellent agreement with exact solution. Low absolute error indicates that our numerical method is effective for solving high order linear boundary value problems.

The aim of this paper is to approximate the solution of system of fractional delay differential equations. Our technique relies on the use of suitable spline functions of polynomial form. We introduce the description of the proposed approximation method. The error analysis and stability of the method are theoretically investigated. Numerical example is given to illustrate the applicability, accuracy and stability of the proposed method.

In this paper, a numerical method based on exponential spline for solving one dimensional nonlinear Benjamin-Bona-Mahony-Burgers equation is presented. Stability analysis of the present method is analyzed by means of Von Neumann stability analysis and is proven to be unconditionally stable. Few numerical evidences are given to prove the validation of the proposed method.

In many practical applications, we face the problem of reconstruction of an un- known function sampled at some data points. Among infinitely many possible reconstructions, the thin plate spline interpolation is known to be the least oscillatory one in the Beppo-Levi semi norm, when the data points are sampled in R 2 . The traditional proofs supporting the argu- ment are quite lengthy and complicated, keeping students and researchers off its understanding. In this article, we introduce a simple and short proof for the optimal reconstruction. Our proof is unique and reguires only elementary mathematical background.

In this study, the author presents an algorithm for approximating the contour of a digital planar image by cubic splines. In the authors’ method, a subset of points (called corners) from the contour is selected. These corners are used to segment the contour and each segment is then approximated by a cubic spline. Parameters of the fitted splines are estimated by optimisation methods. The novelty of the proposed approach lies in the way the corners are selected. An initial set of corners are first selected using a process which is called as iterative points-suppression. This initial set is further reduced by a novel technique termed spline-suppression. The result is a very compact cubic spline representation of the contour using few corners on the contour. The effectiveness of the proposed method is demonstrated on two large databases: MPEG7_CE-Shape-1_Part_B database and a database of handwritten characters.

Cubic L 1 spline fits have shown some favorable shape-preserving property for geometric data. To quantify the shape-preserving capability, we consider the basic shape of two parallel line segments in a given window. When one line segment is sufficiently longer than the other, the spline fit can preserve its linear shape in at least half of the window. We propose to use the minimum of such length difference as a shape-preserving metric because it represents the extra information that the spline fits need to preserve the shape. We analytically calculate this metric in a 3-node window for second-derivative-based, first-derivative-based and function-value-based spline fits. In a 5-node window, we compute this metric numerically. In both cases, the shape-preserving metric is rather small, which explains the observed strong shape-preserving capability of spline fits. Moreover, the function-value-based spline fits are indicated to preserve shape better than the other two types of spline fits. This study initiates a quantitative research on shape preservation of L 1 spline fits. We analytically quantify shape-preserving capability of cubic L1 spline fits.We propose a shape-preserving metric for the linear shape of Heaviside step function.We analytically calculate and numerically compute the metric.We find that function-value-based spline fits preserve linear shape best.

In this paper, the effectiveness of the modified Chebyshev and cubic spline׳s iterative methods is comparatively evaluated on geometrically non-linear analysis of truss structures. For the purpose of a comprehensive comparison, we have also proposed an iterative method free from second derivative originated from modified Chebyshev and cubic spline׳s schemes. The method involves a set of predictor–corrector schemes constructed by Chebyshev as the predictor for spline correctors to improve the approximation of the tangential stiffness matrix. The numerical assessment of the proposed method lies on three-step algorithm with satisfactory convergence of results. The analysis of convergence is carried out and is shown that the proposed method is at least third-order convergent. A simple step-by-step algorithm is developed capable of tracing the non-linear equilibrium curve until the first limit point through an incremental approach. The robustness and efficiency of the proposed schemes are comparatively investigated against classical Newton–Raphson׳s method for solving practical non-linear problems. It is concluded that for the large structural systems, where a large-scaled stiffness matrix is being iteratively updated, the best computational time, thus the optimum cost of analysis is accomplished by the proposed algorithm using reasonably less number of incremental loads. Finally, it is demonstrated that the proposed procedure and spline׳s method require considerably less number of iterations to reach the sufficient accuracy.

In this paper, we propose a new two-level implicit method of order two in time and four in space directions, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. We have discussed the derivation of the proposed method in detail and have also discussed the stability analysis for a model problem. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed methods.

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Nu...

In this paper, we propose a new two level implicit method of order two in time and four in space directions, based on spline in compression approximation for the numerical solution of one space dimensional quasi-linear parabolic partial differential equation on a uniform mesh. The derivation and the stability of the proposed method are discussed in details. We have extended the method to non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed method.

A method of generating spherical non-uniform rational basis spline curves based on De Boor’s algorithm is presented in this article. The spherical curve preserves many good properties from non-uniform rational basis spline curves in Euclidean space, such as local modification property, convex hull property, rotation invariant property, knot insertion property, and so on. Construction of closed spherical non-uniform rational basis spline curve will be discussed too. Furthermore, a progressive iterative approximation scheme is proposed to approximate the given points on unit sphere using spherical non-uniform rational basis spline curves. The convergence and curve continuity will be discussed. Since the analytical expression of the spherical non-uniform rational basis spline curve is messy, numerical differentiation is used to test its continuity at interior knots. Finally, several examples are presented to verify the effectiveness of the proposed method. The proposed method is applied on tool path generation of five-axis machining to avoid interference by adjusting fewer directions and obtain smooth tool path simultaneously.

Non-polynomial spline functions of the form Span{1, x, x 2 , x 3 , x 4 , x 5 , cos ( kx ) + e x } , where k can be real or pure imaginary, are used to find the numerical solution of linear fifth-order boundary value problems. The order of convergence of the method is observed to be of O ( h 2 ) . A fifth order convergent method is defined with the help of improved end-conditions. Three examples are considered to show the reliability and efficiency of the method. The numerical results, obtained, endorse the improved order of convergence of the method.

Spline schemes are proposed to simulate compressible flows on non-uniform structured grid in the framework of finite volume methods. The cubic spline schemes in the present paper can achieve fourth and third order accuracy on the uniform and non-uniform grids respectively. Due to the continuity of cubic spline polynomial function, the inviscid flux can be computed directly from the reconstructed spline polynomial without using the Riemann solvers or other flux splitting techniques. Isotropic and anisotropic artificial viscosity models are introduced to damp high frequency numerical disturbances and to enhance the numerical stability. The first derivatives that are used to calculate the viscous flux are directly obtained from the cubic spline polynomials and preserve second order accuracy on both uniform and non-uniform grids. A hybrid scheme, in which the spline scheme is blended with shock-capturing WENO scheme, is developed to deal with flow discontinuities. Benchmark test cases of inviscid/viscous flows are presented to demonstrate the accuracy, robustness and efficiency of the proposed schemes.

In the paper Barrera et al. (2014), a boolean sum differential quadrature method (DQM) was proposed by combining a spline interpolation operator having a fundamental function with minimal compact support and a spline quasi-interpolation operator reproducing the polynomials in the spline space. It is a general framework that provides results that differ from the ones obtained by defining specific schemes with structures which depend on the degree of the B-spline to be considered. The main drawback of these boolean sum DQMs is that the number of evaluation points increases quickly with the degree of the B-spline due to the use of a quasi-interpolation operator. We propose a different construction avoiding this problem and derive explicit results for low degree B-splines.

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.

An integration–rolling–extrusion process is raised for the manufacture of spline shaft in this study. First, the principle and procedures of integration–rolling–extrusion process are described. Next, the finite element model with a simplified sector blank is established to obtain a practical method for the simulation of integration-rolling-extrusion process. Through the simulation results, the plastic forming mechanisms are clearly revealed. During the integration–rolling–extrusion process, the equivalent stress, deformation degree, and material flow behavior mainly distribute on the surface layer of the blank and then gradually decrease along the radial inward direction. In the core region of the blank, there are almost no effective stress distribution, deformation degree, and material flow behavior. Next, the experiments are carried out on a specialized forming equipment to verify the finite element model. The results are measured and compared with finite element results. The finite element results show a good agreement with experiments; thus, the finite element analysis on the integration–rolling–extrusion process is credible. In addition, the measurement results show that the dimensions meet the requirement of heavy truck application. It indicates that the integration–rolling–extrusion process is feasible for the manufacture of spline shaft. However, the surface quality of the formed spline shaft is not satisfying, which needs to be discussed further.

In this study, for lactation curve, spline interpolations, alternative modeling passing through exactly all data points with respect to widely used Wood model applied to lactation data were be discussed. These models are linear spline, quadratic spline and cubic spline. The observed and estimated values according to spline interpolations and Wood model were given with their Error Sum of Squares and also the lactation curves of spline interpolations and widely used Wood model were shown on the same graph. Thus, the differences have been observed. The estimates for some intermediate values were done by using spline interpolations and Wood model. By using spline interpolations, the estimates of intermediate values could be made more precise. Furthermore, by using spline interpolations, the predicted values for missing or incorrect observation were very successful according to the values of Wood model. By using spline interpolations, new ideas and interpretations in addition to the information of the well-kno...

We explain how to use polygonal splines to numerically solve second-order elliptic partial differential equations. The convergence of the polygonal spline method will be studied. Also, we will use this approach to numerically study the solution of some mixed parabolic and hyperbolic partial differential equations. Comparison with standard bivariate spline method will be given to demonstrate that our polygonal splines have some better numerical performance.

Split-step orthogonal spline collocation (SSOSC) scheme is constructed by combining the orthogonal spline collocation method with the split step technique for the complex Ginzburg-Landau equation in two dimensions. The truncation error of the scheme is studied for the plane wave solution, and the stability is also analyzed. Convergence rate is numerically verified. Numerical results show that the SSOSC method is efficient and superior.Split-step orthogonal spline collocation (SSOSC) scheme is constructed by combining the orthogonal spline collocation method with the split step technique for the complex Ginzburg-Landau equation in two dimensions. The truncation error of the scheme is studied for the plane wave solution, and the stability is also analyzed. Convergence rate is numerically verified. Numerical results show that the SSOSC method is efficient and superior.

In this work, we extend the definition of nonic polynomial spline to non-polynomial spline function which depends on arbitrary parameter k. We derived and discussed the formulation and spline relations. Using such non-polynomial spline relations, we developed the classes of numerical methods, for the solution of the problem in calculus of variations. The proposed boundary formulas which are needed to be associated with spline methods are derived. Truncation errors and orders of accuracy of proposed methods are presented. Convergence analysis of the methods are discussed. The present methods have been tested on three examples, to illustrate practical usefulness of our method.

A spline smoothing stabilized Newton method for finite minimax problems is developed. The spline smoothing technique uses a smooth cubic spline instead of a max function, and, at any fixed point, only a few components of the max function are involved; i.e., it introduces an active set technique, so the proposed method is more efficient for minimizing the maximum function of a large number of complicated functions. Some numerical results comparisons with other methods are also given to show the efficiency of the new method.