Non-polynomial Spline Research Articles

On the Convergence of Non-Polynomial Spline Finite Difference Method for Quasi-Linear Elliptic Boundary Value Problems in Two-Space Dimensions

On the Convergence of Non-Polynomial Spline Finite Difference Method for Quasi-Linear Elliptic Boundary Value Problems in Two-Space Dimensions

A new variable mesh method based on non-polynomial spline in compression approximations for 1D quasilinear hyperbolic equations

In this paper, we present a new three-level implicit method of order two in time and three in space on a non-uniform mesh, based on spline in compression approximation for the numerical solution of 1D quasilinear second order hyperbolic partial differential equations. We also discuss the application of the proposed method to a wave equation with singular coefficients. Stability analysis of a linear scheme and convergence analysis of a general nonlinear scheme are also discussed in this paper. Computational results are given to demonstrate the usefulness of the proposed method.

A New Numerical Method Based on Non-Polynomial Spline in Tension Approximations for 1D Quasilinear Hyperbolic Equations on a Variable Mesh

In this paper, we derive a new three level implicit method of order two in time and three in space, based on spline in tension approximation for the numerical solution of one space dimensional quasi-linear second order hyperbolic partial differential equation on a variable mesh. We also study application of the proposed method to wave equation in polar coordinates. High order approximation at first time level is briefly discussed which is applicable to solve problems both on uniform and non-uniform mesh. Numerical results are given to illustrate the usefulness of the proposed method.

A Solution of Second Kind Volterra Integral Equations Using Third Order Non-Polynomial Spline Function

In this paper, third order non-polynomial spline function is used to solve 2nd kind Volterra integral equations. Numerical examples are presented to illustrate the applications of this method, and to compare the computed results with other known methods.

A Solution of Second Kind Volterra Integral Equations Using Third Order Non-Polynomial Spline Function

In this paper, third order non-polynomial spline function is used to solve 2nd kind Volterra integral equations. Numerical examples are presented to illustrate the applications of this method, and to compare the computed results with other known methods.

Numerical solution of third order BVPs by using non-polynomial spline with FDM

Numerical solution of third order BVPs by using non-polynomial spline with FDM

Hierarchical cascade model leading to 7-th order initial value problem

In turbulent flows, local velocity differences often obey a cascade-like hierarchical dynamics, in the sense that local velocity differences at a given scale k are driven by deterministic and random forces from the next-higher scale k−1. Here we consider such a hierarchically coupled model with periodic boundary conditions, and show that it leads to an N-th order initial value problem, where N is the number of cascade steps. We deal in detail with the case N=7 and introduce a non-polynomial spline method that solves the problem for arbitrary driving forces. Several examples of driving forces are considered, and estimates of the numerical precision of our method are given. We show how to optimize the numerical method to obtain a truncation error of order O(h5) rather than O(h2), where h is the discretization step.

Efficient algorithms for fourth and sixth-order two-point non-linear boundary value problems using non-polynomial spline approximations on a geometric mesh

An efficient algorithm for the numerical solution of linear and non-linear higher (even)-order two-point boundary value problems has been developed. The method is third-order accurate and applicable to both singular and non-singular cases. We have used non-polynomial spline basis and geometric mesh finite differences for the generation of new scheme. The irreducibility and monotone property of the iteration matrix have been established, that justify the convergence property of the proposed method. Some computational experiments have been carried out to demonstrate the efficiency in terms of convergence order and maximum absolute error estimates. The numerical results justify the reliability and efficiency of the method both in terms of order and accuracy.

Reproducing Kernel Hilbert Space Method for Solving Bratu’s Problem

In this paper, we use the reproducing kernel Hilbert space method for solving a boundary value problem for the second order Bratu’s differential equation. Convergence analysis of presented method is discussed. The numerical approximations to the exact solution are computed and compared with other existing methods. Our presented method produces more accurate results in comparison with those obtained by Adomian decomposition, Laplace decomposition, B-spline, non-polynomial spline and Lie-group shooting methods. Our yardstick is absolute error. The comparison of the results with exact ones is made to confirm the validity and efficiency.

Non-polynomial splines method for numerical solutions of the regularized long wave equation

In this paper, we construct a numerical method based on cubic splines in tension for solving regularized long wave equation. The truncation error is analysed and the method shows that by choosing suitably parameters we can obtain various accuracy schemes. Numerical stability of the method has been studied by using a linearized stability analysis. Test problems are dealt with. The numerical simulations can validate and demonstrate the advantages of the method.

Some studies on nonpolynomial interpolation and error analysis

In this paper, we propose nonpolynomial and Hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation. The generalized error analysis results for nonpolynomial and Hermite nonpolynomial interpolations are derived. We establish theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series, and propose solution to Runge’s phenomenon through nonpolynomial interpolation. The Hermite nonpolynomial cubic spline, nonpolynomial cubic spline interpolation methods and their error analysis are presented. Numerical simulations are carried out for the analysis of error in cubic spline interpolations. Proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations, and to Empirical Mode Decomposition (EMD) to illustrate practical usefulness of the results.

Generalized T-splines and VMCR T-meshes

The paper considers the extension of the T-spline approach to the Generalized B-splines (GB-splines), a relevant class of non-polynomial splines. The Generalized T-splines (GT-splines) are based both on the framework of classical polynomial T-splines and on the Trigonometric GT-splines (TGT-splines), a particular case of GT-splines. Our study of GT-splines introduces a class of T-meshes (named VMCR T-meshes) for which both the corresponding GT-splines and the corresponding polynomial T-splines are linearly independent. A practical characterization can be given for a sub-class of VMCR T-meshes, which we refer to as weakly dual-compatible T-meshes, which properly includes the class of dual-compatible (equivalently, analysis-suitable) T-meshes for an arbitrary (polynomial) order.

Quartic Non-polynomial Spline Solution of a Third Order Singularly Perturbed Boundary Value Problem

In this study, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems. The convergence analysis is discussed and the method is shown to have second order convergence. The order of convergence is improved up to fourth order using the improved end conditions. Numerical results are given to describe the efficiency of the method and compared with the method developed by Akram (2012), which shows that the present method is better.

Solution of Second Kind Volterra Integral Equations Using Non-Polynomial Spline Functions

In this paper we use non-polynomial spline functions to develop numerical methods to approximate the solution of 2nd kind Volterra integral equations. Numerical examples are presented to illustrate the applications of these method, and to compare the computed results with other known methods.

Solution of Second Kind Volterra Integral Equations Using Non-Polynomial Spline Functions

In this paper we use non-polynomial spline functions to develop numerical methods to approximate the solution of 2nd kind Volterra integral equations. Numerical examples are presented to illustrate the applications of these method, and to compare the computed results with other known methods.

High accuracy non-polynomial spline in compression method for one-space dimensional quasi-linear hyperbolic equations with significant first order space derivative term

In this paper, we propose a new three-level implicit nine point compact non-polynomial spline in compression finite difference method of order two in time and four in space directions, based on non-polynomial spline approximation in x-direction and central difference approximation in t-direction for the numerical solution of one-space dimensional second order quasi-linear hyperbolic partial differential equations with first order space derivative term. We describe the mathematical details of the method and also discuss how our method is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.

Application of non-polynomial SPLAGE algorithm for the numerical solution of singular two point boundary value problems using non-uniform mesh scheme

The purpose of this article is to present non-polynomial spline approximations using non-uniform mesh for the numerical treatment of singular boundary value problems. The numerical method is compact and exhibit homogeneous fourth order of convergence. The resulting nonlinear difference schemes are solved by alternating group explicit parallel algorithm. The utility of new schemes are illustrated by Burger’s equation, Duffing equation and Thomas Fermi model. Computational order of convergence and maximum absolute errors are given to demonstrate the efficiency of the non-uniform mesh approach.

About modelling nonpolynomial integro-dierential appro-ximations.

In this article the modelling of the nonpolynomial integro-dierential splines is discussed. This splines interpolate function and its derivatives in knots of a grid and provide concurrence of integral size from approximated function and size of integral from spline on the set interval. Here explosive, continuous, and continuously dierentiated several times basic splines are constructed. They allow to solve problems of the construction of approximation if we know the values of the function and its derivatives in knots of a grid and in the additional assumption that values of integrals from approached function on net intervals are known. Estimations errors are resulted and examples of splines are given

The reconstruction and decomposition matrixes for linear splines

The aim of the paper is construction of calibration relations in the case of class of coordinate non-polynomial splines connected with refinement of grids. An embed\-ding of spline spaces is established for arbitrary refinement of grids. The reconstruction matrixes in the case of a grid on an open interval and a grid on a segment are constructed. The system of biorthogonal linear functionals to splines is constructed. The decomposition matrixes in the case of a grid on an open interval and a grid on a segment are constructed.

Cubic Hermite Collocation Method for Solving Boundary Value Problems with Dirichlet, Neumann, and Robin Conditions

Cubic Hermite collocation method is proposed to solve two point linear and nonlinear boundary value problems subject to Dirichlet, Neumann, and Robin conditions. Using several examples, it is shown that the scheme achieves the order of convergence as four, which is superior to various well known methods like finite difference method, finite volume method, orthogonal collocation method, and polynomial and nonpolynomial splines and B-spline method. Numerical results for both linear and nonlinear cases are presented to demonstrate the effectiveness of the scheme.