eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
1
12
514998
مقاله های تحقیقی
Some notes on convergence of homotopy based methods for functional equations
Some notes on convergence of homotopy
based methods for functional equations
آ عزیزی
a.azizi@pnu.ac.ir
1
ج سعیدیان
2
ا بابلیان
3
دانشگاه پیام نور تهران
دانشگاه خوارزمی تهران
دانشگاه خوارزمی تهران
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.
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.
http://msj.iau-arak.ac.ir/article_514998_97009e58cebbe0cb936c4ac2ea5d7c14.pdf
Homotopy analysis method
Homotopy perturbation method
Convergence theorem
Banach fixed point theorem
Series solution
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
13
36
514999
مقاله های تحقیقی
Ranking DMUs by ideal points in the presence of fuzzy and ordinal data
Ranking DMUs by ideal points in the
presence of fuzzy and ordinal data
م ایزدیخواه
m-izadikhah@iau-arak.ac.ir
1
ز علی اکبر پور
2
ه شرفی
3
دانشگاه آزاد اراک
دانشگاه آزاد اراک
دانشگاه علوم و تحقیقات تهران
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.
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.
http://msj.iau-arak.ac.ir/article_514999_5c7dc2f73150b39722dfb7edbec16b1b.pdf
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
37
55
515000
مقاله های تحقیقی
Legendre wavelet method for solving Hammerstein integral equations of the second kind
Legendre wavelet method for solving
Hammerstein integral equations of the
second kind
ش جوادی
1
ج سعیدیان
2
ف صفری
3
دانشگاه خوارزمی تهران
دانشگاه خوارزمی تهران
دانشکده ریاضی دانشگاه خوارزمی تهران
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.
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.
http://msj.iau-arak.ac.ir/article_515000_82910dddddcd079c95584dca6566e445.pdf
Legendre wavelets
Fredholm-Hammerstein integral equations
Volterra-Hammerstein integral equations
Newton's method
Operational
matrix
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
57
80
515001
مقاله های تحقیقی
The Operational matrices with respect to generalized Laguerre polynomials and their applications in solving linear dierential equations with variable coecients
The Operational matrices with respect to
generalized Laguerre polynomials and their
applications in solving linear dierential
equations with variable coecients
ز خلته بجدی
1
س احمدی اصل
2
ا امین عطایی
ataei@kntu.ac.ir
3
دانشگاه بیرجند
دانشگاه بیرجند
دانشگاه خواجه نصیر الدین توسی تهران
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.
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.
http://msj.iau-arak.ac.ir/article_515001_cf35c3fcb35ec049ae5b367cc3365f0f.pdf
Operational matrices
Laguerre polynomials
Linear differential
equations with variable coffecients
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
81
100
515072
مقاله های تحقیقی
On the singular fuzzy linear system of equations
On the singular fuzzy linear system of
equations
M Nikuie
nikoie_m@yahoo.com
1
M.K. Mirnia
2
باشگاه پژوهشگران جوان دانشگاه آزاد یزد.
Department of Computer engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran.
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.
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.
http://msj.iau-arak.ac.ir/article_515072_f39d05fb7b4b81f825ee5c5d92280b8b.pdf
Drazin inverse
Singular fuzzy linear system
Minimal solution
Singular matrices
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
101
114
515073
مقاله های تحقیقی
Convergence Theorems for -Nonexpansive Mappings in CAT(0) Spaces
Convergence Theorems for -Nonexpansive
Mappings in CAT(0) Spaces
Savita Rathee
dr.savitarathee@gmail.com
1
R Ritika
math.riti@gmail.com
2
هندوستان
Department of Mathematics, M.D. University, Rohtak (Haryana), India
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.
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.
http://msj.iau-arak.ac.ir/article_515073_2c90b18306b586d0c6a7ef5a9e2fd133.pdf
CAT(0) spaces
-Nonexpansive mapping
-convergence
SP-iteration
Thianwan's iteration
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
115
133
515074
مقاله های تحقیقی
Numerical solution of fuzzy Hunter-Saxton equation by using Adomian decomposition and Homotopy analysis methods
Numerical solution of fuzzy Hunter-Saxton
equation by using Adomian decomposition
and Homotopy analysis methods
ش. صدیق بهزادی
1
Department of Mathematics, Islamic Azad University, Qazvin Branch, Qazvin, Iran
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.
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.
http://msj.iau-arak.ac.ir/article_515074_001127548a53e68b99a5049d12651104.pdf
Hunter-Saxton equation
Adomian decomposition method
Homotopy analysis method
Generalized dierentiability
Hukuhara-
dierence
Fuzzy number
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
135
149
515075
مقاله های تحقیقی
Evaluating the solution for second kind nonlinear Volterra Fredholm integral equations using hybrid method
Evaluating the solution for second kind
nonlinear Volterra Fredholm integral
equations using hybrid method
احمد شهسواران
a.shahsavaran@iaub.ac.ir
1
اکبر شهسواران
2
Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.
Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.
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.
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.
http://msj.iau-arak.ac.ir/article_515075_93d04aac3c8e31dc1bd4cc309ecc4899.pdf
Nonlinear Fredholm Volterra integral equation
Haar wavelet
Haar coecient matrix
Block-Pulse Function
Collocation points
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
151
158
515076
مقاله های تحقیقی
A note on positive deniteness and stability of interval matrices
A note on positive deniteness and
stability of interval matrices
H Veiseh
veisehana@yahoo.com
1
Department of Applied Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran
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.
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.
http://msj.iau-arak.ac.ir/article_515076_fedf0dc7e74268bd7471f07d5e845be9.pdf
Interval matrix
Real eigenvalues
Positive deniteness
Stability
Symmetric matrix
eng
Islamic Azad University
Theory of Approximation and Applications
2538-2217
2538-2217
2013-09-01
9
2
159
170
515077
مقاله های تحقیقی
Multiple solutions of the nonlinear reaction-diusion model with fractional reaction
Multiple solutions of the nonlinear
reaction-diusion model with fractional
reaction
H. Vosoughi
1
E. Shivanian
shivanian@ikiu.ac.ir
2
M. Anbarloei
3
Department of Mathematics, Faculty of Science, Islamshahr Branch, Islamic Azad University, Islamshahr, Tehran, Iran
Department of Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran
Department of Mathematics, Faculty of Science, Islamshahr Branch, Islamic Azad University, Islamshahr, Tehran, Iran
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.
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.
http://msj.iau-arak.ac.ir/article_515077_42386f352e68134a3980600013f1728c.pdf
Predictor homotopy analysis method
Prescribed parameter
Reaction-diusion model, multiple solutions