A unified representation for some interpolation formulas

Analysis ◽  
2020 ◽  
Vol 40 (3) ◽  
pp. 113-125
Author(s):  
Mohammad Masjed-Jamei ◽  
Zahra Moalemi ◽  
Wolfram Koepf
Keyword(s):  
Existence And Uniqueness ◽  
Lagrange Interpolation ◽  
Quadrature Rules ◽  
Numerical Examples ◽  
Weighted Quadrature

AbstractAs an extension of Lagrange interpolation, we introduce a class of interpolation formulas and study their existence and uniqueness. In the sequel, we consider some particular cases and construct the corresponding weighted quadrature rules. Numerical examples are finally given and compared.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals On fractional Langevin equation involving two fractional orders in different intervals

2019 ◽  
Vol 24 (6) ◽  
Author(s):  
Hamid Baghani ◽  
J. Nieto
Keyword(s):  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Numerical Examples ◽  
Fractional Equations ◽  
Fractional Langevin Equation ◽  
Nonlinear Langevin Equation ◽  
Fractional Orders

In this paper, we study a nonlinear Langevin equation involving two fractional orders  α ∈ (0; 1] and β ∈ (1; 2] with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. Some illustrative numerical examples are also discussed. 

scholarly journals Explicit forms of weighted quadrature rules with geometric nodes

2011 ◽  
Vol 53 (5-6) ◽  
pp. 1133-1139 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir V. Milovanović ◽  
M.A. Jafari
Keyword(s):  
Quadrature Rules ◽  
Weighted Quadrature

scholarly journals Classical Lagrange Interpolation Based on General Nodal Systems at Perturbed Roots of Unity

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 498
Author(s):  
Elías Berriochoa ◽  
Alicia Cachafeiro ◽  
Alberto Castejón ◽  
José Manuel García-Amor
Keyword(s):  
Rate Of Convergence ◽  
Unit Circle ◽  
Lagrange Interpolation ◽  
Roots Of Unity ◽  
Smooth Functions ◽  
Numerical Examples ◽  
Practical Applications ◽  
Bounded Interval ◽  
Interesting Approach ◽  
Nodal System

The aim of this paper is to study the Lagrange interpolation on the unit circle taking only into account the separation properties of the nodal points. The novelty of this paper is that we do not consider nodal systems connected with orthogonal or paraorthogonal polynomials, which is an interesting approach because in practical applications this connection may not exist. A detailed study of the properties satisfied by the nodal system and the corresponding nodal polynomial is presented. We obtain the relevant results of the convergence related to the process for continuous smooth functions as well as the rate of convergence. Analogous results for interpolation on the bounded interval are deduced and finally some numerical examples are presented.

scholarly journals Linear differential algebraic equations with constant coefficients

1970 ◽  
Vol 4 (2) ◽  
pp. 28-35 ◽  
Author(s):  
M Sahadet Hossain ◽  
M Mostafizur Rahman
Keyword(s):  
Existence And Uniqueness ◽  
Differential Algebraic Equations ◽  
Canonical Forms ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Constant Coefficients ◽  
Dae Systems ◽  
International University ◽  
Linear Differential ◽  
Modern Mathematics

Differential-algebraic equations (DAEs) arise in a variety of applications. Their analysis and numerical treatment, therefore, plays an important role in modern mathematics. The paper gives an introduction to the topics of DAEs. Examples of DAEs are considered showing their importance for practical problems. Some essential concepts that are really essential for understanding the DAE systems are introduced. The canonical forms of DAEs are discussed widely to make them more efficient and easy for practical use. Also some numerical examples are discussed to clarify the existence and uniqueness of the system's solutions. Keywords: differential-algebraic equations, index concept, canonical forms. DOI: 10.3329/diujst.v4i2.4365 Daffodil International University Journal of Science and Technology Vol.4(2) 2009 pp.28-35

scholarly journals Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Z. Pashazadeh Atabakan ◽  
A. Kazemi Nasab ◽  
A. Kılıçman ◽  
Zainidin K. Eshkuvatov
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Solution ◽  
Homotopy Analysis Method ◽  
Existence And Uniqueness ◽  
Lagrange Interpolation ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution ◽  
Spectral Homotopy Analysis Method

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

scholarly journals Midpoint Derivative-Based Closed Newton-Cotes Quadrature

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Weijing Zhao ◽  
Hongxing Li
Keyword(s):  
Numerical Integration ◽  
Computational Cost ◽  
Point Of View ◽  
Quadrature Rules ◽  
Numerical Examples ◽  
Error Terms

A novel family of numerical integration of closed Newton-Cotes quadrature rules is presented which uses the derivative value at the midpoint. It is proved that these kinds of quadrature rules obtain an increase of two orders of precision over the classical closed Newton-Cotes formula, and the error terms are given. The computational cost for these methods is analyzed from the numerical point of view, and it has shown that the proposed formulas are superior computationally to the same order closed Newton-Cotes formula when they reduce the error below the same level. Finally, some numerical examples show the numerical superiority of the proposed approach with respect to closed Newton-Cotes formulas.

scholarly journals Existence, Uniqueness, and Approximation for Solutions of a Functional-integral Equation in Lp Spaces

2019 ◽  
Vol 20 (3) ◽  
pp. 403
Author(s):  
Suzete M Afonso ◽  
Juarez S Azevedo ◽  
Mariana P. G. Da Silva ◽  
Adson M Rocha
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence And Uniqueness ◽  
Functional Integral ◽  
Successive Approximation Method ◽  
Banach Fixed Point Theorem ◽  
Numerical Examples ◽  
Lp Spaces ◽  
Chebyshev Quadrature

In this work we consider the general functional-integral equation: \begin{equation*}y(t) = f\left(t, \int_{a}^{b} k(t,s)g(s,y(s))ds\right), \qquad t\in [a,b],\end{equation*}and give conditions that guarantee existence and uniqueness of solution in $L^p([a,b])$, with {$1<p<\infty$}.We use  Banach Fixed Point Theorem and employ the successive approximation method and Chebyshev quadrature for approximating the values of integrals. Finally, to illustrate the results of this work, we provide some numerical examples.

scholarly journals Runge-Kutta and Block by Block Methods to Solve Linear Two-Dimensional Volterra Integral Equation with Continuous Kernel

2016 ◽  
Vol 11 (10) ◽  
pp. 5705-5714
Author(s):  
Abeer Majed AL-Bugami
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solution ◽  
Two Dimensional ◽  
Block Method ◽  
Numerical Examples ◽  
Block Methods ◽  
Runge Kutta ◽  
Method R

In this paper, the existence and uniqueness of solution of the linear two dimensional Volterra integral equation of the second kind with Continuous Kernel are discussed and proved.RungeKutta method(R. KM)and Block by block method (B by BM) are used to solve this type of two dimensional Volterra integral equation of the second kind. Numerical examples are considered to illustrate the effectiveness of the proposed methods and the error is estimated.

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