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

Perturbation method is a commonly used method to solve galloping equation of iced transmission lines, but few scholars have studied the influences of perturbation method on the accuracy of approximate solutions of the galloping equation. In order to analyze the accuracy of approximate solutions obtained by perturbation method for galloping equation of iced transmission lines, the partial differential galloping equation of iced transmission lines with quadratic and cubic nonlinear terms is obtained firstly. Then, the partial differential galloping equation is transformed into ordinary differential galloping equation by Galerkin method. Finally, the approximate solutions of the partial differential galloping equation are obtained by averaging method and first-order, second-order, third-order, and fourth-order multiple scales methods, and the results obtained by these methods are compared systematically. By comparing the numerical solutions and the approximate solutions obtained by averaging method, it can be found that, with the increasing in wind velocity and Young’s modulus of iced transmission lines, the nonlinearity of the system would strengthen and the drift of the vibration center of the system would also increase. The larger the drift is, the greater the error between the approximate solutions obtained by averaging method and the numerical solutions will be. And when the wind velocity reaches 32 m/s, the error would arrive at 17.321%. By comparing the numerical solutions and the approximate solutions obtained by the first-order, the second-order, the third-order, and the fourth-order multiple scales methods, it can be concluded that the first-order multiple scales method is less complex computationally. The accuracy of approximate solutions obtained by the fourth-order multiple scales method is better than that obtained by the first-order, the second-order, and the third-order multiple scales methods, and the error between the approximate solutions obtained by the fourth-order multiple scales method and the numerical solutions is less than 0.639%. The conclusions obtained in this paper would be helpful to the solutions of galloping equation of iced transmission lines and could also give some references to practical engineering.

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“…Based on Galerkin method [31], substituting (10) into ( 9) can obtain the nonlinear ordinary differential equation:…”

Section: Numerical Solutionsmentioning

confidence: 99%

Investigation on the Accuracy of Approximate Solutions Obtained by Perturbation Method for Galloping Equation of Iced Transmission Lines

Liu

Min

Cai

et al. 2021

Mathematical Problems in Engineering

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“…This property of EFup functions of the exponential type gives them an advantage over other finite functions (such as algebraic Fups, splines, wavelets), and it is their flexibility that allows them to “adapt” more to the solution. On a similar track, exponential splines that also contain a tension parameter are used in the literature; these are often applied to solve the singularly perturbed boundary problem [ 58 , 59 ]. When using the exponential type of atomic basis functions, the basic task is to determine the value of the parameter

.…”

Section: Numerical Algorithms Using Fup Basis Functionsmentioning

confidence: 99%

Numerical Algorithms for Estimating Probability Density Function Based on the Maximum Entropy Principle and Fup Basis Functions

Kurbaša

Gotovac

Kozulić

et al. 2021

Entropy

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Estimation of the probability density function from the statistical power moments presents a challenging nonlinear numerical problem posed by unbalanced nonlinearities, numerical instability and a lack of convergence, especially for larger numbers of moments. Despite many numerical improvements over the past two decades, the classical moment problem of maximum entropy (MaxEnt) is still a very demanding numerical and statistical task. Among others, it was presented how Fup basis functions with compact support can significantly improve the convergence properties of the mentioned nonlinear algorithm, but still, there is a lot of obstacles to an efficient pdf solution in different applied examples. Therefore, besides the mentioned classical nonlinear Algorithm 1, in this paper, we present a linear approximation of the MaxEnt moment problem as Algorithm 2 using exponential Fup basis functions. Algorithm 2 solves the linear problem, satisfying only the proposed moments, using an optimal exponential tension parameter that maximizes Shannon entropy. Algorithm 2 is very efficient for larger numbers of moments and especially for skewed pdfs. Since both Algorithms have pros and cons, a hybrid strategy is proposed to combine their best approximation properties.

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“…Among the classical numerical methods for solving second-order differential equations, the finite element method (FEM) is presently the most widely used [9]. Nothing better illustrates the challenges of BVP defined by Equation (1) than the fact that, even in recent years, we can find publications where the authors researched methods for solving the non-constant coefficient problem [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

Vizvari

Klincsik

Ódry

et al. 2023

Axioms

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In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

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

Cited by 3 publications

References 17 publications

Investigation on the Accuracy of Approximate Solutions Obtained by Perturbation Method for Galloping Equation of Iced Transmission Lines

Investigation on the Accuracy of Approximate Solutions Obtained by Perturbation Method for Galloping Equation of Iced Transmission Lines

Numerical Algorithms for Estimating Probability Density Function Based on the Maximum Entropy Principle and Fup Basis Functions

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

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