Special Issue "Multivariate Approximation for solving ODE and PDE"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 January 2020.

Special Issue Editor

Guest Editor
Prof. Clemente Cesarano Website E-Mail
Section of Mathematics – International Telematic University UNINETTUNO
Phone: +39.0669207675
Interests: special functions; orthogonal polynomials; fractional calculus; numerical methods; ODE and PDE

Special Issue Information

Dear Colleagues,

Multivariate approximation is an extension of approximation theory and approximation algorithms. In general, approximations can be provided via interpolation, as approximation/polynomials interpolation and approximation/interpolation with radial basis functions or, more in general, with kernel functions. In this Special Issue, we would like to cover the field of spectral problems, exponential integrators for ODE systems, and some applications for the numerical solution of evolutionary PDE, also discretized, by using the concepts and the related formalism of special functions and orthogonal polynomials, which represent a powerful tool to simplify computation. Since the theory of multivariate approximation meets different branches of mathematics and is applied in various areas such as physics, engineering, and computational mechanics, this Special Issue is open to a large community of researchers.

Prof. Clemente Cesarano
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Numerical methods
  • Approximation theory
  • Polynomial interpolation
  • Spectral problems
  • Evolution operators
  • ODE
  • PDE

Published Papers (5 papers)

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Research

Open AccessArticle
Special Class of Second-Order Non-Differentiable Symmetric Duality Problems with (G,αf)-Pseudobonvexity Assumptions
Mathematics 2019, 7(8), 763; https://doi.org/10.3390/math7080763 - 20 Aug 2019
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 ( G [...] 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)
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Open AccessArticle
Viscovatov-Like Algorithm of Thiele–Newton’s Blending Expansion for a Bivariate Function
Mathematics 2019, 7(8), 696; https://doi.org/10.3390/math7080696 - 02 Aug 2019
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)
Open AccessFeature PaperArticle
One-Point Optimal Family of Multiple Root Solvers of Second-Order
Mathematics 2019, 7(7), 655; https://doi.org/10.3390/math7070655 - 21 Jul 2019
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)
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Open AccessFeature PaperArticle
Some New Oscillation Criteria for Second Order Neutral Differential Equations with Delayed Arguments
Mathematics 2019, 7(7), 619; https://doi.org/10.3390/math7070619 - 11 Jul 2019
Cited by 1
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)
Open AccessArticle
An Efficient Derivative Free One-Point Method with Memory for Solving Nonlinear Equations
Mathematics 2019, 7(7), 604; https://doi.org/10.3390/math7070604 - 06 Jul 2019
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)
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