Non-polynomial Spline Functions Research Articles

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

The aim of this work is to use an efficient and accurate numerical technique based on non-polynomial spline for the solution of the PHI-Four and Allen–Cahn equations. A recent discovery suggests that the PHI-Four equation focuses on its implications for particle physics and the behavior of scalar fields in the quantum realm. In materials science, ongoing research involves using the Allen–Cahn equation to understand and predict the evolution of microstructures in various materials as well as in biophysics. It depicts pattern formation in biological systems and the dynamics of spatial organization in tissues. To obtain an approximate solution of both equations, this technique uses forward differences for the time and cubic non-polynomial spline function for spatial descretization. The stability of the suggested technique is addressed using the von Neumann technique. Convergence test is carried out theoretically to show the order of convergence of the scheme. Some numerical tests are carried out to confirm accuracy and efficiency in terms of absolute error LR. Convergence rates for different test problems are also computed numerically. Numerical results and simulations obtained are compared with the existing methods.

Algorithms for Solving Volterra Integral Equations Using Non-Polynomial Spline Function

This paper discusses some algorithms for solving Volterra's integral equations of the second degree using non-boundary spline functions, including (linear and cubic), as the linear spline function presents some laws and algorithms of its own, as well as the cubic spline function has its laws and algorithm. They are applied and combined in a language program MATLAB R2022a to find the numerical solution for them and the difference between them through the error percentage that appears for each of them. The preview is done through the drawing, as shown in the examples below.

Optimized Trigonometric Spline Function with Conjugate GradientMethod for Solving FDEs

We investigate the non-polynomial spline function to solve the fractional differential equations with the conformable conjugate gradient method. The fractional derivative was described using the Caputo fractional derivative to construct the spline scheme with polynomial fractional order. Therefore, transform the problem to an equivalent iterative linear system that can be solved by Gauss-Seidel and conjugate gradient methods. For the given spline function, error bounds were studied and a stability analysis was completed, the error estimation is also calculated as different values of (n) depend on the step size oh (h). Numerical examples with known analytical solutions are shown to verify the method's accuracy. The outcomes are in satisfactory correlation with the exact answers according to the numerical experiments. Moreover, the convergence analysis was investigated with the drive some theorems. Also, the procedure is explained in depth and supported by computational examples and the results show that the fractional spline function which interpolates data is productive and profitable in solving unique problems and compare with the exact solutions.

The Numerical Solution of Volterra Integral Equation of Second Kind Using Quartic Non-Polynomial Spline Function

In this study, the non-polynomial spline function of fourth order is given to solve the second kind of the nonlinear Volterra integral equation. Numerical examples are show to clarify the applications of the proposed method and to compare the calculated results with other methods. In addition, there is a comparison of the absolute value of the error that estimated by of the suggested method is compared by those that estimated through other methods and the exact solution. The obtained results are very encouraging compared by other numerical methods.

Numerical Solution of Volterra Integral Equations with a Weakly Singular Kernel Using the Seventh Order Non-Polynomial Spline Function Method

Numerical Solution of Volterra Integral Equations with a Weakly Singular Kernel Using the Seventh Order Non-Polynomial Spline Function Method

Numerical Solution of Fourth Order Homogeneous Parabolic Partial Differential Equations (PDEs) using Non-Polynomial Cubic Spline Method (NPCSM)

Non-polynomial cubic spline functions are already being used in the field of engineering, computer sciences, and natural sciences to solve ordinary differential equations (ODEs) and partial differential equations (PDEs). However, many of the above-mentioned problems do not have an exact, stable, or convergent exact solution. There are different approximations and methods that can be applied to solve these problems. This study implemented the purposed method on homogeneous parabolic PDEs having different dimensions. The results obtained were compared with the exact solution and results of other existing methods in tabular and graphical form. Mathematica was used to find the mathematical and graphical results.EMATICA.
 Keywords: Adomian decomposition method (ADM), non-polynomial cubic spline method (NPCSM), continuous approximation, finite difference approximations, fourth order homogeneous parabolic partial differential equations (PDEs)
 Copyright(c) The Authors

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.

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.

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.

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

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.

A smooth approximation for non-linear second order boundary value problems using composite non-polynomial spline functions

A different amalgamation of non-polynomial splines is used to find the approximate solution of linear and non-linear second order boundary value problems.Cubic spline functions are assembled with exponential and trigonometric functions to develop the different orders of numerical schemes. Free parameter k of the non-polynomial part is also used to form a new scheme, which elevates the accuracy of the solution. Numerical illustrations are given to validate the applicability and feasibility of the present method and also depicted in the graphs.

Numerical Scheme based on Non-polynomial Spline Functions for the System of Second Order Boundary Value Problems arising in Various Engineering Applications

Several applications of computational science and engineering, including population dynamics, optimal control, and physics, reduce to the study of a system of second-order boundary value problems. To achieve the improved solution of these problems, an efficient numerical method is developed by using spline functions. A non-polynomial cubic spline-based method is proposed for the first time to solve a linear system of second-order differential equations. Convergence and stability of the proposed method are also investigated. A mathematical procedure is described in detail, and several examples are solved with numerical and graphical illustrations. It is shown that our method yields improved results when compared to the results available in the literature.

A high-resolution method based on off-step non-polynomial spline approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations

PurposeThis paper aims to develop a new high accuracy numerical method based on off-step non-polynomial spline in tension approximations for the solution of Burgers-Fisher and coupled nonlinear Burgers’ equations on a graded mesh. The spline method reported here is third order accurate in space and second order accurate in time. The proposed spline method involves only two off-step points and a central point on a graded mesh. The method is two-level implicit in nature and directly derived from the continuity condition of the first order space derivative of the non-polynomial tension spline function. The linear stability analysis of the proposed method has been examined and it is shown that the proposed two-level method is unconditionally stable for a linear model problem. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, the authors have solved the generalized Burgers-Huxley equation, generalized Burgers-Fisher equation, coupled Burgers-equations and parabolic equation in polar coordinates. The authors show that the proposed method enables us to obtain the high accurate solution for high Reynolds number.Design/methodology/approachIn this method, the authors use only two-level in time-direction, and at each time-level, the authors use three grid points for the unknown function u(x,t) and two off-step points for the known variable x in spatial direction. The methodology followed in this paper is the construction of a non-polynomial spline function and using its continuity properties to obtain consistency condition, which is third order accurate on a graded mesh and fourth order accurate on a uniform mesh. From this consistency condition, the authors derive the proposed numerical method. The proposed method, when applied to a linear equation is shown to be unconditionally stable. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.FindingsThe paper provides a third order numerical scheme on a graded mesh and fourth order spline method on a uniform mesh obtained directly from the consistency condition. In earlier methods, consistency conditions were only second order accurate. This brings an edge over other past methods. Also, the method is directly applicable to physical problems involving singular coefficients. So no modification in the method is required at singular points. This saves CPU time and computational costs.Research limitations/implicationsThere are no limitations. Obtaining a high accuracy spline method directly from the consistency condition is a new work. Also being an implicit method, this method is unconditionally stable.Practical implicationsPhysical problems with singular and non-singular coefficients are directly solved by this method.Originality/valueThe paper develops a new method based on non-polynomial spline approximations of order two in time and three (four) in space, which is original and has lot of value because many benchmark problems of physical significance are solved in this method.

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.

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.

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.

Quintic non-polynomial spline methods for third order singularly perturbed boundary value problems

In this paper, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems of the reaction–diffusion equation type. The convergence analysis is discussed and the method is shown to have fourth order convergence. To validate the applicability of the method, two model examples have been solved for different values of the perturbation parameter and mesh sizes. The numerical results have been tabulated and also presented in graphs. It can be observed from the results that the present method approximates the exact solution very well.

Numerical Solution Of Volterra Integral Equation With Delay By Using Non-Polynomial Spline Function

Numerical Solution Of Volterra Integral Equation With Delay By Using Non-Polynomial Spline Function

Computational Non-Polynomial Spline Function for Solving Fractional Bagely-Torvik Equatio

Computational Non-Polynomial Spline Function for Solving Fractional Bagely-Torvik Equatio