Non-polynomial Spline Research Articles

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.

Composite non-polynomial spline solution of boundary value problems in plate deflection theory

This article presents the numerical solution of a system of linear fourth-order boundary value problems using a different amalgamation of non-polynomial splines. A novel approach was developed using quintic spline function together with exponential and trigonometric functions. Our method is convergent and second-order accurate. Numerical examples show that the method congregates with sufficient accuracy to the exact solutions. Our methodology has the advantages over some existing quintic spline method, direct method, and finite difference method.

Non-polynomial quintic spline for numerical solution of fourth-order time fractional partial differential equations

This paper presents a novel approach to numerical solution of a class of fourth-order time fractional partial differential equations (PDEs). The finite difference formulation has been used for temporal discretization, whereas the space discretization is achieved by means of non-polynomial quintic spline method. The proposed algorithm is proved to be stable and convergent. In order to corroborate this work, some test problems have been considered, and the computational outcomes are compared with those found in the exiting literature. It is revealed that the presented scheme is more accurate as compared to current variants on the topic.

On Implementation of Non-Polynomial Spline Approximation

In this paper, different variants of processing of number flows using Lagrange and Hermite non-polynomial splines are studied. The splines are constructed from approximate relations including a generating vector function with components of different character, including non-polynomial. Approximations by first-order Lagrange and third-order Hermite splines are considered. The efficiency of the approximations constructed is demonstrated on the examples of flows of the values of a function and flows of the values of a function and its derivative. The advantages of the splines considered are the simplicity of construction, maximum smoothness, interpolation and approximation properties, and the accuracy on a priori given functions (on the components of the generating vector function).

Quintic Non-polynomial Spline Method for Solving the Time Fractional Biharmonic Equation

Quintic Non-polynomial Spline Method for Solving the Time Fractional Biharmonic Equation

Convergence analysis of non-polynomial spline functions for the Fredholm integral equation

ABSTRACTIn our knowledge so far, the Non-polynomial Spline functions (NPS) have not been yet applied for approximating the integral equations. In this article, we want to use such functions for obtaining numerical solutions of Fredholm integral equations of the second kind. In our approach, the coefficients of the non polynomial spline are obtained by solving a system of linear equations. Finally, these functions are utilized to reduce the Fredholm integral equations to the solution of algebraic equations. Analysis of convergence is investigated. At the end, our test examples are presented to show the effectiveness of the method.

Non-polynomial spline approach in two-dimensional fractional sub-diffusion problems

In this paper, we propose a new numerical scheme for two-dimensional fractional sub-diffusion problems using non-polynomial spline. The solvability, stability and convergence of the proposed method are established using the well known discrete energy methodology. It is shown that the spatial convergence order is at least 4.5 which improves the best result achieved to date. We also carry out simulation to demonstrate the accuracy and efficiency of the proposed scheme and to compare with other methods.

Fourth order computational method for two parameters singularly perturbed boundary value problem using non-polynomial cubic spline

In this paper, we proposed a fourth order finite difference scheme using non-polynomial cubic spline for the solution of two parameters singularly perturbed two-point boundary value problem having dual boundary layer on a uniform mesh. In this method, the first order derivatives in the non-polynomial cubic spline finite difference scheme are replaced by the higher order finite differences to get the discretisation equation for the problem. The discretisation equation is solved by the tridiagonal solver discrete invariant imbedding. The proposed method is analysed for convergence and a fourth order rate of convergence is proved. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.

Spaces of polynomial and nonpolynomial spline-wavelets

This paper, discusses spaces of polynomial and nonpolynomial splines suitable for solving the Hermite interpolation problem (with first-order derivatives) and for constructing a wavelet decomposition. Such splines we call Hermitian type splines of the first level. The basis of these splines is obtained from the approximation relations under the condition connected with the minimum of multiplicity of covering every point of (α, β) (almost everywhere) with the support of the basis splines. Thus these splines belong to the class of minimal splines. Here we consider the processing of flows that include a stream of values of the derivative of an approximated function which is very important for good approximation. Also we construct a splash decomposition of the Hermitian type splines on a non-uniform grid.

Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations

In this paper, a reduced spline (RS) method based on a proper orthogonal decomposition (POD) technique for numerical solution of time fractional sub-diffusion equations is investigated. Combining POD with polynomial and non-polynomial spline methods yield a new model with lower dimensions and sufficiently high accuracy, so that the amount of computations and the calculation time decrease in comparison with usual spline methods. Although, the classical L1 scheme on uniform meshes is one of the most successful methods for approximation of the Caputo fractional derivative in time, due to the fact that the solution of time fractional sub-diffusion equations typically has a singularity at the origin, a weak convergence will conclude for such scheme. To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution. Also, the error estimates between the POD approximate solution of the RS scheme and the exact solution of fractional sub-diffusion equation are established for two types of uniform and graded meshes in time. Numerical examples are given to illustrate the feasibility and efficiency of the proposed method.

Mid-knot cubic non-polynomial spline for a system of second-order boundary value problems

In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature.

Cubic Spline Solution of linear fourteenth order boundary value problems

As higher order differential equations have constantly been tiresome and problematic to resolve for the mathematicians and engineers so diverse numerical procedures were conceded out to acquire numerical estimates to such problems. In this paper an innovative numerical procedure is developed to estimate the fourteenth-order boundary value problems (BVPs) using Polynomial and Non-Polynomial Cubic spline. The procedures adopted in our work are based on cubic polynomial and non-polynomial spline method together with the decomposition procedure. In this paper polynomial and non-polynomial cubic splines along with the finite difference approximations will be used to squeeze the system of second order Boundary Value Problems in such a way that it will be converted into to a system consists of linear algebraic equations along with boundary conditions. These strategies will be operated on two problems to evidence the handiness of the technique by means of step size h = 1/5. The exactness of this method for detailed investigation is equated with the precise solution and conveyed through tables.

Non-polynomial spline method for the time-fractional nonlinear Schr\xf6dinger equation

In this paper, we propose a cubic non-polynomial spline method to solve the time-fractional nonlinear Schrödinger equation. The method is based on applying the L_{1} formula to approximate the Caputo fractional derivative and employing the cubic non-polynomial spline functions to approximate the spatial derivative. By considering suitable relevant parameters, the scheme of order O(tau^{2-alpha }+h^{4}) has been obtained. The unconditional stability of the method is analyzed by the Fourier analysis. Numerical experiments are given to illustrate the effectiveness and accuracy of the proposed method.

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.

Sinc-Galerkin solution to eighth-order boundary value problems

Many problems that arise in astrophysics, hydrodynamic and hydromagnetic stability, fluid dynamics, astronomy, beam and long wave theory are modeled as eighth-order boundary-value problems. In this paper we show that the sinc-Galerkin method is an efficient and accurate numerical scheme for solving these problems. The inner product approximations to eighth and seventh order derivatives and the corresponding error estimates are derived. Compared to the non-polynomial spline technique and the reproducing kernel space method, we show that the sinc-Galerkin method provides more accurate results. In addition, for problems with singular solutions or singular source functions, the sinc-Galerkin method is shown to maintain the exponential convergence rate.

A New Two-Level Implicit Scheme for the System of 1D Quasi-Linear Parabolic Partial Differential Equations Using Spline in Compression Approximations

In this article, we proposed a new two-level implicit method of accuracy two in time and four in space based on spline in compression approximations using two half-step points and a central point on a uniform mesh for the numerical solution of the system of 1D quasi-linear parabolic partial differential equations subject to appropriate initial and natural boundary conditions prescribed. The proposed method is derived directly from the continuity condition of the first order derivative of the non-polynomial compression spline function. The stability analysis for a model problem is discussed. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, we have solved generalized Burgers–Huxley equation, generalized Burgers–Fisher equation, coupled Burgers-equations and parabolic equations with singular coefficients. We show that the proposed method enables us to obtain high accurate solution for high Reynolds number.

An efficient nonpolynomial spline method for distributed order fractional subdiffusion equations

In this paper, we propose an efficient numerical method for a distributed order fractional subdiffusion problem using nonpolynomial spline approach. The solvability, stability, and convergence of the scheme are established rigorously, and it is shown that the spatial convergence order improves some previous work done. Simulation is then conducted to verify the accuracy of the proposed scheme as well as to compare with earlier work.

Solution of fourth-order two-point BVPs with cubic non-polynomial spline and SOR iterative method

No Abstract Keywords: successive over-relaxation; cubic non-polynomial spline; fourth-order two-point boundary value problem

Non-polynomial spline functions and Quasi-linearization to approximate nonlinear Volterra integral equation

In this work, we want to use the Non-polynomial spline basis and Quasi-linearization method to solve the nonlinear Volterra integral equation. When the iterations of the Quasilinear technique employed in nonlinear integral equation we obtain a linear integral equation then by using the Non-polynomial spline functions and collocation method the solution of the integral equation can be approximated. Analysis of convergence is investigated. At the end, some numerical examples are presented to show the effectiveness of the method.

Non-polynomial spline finite difference method for solving second order boundary value problem

Second -order two-point boundary value problems (BVPs) were solved based on a of non-polynomial spline general functions with finite difference method.In this paper we discusses a method depended on using special finite-difference approximations for derivatives and formatting a formula that can be deal with endpoints that exceed the usual finite-difference formula for derivatives. Convergence analysis of the method is discussed. The numerical description of the method is shown by four examples. So the results obtained by our method are very encouraging over other existing methods.