Research

19 pages, 3139 KiB

Open AccessArticle

by Marta Gatto and Fabio Marcuzzi

Mathematics 2020, 8(6), 982; https://doi.org/10.3390/math8060982 - 16 Jun 2020

Cited by 2 | Viewed by 1636

In this paper we analyze the bias in a general linear least-squares parameter estimation problem, when it is caused by deterministic variables that have not been included in the model. We propose a method to substantially reduce this bias, under the hypothesis that [...] Read more.

In this paper we analyze the bias in a general linear least-squares parameter estimation problem, when it is caused by deterministic variables that have not been included in the model. We propose a method to substantially reduce this bias, under the hypothesis that some a-priori information on the magnitude of the modelled and unmodelled components of the model is known. We call this method Unbiased Least-Squares (ULS) parameter estimation and present here its essential properties and some numerical results on an applied example. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

► Show Figures

Figure 1

10 pages, 280 KiB

Open AccessArticle

Improved Conditions for Oscillation of Functional Nonlinear Differential Equations

by Omar Bazighifan and Mihai Postolache

Mathematics 2020, 8(4), 552; https://doi.org/10.3390/math8040552 - 09 Apr 2020

Cited by 28 | Viewed by 1930

Abstract

The aim of this work is to study oscillatory properties of a class of fourth-order delay differential equations. New oscillation criteria are obtained by using generalized Riccati transformations. This new theorem complements and improves a number of results reported in the literature. Some [...] Read more.

The aim of this work is to study oscillatory properties of a class of fourth-order delay differential equations. New oscillation criteria are obtained by using generalized Riccati transformations. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

23 pages, 357 KiB

Open AccessArticle

Approximation of Finite Hilbert and Hadamard Transforms by Using Equally Spaced Nodes

by Frank Filbir, Donatella Occorsio and Woula Themistoclakis

Mathematics 2020, 8(4), 542; https://doi.org/10.3390/math8040542 - 07 Apr 2020

Cited by 8 | Viewed by 1522

Abstract

In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function f, supposing to know only the samples of f at equidistant points. As reference interval we consider [...] Read more.

In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function f, supposing to know only the samples of f at equidistant points. As reference interval we consider

[- 1, 1]

and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

12 pages, 775 KiB

Open AccessArticle

Oscillation Criteria of Higher-order Neutral Differential Equations with Several Deviating Arguments

by Osama Moaaz, Ioannis Dassios and Omar Bazighifan

Mathematics 2020, 8(3), 412; https://doi.org/10.3390/math8030412 - 13 Mar 2020

Cited by 26 | Viewed by 2008

Abstract

This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using the technique of Riccati transformation and comparison principles with the second-order differential equations, we obtain a new Philos-type criterion. Our results extend and improve some known [...] Read more.

This work is concerned with the oscillatory behavior of solutions of even-order neutral differential equations. By using the technique of Riccati transformation and comparison principles with the second-order differential equations, we obtain a new Philos-type criterion. Our results extend and improve some known results in the literature. An example is given to illustrate our main results. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

19 pages, 3111 KiB

Open AccessArticle

Alternating Asymmetric Iterative Algorithm Based on Domain Decomposition for 3D Poisson Problem

by Qiuyan Xu and Zhiyong Liu

Mathematics 2020, 8(2), 281; https://doi.org/10.3390/math8020281 - 19 Feb 2020

Viewed by 1926

Abstract

Poisson equation is a widely used partial differential equation. It is very important to study its numerical solution. Based on the strategy of domain decomposition, the alternating asymmetric iterative algorithm for 3D Poisson equation is provided. The solution domain is divided into several [...] Read more.

Poisson equation is a widely used partial differential equation. It is very important to study its numerical solution. Based on the strategy of domain decomposition, the alternating asymmetric iterative algorithm for 3D Poisson equation is provided. The solution domain is divided into several sub-domains, and eight asymmetric iterative schemes with the relaxation factor for 3D Poisson equation are constructed. When the numbers of iteration are odd or even, the computational process of the presented iterative algorithm are proposed respectively. In the calculation of the inner interfaces, the group explicit method is used, which makes the algorithm to be performed fast and in parallel, and avoids the difficulty of solving large-scale linear equations. Furthermore, the convergence of the algorithm is analyzed theoretically. Finally, by comparing with the numerical experimental results of Jacobi and Gauss Seidel iterative algorithms, it is shown that the alternating asymmetric iterative algorithm based on domain decomposition has shorter computation time, fewer iteration numbers and good parallelism. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

► Show Figures

Figure 1

28 pages, 319 KiB

Open AccessArticle

Weighted Fractional Iyengar Type Inequalities in the Caputo Direction

by George A. Anastassiou

Mathematics 2019, 7(11), 1119; https://doi.org/10.3390/math7111119 - 16 Nov 2019

Cited by 1 | Viewed by 1612

Abstract

Here we present weighted fractional Iyengar type inequalities with respect to

L_{p}

norms, with

1 \leq p \leq \infty

. Our employed fractional calculus is of Caputo type defined with respect to another function. Our results provide quantitative estimates for the approximation [...] Read more.

Here we present weighted fractional Iyengar type inequalities with respect to

L_{p}

norms, with

1 \leq p \leq \infty

. Our employed fractional calculus is of Caputo type defined with respect to another function. Our results provide quantitative estimates for the approximation of the Lebesgue–Stieljes integral of a function, based on its values over a finite set of points including at the endpoints of its interval of definition. Our method relies on the right and left generalized fractional Taylor’s formulae. The iterated generalized fractional derivatives case is also studied. We give applications at the end. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

12 pages, 271 KiB

Open AccessArticle

Duality for Unified Higher-Order Minimax Fractional Programming with Support Function under Type-I Assumptions

by Ramu Dubey, Vishnu Narayan Mishra and Rifaqat Ali

Mathematics 2019, 7(11), 1034; https://doi.org/10.3390/math7111034 - 03 Nov 2019

Cited by 7 | Viewed by 1556

Abstract

This article is devoted to discussing the nondifferentiable minimax fractional programming problem with type-I functions. We focus our study on a nondifferentiable minimax fractional programming problem and formulate a higher-order dual model. Next, we establish weak, strong, and strict converse duality theorems under [...] Read more.

This article is devoted to discussing the nondifferentiable minimax fractional programming problem with type-I functions. We focus our study on a nondifferentiable minimax fractional programming problem and formulate a higher-order dual model. Next, we establish weak, strong, and strict converse duality theorems under generalized higher-order strictly pseudo

(V, α, ρ, d)

-type-I functions. In the final section, we turn our focus to study a nondifferentiable unified minimax fractional programming problem and the results obtained in this paper naturally unify. Further, we extend some previously known results on nondifferentiable minimax fractional programming in the literature. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

16 pages, 1203 KiB

Open AccessArticle

Special Class of Second-Order Non-Differentiable Symmetric Duality Problems with (G,α_f)-Pseudobonvexity Assumptions

by Ramu Dubey, Lakshmi Narayan Mishra and Rifaqat Ali

Mathematics 2019, 7(8), 763; https://doi.org/10.3390/math7080763 - 20 Aug 2019

Cited by 13 | Viewed by 2547

Abstract

In this paper, we introduce the various types of generalized invexities, i.e.,

α_{f}

-invex/

α_{f}

-pseudoinvex and

(G, α_{f})

-bonvex/

(G, α_{f})

-pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of [...] Read more.

In this paper, we introduce the various types of generalized invexities, i.e.,

α_{f}

-invex/

α_{f}

-pseudoinvex and

(G, α_{f})

-bonvex/

(G, α_{f})

-pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of

(G, α_{f})

-bonvexity/

(G, α_{f})

-pseudobonvexity, which is neither

α_{f}

-bonvex/

α_{f}

-pseudobonvex nor

α_{f}

-invex/

α_{f}

-pseudoinvex with the same

η

. Further, we formulate a pair of second-order non-differentiable symmetric dual models and prove the duality relations under

α_{f}

-invex/

α_{f}

-pseudoinvex and

(G, α_{f})

-bonvex/

(G, α_{f})

-pseudobonvex assumptions. Finally, we construct a nontrivial numerical example justifying the weak duality result presented in the paper. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

► Show Figures

Figure 1

15 pages, 827 KiB

Open AccessArticle

Viscovatov-Like Algorithm of Thiele–Newton’s Blending Expansion for a Bivariate Function

by Shengfeng Li and Yi Dong

Mathematics 2019, 7(8), 696; https://doi.org/10.3390/math7080696 - 02 Aug 2019

Cited by 3 | Viewed by 1905

Abstract

In this paper, Thiele–Newton’s blending expansion of a bivariate function is firstly suggested by means of combining Thiele’s continued fraction in one variable with Taylor’s polynomial expansion in another variable. Then, the Viscovatov-like algorithm is given for the computations of the coefficients of [...] Read more.

In this paper, Thiele–Newton’s blending expansion of a bivariate function is firstly suggested by means of combining Thiele’s continued fraction in one variable with Taylor’s polynomial expansion in another variable. Then, the Viscovatov-like algorithm is given for the computations of the coefficients of this rational expansion. Finally, a numerical experiment is presented to illustrate the practicability of the suggested algorithm. Henceforth, the Viscovatov-like algorithm has been considered as the imperative generalization to find out the coefficients of Thiele–Newton’s blending expansion of a bivariate function. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

11 pages, 666 KiB

Open AccessFeature PaperArticle

One-Point Optimal Family of Multiple Root Solvers of Second-Order

by Deepak Kumar, Janak Raj Sharma and Clemente Cesarano

Mathematics 2019, 7(7), 655; https://doi.org/10.3390/math7070655 - 21 Jul 2019

Cited by 4 | Viewed by 2320

Abstract

This manuscript contains the development of a one-point family of iterative functions. The family has optimal convergence of a second-order according to the Kung-Traub conjecture. This family is used to approximate the multiple zeros of nonlinear equations, and is based on the procedure [...] Read more.

This manuscript contains the development of a one-point family of iterative functions. The family has optimal convergence of a second-order according to the Kung-Traub conjecture. This family is used to approximate the multiple zeros of nonlinear equations, and is based on the procedure of weight functions. The convergence behavior is discussed by showing some essential conditions of the weight function. The well-known modified Newton method is a member of the proposed family for particular choices of the weight function. The dynamical nature of different members is presented by using a technique called the “basin of attraction”. Several practical problems are given to compare different methods of the presented family. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

► Show Figures

Figure 1

9 pages, 254 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Some New Oscillation Criteria for Second Order Neutral Differential Equations with Delayed Arguments

by Omar Bazighifan and Clemente Cesarano

Mathematics 2019, 7(7), 619; https://doi.org/10.3390/math7070619 - 11 Jul 2019

Cited by 41 | Viewed by 2250

Abstract

In this paper, we study the oscillation of second-order neutral differential equations with delayed arguments. Some new oscillatory criteria are obtained by a Riccati transformation. To illustrate the importance of the results, one example is also given. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

10 pages, 276 KiB

Open AccessArticle

An Efficient Derivative Free One-Point Method with Memory for Solving Nonlinear Equations

by Janak Raj Sharma, Sunil Kumar and Clemente Cesarano

Mathematics 2019, 7(7), 604; https://doi.org/10.3390/math7070604 - 06 Jul 2019

Cited by 7 | Viewed by 1923

Abstract

We propose a derivative free one-point method with memory of order 1.84 for solving nonlinear equations. The formula requires only one function evaluation and, therefore, the efficiency index is also 1.84. The methodology is carried out by approximating the derivative in Newton’s iteration [...] Read more.

We propose a derivative free one-point method with memory of order 1.84 for solving nonlinear equations. The formula requires only one function evaluation and, therefore, the efficiency index is also 1.84. The methodology is carried out by approximating the derivative in Newton’s iteration using a rational linear function. Unlike the existing methods of a similar nature, the scheme of the new method is easy to remember and can also be implemented for systems of nonlinear equations. The applicability of the method is demonstrated on some practical as well as academic problems of a scalar and multi-dimensional nature. In addition, to check the efficacy of the new technique, a comparison of its performance with the existing techniques of the same order is also provided. Full article

(This article belongs to the Special Issue Multivariate Approximation for solving ODE and PDE)

Journal Menu

Journal Browser

Multivariate Approximation for solving ODE and PDE

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Published Papers (12 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI