2013
9
2
2
130
Some notes on convergence of homotopy based methods for functional equations
Some notes on convergence of homotopy
based methods for functional equations
2
2
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.
1
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.
1
12
A.
Azizi
آ
عزیزی
دانشگاه پیام نور تهران
دانشگاه پیام نور تهران
Iran
a.azizi@pnu.ac.ir
J.
Saiedian
ج
سعیدیان
دانشگاه خوارزمی تهران
دانشگاه خوارزمی تهران
Iran
E.
Babolian
ا
بابلیان
دانشگاه خوارزمی تهران
دانشگاه خوارزمی تهران
Iran
[[1] S.J. Liao, Proposed homotopy analysis technique for the solution##of nonlinear problems, Ph.D. dissertation, Shanghai Jiao Tong##University, (1992).##[2] J.H. He, Homotopy perturbation technique, Comp. Meth. Appl.##Mech. Eng., 178 (1999) 257-262. ##[3] S.J. Liao, E. Magyari, Exponentially decaying boundary layers as##limiting cases of families of algebraically decaying ones, ZAMP, 57##(2006) 777-792.##[4] S.J. Liao, A new branch of solutions of boundary-layer ows over##a permeable stretching plate, Int. J. Non-Linear Mech., 42 (2007)##[5] S. Abbasbandy, Y. Tan and S.J. Liao, Newton-homotopy analysis##method for nonlinear equations, Appl. Math. Comput., 188 (2007)##1794-1800.##[6] S.J. Liao, On the relationship between the homotopy analysis##method and Euler transform, Commun. Nonlin. Sci. Num. Simul.,##18 (2010) 1421-1431.##[7] S. Abbasbandy, Application of He's homotopy perturbation method##for Laplace transform, Chaos, Solitons and Fractals, 30 (2006) 1206-##[8] M. A. Rana, A. M. Siddiqui, Q. K. Ghori and R. Qamar, Application##of He's homotopy perturbation method to Sumudu transform, Int.##J. Nonlinear Sci. Numer. Simul., 8 (2008) 185-190.##[9] E. Babolian, J. Saeidian, M. Paripour, Computing the Fourier##Transform via Homotopy Perturbation Method, Z. Naturforsch., A:##Phys. Sci., 64a (2009) 671-675.##[10] E. Babolian, J. Saeidian, New application of HPM for quadratic##riccati dierential equation: a comparative study, Math. Sci. J., 3##[11] M. Ghasemi, M. Tavassoli Kajani, A. Azizi, The application of##homotopy perturbation method for solving Schrodinger equation,##Math. Sci. J., 1 (2009)##[12] M. Ghasemi, A. Azizi, M. Fardi, Numerical solution of seven-order##Sawada-Kotara equations by homotopy perturbation method, Math.##Sci. J., 7 (2011) 69-77.##[13] J. Biazar, H. Ghazvini, Convergence of the homotopy perturbation##method for partial dierential equations, Nonlinear Anal. Real##World Appl., 10 (2009) 2633-2640. ##[14] J. Biazar, H. Aminikhah, Study of convergence of homotopy##perturbation method for systems of partial dierential equations,##Comput. Math. Appl., 58 (2009) 2221-2230.##[15] Z. Odibat, A study on the convergence of homotopy analysis##method, Appl. Math. Comput., 217 (2010) 782-789.##[16] S.J. Liao, Beyond Perturbation: An Introduction to Homotopy##Analysis Method, Chapman Hall/CRC Press, Boca Raton, 2003.##[17] S.J. Liao, Y. Tan, A general approach to obtain series solutions of##nonlinear dierential equations, Stud. Appl. Math., 119 (2007) 297-##[18] E. Babolian, A. Azizi, J. Saeidian, Some notes on using##the homotopy perturbation method for solving time-dependent##dierential equations, Math. Comput. Model., 50 (2009) 213-224.##]
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
2
2
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.
1
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.
13
36
M.
Izadikhah
م
ایزدیخواه
دانشگاه آزاد اراک
دانشگاه آزاد اراک
Iran
m-izadikhah@iau-arak.ac.ir
Z.
Aliakbarpoor
ز
علی اکبر پور
دانشگاه آزاد اراک
دانشگاه آزاد اراک
Iran
H.
Sharafi
ه
شرفی
دانشگاه علوم و تحقیقات تهران
دانشگاه علوم و تحقیقات تهران
Iran
[[1] Adler, N., Friedman, L., Sinuany-Stern, Z.. Review of ranking methods##in the data envelopment analysis context. European Journal of Operational##Research, 140(2) (2002) 249-265.##[2] Andersen, P., Petersen, N. C. A procedure for ranking ecient units in##data envelopment analysis.Management Science,39(10) (1993), 1261-1264.##[3] A. Charnes, W.W. Cooper, A.Y. Lewin and L.M. Seiford, Data##Envelopment analysis: Theory ,Methodology and Applications Kluwer##Academic Publishers, Norwell, MA,1994.##[4] A. Charnes, W.W. Cooper, E. Rhodes, Measuring the eciency of decision##making units, Euro. J. Operat. Res., 2 (1978) 429-444.##[5] G.X. Chen and M. Deng, A cross-dependence based ranking system for##ecient and inecient units in DEA, Expert Systems with Applications,38##(2011) 9648-9655.##[6] W.D. Cook, J. Doyle, R. Green and M. Kress, Multiple criteria modeling##and ordinal data European Journal of the Operational Research, 98 (1997)##[7] W.D. Cook and M. Kress, A multiple criteria decision model with ordinal##preference data, European Journal of Operational Research, 54(1991) 191-##[8] W.D. Cook, M. Kress and L. Seiford, On the use of ordinal data in Data##envelopment analysis Journal of the Operational Research Society, 44 (1993)##[9] W.W. Cooper, K.S. Park and G. Yu, IDEA and AR-IDEA: models for##dealing with imprecise data in DEA, Management Science 45 (1999) 597-##[10] W.W. Cooper, K.S. Park and G. Yu, An illustrative application##of IDEA (imprecise data envelopment analysis) to a Korean mobile##telecommunication company, Operations Research 49 (2001) 807-820.##[11] W.W. Cooper, K.S. Park and G. Yu, IDEA (imprecise data envelopment##analysis) with CMDs (column maximum decision making units), Journal of##the Operational Research Society 52 (2001) 176-181.##[12] D.K. Despotis and Y.G. Smirlis, Data envelopment analysis with imprecise##data, Eur. J. Oper. Res. 140 (2002) 24-36. ##[13] D.Dubois and H. Prade, Operations on fuzzy numbers, Internat. J.##Systems Sci., 9 (1978) 626-631.##[14] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Application,##Academic Press, New York, 1980##[15] T. Entani, Y. Maeda and H. Tanaka, Dual models of interval DEA and##its extension to interval data, Eur. J. Oper. Res. 136 (2002) 32-45.##[16] P. Guo and H. Tanaka, Fuzzy DEA: a perceptual evaluation method, Fuzzy##Sets and Systems 119 (2001) 149-160.##[17] F. Hosseinzadeh Lot, R. Fallahnejad and N. Navidi, Ranking Ecient##Units in DEA by Using TOPSIS Method, Applied Mathematical Sciences,##5 (17) (2011) 805-815.##[18] F. Hosseinzadeh Lot, M. Izadikhah, R. Roostaee, M. Rostamy##Malkhalifeh, A goal programming procedure for ranking decision making##units in DEA, Mathematics Scientic Journal, Vol. 7, No. 2, (2012), 19-38.##[19] M. Izadikhah, Ranking units with interval data based on ecient and##inecient frontiers, Mathematics Scientic Journal, 3, 2 (2007) 49-57.##[20] G.R. Jahanshahloo, F. Hosseinzadeh Lot, M. Khanmohammadi, M.##Kazemimanesh and V. Rezaie, Ranking of units by positive ideal DMU with##common weights, Expert Systems with Applications, 37 (2010) 7483-7488.##[21] G.R. Jahanshahloo, F. Hosseinzadeh Lot, V. Rezaie and M.##Khanmohammadi, Ranking DMUs by ideal points with interval data in##DEA, Applied Mathematical Modelling 35 (2011) 218-229.##[22] G.R. Jahanshahloo, F. Hosseinzadeh Lot, M. Sanei and M. Fallah##Jelodar, Review of Ranking Models in Data Envelopment Analysis, Applied##Mathematical Sciences, 2 (29) (2008) 1431-1448.##[23] G.R. Jahanshahloo, F. Hosseinzadeh Lot, N. Shoja, G. Tohidi and S.##Razavyan, Ranking using l1-norm in data envelopment analysis, Applied##Mathematics and Computation, 153 (2004) 215-224.##[24] G.R. Jahanshahloo, H.V. Junior, F. Hosseinzadeh Lot and D. Akbarian,##A new DEA ranking system based on changing the reference set, European##Journal of Operational Research, 181 (2007) 331-337.##[25] G.R. Jahanshahloo, M. Soleimani-damaneh, and E. Nasrabadi, Measure##of eciency in DEA with fuzzy input-output levels: A methodology##for assessing, ranking and imposing of weights restrictions. Applied##Mathematics and Computation, 156 (2004) 175-187.##[26] C. Kao, Interval eciency measures in data envelopment analysis with##imprecise data, European Journal of Operational Research 174 (2006)1087-##[27] C. Kao and S.T. Liu, Fuzzy eciency measures in data envelopment##analysis, Fuzzy Sets and Systems 119 (2000) 149-160. ##[28] S.H. Kim, C.G. Park and K.S. Park, An application of data envelopment##analysis in telephone oces evaluation with partial data, Comput. Oper.##Res. 26 (1999) 59-72.##[29] Y.K. Lee, K.S. Park and S.H. Kim, Identication of ineciencies in an##additive model based IDEA (imprecise data envelopment analysis), Comput.##Oper. Res. 29 (2002) 1661-1676.##[30] S. Lertworasirikul, S.C. Fang, J.A. Joines and H.L.W. Nuttle, Fuzzy data##envelopment analysis (DEA): a possibility approach, Fuzzy Sets and Systems##139 (2003) 379-394.##[31] S. Li, G.R. Jahanshahloo and M. Khodabakhshi, A super-eciency##model for ranking ecient units in data envelopment analysis, Applied##Mathematics and Computation, 184 (2007) 638-648.##[32] B. Liu and Y.K. Liu, Expected value of fuzzy variable and fuzzy expected##value models, IEEE Transactions on Fuzzy Systems, 10 (4) (2002) 445-450.##[33] F-H.F. Liu and H.H. Peng, Ranking of units on the DEA frontier with##common weights, Computers and Operations Research, 35 (2008) 1624-1637.##[34] S. Mehrabian, M.R. Alirezaee, G.R. Jahanshahloo, A complete eciency##ranking of decision making units in data envelopment analysis, Comput.##Optim. Appl. 14 (1999) 261-266.##[35] F. Rezai Balf, H. Zhiani Rezai, G.R. Jahanshahloo and F. Hosseinzadeh##Lot, Ranking eecient DMUs using the Tchebychev norm, Applied##Mathematical Modelling, 36 (2012) 46-56.##[36] M. Rostamy-Malkhalifeh, N. Aghayi, Two Ranking of Units on the Overall##Prot Eciency with Interval Data, Mathematics Scientic Journal, Vol. 8,##No. 2, (2013), 73-93.##[37] A. Saeidifar, Application of weighting functions to the ranking of fuzzy##numbers, Computers and Mathematics with Applications, 62 (2011) 2246-##[38] J.K. Sengupta, A fuzzy systems approach in data envelopment analysis.##Computers and Mathematics with Applications, 24 (1992) 259-266.##[39] A.H. Shokouhi, A. Hatami-Marbini, M. Tavana and S. Saati, A robust##optimization approach for imprecise data envelopment analysis, Computers##and Industrial Engineering 59 (2010) 387-397.##[40] Y.M. Wang, R. Greatbanks and J.B. Yang, Interval eciency assessment##using data envelopment analysis, Fuzzy Sets and Systems 153 (2005) 347-##[41] Y-M. Wang, Y. Luo and Y-X. Lan, Common weights for fully##ranking decision making units by regression analysis. Expert Systems with##Applications, 38 (2011) 9122-9128.##[42] Y-M. Wang, Y. Luo and L. Liang, Ranking decision making units by##imposing a minimum weight restriction in the data envelopment analysis.##Journal of Computational and Applied Mathematics, 223 (2009) 469-484. ##[43] H.J. Zimmermann, Description and optimization of fuzzy system.##International Journal of General System, 2 (1976) 209-216.##[44] H.J. Zimmermann, Fuzzy set theory and its applications. Boston: Kluwer-##Nijho, (1996).##[45] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets##and Systems, 1 (1978) 3-28.##[46] J. Zhu, Eciency evaluation with strong ordinal input and output##measures , European Journal of Operational Research, 146 (2003)477-485.##]
Legendre wavelet method for solving Hammerstein integral equations of the second kind
Legendre wavelet method for solving
Hammerstein integral equations of the
second kind
2
2
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.
1
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.
37
55
Sh.
Javadi
ش
جوادی
دانشگاه خوارزمی تهران
دانشگاه خوارزمی تهران
Iran
J.
Saiedian
ج
سعیدیان
دانشگاه خوارزمی تهران
دانشگاه خوارزمی تهران
Iran
F.
Safari
ف
صفری
دانشکده ریاضی دانشگاه خوارزمی تهران
دانشکده ریاضی دانشگاه خوارزمی تهران
Iran
[[1] K. E. Atkinson, The Numerical Solution of Integral Equations of The##Second Kind, Cambridge University Press, Cambridge, 1997. ##[2] S. M. Berman, A. L. Stewart, A Nonlinear Integral Equation for Visual##Impedance, Biol. Cybernetics 33 (1979) 137{141.##[3] A.M. Bica, M. Curila, S. Curila, About a numerical method of successive##interpolations for functional Hammerstein integral equations, J. Comput.##Appl. Math. 236 (2012) 2005{2024.##[4] A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen,##Acta Math. 54 (1930) 117{176.##[5] H. R. Thieme, On a class of Hammerstein integral equations, Manuscripta##Math. 29 (1979) 49{84.##[6] J. Banas, J. Rocha Martin, K. Sadarangani, On solutions of a quadratic##integral equation of Hammerstein type, Math. Comput. Model. 43 (2006)##[7] J. Banas, Integrable solutions of Hammerstein and Uryshon integral##equations, J. Austral. Math. Soc. (A) 46 (1989) 61{68.##[8] D. ORegan, Existence results for nonlinear integral equations, J. Math.##Anal. Appl. 192 (1995) 705{726.##[9] D. ORegan, M. Meehan, Existence Theory for Nonlinear Integral and##Integro-dierential Equations, Kluwer Academic Publishers, Dordrecht,##[10] K. Atkinson, A survey of numerical methods for solving nonlinear integral##equations, J. Int. Eqns. Applics. 4 (1992) 15{46.##[11] M. M. Shamivand, A. Shahsavaran, Numerical solution of Hammerstein##Fredholm and Volterra integral equations of the second kind using block##pulse functions and collocation method, Math. Sci. J.,7 (2011) 93-103.##[12] A. Shahsavaran, E. Babolian, Computational method for solving nonlinear##Fredholm integral equations of Hammerstein type based on Lagrange##interpolation and quadrature method, Math. Sci. J.,5 (2009) 137-145.##[13] S. Kumar, I. Sloan, A new collocation-type method for Hammerstein##equations, Math. Comp. 48 (1987) 585{593.##[14] G. N. Elnagar, M. Kazemi, Chebyshev spectral solution of nonlinear##Volterra-Hammerstein integral equations, J. Comput. Appl. Math. 76##(1996) 147{158.##[15] G. N. Elnagar, M. Kazemi, A cell-averaging chebyshev spectral method##for nonlinear fredholm-hammerstein integral equations, Int. J. Comput.##Math. 60 (1996) 91{104.##[16] H. Kaneko, R. D. Noren, B. Novaprateep, Wavelet applications to the##PetrovGalerkin method for Hammerstein equations, Appl. Numer. Math.##45 (2003) 255-273.##[17] K. Maleknejad, H. Derili, The collocation method for Hammerstein##equations by Daubechies wavelets, Appl. Math. Comput. 172 (2006) 846{##[18] S. Youse, M. Razzaghi, Legendre wavelets method for the nonlinear##VolterraFredholm integral equations, Math. Comput. Simul. 70 (2005) 1{##[19] E. Babolian, F. Fattahzadeh, E. Golpar Raboky, A Chebyshev##approximation for solving nonlinear integral equations of Hammerstein##type, Appl. Math. Comput. 189 (2007) 641{646.##[20] Y. Ordokhani, Solution of nonlinear VolterraFredholmHammerstein##integral equations via rationalized Haar functions, Appl. Math. Comput.##180 (2006) 436{443.##[21] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. ##[22] M. Razzaghi, S. Youse, The Legendre wavelets operational matrix of##integration, Int. J. Syst. Sci. 32 (2001) 495{502.##[23] M. Rehman, R. A. Khan, The Legendre wavelet method for solving##fractional dierential equations, Commun. Nonlinear Sci. Numer.##Simulat. 16 (2011) 4163{4173.##[24] M. Razzaghi, S. Youse, Legendre wavelets method for the solution of##nonlinear problems in the calculus of variations, Math. Comput. Model.##34 (2001) 45{54.##[25] F. Awawdeh, A. Adawi, A numerical method for solving nonlinear integral##equations, Int. Math. Forum 4 (2009) 805{817.##[26] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of##nonlinear integral equation, Appl. Math. Comput. 167 (2005) 1119{1129.##[27] E. Babolian, A. Shahsavaran, Numerical solution of nonlinear Fredholm##and Volterra integral equations of the second kind using Haar wavelets and##collocation method, J. Sci. Tarbiat Moallem University, 7 (2007) 213{222.##[28] K. Maleknejad, K. Nedaiasl, Application of Sinc-collocation method for##solving a class of nonlinear Fredholm integral equations, Comput. Math.##Appl. 62 (2011) 3292{3303.##]
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
2
2
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.
1
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.
57
80
Z.
Khalteh Bojdi
ز
خلته بجدی
دانشگاه بیرجند
دانشگاه بیرجند
Iran
S.
Ahmadi-Asl
س
احمدی اصل
دانشگاه بیرجند
دانشگاه بیرجند
Iran
A.
Amin Ataei
ا
امین عطایی
دانشگاه خواجه نصیر الدین توسی تهران
دانشگاه خواجه نصیر الدین توسی تهران
Iran
ataei@kntu.ac.ir
[[1] W. Gautschi, Orthogonal Polynomials (Computation and##Approximation), Oxford University Press, 2004.##[2] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables,##Cambridge University Press, 2001.##[3] F. Marcellan, W.V. Assche, Orthogonal Polynomials and Special##Functions (a Computation and Applications), Springer-Verlag Berlin##Heidelberg, 2006.##[4] R. Askey, Orthogonal Polynomials and Special Functions, SIAM-CBMS,##Philadelphia, 1975.##[5] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory##and Applications, SIAM-CBMS, Philadelphia, 1977.##[6] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications,##Inc, New York, 2000.##[7] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods:##Fundamentals in Single Domains, Springer-Verlag, 2006.##[8] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Method in##Fluid Dynamics, Prentice Hall, Engelwood Clis, NJ, 1984.##[9] L.N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA,##[10] J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-##Dependent Problems, Cambridge University, 2009.##[11] G. Ben-yu, The State of Art in Spectral Methods. Hong Kong University,##[12] J. Shen, T. Tang, L.L. Wang, Spectral Methods Algorithms, Analysis and##Applications, Springer, 2011.##[13] D. Funaro, Polynomial Approximations of Dierential Equations,##Springer-Verlag, 1992.##[14] R.P. Agraval, D.O. Oregan, Odinary and Partial Dierential Equations,##Springer, 2009.##[15] A.C. King, J. Bilingham, S.R. Otto, Dierential Equations (Linear,##Nonlinear, Integral, Partial), Cambridge University, 2003.##[16] A.M. Wazwaz, The combined Laplace transform-Adomian decomposition##method for handling nonlinear Volterra integro-dierential equations,##Appl. Math. Comput., 216 (2010) 1304-1309.##[17] A. Aminataei, S.S. Hussaini, The comparison of the stability of##decomposition method with numerical methods of equation solution, Appl.##Math. Comput., 186 (2007) 665-669.##[18] A. Aminataei, S.S. Hussaini, The barrier of decomposition method, Int.##J. Contemp. Math. Sci., 5 (2010) 2487-2494.##[19] M. Gulsu, M. Sezer, Z. Guney, Approximate solution of general high-order##linear non-homogenous dierence equations by means of Taylor collocation##method, Appl. Math. Comput., 173 (2006) 683-693.##[20] M. Gulsu, M. Sezer, A Taylor polynomial approach for solving dierential-##dierence equations, Comput. Appl. Math., 186 (2006) 349-364.##[21] M. Sezer, M. Gulsu, Polynomial solution of the most general linear##Fredholm integro-dierential-dierence equation by means of Taylor##matrix method, Int. J. Complex Variables., 50 (2005) 367-382.##[22] M. Gulsu, M. Sezer, A method for the approximate solution of the high-##order linear dierence equations in terms of Taylor polynomials, Int. J.##Comput. Math., 82 (2005) 629-642.##[23] K. Maleknejad, F. Mirzaee, Numerical solution of integro-dierential##equations by using rationalized Haar functions method, Kyber. Int. J.##Syst. Math., 35 (2006) 1735-1744.##[24] M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving##Fredholm and Volterra integral equations, Comput. Appl. Math, 200##(2007) 12-20. ##[25] E.L. Ortiz, L. Samara, An operational approach to the Tau method for##the numer- ical solution of nonlinear dierential equations, Computing,##27 (1981) 15-25.##[26] E.L. Ortiz, On the numerical solution of nonlinear and functional##dierential equa- tions with the Tau method, in: Numerical Treatment##of Dierential Equations in Applications, in: Lecture Notes in Math., 679##(1978) 127-139.##[27] H. Danfu, S. Xufeng, Numerical solution of integro-dierential equations##by using CAS wavelet operational matrix of integration, Appl. Math.##Comput., 194 (2007) 460-466.##[28] C.H. Hsiao, Hybrid function method for solving Fredholm and Volterra##integral equations of the second kind, Comput. Appl. Math., 230 (2009)##[29] M. Razzaghi, S.A. Youse, Legendre wavelets method for the nonlinear##Volterra- Fredholm integral equations, Math. Comput. Simul., 70 (2005)##[30] A. Imani, A. Aminataei, A. Imani, Collocation method via Jacobi##polynomials for solving nonlinear ordinary dierential equations, Int. J.##Math. Math. Sci., Article ID 673085, 11P, 2011.##[31] M. Sezer, A.A. Dascioglu, Taylor polynomial solutions of general linear##dierential-dierence equations with variable coecients, Appl. Math.##Comput., 174 (2006) 1526-1538.##[32] T. Akkaya, S. Yalcinbas, Boubaker polynomial approach for solving high-##order linear dierential-dierence equations, AIP Conference Proceedings##of 9th international conference on mathematical problems in engineering,##56 (2012) 26-33.##[33] K. Erdem, S. Yalcinbas, Bernoulli polynomial approach to high-order##linear dierential-dierence equations, AIP Conference Proceedings of##Numerical Analysis and Applied Mathematics, 73 (2012) 360-364.##[34] M.R. Eslahchi, M. Dehghan, Application of Taylor series in obtaining##the orthogonal operational matrix, Computers and Mathematics with##Applications, 61 (2011) 2596-2604.##[35] M. Razzaghi, Y. Ordokhani, Solution of nonlinear Volterra Hammerstein##integral equations via rationalized Haar functions, Math. Prob. Eng., 7##(2001) 205-219. ##[36] F. Khellat, S. A. Youse, The linear Legendre wavelets operational matrix##of integration and its application, J. Frank. Inst., 343 (2006) 181-190.##[37] C. Kesan, Taylor polynomial solutions of linear dierential equations,##Appl. Math. Comput., 142 (2003) 155-165.##[38] N. Kurt, M. Sezer, Polynomial solution of high-order linear Fredholm##integro-dierential equations with constant coecients, J. Frank. Inst.,##345 (2008) 839-850.##[39] A. Golbabai, M. Javidi, Application o f homotopy perturbation method##for solving eighth-order boundary value problems, Appl. Math. Comput.,##213 (2007) 203-214.##]
On the singular fuzzy linear system of equations
On the singular fuzzy linear system of
equations
2
2
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.
1
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.
81
100
M.
Nikuie
M
Nikuie
باشگاه پژوهشگران جوان دانشگاه آزاد یزد.
باشگاه پژوهشگران جوان دانشگاه آزاد یزد.
Iran
nikoie_m@yahoo.com
M. K.
Mirnia
M.K.
Mirnia
Department of Computer engineering, Tabriz Branch, Islamic Azad
University, Tabriz, Iran.
Department of Computer engineering, Tabriz
Iran
[[1] B.C.Tripathy., A. Baruah, Notlund and Riesz mean of sequences of fuzzy##real numbers, Applied Math.Letters, 23(2010) 651-655.##[2] B.C. Tripathy, S.Borgohain, Some classes of dierence sequence spaces##of fuzzy real numbers dened by Orlicz function, Advances in Fuzzy##Systems,( 2011) Article ID216414, 6pages.##[3] B.C.Tripathy, M.Sen and S.Nath, I-Convergence in probabilistic Nnormed##space, Soft Comput, 2012, 16, 1021-1027, DOI 10.1007/s00500-##011-0799-8.##[4] B.C.Tripathy, P.C.Das, On Convergence of seriries of fuzzy real numbers##, J.Kuwait J Sci Eng, 39(1A)(2012) 57-70. ##[5] B.C.Tripathy, B.Sarma, I-Convergence double sequences of fuzzy real##numbers, Kyungpook Math Journal, 52(2)(2012) 189-200.##[6] B.C.Tripathy, A.Baruah ,M. Et and M.Gungor , On almost statistical##convergence of new type o generalized dierence sequence of fuzzy##number, Iranian Journal of Science and Technology, Transacations A:##Science , 36(2)(2012) 147-155.##[7] B.C.Tripathy, G.C.Ray, On Mixed fuzzy topological spaces and##countability, Soft Computing, 16(10)(2012) 1691-1695.##[8] S.Abbasbandy, R.Ezzati, A.Jafarian, LU decomposition method for##solving fuzzy symmetric positive denite system of linear equations,##Appl.Math.Comp, 172 (2006), 633-643.##[9] M.Friedman, M.Ming,A. Kandel, Fuzzy Linear Systems, Fuzzy Sets and##Systems, 96(1998), 201-209.##[10] S.M.Hashemi, M. K.Mirnia, S.Shahmorad, Solving fuzzy linear system by##using the Schur complement when coecient matrix is an M-Matrix, IJFS,##5 (2008) , 15-30.##[11] S.Abbasbandy, M.Amirfakhrian, Approximation of fuzzy functions by##distance method, Mathematics Scientic Journal(Islamic Azad University-##Arak Branch),2( 2006), 15-28.##[12] T.Allahviranloo, The adomain decomposition methods for solving fuzzy##systems of linear equations, Applied Mathematics and Computation, 163##(2005), 553-563,##[13] A.M.Al-Dubiban,S.M.El-Sayed, On the Positive Denite Solutions of the##Nonlinear Matrix Equation XA*XsAB*XtB = I, Communications in##numerical analysis,(2012), IDcna00116.##[14] A.Jafarian,S. Measoomy Nia, Utilizing a new feed-back fuzzy neural##network for solving a system of fuzzy equations , Communications in##Numerical Analysis, (2012), IDcna-00096.##[15] S.Abbasbandy, M.Otadi, M.Mosleh, Minimal solution of general dual##fuzzy linear systems, Chaos, Solutions and Fractals, 37 (2008) ,1113-1124.##[16] B.Asady, P.Mansouri, Numerical Solution of Fuzzy Linear Systems ,##International journal computer mathematics ,86( 2009),151-162. ##[17] R.Ezzati, Solving fuzzy linear systems, Soft Comput , (2010)DOI##10.1007/s00500-009-0537-7.##[18] M.Nikuie, M.K. Mirnia, Normal equations for singular fuzzy linear##systems, Journal of Mathematical Extension, 6(1)(2012) 29-42.##[19] S.Campbell ,C.Meyer, Generalized Inverses of Linear Transformations,##Pitman London,1979.##[20] M.Nikuie, M.K.Mirnia, A method for solving singular linear system of##equations, Appl Math Islamic Azad University-Lahijan Branch, 7(2)(2010)##[21] B.Detta, Numerical Linear Algebra and Application, Thomson, 1994.##[22] M.Nikuie, M.K.Mirnia, On the consistency of singular fuzzy linear system##of equations, 41th Iranian International Conference on Mathematics, 2010.##[23] S.Abbasbandy, M.Alavi, A new method for solving symmetric fuzzy linear##systems, Mathematics Scientic Journal(Islamic Azad University Arak##Branch), 1(2005) 55-62.##[24] M.Saravi,E. Babolian, M.Rastegari, Systems of linear dierential##equations and collocation method , Mathematics Scientic Journal(Islamic##Azad University-Arak Branch),5(2010) 79-87.##[25] L.Zheng, A characterization of the Drazin inverse, Linear Algebra and its##Application, 335 (2001) 183-188.##[26] M.Nikuie, M.K. Mirnia, Mahmoudi.Y., Some results about the index##of matrix and Drazin inverse, Math Sci. Islamic Azad University-Karaj##Branch,4(3)(2010), 283-294.##[27] D.Kincaid, W.Cheney, Numerical analysis mathematics of scientic##Computing California,1990.##[28] K.Homan, R.Kunze, Linear algebra, Prentice-Hall,1971.##]
Convergence Theorems for -Nonexpansive Mappings in CAT(0) Spaces
Convergence Theorems for -Nonexpansive
Mappings in CAT(0) Spaces
2
2
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.
1
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.
101
114
Savita
Rathee
Savita
Rathee
هندوستان
هندوستان
Iran
dr.savitarathee@gmail.com
R.
Ritika
R
Ritika
Department of Mathematics, M.D. University, Rohtak (Haryana), India
Department of Mathematics, M.D. University,
Iran
math.riti@gmail.com
[[1] K. Aoyama and F. Kohsaka, Fixed point theorem for -nonexpansive##mappings in Banach spaces, Nonlinear Analysis 74 (2011), 4387-4391.##[2] M. Bridson and A. Hae iger, Metric Spaces of Non-Positive Curvature,##Springer-Verlag, Berlin, Heidelberg, (1999).##[3] F. Bruhat and J. Tits, Groupes reductifs sur un corps local. I. Donnees##radicielles values, Inst. Hautes Etudes Sci. Publ. Math. 41 (1972), 5{251.##[4] S. Dhompongsa, W.A. Kirk and B. Panyanak, Non-expansive set-valued##mappings in metric and Banach spaces, Journal of Nonlinear and Convex##Analysis 8 (2007), 35{45.##[5] S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly##lipschitzian mappings, Nonlinear Analysis: TMA 65 (2006), 762{772.##[6] S. Dhompongsa and B. Panyanak, On -convergence theorems in CAT(0)##spaces, Computer and Mathematics with Applications 56 (2008), 2572{##[7] W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces,##Nonlinear Analysis: TMA 68 (2008), 3689{3696.##[8] T.C. Lim, Remarks on some xed point theorems, Proc. Amer. Math. Soc.##60 (1976), 179{182. ##[9] E. Naraghirad, N.C. Wong and J.C. Yao, Approximating xed points##of -nonexpansive mappings in uniformly convex Banach spaces and##CAT(0) spaces, Fixed Point Theory and Applications (2013), 2013:57,##doi:10.1186/1687-1812-2013-57.##[10] W. Phuengrattana and S. Suantai, On the rate of convergence of Mann,##Ishikawa, Noor and SP iterations for continuous functions on an arbitrary##interval, Journal of Computational and Applied Mathematics 235 (2011),##3006{3014.##[11] S. Shabani and S.J.H. Ghoncheh, Approximating xed points##of generalized nonexpansive nonself mappings in CAT(0) spaces,##Mathematics Scientic Journal 7(1) (2011), 89{95.##[12] S. Thianwan, Common xed points of new iterations for two##asymptotically non-expansive non-self mappings in Banach spaces,##Journal of Computational and Applied Mathematics (2009), 688{695.##]
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
2
2
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.
1
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.
115
133
Sh.
Sadigh Behazadi
ش.
صدیق بهزادی
Department of Mathematics, Islamic Azad University, Qazvin Branch,
Qazvin, Iran
Department of Mathematics, Islamic Azad University
Iran
[[1] Allahviranloo. T, Ahmady. N, Ahmady. E (2009). A method for##solving n-th order fuzzy linear dierential equations: Comput. Math.##Appl 86: 730-742.##[2] Abbasbandy. S, Allahviranloo. T (2002). Numerical solutions of##fuzzy dierential equations by taylor method: Comput. Methods##Appl. Math. 2: 113-124.##[3] Abbasbany. S (2008). Homptopy analysis method for generalized##Benjamin-Bona-Mahony equation: Zeitschri fur angewandte##Mathematik und Physik ( ZAMP). 59: 51-62.##[4] Abbasbany. S (2010). Homptopy analysis method for the Kawahara##equation: Nonlinear Analysis: Real Wrorld Applications. 11: 307-##[5] Bressan. A, Constantin. A (2005). Global solutions of the Hunter-##Saxton equation: SIAM J. Math. Anal. 37: 996-1026.##[6] Beals. R, Sattinger. D, Szmigielski. J (2001). Inverse scattering##solutions of the Hunter-Saxton equation: Applicable Analysis. 78##: 255-269.##[7] Behriy.S.H, Hashish.H, E-Kalla.I.L , A.Elsaid (2007). A new##algorithm for the decomposition solution of nonlinear dierential##equations: Appl. Math. Comput. 54: 459-466.##[8] El-KallaI.L (2008). Convergence of the Adomian method applied##to a class of nonlinear integral equations: Appl.Math.Comput. 21:##[9] Fariborzi Araghi M.A, Sadigh Behzadi.Sh (2009). Solving nonlinear##Volterra-Fredholm integral dierential equations using the modied##Adomian decomposition method: Comput. Methods in Appl. Math.##[10] Fariborzi Araghi.M.A ,##Sadigh Behzadi.Sh (2010). Numerical solution of nonlinear Volterra-##Fredholm integro-dierential equations using Homotopy analysis##method: Journal of Applied Mathematics and Computing, DOI:##10.1080/00207161003770394. ##[11] Guan.C (2012). Global weak solutions for a periodic two-##component PERIODIC Hunter-Saxton system: Quarterly of##Applied Mathematics. 2: 285-297.##[12] Hunter.J.K, Saxton.R (1991). Dynamics of director elds: SIAM J.##Appl. Math. 51: 1498-1521.##[13] Li.J, Zhang.K (2011). Global existence of dissipative solutions to##the Hunter-Saxton equation via vanishing viscosity: J. Dierential##Equations. 250: 1427-1447.##[14] Khesin.B, Lenells.J, Misiolek.G (2013). Generalized HunterSaxton##equation and the geometry of the group of circle dieomorphisms:##Math. Ann, DOI 10.1007/s00208-008-0250-3.##[15] Lenells.J (2008). Poisson structure of a modied Hunter-Saxton##equation: J. Phys. A: Math. Theor. 41: 1-9.##[16] Lenells.J (2007). The Hunter-Saxton equation describes the geodesic## ow on a sphere: Journal of Geometry and Physics. 57: 2049-2064.##[17] LiaoS.J (2003). Beyond Perturbation: Introduction to the Homotopy##Analysis Method: Chapman and Hall/CRC Press,Boca Raton.##[18] LiaoS.J (2009). Notes on the homotopy analysis method: some##denitions and theorems: Communication in Nonlinear Science and##Numerical Simulation. 14:983-997.##[19] Nadjakhah.M, Ahangari.F (2013). Symmetry Analysis and##Conservation Laws for the Hunter-Saxton Equation: Commun.##Theor. Phys, doi:10.1088/0253-6102/59/3/16.##[20] PenskoiA. V (2002). Lagrangian time-discretization of the Hunter-##Saxton equation: Physics Letters A. 304: 157-167.##[21] BehzadiSh.S (2010). The convergence of homotopy methods##for nonlinear Klein-Gordon equation: J.Appl.Math.Informatics.##28:1227-1237.##[22] behzadiSh.S , Fariborzi Araghi.M.A (2011).##The use of iterative methods for solving Naveir-Stokes equation:##J.Appl.Math.Informatics. 29: 1-15. ##3] BehzadiSh.S, Fariborzi AraghiM.A (2011). Numerical solution##for solving Burger's-Fisher equation by using Iterative Methods:##Mathematical and Computational Applications. 16:443-455.##[24] BehzadiSh.S (2011). Numerical solution of fuzzy Camassa-Holm##equation by using homotopy analysis method: Joournal of Applied##Analysis and Computations. 1:1-9.##[25] BehzadiSh.S (2011). Numerical solution##of Hunter-Saxton equation by using iterative methods: Journal of##Information and Mathematical Sciences. 3: 127-143.##[26] BehzadiSh.S (2011). Solving Schrodinger equation by using modied##variational iteration and homotopy analysis methods: Journal of##Applied Analysis and Computations. 4: 427-437.##[27] Wazwaz.A.M (2001). Construction of solitary wave solution and##rational solutions for the KdV equation by ADM.: Chaos,Solution##and fractals. 12: 2283-2293.##[28] YinZ.Y (2004). On the structure of solutions to the periodic Hunter-##Saxton equation: SIAM J. Math. Anal. 36: 272-283.##]
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
2
2
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.
1
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.
135
149
Ahmad
Shahsavaran
احمد
شهسواران
Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.
Islamic Azad University, Boroujerd Branch,
Iran
a.shahsavaran@iaub.ac.ir
Akbar
Shahsavaran
اکبر
شهسواران
Islamic Azad University, Boroujerd Branch, Boroujerd, Iran.
Islamic Azad University, Boroujerd Branch,
Iran
[[1] E. Babolian, A. Shahsavaran, Numerical solution of nonlinear Fredholm##integral equations of the second kind using Haar wavelets, J. Comput. Appl.##Math. 225 (2009) 87{95.##[2] C. F. Chen, C. H. Hsiao, Haar wavelet method for solving lumped and##distributed-parameter systems, IEE Proc. Control Theory Appl. 144 (1997)##[3] M. Ghasemi, A. Azizi and M. Fardi, Numerical solution of seven order##Sawada Kotara equations by homotopy perturbation method, Mathematics##Scientic Journal 1 (2011) 69{77.##[4] L. Hooshangian, D. Mirzaei, A Legendre spectral scheme for solution##of nonlinear system of Volterra Fredholm integral equations, Mathematics##Scientic Journal 1 (2012) 1{14.##[5] U. Lepik, E. Tamme, Application of the Haar wavelets for solution of linear##integral equations, in:Dynamical systems and applications 1 (2005) 395{407.##[6] K. Maleknejad, T. Lot, Numerical expansion method for solving integral##equation by interpolation and Gauss quadrature rules, Appl. Math. Comput.##168 (2005) 111{124.##[7] K. Maleknejad, M. T. Kajani, Y. Mahmoudi, Numerical solution of the##second kind integral equation by using Legendre wavelets, J. Kybornet In##[8] K. Maleknejad, M. Karami, Numerical solution of non-linear Fredholm##integral Equations by using multiwavelet in Petrov-Galerkin method, Appl.##Math. Comput. 168 (2005) 102{110.##[9] K. Maleknejad, Y. Mahmoudi, Numerical solution of linear Fredholm##integral equation by using hybrid Taylor and Block-Pulse functions, Appl.##Math. Comput. 149 (2004) 799{806.##[10] K. Maleknejad, M. Shahrezaee, Using Runge-Kutta method for numerical##solution of the system of Volterra integral equation, Appl. Math. Comput.##149 (2004) 399-410.##[11] K. Maleknejad, M. T. Kajani, Solving second kind integral equations by##Galerkin method with hybrid Legendre and Block-Pulse functions, Appl.##Math. Comput. 145 (2003) 623{629. ##[12] Y. Mahmoudi, Wavelet Galerkin method for numerical solution of##nonlinear integral equation, Appl. Math. Comput. 167 (2005) 1119{1129.##[13] K. Maleknejad, F. Mirzaee, Using rationalized Haar wavelet for solving##linear integral equations, Appl. Math. Comput. 160 (2005) 579{587.##[14] K. Maleknejad, F. Mirzaee, Numerical solution of linear Fredholm integral##equations system by rationalized Haar functions method, Int. J. Comput.##Math. 8 (2003) 1397{1405.##[15] A. Shahsavaran, Numerical solution of linear Volterra and Fredholm##integro dierential equations using Haar wavelets, Mathematics Scientic##Journal 1 (2010) 85{96.##[16] C. P. Rao, Piecewise constant orthogonal functions and their applications##to system and control, Springer, Berlin 1983.##[17] P. Wojtaszczyk, A mathematical introduction to wavelets, Cambridge##University Press 1997.##]
A note on positive deniteness and stability of interval matrices
A note on positive deniteness and
stability of interval matrices
2
2
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.
1
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.
151
158
H.
Veiseh
H
Veiseh
Department of Applied Mathematics, Hamedan Branch, Islamic Azad
University, Hamedan, Iran
Department of Applied Mathematics, Hamedan
Iran
veisehana@yahoo.com
[[1] J. Rohn, Checking positive deniteness or stability of symmetric interval##matrices is NP-hard, Commentationes Mathematicae Universitatis##Carolinae. 35 (1994) 795{797. ##[2] J. Rohn, Checking properties of interval matrices, Technical Report 686,##Institute of Computer Science, Academy of Sciences of the Czech Republic,##Prague, September 1996.##[3] R. Farhadsefat, T. Lot, J. Rohn, A note on regularity and positive##deniteness of interval matrices, Cent. Eur. J. Math. 10(1) (2012) 322{##[4] M. Mansour, Robust stability of interval matrices, Proceeding of the 28th##Conference on Decision and Control, Tampa, FL. (1989) 46{51.##[5] J. Rohn, A Handbook of Results on Interval Linear Problems, Prague:##Czech Academy of Sciences, 2005.##[6] J. Rohn, Positive deniteness and stability of interval matrices, SIAM J.##Matrix Anal. Appl. 15(1) (1994) 175{184.##[7] S. Poljak, J. Rohn, Checking roboust nonsingularity is NP-hard, Math.##Control Signals Syst. 6(1) (1993) 1{9.##[8] A. S. Deif, The interval eigenvalue problem, Z. Angew. Math. Mech. 71(1)##(1991) 61{64.##[9] M. Hladik, D. Daney, E. P. Tsigaridas, Bounds on real eigenvalues and##singular values of interval matrices, SIAM J. Matrix Anal. Appl. 31(4)##(2010) 2116{2129.##[10] J. Rohn, Bounds on eigenvalues of interval matrices, ZAMM, Z. Angew.##Math. Mech. 78(Suppl. 3) (1998) S1049{S1050.##[11] J. Rohn, Interval matrices: Singularity and real eigenvalues, SIAM##Journal on Matrix Analysis and Applications. 14(1) (1993) 82{91.##[12] J. Rohn, A. Deif, On the range of eigenvalues of an interval matrix,##Comput. 47(3-4) (1992) 373{377.##[13] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge: Cambridge##University Press, 1985.##[14] J. Stoer and R. Bulrisch, Introduction to Numerical Analysis, Springer-##Verlag, Berlin, 1980.##]
Multiple solutions of the nonlinear reaction-diusion model with fractional reaction
Multiple solutions of the nonlinear
reaction-diusion model with fractional
reaction
2
2
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.
1
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.
159
170
H.
Vosoughi
H.
Vosoughi
Department of Mathematics, Faculty of Science, Islamshahr Branch,
Islamic Azad University, Islamshahr, Tehran, Iran
Department of Mathematics, Faculty of Science,
Iran
E.
Shivanian
E.
Shivanian
Department of Mathematics, Imam Khomeini International University,
Qazvin, 34149-16818, Iran
Department of Mathematics, Imam Khomeini
Iran
shivanian@ikiu.ac.ir
M.
Anbarloei
M.
Anbarloei
Department of Mathematics, Faculty of Science, Islamshahr Branch,
Islamic Azad University, Islamshahr, Tehran, Iran
Department of Mathematics, Faculty of Science,
Iran
[[1] A.J. Ellery, M.J. Simpson, An analytical method to solve a general class##of nonlinear reactive transport models, Chem. Eng. J. 169 (2011) 313-318.##[2] S. Abbasbandy, Approximate solution for the nonlinear model of diusion##and reaction in porous catalysts by means of the homotopy analysis##method, Chem. Eng. J. 136 (2008) 144-150.##[3] S. Abbasbandy, E. Magyari, E. Shivanian, The homotopy analysis method##for multiple solutions of nonlinear boundary value problems, Commun.##Nonlinear Sci. Numer. Simulat. 14 (2009) 3530-3536.##[4] Y.P. Sun and S.B. Liu and K. Scott, Approximate solution for the##nonlinear model of diusion and reaction in porous catalysts by the##decomposition method, Chem. Eng. J. 101 (2004) 1-10.##[5] S. Abbasbandy and E. Shivanian, Exact analytical solution of a nonlinear##equation arising in heat transfer, Phys. Lett. A, 374 (2010) 567-574.##[6] J.E. Bailey, D.E. Ollis, Biochemical Engineering Fundamentals, second##edition, McGrawHill, 1986. ##7] T. P. Clement, Y. Sun, B. S. Hooker, J. N. Peterson, Modeling##multispecies reactive transport in ground water, Groundwater Monitoring##and Remediation 18 (1998) 7992.##[8] C. Zheng, G.D. Bennett, Applied Contaminant Transport Modelling,##second edition, Wiley Interscience, New York, 2002.##[9] A. Aris, The mathematical theory of diusion and reaction in permeable##catalysts, Volume 1 The Theory of Steady State, Oxford, 1975.##[10] E.J. Henley, E.M. Rosen, Material and Energy Balance Computations,##John Wiley and Sons, New York, 1969.##[11] S. Abbasbandy, E. Shivanian, Predictor homotopy analysis method and its##application to some nonlinear problems, Commun. Nonlinear Sci. Numer.##Simulat. 16 (2011) 2456-2468.##[12] S. Abbasbandy, E. Shivanian, Prediction of multiplicity of solutions##of nonlinear boundary value problems: Novel application of homotopy##analysis method, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010)##3830-3846.##[13] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis##Method, Chapman Hall CRC/Press, Boca Raton, 2003.##[14] S. J. Liao, Homotopy Analysis Method in Nonlinear Dierential##Equations, Springer, 2012.##[15] T. Hayat, T. Javed, M. Sajid, Analytic solution for rotating ow and heat##transfer analysis of a third-grade uid, Acta. Mech. 191 (2007) 219-29.##[16] T. Hayat, M. Khan, M. Sajid, S. Asghar, Rotating ow of a third-grade## uid in a porous space with hall current, Nonlinear Dyn. 49 (2007) 83-91.##[17] T. Hayat, Z. Abbas, M. Sajid, S. Asghar, The in uence of thermal##radiation on MHD ow of a second grade uid, Int. J. Heat. Mass. Transf.##50 (2007) 931-41.##[18] T. Hayat, N. Ahmed, M. Sajid, S. Asghar, On the MHD ow of a second##grade uid in a porous channel, Comput. Math. Appl. 54 (2007) 14-40.##[19] T. Hayat, M. Khan, M. Ayub, The eect of the slip condition on ows of##an Oldroyd 6 constant uid, J. Comput. Appl. 202 (2007) 402-13. ##[20] M. Sajid, A. Siddiqui, T. Hayat, Wire coating analysis using MHD##Oldroyd 8-constant uid, Int. J. Eng. Sci. 45 (2007) 381-92.##[21] M. Sajid, T. Hayat, S. Asghar, Non-similar analytic solution for MHD## ow and heat transfer in a third-order uid over a stretching sheet. Int.##J. Heat Mass. Transf. 50 (2007) 1723-36.##[22] L. Song, HQ. Zhang, Application of homotopy analysis method to##fractional KdV-Burgers-Kuramoto equation, Phys. Lett. A. 367 (2007)##[23] J. Cheng, S. J. Liao, RN. Mohapatra, K. Vajravelu, Series solutions of##nano boundary layer ows by means of the homotopy analysis method, J.##Math. Anal. Appl. 343 (2008) 233-45.##[24] S. Abbasbandy, The application of the homotopy analysis method to##nonlinear equations arising in heat transfer, Phys. Lett. A. 360 (2006)##[25] SP. Zhu, An exact and explicit solution for the valuation of American put##options, Quant. Fin. 6 (2006) 229-42.##[26] Y. Wu, KF. Cheung, Explicit solution to the exact Riemann problem and##application in nonlinear shallow-water equations, Int. J. Numer. Meth.##Fluids. 57 (2008) 1649-68.##[27] M. Yamashita, K. Yabushita, K. Tsuboi, An analytic solution of projectile##motion with the quadratic resistance law using the homotopy analysis##method, J. Phys. A. 40 (2007) 840316.##[28] Y. Bouremel, Explicit series solution for the Glauert-jet problem by##means of the homotopy analysis method, Commun. Nonlinear. Sci. Numer.##Simulat. 12(5) (2007) 714-24.##[29] L. Tao, H. Song, Chakrabarti S. Nonlinear progressive waves in water of##nite depth-an analytic approximation, Clastal. Eng. 54 (2007) 825-34.##[30] H. Song, L. Tao, Homotopy analysis of 1D unsteady, nonlinear##groundwater ow through porous media, J. Coastal. Res. 50 (2007) 292-5.##[31] A. Molabahrami, F. Khani, The homotopy analysis method to solve the##Burgers-Huxley equation. Nonlinear Anal. B: Real World Appl. 10 (2009)##[32] A. S. Bataineh, M. S. Noorani, I. Hashim, Solutions of time-dependent##EmdenFowler type equations by homotopy analysis method, Phys. Lett.##A. 371 (2007) 7282.##[33] Z. Wang, L. Zou, H. Zhang, Applying homotopy analysis method for##solving dierential-dierence equation, Phys. Lett. A. 369 (2007) 77-84.##[34] M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann##boundary conditions by the homotopy analysis method, Phys. Lett. A. 365##(2007) 412-5.##[35] W. H. Cai, Nonlinear dynamics of thermal-hydraulic networks. Ph.D.##thesis, University of Notre Dame; 2006.##[36] T. T. Zhang, L. Jia, Z. C. Wang, X. Li, The application of homotopy##analysis method for 2-dimensional steady slip ow in microchannels, Phys.##Lett. A. 372 (2008) 32237.##[37] A. K. Alomari, M. S. Noorani, R. Nazar, Adaptation of homotopy analysis##method for the numeric-analytic solution of Chen system. Commun.##Nonlinear Sci. Numer. Simul. 4 (2009) 2336-46.##[38] M. M. Rashidi, S. Dinarvand, Purely analytic approximate solutions##for steady three-dimensional problem of condensation lm on inclined##rotating disk by homotopy analysis method, Nonlinear Anal. B: Real##World Appl. 10 (2009) 2346-2356.##[39] Z. Odibat, S. Momani, H. Xu, A reliable algorithm of homotopy analysis##method for solving nonlinear fractional dierential equations, Applied##Mathematical Modelling 2010;34:593-600.##[40] S. Xinhui, Z. Liancun, Z. Xinxin, Y. Jianhong, Homotopy analysis method##for the heat transfer in a asymmetric porous channel with an expanding##or contracting wall, Appl. Math. Modell. 35 (2011) 4321-4329.##[41] R. A. Van Gorder, K. Vajravelu, Analytic and numerical solutions to the##Lane-Emden equation, Phys. Lett. A. 372 (372) 6060-5.##[42] Q. Wang, The optimal homotopy analysis method for Kawahara equation,##Nonlinear Anal. B: Real World Appl. 12(3) (2011) 1555-1561.##[43] A. R. Ghotbi, A. Bararni, G. Domairry, A. Barari, Investigation of a##powerful analytical method into natural convection boundary layer ow,##Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 2222-2228. ##[44] M. Ayub, H. Zaman, M. Ahmad, Series solution of hydromagnetic ow##and heat transfer with Hall eect in a second grade uid over a stretching##sheet, Cent. Eur. J. Phys. 8 (2010) 135-49.##[45] H. Vosughi, E. Shivanian, S. Abbasbandy, A new analytical technique to##solve Volterra's integral equations, Mathematical methods in the applied##sciences, 10(34) (2011) 1243-1253.##[46] M. Ghasemi, A. Azizi, M. Fardi, Numerical solution of seven-order##Sawada-Kotara equations by homotopy perturbation method, Math. Sc.##J. 7(1) (2011) 69-77.##[47] L. Hooshangian, D. Mirzaei, A Legendre-spectral scheme for solution of##nonlinear system of Volterra-Fredholm integral equations, Math. Sc. J.##8(1) (2012) 1-14.##[48] S. Abbasbandy, E. Shivanian, K. Vajravelu, Mathematical properties of##~-curve in the frame work of the homotopy analysis method, Commun.##Nonlinear Sci. Numer. Simulat. 16 (2011) 4268-4275.##]