Some notes on convergence of homotopy
based methods for functional equations
A.
Azizi
Department of Mathematics, Payame Noor university, 19395-4697, Tehran,
I. R. of Iran.
author
J.
Saiedian
Faculty of Mathematical Sciences and Computer, Kharazmi University, 599
Taleghani avenue, Tehran 1561836314, Iran.
author
E.
Babolian
Faculty of Mathematical Sciences and Computer, Kharazmi University, 599
Taleghani avenue, Tehran 1561836314, Iran.
author
text
article
2014
per
Although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. Some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.Here, after criticizing previous works, a sucient condition for convergence ofhomotopy methods is presented. Finally, examples are given to show that evenif the homotopy method leads to a convergent series, it may not converge tothe exact solution of the equation under consideration.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
1
12
http://msj.iau-arak.ac.ir/article_514998_97009e58cebbe0cb936c4ac2ea5d7c14.pdf
Ranking DMUs by ideal points in the
presence of fuzzy and ordinal data
M.
Izadikhah
Department of Mathematics, College of Science, Arak-Branch, Islamic Azad
University, Arak, Iran
author
Z.
Aliakbarpoor
Department of Mathematics, College of Science, Arak-Branch, Islamic Azad
University, Arak, Iran
author
H.
Sharafi
Department of Mathematics, Science and Research Branch, Islamic Azad
University, Tehran, Iran
author
text
article
2014
per
Envelopment Analysis (DEA) is a very eective method to evaluate the relative eciency of decision-making units (DMUs). DEA models divided all DMUs in two categories: ecient and inecientDMUs, and don't able to discriminant between ecient DMUs. On the other hand, the observedvalues of the input and output data in real-life problems are sometimes imprecise or vague, suchas interval data, ordinal data and fuzzy data. This paper develops a new ranking system under thecondition of constant returns to scale (CRS) in the presence of imprecise data, In other words, inthis paper, we reformulate the conventional ranking method by ideal point as an imprecise dataenvelopment analysis (DEA) problem, and propose a novel method for ranking the DMUs when theinputs and outputs are fuzzy and/or ordinal or vary in intervals. For this purpose we convert alldata into interval data. In order to convert each fuzzy number into interval data we use the nearestweighted interval approximation of fuzzy numbers by applying the weighting function and also weconvert each ordinal data into interval one. By this manner we could convert all data into intervaldata. The numerical example illustrates the process of ranking all the DMUs in the presence of fuzzy,ordinal and interval data.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
13
36
http://msj.iau-arak.ac.ir/article_514999_5c7dc2f73150b39722dfb7edbec16b1b.pdf
Legendre wavelet method for solving
Hammerstein integral equations of the
second kind
Sh.
Javadi
Faculty of Mathematical Sciences and Computer, Kharazmi University, 50
Taleghani avenue, Tehran 15618-36314, Iran
author
J.
Saiedian
Faculty of Mathematical Sciences and Computer, Kharazmi University, 50
Taleghani avenue, Tehran 15618-36314, Iran
author
F.
Safari
Faculty of Mathematical Sciences and Computer, Kharazmi University, 50
Taleghani avenue, Tehran 15618-36314, Iran
author
text
article
2014
per
An ecient method, based on the Legendre wavelets, is proposed to solve thesecond kind Fredholm and Volterra integral equations of Hammerstein type.The properties of Legendre wavelet family are utilized to reduce a nonlinearintegral equation to a system of nonlinear algebraic equations, which is easilyhandled with the well-known Newton's method. Examples assuring eciencyof the method and its superiority are presented.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
37
55
http://msj.iau-arak.ac.ir/article_515000_82910dddddcd079c95584dca6566e445.pdf
The Operational matrices with respect to
generalized Laguerre polynomials and their
applications in solving linear dierential
equations with variable coecients
Z.
Khalteh Bojdi
Department of Mathematics, Birjand University, Birjand, Iran.
author
S.
Ahmadi-Asl
Department of Mathematics, Birjand University, Birjand, Iran.
author
A.
Amin Ataei
Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box
16315-1618, Tehran, Iran.
author
text
article
2014
per
In this paper, a new and ecient approach based on operational matrices with respect to the gener-alized Laguerre polynomials for numerical approximation of the linear ordinary dierential equations(ODEs) with variable coecients is introduced. Explicit formulae which express the generalized La-guerre expansion coecients for the moments of the derivatives of any dierentiable function in termsof the original expansion coecients of the function itself are given in the matrix form. The mainimportance of this scheme is that using this approach reduces solving the linear dierential equationsto solve a system of linear algebraic equations, thus greatly simplify the problem. In addition, severalnumerical experiments are given to demonstrate the validity and applicability of the method.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
57
80
http://msj.iau-arak.ac.ir/article_515001_cf35c3fcb35ec049ae5b367cc3365f0f.pdf
On the singular fuzzy linear system of
equations
M.
Nikuie
Young Researchers and Elite Club, Tabriz Branch, Islamic Azad University,
Tabriz, Iran.
author
M. K.
Mirnia
Department of Computer engineering, Tabriz Branch, Islamic Azad
University, Tabriz, Iran.
author
text
article
2014
per
The linear system of equations Ax = b where A = [aij ] in Cn.n is a crispsingular matrix and the right-hand side is a fuzzy vector is called a singularfuzzy linear system of equations. In this paper, solving singular fuzzy linearsystems of equations using generalized inverses such as Drazin inverse andpseudo-inverse are investigated.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
81
100
http://msj.iau-arak.ac.ir/article_515072_f39d05fb7b4b81f825ee5c5d92280b8b.pdf
Convergence Theorems for -Nonexpansive
Mappings in CAT(0) Spaces
Savita
Rathee
Department of Mathematics, M.D. University, Rohtak (Haryana), India
author
R.
Ritika
Department of Mathematics, M.D. University, Rohtak (Haryana), India
author
text
article
2014
per
In this paper we derive convergence theorems for an -nonexpansive mappingof a nonempty closed and convex subset of a complete CAT(0) space for SP-iterative process and Thianwan's iterative process.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
101
114
http://msj.iau-arak.ac.ir/article_515073_2c90b18306b586d0c6a7ef5a9e2fd133.pdf
Numerical solution of fuzzy Hunter-Saxton
equation by using Adomian decomposition
and Homotopy analysis methods
Sh.
Sadigh Behazadi
Department of Mathematics, Islamic Azad University, Qazvin Branch,
Qazvin, Iran
author
text
article
2014
per
In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian'sdecomposition method (ADM) and homotopy analysis method (HAM). Theapproximation solution of this equation is calculated in the form of series whichits components are computed by applying a recursive relation. The existenceand uniqueness of the solution and the convergence of the proposed methodsare proved. A numerical example is studied to demonstrate the accuracy ofthe presented methods.
In this paper, a fuzzy Hunter-Saxton equation is solved by using the Adomian'sdecomposition method (ADM) and homotopy analysis method (HAM). Theapproximation solution of this equation is calculated in the form of series whichits components are computed by applying a recursive relation. The existenceand uniqueness of the solution and the convergence of the proposed methodsare proved. A numerical example is studied to demonstrate the accuracy ofthe presented methods.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
115
133
http://msj.iau-arak.ac.ir/article_515074_001127548a53e68b99a5049d12651104.pdf
Evaluating the solution for second kind
nonlinear Volterra Fredholm integral
equations using hybrid method
Ahmad
Shahsavaran
Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.
author
Akbar
Shahsavaran
Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.
author
text
article
2014
per
In this work, we present a computational method for solving second kindnonlinear Fredholm Volterra integral equations which is based on the use ofHaar wavelets. These functions together with the collocation method are thenutilized to reduce the Fredholm Volterra integral equations to the solution ofalgebraic equations. Finally, we also give some numerical examples that showsvalidity and applicability of the technique.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
135
149
http://msj.iau-arak.ac.ir/article_515075_93d04aac3c8e31dc1bd4cc309ecc4899.pdf
A note on positive deniteness and
stability of interval matrices
H.
Veiseh
Department of Applied Mathematics, Hamedan Branch, Islamic Azad
University, Hamedan, Iran
author
text
article
2014
per
It is proved that by using bounds of eigenvalues of an interval matrix, someconditions for checking positive deniteness and stability of interval matricescan be presented. These conditions have been proved previously with variousmethods and now we provide some new proofs for them with a unity method.Furthermore we introduce a new necessary and sucient condition for checkingstability of interval matrices.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
151
158
http://msj.iau-arak.ac.ir/article_515076_fedf0dc7e74268bd7471f07d5e845be9.pdf
Multiple solutions of the nonlinear
reaction-diusion model with fractional
reaction
H.
Vosoughi
Department of Mathematics, Faculty of Science, Islamshahr Branch,
Islamic Azad University, Islamshahr, Tehran, Iran
author
E.
Shivanian
Department of Mathematics, Imam Khomeini International University,
Qazvin, 34149-16818, Iran
author
M.
Anbarloei
Department of Mathematics, Faculty of Science, Islamshahr Branch,
Islamic Azad University, Islamshahr, Tehran, Iran
author
text
article
2014
per
The purpose of this letter is to revisit the nonlinear reaction-diusion modelin porous catalysts when reaction term is fractional function of the concen-tration distribution of the reactant. This model, which originates also in uidand solute transport in soft tissues and microvessels, has been recently givenanalytical solution in terms of Taylors series for dierent family of reactionterms. We apply the method so-called predictor homotopy analysis method(PHAM) which has been recently proposed to predict multiplicity of solutionsof nonlinear BVPs. Consequently, it is indicated that the problem for somevalues of the parameter admits multiple solutions. Also, error analysis of thesesolutions are given graphically.
Theory of Approximation and Applications
Islamic Azad University
2538-2217
9
v.
2
no.
2014
159
170
http://msj.iau-arak.ac.ir/article_515077_42386f352e68134a3980600013f1728c.pdf