Mohammad Reza Eslahchi - Academia.edu (original) (raw)

Papers by Mohammad Reza Eslahchi

Numerical Methods for Partial Differential Equations

The main target of this paper is to present an efficient method to solve a nonlinear free boundar... more The main target of this paper is to present an efficient method to solve a nonlinear free boundary mathematical model of prostate tumor. This model consists of two parabolics, one elliptic and one ordinary differential equations that are coupled together and describe the growth of a prostate tumor. We start our discussion by using the front fixing method to fix the free domain. Then, after employing a nonclassical finite difference and the collocation methods on this model, their stability and convergence are proved analytically. Finally, some numerical results are considered to show the efficiency of the mentioned methods.

Applied Numerical Mathematics

arXiv (Cornell University), Feb 25, 2022

Mathematical Sciences, 2021

Neural Computing and Applications, 2022

Iranian Journal of Mathematical Sciences and Informatics, 2012

In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn... more In this paper we obtain the explicit form of the best uniform polynomial approximations out of Pn of two classes of rational functions using properties of Chebyshev polynomials. In this way we present some new theorems and lemmas. Some examples will be given to support the theoretical results.

$Research paper thumbnail of Image denoising by a novel variable‐order total fractional variation model$

Mathematical Methods in the Applied Sciences, 2021

The total variation model performs very well for removing noise while preserving edges. However, ... more The total variation model performs very well for removing noise while preserving edges. However, it gives a piecewise constant solution which often leads to the staircase effect, consequently small details such as textures are filtered out in the denoising process. Fractional‐order total variation method is one of the major approaches to overcome such drawbacks. Unlike their good quality of fractional order, all these methods use a fixed fractional order for the whole of the image. In this paper, a novel variable‐order total fractional variation model is proposed for image denoising, in which the order of fractional derivative will be allocated automatically for each pixel based on the context of the image. This kind of selection is able to capture the edges and texture of the image simultaneously. In this regard, we prove the existence and uniqueness of the presented model. The split Bregman method is adapted to solve the model. Finally, the results illustrate the efficiency of the proposed model that yielded good visual effects and a better signal‐to‐noise ratio.

Applied Numerical Mathematics, 2020

This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

$Research paper thumbnail of Extension of Tikhonov regularization method using linear fractional programming$

Journal of Computational and Applied Mathematics, 2019

Numerical Algorithms, 2019

In this paper, we present two families of modified three-term conjugate gradient methods for solv... more In this paper, we present two families of modified three-term conjugate gradient methods for solving unconstrained large-scale smooth optimization problems. We show that our new families satisfy the Dai-Liao conjugacy condition and the sufficient descent condition under any line search technique which guarantees the positiveness of y T k s k. For uniformly convex functions, we indicate that our families are globally convergent under weak-Wolfe-Powell line search technique and standard conditions on the objective function. We also establish a weaker global convergence theorem for general smooth functions under similar assumptions. Our numerical experiments for 260 standard problems and seven other recently developed conjugate gradient methods illustrate that the members of our families are numerically efficient and effective.

Numerical Algorithms, 2017

This paper presents a new approach to improve the order of approximation of the Bernstein operato... more This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically.

Journal of Computational Physics, 2017

This paper is intended to provide exponentially accurate Galerkin, Petrov-Galerkin and pseudo-spe... more This paper is intended to provide exponentially accurate Galerkin, Petrov-Galerkin and pseudo-spectral methods for fractional differential equations on a semi-infinite interval. We start our discussion by introducing two new non-classical Lagrange basis functions: NLBFs-1 and NLBFs-2 which are based on the two new families of the associated Laguerre polynomials: GALFs-1 and GALFs-2 obtained recently by the authors in [28]. With respect to the NLBFs-1 and NLBFs-2, two new non-classical interpolants based on the associated-Laguerre-Gauss and Laguerre-Gauss-Radau points are introduced and then fractional (pseudo-spectral) differentiation (and integration) matrices are derived. Convergence and stability of the new interpolants are proved in detail. Several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives (and integrals) of some functions. Moreover, the pseudo-spectral, Galerkin and Petrov-Galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones. Some useful comments from the numerical point of view on Galerkin and Petrov-Galerkin methods are listed at the end.

Mathematical Sciences, 2015

In this paper, we use a general identity for generalized hypergeometric series to obtain some new... more In this paper, we use a general identity for generalized hypergeometric series to obtain some new applications. The first application is a hypergeometric-type decomposition formula for elementary special functions and the second one is a generalization of the well-known Euler identity e i x ¼ cos x þ i sin x and an extension of hyperbolic functions in the sequel. Applying the mentioned identity on classical hypergeometric orthogonal polynomials and deriving summation formulae for some classical summation theorems are two further applications of this identity.

Journal of Numerical Mathematics, 2015

In this research using properties of Chebyshev polynomialswe explicitly determine the best unifor... more In this research using properties of Chebyshev polynomialswe explicitly determine the best uniform polynomial approximation of some classes of functions. In this way we present some new theorems about the best approximation of these classes.

Journal of Computational and Applied Mathematics, 2016

In this paper we introduce a modified spectral method for solving the linear operator equation Lu... more In this paper we introduce a modified spectral method for solving the linear operator equation Lu = f, L : D(L) ⊆ H 1 → H 2 , where H 1 and H 2 are normed vector spaces with norms ∥.∥ 1 and ∥.∥, respectively and D(L) is the domain of L. Also for each h ∈ H 2 , ∥h∥ 2 = (h, h) where (., .) is an inner product on H 2. In this method we make a new set {ψ n } ∞ n=0 for H 1 using L and two sets in H 1 and H 2. Then using the new set {ψ n } ∞ n=0 we solve this linear operator equation. We show that this method doesn't have some shortcomings of spectral method, also we prove the stability and convergence of the new method. After introducing the method we give some conditions that under them the nonlinear operator equation Lu + N u = f can be solved. Some examples are considered to show the efficiency of method.

Mathematical and Computer Modelling, 2012

In this study, using the properties of third and fourth kinds of Chebyshev polynomials, we explic... more In this study, using the properties of third and fourth kinds of Chebyshev polynomials, we explicitly determine the best uniform polynomial approximation out of P n to classes of functions that are obtained from their generating function and their derivatives. Efficiency of these polynomials is demonstrated by some examples.

International Journal of Computer Mathematics, 2006

ABSTRACT Due to having the minimax property, Chebyshev polynomials are used today to economize th... more ABSTRACT Due to having the minimax property, Chebyshev polynomials are used today to economize the arbitrary polynomial functions. In this work, we present a statistical approach to show that, contrary to current thought, the Chebyshev polynomials of the first kind are not appropriate for economizing these polynomials if one uses this statistical approach. In this way, a numerical results section is also given to clearly prove our claim.

Computers & Mathematics with Applications, 2011

In this research first we explicitly obtain the relation between the coefficients of the Taylor s... more In this research first we explicitly obtain the relation between the coefficients of the Taylor series and Jacobi polynomial expansions. Then we present a new method for computing classical operational matrices (derivative, integral and product) for general Jacobi orthogonal functions (polynomial and rational). This method can be used for many classes of orthogonal functions.

Computers & Mathematics with Applications, 2010

In this research paper using the Chebyshev expansion, we explicitly determine the best uniform po... more In this research paper using the Chebyshev expansion, we explicitly determine the best uniform polynomial approximation out of P qn (the space of polynomials of degree at most qn) to a class of rational functions of the form 1/(T q (a) ± T q (x)) on [−1, 1], where T q (x) is the first kind of Chebyshev polynomial of degree q and a 2 > 1. In this way we give some new theorems about the best approximation of this class of rational functions. Furthermore we obtain the alternating set of this class of functions.

Computational Economics, 2007

We investigate a class of optimal control problems that exhibit constant exogenously given delays... more We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a stateof-the-art direct method by applying Bock's direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.