Non-polynomial Spline Research Articles

Solution of Parabolic Partial Differential Equations Via Non-Polynomial Cubic Spline Technique

The discovery of parabolic partial differential equation (PDE) has made a profound impact on the scientific, engineering and technological community. A vast amount of research has been conducted to find the solution of parabolic PDEs. In this research, we introduced a novel technique to find the numerical solution of the fourth order PDEs. The novel technique is based upon the polynomial cubic spline method (PCSM) used along with Adomian decomposition method (ADM). The constraint of the alternative variables was decomposed by ADM to achieve successive approximation. Additionally, a numerical test problem of parabolic PDEs was solved by the proposed technique to check its viability.
 Copyright (c) The Author

Splines of the Fourth Order Approximation and the Volterra Integral Equations

This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.

Fractal non-polynomial spline method for the solution of fourth-order boundary value problems in plate deflection theory

Fractal non-polynomial spline method for the solution of fourth-order boundary value problems in plate deflection theory

Semi-analytical solutions for the hydrodynamic stability based nonlinear fourteenth order differential problem

This research article is concerned with the solution of hydrodynamic stability based linear and nonlinear fourteenth order differential problem, which has great significance in applied physics, astrophysics, applied mathematics, engineering departments. The homotopy perturbation method (HPM) and optimal homotopy asymptotic method (OHAM) are applied for the solution of the existed problem. These semi analytical techniques are continuously evolved to solve diverse range of linear and nonlinear problems with effective approximate agents which is a rapid approach to the exact solutions. This approach is effectively proposed with different numerical examples, which are taken from literature. Numerical results are accomplished by phrase of convergent series solutions and approach to the accurate solutions only by taking minimum steps. The numerical results are exercised with exact solutions, cubic polynomial spline technique (CPST) and cubic non-polynomial spline technique (CNPST), excellent agreement has been observed. The observations suggested that OHAM and HPM performed excellent in comparison to the CPST and CNPST in terms of solution, which demonstrated the effectiveness, potential and validity of suggested schemes in reality and acquired results are of top-level perfection.

Numerical approach for differential-difference equations having layer behaviour with small or large delay using non-polynomial spline

A numerical approach is suggested for the layer behaviour differential-difference equations with small and large delays in the differentiated term. Using the non-polynomial spline, the numerical scheme is derived. The discretization equation is constructed using the first-order derivative continuity at non-polynomial spline internal mesh points. A fitting parameter is introduced into the scheme with the help of the singular perturbation theory to minimize the error in the solution. The maximum errors in the solution are tabulated to verify the competence of the numerical method relative to the other methods in literature. We also focussed on the impact of large delays on the layer behaviour or oscillatory behaviour of solutions using a special mesh with and without fitting parameter in the proposed scheme. Graphs show the effect of the fitting parameter on the solution layer.

Numerical Study of the System of Nonlinear Volterra Integral Equations by Using Spline Method

The second order non-polynomial spline function for solving system of two nonlinear Volterra integral equations is proposed in this paper. An algorithm introduced as well to numerical examples to illustrate carry out of this method. Also, we compare the absolute error of quadratic non-polynomial spline method with absolute error of linear non-polynomial spline method and the exact solution.

Image Compression and Enlargement Algorithms

In some cases, there are problems associated with the compression and enlargement of images. The use of splines is quite effective in some cases. In this paper, a new image compression algorithm is presented. The features of increasing the size of an image when using local polynomial or non-polynomial splines are considered. The main method for enlarging an image is based on the use of splines of the second and third order of approximation. Polynomial and trigonometric splines are considered. To speed up the process of enlarging the image, we used the parallelization techniques

QSSOR and cubic non-polynomial spline method for the solution of two-point boundary value problems

Two-point boundary value problems are commonly used as a numerical test in developing an efficient numerical method. Several researchers studied the application of a cubic non-polynomial spline method to solve the two-point boundary value problems. A preliminary study found that a cubic non-polynomial spline method is better than a standard finite difference method in terms of the accuracy of the solution. Therefore, this paper aims to examine the performance of a cubic non-polynomial spline method through the combination with the full-, half-, and quarter-sweep iterations. The performance was evaluated in terms of the number of iterations, the execution time and the maximum absolute error by varying the iterations from full-, half- to quarter-sweep. A successive over-relaxation iterative method was implemented to solve the large and sparse linear system. The numerical result showed that the newly derived QSSOR method, based on a cubic non-polynomial spline, performed better than the tested FSSOR and HSSOR methods.

An Approximate Solution of the Space Fractional-Order Heat Equation by the Non-Polynomial Spline Functions

The linear non-polynomial spline is used here to solve the fractional partial differential equation (FPDE). The fractional derivatives are described in the Caputo sense. The tensor products are given for extending the one-dimensional linear non-polynomial spline to a two-dimensional spline to solve the heat equation. In this paper, the convergence theorem of the method used to the exact solution is proved and the numerical examples show the validity of the method. All computations are implemented by Mathcad15.

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.

Approximate Solution of 2-Dimensional VO Linear Fractional Partial Differential Equation

The non-polynomial spline method has been used to solving 2-dimensional variable-order(VO) fractional partial differential equations (FPDE). For VO fractional derivative, described in the sense of the Caputo. The main objective of this study and advantage of the proposed method is to investigate a public approximation for the frequency of the trigonometric functions of the non-polynomial part of the spline function. The powerful algorithm of the proposed method gives high accuracy results.

Fifth order of non-polynomial spline functions to find a second type integral numerical solution linear Volterra with weakly singular kernel

In this work, the fifth order non-polynomial spline functions are used to solve linear Volterra integral equations with weakly singular kernel. Numerical example is presented to illustrate the applications of this method and to compare the computed results with other numerical methods. we take integral equation when it has the kernel and it is non-linear. The results are the approximations that depended on Volterra integral equation. Our work calculated the results for the Volterra integral equation, which are approximate that the error is found.

Robust numerical method for singularly perturbed differential equations having both large and small delay

PurposeThe purpose of this study is to develop stable, convergent and accurate numerical method for solving singularly perturbed differential equations having both small and large delay.Design/methodology/approachThis study introduces a fitted nonpolynomial spline method for singularly perturbed differential equations having both small and large delay. The numerical scheme is developed on uniform mesh using fitted operator in the given differential equation.FindingsThe stability of the developed numerical method is established and its uniform convergence is proved. To validate the applicability of the method, one model problem is considered for numerical experimentation for different values of the perturbation parameter and mesh points.Originality/valueIn this paper, the authors consider a new governing problem having both small delay on convection term and large delay. As far as the researchers' knowledge is considered numerical solution of singularly perturbed boundary value problem containing both small delay and large delay is first being considered.

Nonpolynomial twin parameter spline approach to treat boundary-value problems arising in engineering problems

This study puts forward construction of an efficient nonpolynomial twin parameter cubic spline-based numerical scheme for approximations to the solution of heat transfer and defection in cables problems represented as system of second-order boundary-value problems. The introduction of an additional parameter in trigonometric part of nonpolynomial cubic spline makes this scheme a better one as compared to other existing numerical methods. The Icing on the cake is the applicability of the proposed scheme for unequal step size. The present algorithm gives better approximations in comparison to other spline, collocation, and finite-difference methods. The convergence analysis of the proposed algorithm is talked about to make a strong foundation to the proposed algorithm. Practical usefulness of the proposed method is illustrated through numerical examples.

Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

The present paper is devoted to the application of local polynomial and non-polynomial interpolation splines of the third order of approximation for the numerical solution of the Volterra integral equation of the second kind. Computational schemes based on the use of the splines include the ability to calculate the integrals over the kernel multiplied by the basis function which are present in the computational methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra integral equations is also discussed. The results of the numerical experiments are presented.

On the Solution of Fredholm Integral Equations of the First Kind

As it is well known the problem of solving the Fredholm integral equation of the first kind belongs to the class of ill-posed problems. The Tikhonov regularization method is well known. This method is usually applied to an integral equation and a system of linear algebraic equations. The authors firstly propose to reduce the integral equation of the first kind to a system of linear algebraic equations. This system is usually extremely ill-posed. Therefore, it is necessary to carry out the Tikhonov regularization for the system of equations. In this paper, to form a system of linear algebraic equations, local polynomial and non-polynomial spline approximations of the second order of approximation are used. The results of numerical experiments are presented.

Non-Polynomial Spline Function for a Class of Weakly Singular Volterra Integral Equations of the Second Kind

Non-Polynomial Spline Function for a Class of Weakly Singular Volterra Integral Equations of the Second Kind

Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical Systems

This paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented

Solution of Singularly Perturbed Boundary Value Problems with Singularity Using Variable Mesh Finite Difference Method

We use non-polynomial spline with variable mesh to establish a numerical scheme for the solution of boundary value problem with singularity. The discrete equation of the problem is developed based on the condition of the class C 1 of non-polynomial spline at the inner nodes and it is not valid for singularity. At singularity t = 0, the problem is modified in order to have a three term relationship. The method’s tridiagonal scheme is analyzed using the well-known discrete imbedding invariant algorithm. We discuss the error analysis of the scheme and two examples with layer at one end of the boundary are consider to demonstrate the practical utility of the scheme. Maximum absolute errors are present in tabular form to show the efficiency of the proposed method.

Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions

Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree <i>n</i> joined together at the break points with <i>n</i>-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.