G. Chandhini

Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, … Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter is analogous to that in the Dirichlet mollifier, which is detailed by analyzing the numerical solution. Further, they are extended to linear scalar conservation law problems.

Unified convergence analysis of a class of iterative methods

2024-07-24

M Muniyasamy , Santhosh George , G. Chandhini

A Procedure for Increasing the Convergence Order of Iterative Methods from p to 5p for Solving Nonlinear System

2024-12-01

Santhosh George , M Muniyasamy , Manjusree Gopal +2 more

Regularization of Highly Ill-Conditioned RBF Asymmetric Collocation Systems in Fractional Models

2019-01-01

K. S. Prashanthi , G. Chandhini

Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method

2019-08-02

Ioannis K. Argyros , Santhosh George , G. Chandhini +1 more

Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear … Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems. The goal of this article is two-fold: On the one hand, we present a correct convergence criterion for Newton–Hermitian splitting (NHSS) method under the Kantorovich theory, since the criterion given in Numer. Linear Algebra Appl., 2011, 18, 299–315 is not correct. Indeed, the radius of convergence cannot be defined under the given criterion, since the discriminant of the quadratic polynomial from which this radius is derived is negative (See Remark 1 and the conclusions of the present article for more details). On the other hand, we have extended the corrected convergence criterion using our idea of recurrent functions. Numerical examples involving convection–diffusion equations further validate the theoretical results.

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Iterative methods for a fractional-order Volterra population model

2019-04-01

Rupsha Roy , V. Antony Vijesh , G. Chandhini

We prove an existence and uniqueness theorem for a fractional-order Volterra population model via an efficient monotone iterative scheme. By coupling a spectral method with the proposed iterative scheme, the … We prove an existence and uniqueness theorem for a fractional-order Volterra population model via an efficient monotone iterative scheme. By coupling a spectral method with the proposed iterative scheme, the fractional-order integrodifferential equation is solved numerically. The numerical experiments show that the proposed iterative scheme is more efficient than an existing iterative scheme in the literature, the convergence of which is very sensitive to various parameters, including the fractional order of the derivative. The spectral method based on our proposed iterative scheme shows greater flexibility with respect to various parameters. Sufficient conditions are provided to select the initial guess that ensures the quadratic convergence of the quasilinearization scheme.

Filtering in Time-Dependent Problems

2023-01-01

P. Megha , G. Chandhini

Mollification of Fourier Spectral Methods with Polynomial Kernels

2023-04-03

G. Chandhini , P Megha

Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, … Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter P is analogous to that in the Dirichlet mollifier, which is detailed by analysing the numerical solution. Further, they are extended to linear scalar conservation law problems.

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Unified Convergence Analysis of Newton Midpoint-type Iterative Methods

2024-03-27

G. Chandhini , M Muniyasamy , Santhosh George +2 more

<title>Abstract</title> In this paper, unified convergence analyses for Newton midpoint-based iterative methods of order three and five are studied to solve the system of nonlinear equations in Banach space settings. … <title>Abstract</title> In this paper, unified convergence analyses for Newton midpoint-based iterative methods of order three and five are studied to solve the system of nonlinear equations in Banach space settings. Our analysis uses assumptions only on the first two derivatives of the involved operator and gives the number of iterations needed to achieve the given accuracy. Numerical examples are illustrated to show the performance of the methods considered. MSC Classification: 65H10 , 65J15 , 65J05

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Spectral error bound in the mollification of Fourier approximation using Gegenbauer polynomial based mollifier

2025-04-25

P. Megha , G. Chandhini

Enhancing the applicability of Jarratt-type fourth-order and sixth-order iterative methods

2025-05-10

M Muniyasamy , G. Chandhini , Santhosh George

Enhancing the applicability of Jarratt-type fourth-order and sixth-order iterative methods

2025-05-10

M Muniyasamy , G. Chandhini , Santhosh George

Spectral error bound in the mollification of Fourier approximation using Gegenbauer polynomial based mollifier

2025-04-25

P. Megha , G. Chandhini

A Procedure for Increasing the Convergence Order of Iterative Methods from p to 5p for Solving Nonlinear System

2024-12-01

Santhosh George , M Muniyasamy , Manjusree Gopal +2 more

Jarratt-type methods and their convergence analysis without using Taylor expansion

2024-10-08

Indra Bate , Kedarnath Senapati , Santhosh George +2 more

Unified convergence analysis of a class of iterative methods

2024-07-24

M Muniyasamy , Santhosh George , G. Chandhini

On obtaining convergence order of a fourth and sixth order method of Hueso et al. without using Taylor series expansion

2024-07-14

M Muniyasamy , G. Chandhini , Santhosh George +2 more

Enhancing the applicability of Chebyshev-like method

2024-04-17

Santhosh George , Indra Bate , M Muniyasamy +2 more

Unified Convergence Analysis of Newton Midpoint-type Iterative Methods

2024-03-27

G. Chandhini , M Muniyasamy , Santhosh George +2 more

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Mollification of Fourier spectral methods with polynomial kernels

2024-01-17

Megha Puthukkudi , G. Chandhini

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

2023-06-20

J. M. Revathy , G. Chandhini

Mollification of Fourier Spectral Methods with Polynomial Kernels

2023-04-03

G. Chandhini , P Megha

Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, … Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter P is analogous to that in the Dirichlet mollifier, which is detailed by analysing the numerical solution. Further, they are extended to linear scalar conservation law problems.

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Filtering in Time-Dependent Problems

2023-01-01

P. Megha , G. Chandhini

Extended Convergence Analysis of the Newton–Hermitian and Skew–Hermitian Splitting Method

2019-08-02

Ioannis K. Argyros , Santhosh George , G. Chandhini +1 more

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Direct and integrated radial functions based quasilinearization schemes for nonlinear fractional differential equations

2019-07-04

G. Chandhini , K. S. Prashanthi , V. Antony Vijesh

Iterative methods for a fractional-order Volterra population model

2019-04-01

Rupsha Roy , V. Antony Vijesh , G. Chandhini

Regularization of Highly Ill-Conditioned RBF Asymmetric Collocation Systems in Fractional Models

2019-01-01

K. S. Prashanthi , G. Chandhini

A radial basis function method for fractional Darboux problems

2017-11-01

G. Chandhini , Prashanthi K.S. , V. Antony Vijesh

A modified quasilinearization method for fractional differential equations and its applications

2015-06-18

V. Antony Vijesh , Rupsha Roy , G. Chandhini

Coauthor	Papers Together
Santhosh George	8
M Muniyasamy	7
Indra Bate	4
V. Antony Vijesh	4
Kedarnath Senapati	4
P. Megha	2
Rupsha Roy	2
K. S. Prashanthi	2
Ioannis K. Argyros	2
Manjusree Gopal	1
P Megha	1
Á. Alberto Magreñán	1
J. M. Revathy	1
Megha Puthukkudi	1
Prashanthi K.S.	1

Chebyshev's approximation algorithms and applications

2001-02-01

M.A. Hernández

Convergence, efficiency and dynamics of new fourth and sixth order families of iterative methods for nonlinear systems

2014-06-16

José L. Hueso , Eulalia Martı́nez , Carles Teruel

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Dynamics of Newton-like root finding methods

2022-12-15

B. Campos , Jordi Canela , P. Vindel

Abstract When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z 2 − c has the same … Abstract When exploring the literature, it can be observed that the operator obtained when applying Newton-like root finding algorithms to the quadratic polynomials z 2 − c has the same form regardless of which algorithm has been used. In this paper, we justify why this expression is obtained. This is done by studying the symmetries of the operators obtained after applying Newton-like algorithms to a family of degree d polynomials p ( z ) = z d − c . Moreover, we provide an iterative procedure to obtain the expression of new Newton-like algorithms. We also carry out a dynamical study of the given generic operator and provide general conclusions of this type of methods.

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Convergence and Applications of Newton-type Iterations

2008-01-01

Ioannis K. Argyros

Graphic and numerical comparison between iterative methods

2002-12-01

Juan Malumbres

Enhancing the applicability of Chebyshev-like method

2024-04-17

Santhosh George , Indra Bate , M Muniyasamy +2 more

On the Order of Convergence of the Noor–Waseem Method

2022-12-01

Santhosh George , Ramya Sadananda , P. Jidesh +1 more

In 2009, Noor and Waseem studied an important third-order iterative method. The convergence order is obtained using Taylor expansion and assumptions on the derivatives of order up to four. In … In 2009, Noor and Waseem studied an important third-order iterative method. The convergence order is obtained using Taylor expansion and assumptions on the derivatives of order up to four. In this paper, we have obtained convergence order three for this method using assumptions on the first and second derivatives of the involved operator. Further, we have extended the method to obtain a fifth- and a sixth-order methods. The dynamics of the methods are also provided in this study. Numerical examples are included. The same technique can be used to extend the utilization of other single or multistep methods.

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Enhancing the practicality of Newton–Cotes iterative method

2023-06-17

Ramya Sadananda , Santhosh George , Ajil Kunnarath +2 more

A modified Newton method with cubic convergence: the multivariate case

2004-03-20

Herbert H. H. Homeier

Construction of Lanczos type filters for the Fourier series approximation

2008-03-05

Beong In Yun , Kyung Soo Rim

Order of Convergence, Extensions of Newton–Simpson Method for Solving Nonlinear Equations and Their Dynamics

2023-02-06

Santhosh George , Ajil Kunnarath , Ramya Sadananda +2 more

Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS … Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS can converge. For this reason, we recover the convergence order based only on the first two derivatives. Moreover, the semilocal convergence of NS and some of its extensions not given before is developed. Furthermore, the dynamics are explored for these methods with many illustrations. The study contains examples verifying the theoretical conditions.

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Idempotent filtering in spectral and spectral element methods

2006-07-11

Alex Kanevsky , Mark H. Carpenter , Jan S. Hesthaven

Towards the resolution of the Gibbs phenomena

2003-10-28

Bernie D. Shizgal , Jae-Hun Jung

Spectral methods in the presence of discontinuities

2019-04-11

Joanna M. Piotrowska , Jonah Miller , Erik Schnetter

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Pseudospectral Fourier reconstruction with the modified Inverse Polynomial Reconstruction Method

2009-10-23

Tomasz Hrycak , Karlheinz Gröchenig

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Recovering Exponential Accuracy in Fourier Spectral Methods Involving Piecewise Smooth Functions with Unbounded Derivative Singularities

2015-03-25

Zheng Chen , Chi‐Wang Shu

Optimal filter and mollifier for piecewise smooth spectral data

2006-01-23

Jared Tanner

We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities … We discuss the reconstruction of piecewise smooth data from its (pseudo-) spectral information. Spectral projections enjoy superior resolution provided the function is globally smooth, while the presence of jump discontinuities is responsible for spurious <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal O}(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Gibbs’ oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order. Classical filters and mollifiers are constructed to have compact support in the Fourier (frequency) and physical (time) spaces respectively, and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region. Here we construct a noncompactly supported filter and mollifier with optimal <italic>joint</italic> time-frequency localization for a given number of vanishing moments, resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains. Not giving preference to either space allows for a more balanced error decomposition, which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters, yet obtain true exponential accuracy.

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Adaptive Mollifiers for High Resolution Recovery of Piecewise Smooth Data from its Spectral Information

2002-01-01

Eitan Tadmor , Jared Tanner

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Exclusive robustness of Gegenbauer method to truncated convolution errors

2021-12-27

Ehsan Faghihifar , Mahmood Akbari

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Spectral methods for the computation of discontinuous solutions

1989-03-01

Nira Gruberger

Space-fractional advection–dispersion equations by the Kansa method

2014-07-23

Guofei Pang , Wen Chen , Zhuojia Fu

Recovering Pointwise Values of Discontinuous Data within Spectral Accuracy

1985-01-01

David Gottlieb , Eitan Tadmor

Let f(x) be a bounded 2π-periodic function whose Fourier coefficients are given by 1.1 $${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}(k) = {{\frac{1}{2}}_{\pi }}\int\limits_{{ - \pi }}^{\pi } {f(y){{e}^{{ - ik \bullet y}}}dy, - \infty < … Let f(x) be a bounded 2π-periodic function whose Fourier coefficients are given by 1.1 $${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}(k) = {{\frac{1}{2}}_{\pi }}\int\limits_{{ - \pi }}^{\pi } {f(y){{e}^{{ - ik \bullet y}}}dy, - \infty < k < \infty .} $$ It is well-known that whenever f is a smooth function, then its spectral approximation — consisting of the partial sums 1.2 $$ {{S}_{N}}f(x) \equiv {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}}_{N}}(x) = \sum\limits_{{|k| \leqslant N}} {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}}(k){{e}^{{ik\bullet x}}},} $$ converges pointwise to f(x). A typical error estimate in this case, asserts that for any x in the domain we have 1.3 $$ |{\text{f(x)}}\;{\text{ - }}\;{{\hat{f}}_{{\text{N}}}}({\text{x}})|\; \leqslant \;{{{\text{C}}}_{{\text{S}}}}|\,|\;{\text{f}}\;|\,{{|}_{{({\text{S}})}}}\bullet {{{\text{N}}}^{{ - {\text{S + 1}}}}},\quad {\text{s}}\;{\text{ > }}\;{\text{1}}{\text{.}} $$ Here and below, CS stands for (possibly different) generic constant bounds, and ||f||(S) denotes the largest maximum norm of f and its first s derivatives, the maximum taken over the whole domain.

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On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function

1992-11-01

David Gottlieb , Chi‐Wang Shu , Alex Solomonoff +1 more

Enhanced spectral viscosity approximations for conservation laws

2000-05-01

Anne Gelb , Eitan Tadmor

Family of spectral filters for discontinuous problems

1991-06-01

Hervé Vandeven

A radial basis functions method for fractional diffusion equations

2012-12-08

Cécile Piret , Emmanuel Hanert

Filtering non-periodic functions

1994-01-01

Sidi Mahmoud Kaber

Convergence acceleration of modified Fourier series in one or more dimensions

2010-08-13

Ben Adcock

Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on $d$-variate cubes. Such approximations offer a number of … Modified Fourier series have recently been introduced as an adjustment of classical Fourier series for the approximation of nonperiodic functions defined on $d$-variate cubes. Such approximations offer a number of advantages, including uniform convergence. However, like Fourier series, the rate of convergence is typically slow. In this paper we extend Eckhoff's method to the convergence acceleration of multivariate modified Fourier series. By suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order. Moreover, we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions. In the univariate setting it is known that Eckhoff's method exhibits an auto-correction phenomenon. We extend this result to the multivariate case. Finally, we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set.

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The Chebyshev spectral viscosity method for the time dependent Eikonal equation

2010-02-09

Mehdi Dehghan , Rezvan Salehi

Fractional diffusion equations by the Kansa method

2009-09-05

Wen Chen , Linjuan Ye , HongGuang Sun

Order of Convergence and Dynamics of Newton–Gauss-Type Methods

2023-02-13

Ramya Sadananda , Santhosh George , Ioannis K. Argyros +1 more

On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in … On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in this paper. The study of high-convergence-order iterative methods under weak conditions is of extreme importance, because higher order means that fewer iterations are carried out to achieve a predetermined error tolerance. In order to enhance the practicality of these methods by Zhongli Liu, the convergence analysis is carried out without the application of Taylor expansion and requires the operator to be only two times differentiable, unlike the earlier studies. A semilocal convergence analysis is provided. Furthermore, numerical experiments verifying the convergence criteria, comparative studies and the dynamics are discussed for better interpretation.

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Detection of Edges in Spectral Data III—Refinement of the Concentration Method

2007-12-07

Anne Gelb , Dennis M. Cates

Error estimate in fractional differential equations using multiquadratic radial basis functions

2012-12-27

B. Fakhr Kazemi , F. Ghoreishi

Detailed Error Analysis for a Fractional Adams Method

2004-05-01

Kai Diethelm , Neville J. Ford , Alan D. Freed

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Series of Bessel and Kummer-Type Functions

2017-01-01

Árpád Baricz , Dragana Jankov Maširević , Tibor K. Pogány

Complete algebraic reconstruction of piecewise-smooth functions from Fourier data

2015-02-19

Dmitry Batenkov

In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a priori known smoothness and a number of discontinuities, from their Fourier coefficients, possessing the maximal possible asymptotic … In this paper we provide a reconstruction algorithm for piecewise-smooth functions with a priori known smoothness and a number of discontinuities, from their Fourier coefficients, possessing the maximal possible asymptotic rate of convergence—including the positions of the discontinuities and the pointwise values of the function. This algorithm is a modification of our earlier method, which is in turn based on the algebraic method of K. Eckhoff proposed in the 1990s. The key ingredient of the new algorithm is to use a different set of Eckhoff's equations for reconstructing the location of each discontinuity. Instead of consecutive Fourier samples, we propose to use a "decimated" set which is evenly spread throughout the spectrum.

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To the question of efficiency of iterative methods

2016-11-19

Tamara Kogan , Luba Sapir , Amir Sapir +1 more

Singularity preserving spectral collocation method for nonlinear systems of fractional differential equations with the right-sided Caputo fractional derivative

2021-02-09

Ibrahem G. Ameen , Mahmoud A. Zaky , E. H. Doha

A History of Algorithms

1999-01-01

Jean-Luc Chabert

Adaptive filters for piecewise smooth spectral data*

2005-08-24

Eitan Tadmor , Jared Tanner

Journal Article Adaptive filters for piecewise smooth spectral data* Get access Eitan Tadmor, Eitan Tadmor **Email: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Jared … Journal Article Adaptive filters for piecewise smooth spectral data* Get access Eitan Tadmor, Eitan Tadmor **Email: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Jared Tanner Jared Tanner Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 25, Issue 4, October 2005, Pages 635–647, https://doi.org/10.1093/imanum/dri026 Published: 01 October 2005

The complex dynamics of Newton's method for a double root

1991-01-01

William J. Gilbert

Gibbs phenomenon removal by adding Heaviside functions

2011-11-28

Kyung Soo Rim , Beong In Yun

Differential Equations with Applications and Historical Notes.

1975-10-01

S. H. Cox , George F. Simmons

The book The book

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Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method

2013-04-17

Mehdi Dehghan , Ahmad Nikpour

Generalized quasilinearization for fractional differential equations

2009-06-25

J. Vasundhara Devi , F.A. McRae , Z. Drici

New variants of Jarratt’s method with sixth-order convergence

2009-05-29

Hongmin Ren , Qingbiao Wu , Weihong Bi

Haar wavelet–quasilinearization technique for fractional nonlinear differential equations

2013-08-03

Umer Saeed , Mujeeb ur Rehman

Variants of Newton’s Method using fifth-order quadrature formulas

2007-02-05

Alicia Cordero , Juan R. Torregrosa

An optimization of Chebyshev’s method

2009-04-09

J.A. Ezquerro , M.A. Hernández

Spectral Simulation of Supersonic Reactive Flows

1998-12-01

Wai Sun Don , David Gottlieb

We present in this paper numerical simulations of reactive flows interacting with shock waves. We argue that spectral methods are suitable for these problems and review the recent developments in … We present in this paper numerical simulations of reactive flows interacting with shock waves. We argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long-term integrations of complicated flows, even in the presence of shock waves. A spectral code is described in detail, and the theory that leads to each of its components is explained. Results of interactions of hydrogen jetswith shock waves are presented and analyzed, and comparisons with ENO finite difference schemes are carried out.