Exponential Spline Functions Research Articles

An Algorithm for Solving Nonlinear Third-Order Differential Equations Using Exponential Spline Functions

This research dealt with the study of the boundary values associated with differential equations, which are of the non-linear and third-order type. A new algorithm was created that uses exponential spline functions to study and address boundary value problems of a general nature. We have demonstrated that the numerical approach used, which was built using exponential spline functions, gives us good and accurate results for these problems, which have been compared to existing numerical methods. We found that the proposed method is accurate and effective compared to other Spalline methods.

A New Approach for Solving Singularly Perturbed Boundary Value Problems by Using Exponential Spline Method

This article presents the development of families of approaches for numerically solving singularly perturbed two-point boundary-value problems using exponential spline functions. The proposed approaches exhibit second-order and fourth-order accuracy and are suitable for both singular and non-singular problem scenarios. Numerical data are presented to demonstrate the efficacy of our methodologies and are compared with those proposed by various writers.

Exponential spline method for singularly perturbed third-order boundary value problems

Abstract The exponential spline function is presented to find the numerical solution of third-order singularly perturbed boundary value problems. Convergence analysis of the method is briefly discussed, and it is shown to be sixth order convergence. To validate the applicability of the method, some model problems are solved for different values of the perturbation parameter, and the numerical results are presented both in tables and graphs. Furthermore, the present method gives more accurate solution than some methods existing in the literature.

Numerical Solution of Singularly Perturbed Two Point Boundary Value Problems by Using Non-Polynomial Exponential Spline Functions

Numerical Solution of Singularly Perturbed Two Point Boundary Value Problems by Using Non-Polynomial Exponential Spline Functions

An Exponential Spline Difference Scheme for Solving a Class of Boundary Value Problems of Second-Order Ordinary Differential Equations

In this paper, we mainly study an exponential spline function space, construct a basis with local supports, and present the relationship between the function value and the first and the second derivative at the nodes. Using these relations, we construct an exponential spline-based difference scheme for solving a class of boundary value problems of second-order ordinary differential equations (ODEs) and analyze the error and the convergence of this method. The results show that the algorithm is high accurate and conditionally convergent, and an accuracy of 1/240h6 was achieved with smooth functions.

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015).

An exponential spline approximation for fractional Bagley\u2013Torvik equation

In this paper, we approximate the solution of fractional Bagley–Torvik equation by using the exponential spline function and the shifted Grünwald difference operator. The proposed methods reduce to the system of algebraic equations. The convergence analysis of the methods has been discussed. The numerical examples are presented to illustrate the applications of the methods and to compare the computed results with the other methods.

Exponential spline method for approximation solution of Fredholm integro-differential equation

ABSTRACT In this paper, we approximate the solution of Fredholm integro-differential equations of the second kind by using exponential spline function. The proposed method reduces to the system of algebraic equations. The convergence analysis of the method has been discussed. The numerical examples are presented to illustrate the applications of the method and to compare the computed results with the method in Chen et al. [A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-differential equation, J. Comput. Appl. Math. 290 (2015), pp. 633–640].

Partially penetrating slug tests in an unweathered till layer

This research improves field based estimates of aquitard compressibility and permeability. A semianalytical model of partially penetrating, overdamped slug tests achieves this objective. The short term solution is an existing fully penetrating model, the long term solution is the polar residue of an inverse Laplace transform, and an exponential spline function patches the solutions together. Large amplitude slug test data from ten pairs of partially penetrating monitoring wells installed in an unweathered till at Scituate Hill in eastern Massachusetts calibrate the model. The deposit is bound by weathered till and the Dedham Granite fracture zone, and both are far more permeable than the unweathered till. The calibrated till permeability of 8.4 × 10–16 m2 is about 25% less than existing model calibrations that include boundary recharge in permeability values. The calibrated till compressibility of 5.1 × 10–10 Pa–1 reflects the proper inclusion of recharge as a long term source of groundwater, rather than the unrealistically large compressibility calibrations required by fully penetrating models.

An exponential spline technique for solving fractional boundary value problem

Differential equations of fractional order are widely used in physics, chemistry as well as engineering fields, this is the main reason that the approximate solution of fractional differential equations becomes a hot topic. In this paper, a numerical scheme for a class of fractional boundary value problems (FBVPs) is presented. In this approach, the FBVPs are expressed in terms of Caputo's fractional derivative. This scheme is based on exponential spline functions consisting of a polynomial part of degree one and an exponential part. For convergence analysis of this method, it is assumed that the exact solution of fractional boundary value problem belongs to a class of $$C^{6}$$C6-functions. Numerical examples are considered to illustrate the practical usefulness of this method and comparison show that this scheme is more accurate than the existing method Zahra and Elkholy (Numer Algorithms 59:373---391, 2012).

The Property of a Special Type of Exponential Spline Function

Approximation theory experienced a long term history. Since 50’ last century, the rise of spline function as well as the advance of calculation promotes the growth of classical approximation theory and makes them develop a profound theory in maths, and application values have shown among the field of scientific calculation and engineering technology and etc. At present, the study of spline function had made a great progress and had a lot of fruits, as for that, the reader could look up the book [1] or [2]. Nevertheless, the research staff pays less attention to exponential spline function, since polynomial spline function is a special case of that, so it is much essential and meaningful for one to explore the nature of exponential spline function.

An exponential spline solution of nonlinear Schrödinger equations with constant and variable coefficients

In this study, the exponential spline scheme is implemented to find a numerical solution of the nonlinear Schrödinger equations with constant and variable coefficients. The method is based on the Crank–Nicolson formulation for time integration and exponential spline functions for space integration. The error analysis, existence, stability, uniqueness and convergence properties of the method are investigated using the energy method. We show that the method is unconditionally stable and accurate of orders O(k+kh+h2) and O(k+kh+h4). This method is tested on three examples by using the cubic nonlinear Schrödinger equation with constant and variable coefficients and the Gross–Pitaevskii equation. The computed results are compared wherever possible with those already available in the literature. The results show that the derived method is easily implemented and approximate the exact solution very well.

Finite-difference technique based on exponential splines for the solution of obstacle problems

In this paper, we introduce a new technique based on cubic exponential spline functions for computing approximations to the solution of a system of fourth-order boundary value problems associated with obstacle, unilateral and contact problems. It is shown that the present method is of order two and four and gives approximations which are better than those produced by other collocation, finite difference and spline methods. Numerical evidence is presented to illustrate the applicability of the new methods.

A smooth approximation based on exponential spline solutions for nonlinear fourth order two point boundary value problems

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.

Spline-functions for investigating and forecasting systems

An explicit formula of a multivariate B-spline, some useful investigations in the field of linear transformations of independent exponentially distributed random variables, representation of theirs density functions with the help of multivariate exponential spline functions, and their usage are considered. The consideration is illustrated by an appropriate example.

A method of organization of a parallel-hierarchical network for image recognition

An explicit formula of a multivariate B-spline, some useful investigations in the field of linear transformations of independent exponentially distributed random variables, representation of theirs density functions with the help of multivariate exponential spline functions, and their usage are considered. The consideration is illustrated by an appropriate example.

Exponential spline solutions for a class of two point boundary value problems over a semi-infinite range

In this paper, we developed numerical methods of order O(h 2) and O(h 4) based on exponential spline function for the numerical solution of class of two point boundary value problems over a Semi-infinite range. The present approach gives better approximations over all the existing finite difference methods. Properties of the infinite linear system are established. Convergence analysis and a bound on the approximate solution are discussed. Test problem with various kinds of boundary conditions is included to illustrate the practical usefulness and superiority of our methods.

Improvement of the repulsive part of the classical interatomic potential for SiC

In order to enable a better description of ballistic and athermal processes occurring in the initial stage of ion-beam-induced defect formation, the repulsive part of the interatomic potentials of Gao and Tersoff is improved. The first modification concerns the two-body part of the potentials. At small interatomic distances it is replaced by the well-tested potential of Ziegler, Biersack and Littmark (ZBL). For repulsive interactions between zero and some 10 electron volt, an exponential spline function is employed to connect the ZBL potential with the two-body part of the Tersoff and the Gao potential. The modified two-body potentials and their first derivatives are continuous and monotonic over the whole range of repulsive interaction. They are in good agreement with data obtained by density-functional-theory calculations. Furthermore, the three-body part of the Tersoff and the Gao potential is modified in order to avoid the strong dependence of repulsive interactions between two atoms on the bond-order parameter, i.e. on their coordinations. The modification is performed in such a manner that the total potential and its first derivative remain continuous and monotonic for all repulsive interactions.