2015
5
3
3
0
APPLICATION OF DIFFERENTIAL TRANSFORM METHOD TO SOLVE HYBRID FUZZY DIFFERENTIAL EQUATIONS
2
2
In this paper, we study the numerical solution of hybrid fuzzy differential equations by using differential transformation method (DTM). This is powerful method which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. Several numerical examples are given and by comparing the numerical results obtained from DTM and predictor corrector method (PCM), we have studied their accuracy.
1

203
217


Mahmoud
Paripour
Department of Mathematics, Hamedan University of Technology, Hamedan, 65156579, Iran
Iran, Islamic Republic of
Department of Mathematics, Hamedan University
Iran


Homa
Heidari
Iran


Elahe
Hajilou
Iran
Hybrid systems
Fuzzy Differential Equations
Differential transformation method
SOLVING NONLINEAR TWODIMENSIONAL VOLTERRA INTEGRAL EQUATIONS OF THE FIRSTKIND USING BIVARIATE SHIFTED LEGENDRE FUNCTIONS
2
2
In this paper, a method for ﬁnding an approximate solution of a class of twodimensional nonlinear Volterra integral equations of the ﬁrstkind is proposed. This problem is transformedto a nonlinear twodimensional Volterra integral equation of the secondkind. The properties ofthe bivariate shifted Legendre functions are presented. The operational matrices of integrationtogether with the product operational matrix are utilized to reduce the solution of the secondkind equation to the solution of a system of linear algebraic equations. Finally, a system of nonlinear algebraic equations is obtained to give an approximate solution of the main problem.Also, numerical examples are included to demonstrate the validity and applicability of themethod.
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219
230


Somayeh
Nemati
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
Department of Mathematics, Faculty of Mathematical
Iran


Y.
Ordokhani
Iran
AN M/G/1 QUEUE WITH REGULAR AND OPTIONAL PHASE VACATION AND WITH STATE DEPENDENT ARRIVAL RATE
2
2
We consider an M/G/1 queue with regular and optional phase vacation and withstate dependent arrival rate. The vacation policy is after completion of service if there are no customers in the system, the server takes vacation consisting of K phases, each phase is generally distributed. Here the first phase is compulsory where as the other phases are optional. For this model the supplementary variable technique has been applied to obtain the probability generating functions of number of customers in the queue at the different server states. Some particular models are obtained and a numerical study is also carried out.
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231
244


Rathinasabapathy
Kalyanaraman
Professor of Mathematics, Annamalai University
India
Professor of Mathematics 2nd
Professor of Mathematics, Annamalai University
In
Iran


Shanthi
R
Assistant professor Annamalai University
India
Research Scholar
Assistant professor Annamalai University
India
R
Iran
Vacation queue
Supplementary variable
Probability generating function
Performance Measures
[Chen, H. Y., JiHong Li and NaiShuo Tian, The GI/M/1 queue with phasetype working vacations and vacation interruptions, J. Appl. Math. Comput., 30, 121141, 2009. ##Doshi, B. T., Queueing systems with vacations  a survey, Queueing System, 1, 2966, 1986. ##Doshi, B. T., Single server queues with vacations, In: Takagi, H.(ed.), Stochastic Analysis of the computer and communication systems, 217264, North Holland/Elsevier, Amsterdam, 1990. ##Gross, D. and Harris, C. M., Fundamentals of queueing theory, 3rd edn, Wiley, New York, 1998. ##Ke, J. C., The analysis of general input queue with N policy and exponential vacations, Queueing Syst., 45, 13516, 1986. ##Ke, J. C., The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns, Computers and Industrial Engineering, Vol. 44, 567579, 2003. ##Ke, J. C., and Wang, K. H., Analysis of operating characteristics for the heterogeneous batch arrival queue with server startup and breakdowns, RAIROOper. Res., Vol.37, 157177, 2003. ##Keilson, J. and Servi, L. D., Dynamic of the M/G/1 vacation model, Operations Research, Vol.35 (4),575582, 1987. ##Takagi, H., Vacation and Priority Systems Part1. Queueing Analysis: A Foundation of Performance Evaluation, Vol.1., NorthHolland/Elsevier, Amsterdam, 1991. ##Tian, N., and Zhang, Z. G., The discrete time GI/Geo/1 queue with multiple vacations, Queueing Syst., 40, 283294, 2002. ##Tian, N., and Zhang, Z. G., A note on GI/M/1 queues with phasetype setup times or server vacations, INFOR, 41,341351, 2003. ##Tian, N., and Zhang, Z. G., Vacation queueing models: Theory and Applications, Springer, New York, 2006.##]
A NOTE ON "A SIXTH ORDER METHOD FOR SOLVING NONLINEAR EQUATIONS"
2
2
In this study, we modify an iterative nonoptimal without memory method, in such a way that is becomes optimal. Therefore, we obtain convergence order eight with the some functional evaluations. To justify our proposed method, some numerical examples are given.
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245
249


Paria
Assari
ORCID iD Islamic Azad University, Hamedan Branch
Iran, Islamic Republic of
ORCID iD Islamic Azad University, Hamedan
Iran


Taher
Lotfi
Islamic Azad University, Hamedan Branch
Iran, Islamic Republic of
Islamic Azad University, Hamedan Branch
Iran,
Iran
Nonlinear equation
Multipoint method
Convergence order
optimal method
[Cordero. A, Lotfi. T, Bakhtiari. P and Torregrosa. J. R, An efficient twoparametric family with memory for nonlinear equations, Numer Algor. DOI 10.1007/s1107501498468. ##Cordero. A, Lotfi. T, Mahdiani. K and Torregrosa. J. R, Two optimal general classes of iterative methods with eighthOrder, Acta Appl Math. DOI 10.1007/s1044001498690. ##Cordero. A, Lotfi. T, Torregrosa. J. R, Assari. P and Mahdiani. K, Some new biaccelerator twopoint methods for solving nonlinear equations, Comp. Appl. Math. DOI 10.1007/s4031401401921. ##Kung. H.T and Traub. J. F, Optimal order of onepoint and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634651. ##Lotfi. T and Assari. P, A new calss of two step methods with memory for solving nonlinear equation with high efficiency index, International Journal of Mathematical Modelling and Computations. 4 (2014) 277288. ##Lotfi. T, Magrenan. A. A, Mahdiani. K and Rainer. J. J, A variant of Steffensen–King’s type family with accelerated sixthorder convergence and high efficiency index: Dynamic study and approach, Applied Mathematics and Computation. 252 (2015) 347–353. ##Lotfi. T, Soleymani. F, Shateyi. S, Assari. P and Khaksar Haghani. F, New mono and biaccelerator iterative methods with memory for nonlinear equations, Abstract and Applied Analysis. Volume 2014, Article ID 705674, 8 pages. ##Lotfi. T and Tavakoli. E, On construction a new efficient Steffensenlike iterative class by applying a suitable selfaccelerator parameter. ##Mirzaee. F and Hamzeh. A, A sixth order method for solving nonlinear equations, International Journal of ##Mathematical Modelling and Computations. 4 (2014) 5560. ##Ostrowski. A. M, Solution of Equations and Systems of Equations, Academic Press, New York, 1960. ##Traub. J. F, Iterative Methods for the Solution of Equations, Prentice Hall, New York, 1964. ##Weerakoon. S and Fernando. T. G. I, A variant of Newton's method with accelerated thirdorder convergence, J. Appl. Math. Lett. 13 (8) (2000) 8793.##]
A STRONG COMPUTATIONAL METHOD FOR SOLVING OF SYSTEM OF INFINITE BOUNDARY INTEGRODIFFERENTIAL EQUATIONS
2
2
The introduced method in this study consists of reducing a system of
infinite boundary integrodifferential equations (IBIDE) into a system of al
gebraic equations, by expanding the unknown functions, as a series in terms
of Laguerre polynomials with unknown coefficients. Properties of these polynomials and operational matrix of integration are rst presented. Finally, two examples illustrate the simplicity and the effectiveness of the proposed method have been presented.
1

251
258


M.
Matinfar
University of Mazandaran
Iran, Islamic Republic of
University of Mazandaran
Iran, Islamic Republic
Iran


Abbas
Riahifar
University of Mazandaran
Iran, Islamic Republic of
University of Mazandaran
Iran, Islamic Republic
Iran


H.
Abdollahi
University of Mazandaran
Iran, Islamic Republic of
University of Mazandaran
Iran, Islamic Republic
Iran
Systems of infinite boundary integrodifferential equations
Laguerre polynomial
Operational matrix
[F. M. Maalek Ghaini, F. Tavassoli Kajani, M. Ghasemi, Solving boundary integral equation using Laguerre polynomials, World Applied Sciences Journal., 7(1) (2009) 102104. ##N. M. A. Nik Long, Z. K. Eshkuvatov, M. Yaghobifar, M. Hasan, Numerical solution of infinite boundary integral equation by using Galerkin method with Laguerre polynomials, World Academy of Science, Engineering and Technology., 47 (2008) 334337. ##D. G. Sanikidze, On the numerical solution of a class of singular integral equations on an infinite interval, Differential Equations., 41(9) (2005) 13531358. ##M. Gulsu, B. Gurbuz, Y. Ozturk, M. Sezer, Laguerre polynomials approach for solving linear delay difference equations, Applied Mathematics Computation., ##(2011) 67656776. ##J. PourMahmoud, M. Y. RahimiArdabili, S. Shahmorad, Numerical solution of the system of Fredholm integrodifferential equations by the Tau method., Applied Mathematics and Computation., 168 (2005) 465478. ##J. Biazar, H. Ghazvini, M. Eslami, He's homotopy perturbation method for systems of integrodifferential equations, Chaos, Solitions and Fractals., 39(3) ##(2009) 12531258. ##K. Maleknejad, F. Mirzaee, S. Abbasbandy, Solving linear integrodifferential equations system by using rationalized Haar functions method, Applied Mathematics and Computation., 155(2) (2004) 317328.##]
NONSTANDARD FINITE DIFFERENCE METHOD FOR NUMERICAL SOLUTION OF SECOND ORDER LINEAR FREDHOLM INTEGRODIFFERENTIAL EQUATIONS
2
2
In this article we have considered a nonstandard finite difference method for the solution of second order Fredholm integro differential equation type initial value problems. The nonstandard finite difference method and the composite trapezoidal quadrature method is used to transform the Fredholm integrodifferential equation into a system of equations. We have also developed a numerical method for the numerical approximation of the derivative of the solution of the problems. The numerical results in experiment on some model problems show the simplicity and efficiency of the method. Numerical results showed that the proposed method is convergent and at least second order of accurate.
1

259
266


Pramod Kumar
Pandey
Dyal Singh College (University of Delhi)
India
Department of Mathematics
Dyal Singh College (University of Delhi)
India
D
Iran
[Delves, L. M. and Mohamed, J. L., Computational Methods for Integral Equations. Cambridge University Press, Cambridge (1985). ##Liz, E. and Nieto, J. J., Boundary value problems for second order integrodifferential equations of Fredholm type. J. Comput. Appl. Math. 72, 215225 (1996). ##Zhao, J. and Corless, R.M. Corless, Compact finite difference method has been used for integrodifferential equations. Appl. Math. Comput., ##: 271288 (2006). ##Chang, S.H., On certain extrapolation methods for the numerical solution of integrodifferential equations. J. Math. Comp., ##: 165171 (1982). ##Yalcinbas, S., Taylor polynomial solutions of nonlinear VolterraFredholm integral equations.Appl. Math. Comput., 127: 195206 (2002). ##Phillips, D.L., A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach, 9, 84–96 ##Tikhonov, A.N., On the solution of incorrectly posed problem and the method of regularization. Soviet Math, 4, 1035–1038 (1963). ##He, J.H., Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput., 114(2/3), 115–123 (2000). ##Wazwaz, A.M., A reliable modification of the Adomian decomposition method. Appl. Math. Comput., 102, 77–86 (1999). ##Saadati, R.,Raftari, B., Abibi, H., Vaezpour , S.M. and Shakeri, S., A Comparison Between the Variational Iteration Method and Trapezoidal Rule for Solving Linear IntegroDifferential Equations. World Applied Sciences Journal, 4: 321325 (2008). ##Hu, S., Wan, Z. and Khavanin, M., On the existence and uniqueness for nonlinear integro  differential equations. Jour. Math Phy. Sci., 21, no. 2, 93  103 (1987). ##Hairer, E., Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I Nonstiff Problems (Second Revised Edition). SpringerVerlag New York, Inc. New York, NY, USA (1993). ##Van Niekerk, F. D., Rational one step method for initial value problem. Comput. Math. Applic. Vol.16, No.12, 10351039 (1988). ##Pandey, P. K., Nonlinear Explicit Method for first Order Initial Value Problems. Acta Technica Jaurinensis, Vol. 6, No. 2, 118125 (2013). ##Ramos, H., A nonstandard explicit integration scheme for initial value problems. Applied Mathematics and Computation. 189, no.1,710718 (2007). ##Jain, M.K., Iyenger, S. R. K. and Jain, R. K., Numerical Methods for Scientific and Engineering Computation {(2/e)}. Willey Eastern Limited, New Delhi, (1987). ##Lambert, J. D., Numerical Methods for Ordinary Differential Systems. John Wiley, England, 1991. ##Pandey, P. K. and Jaboob, S. S. A., Explicit Method in Solving Ordinary Differential Equations of the Second Order”. Int. J. of Pure and Applied Mathematics, vol. 76, no.2, pp.233239 (2012). ##Saadatmandia, A. and Dehghanb, M., Numerical solution of the higherorder linear Fredholm integrodifferentialdifference equation with variable coefficients. Computers & Mathematics with Applications, Vol. 59, Issue 8, 2996–3004 (2010). ##Jaradat, H. M., Awawdeh, F., Alsayyed, O., Series Solutions to the Highorder Integrodifferential Equations. Analele Universitatii Oradea Fasc. Matematica, Tom XVI, pp. 247257 (2009).##]
OPTIMUM GENERALIZED COMPOUND LINEAR PLAN FOR MULTIPLESTEP STEPSTRESS ACCELERATED LIFE TESTS
2
2
In this paper, we consider an i.e., multiple stepstress accelerated life testing (ALT) experiment with unequal duration of time . It is assumed that the time to failure of a product follows Rayleigh distribution with a loglinear relationship between stress and lifetime and also we assume a generalized KhamisHiggins model for the effect of changing stress levels. Taking into account that the problem of choosing the optimal time for 3step stepstress tests under compound linear plan was initially attempted by Khamis and Higgins [16]. We ever first have developed a generalized compound linear plan for multiplestep stepstress setting using varianceoptimality criteria. Some numerical examples are discussed to illustrate the proposed procedures.
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267
275


Navin
Chandra
Pondicherry University
India
Department of Statistics
Pondicherry University
India
Department
Iran


Mashroor
Ahmad Kha
Iran
accelerated life testing
Rayleigh distribution
cumulative exposure model
maximum likelihood estimate
generalized compound linear plan
[AlHaj Ebrahem M., AlMasri A., Optimum simple stepstress plan for log logistic cumulative exposure model. Metron LXV(1) (2007) 2334. ##Bai D.S., Kim M.S., Lee S.H., Optimum Simple StepStress Accelerated Life Tests with Censoring. IEEE transactions on reliability 38(5) (1989) 528–532. ##Balakrishnan N., A synthesis of exact inferential results for exponential stepstress models and associated optimal accelerated lifetests. Metrika 69(2009) 351–396. ##Balakrishnan N., Han D., Optimal stepstress testing for progressively TypeI censored data from exponential distribution. Journal of statistical planning and inference 139(2009) 1782–1798. ##Balakrishnan N., Xie Q., Exact inference for a simple stepstress model with TypeI hybrid censored data from the exponential distribution. Journal of statistical planning and inference 137: (2007a) 3268–3290. ##Balakrishnan N., Xie Q., Exact inference for a simple stepstress model with TypeII hybrid censored data from the exponential distribution. Journal of statistical planning and inference 137(2007b) 2543–2563 ##Chandra N., Khan M.A., A New Optimum Test Plan for Simple StepStress Accelerated Life Testing, in Applications of Reliability Theory and Survival Analysis. N. Chandra and G. Gopal, eds., Bonfring Publication, Coimbatore, India (2012) 5765. ##Chandra N., Khan M.A., Optimum plan for stepstress accelerated life testing model under typeI censored samples. Journal of modern mathematics and statistics 7(5) (2013) 5862. ##Chandra N., Khan M.A., Pandey M., Optimum test plan for 3step stepstress accelerated life tests. International Journal of Performability Engineering 10(1) (2014) 0314. ##Fan T.H., Wang W.L., Balakrishnan N., Exponential progressive stepstress lifetesting with link function based on Box–Cox transformation. Journal of statistical planning and inference 138(2008) 2340–2354. ##Fard N., Li C., Optimal simple step stress accelerated life test design for reliability prediction. Journal of statistical planning and inference 139(5) (2009) 17991808. ##Gouno E., Sen A., Balakrishnan N., Optimal stepstress test under progressive Type I censoring. IEEE transactions on reliability 53(2004) 83–393 ##Guan Q., Tang Y., Optimal stepstress test under TypeI censoring for multivariate exponential distribution. Journal of statistical planning and inference 142(7) (2012) 19081923. ##Hassan A.S., AlGhamdi A.S., Optimum stepstress accelerated life testing for Lomax distribution. Journal of Applied Sciences Research 5(12) (2009) 21532164. ##Khamis I.H., Optimum Mstep stepstress design with k stress variables. Communications in Statistics  Simulation and Computation 26(4) (1997) 13011313. ##Khamis I.H., Higgins J.J., Optimum 3step stepstress tests. IEEE transactions on reliability 45(2) (1996) 341345. ##Lin C.T., Chou C.C., Balakrishnan N., Planning stepstress test plans under TypeI censoring for the loglocationscale case. Journal of statistical computation and simulations 83(10) (2013) 18521867. ##Meeker W.Q., Escobar L.A., Statistical Methods for Reliability Data. Wiley, New York, (1998). ##Miller R., Nelson W.B., Optimum simple stepstress plans for accelerated life testing. IEEE transactions on reliability R32(1) (1983) 59–65. ##Nelson W.B., Accelerated life testing stepstress models and data analysis. IEEE Trans on Reliability 29(1980) 103–108. ##Shen K.F., Shen Y.J., Leu L.Y., Design of optimal step–stress accelerated life tests under progressive typeI censoring with random removals. Quality & Quantity 45(3) (2011) 587597. ##Srivastava P.W., Shukla R., Optimum LogLogistic StepStress Model with Censoring. International Journal of Quality & Reliability Management 25(9) (2008) 968976. ##Wang B.X., Interval estimation for exponential progressive TypeII censored stepstress accelerated lifetesting. Journal of statistical planning and inference 140(2010) 2706–2718. ##Wu S.J., Lin Y.P., Chen Y.J., Planning stepstress life test with progressively TypeI groupcensored exponential data. Statist Neerlandica 60(2006) 46–56. ##Wu S.J., Lin Y.P., Chen Y.J., Optimal stepstress test under TypeI progressive groupcensoring with random removals. Journal of statistical planning and inference 138(2008) 817–826. ##Xiong C., Inference on a simple stepstress model with TypeII censored exponential data. IEEE transactions on reliability 47(1998) 142–146.##]
EFFECT OF COUNTERPROPAGATING CAPILLARY GRAVITY WAVE PACKETS ON THIRD ORDER NONLINEAR EVOLUTION EQUATIONS IN THE PRESENCE OF WIND FLOWING OVER WATER
2
2
Asymptotically exact and nonlocal third order nonlinear evolution equations are derivedfor two counterpropagating surface capillary gravity wave packets in deep water in thepresence of wind flowing over water.From these evolution equations stability analysis ismade for a uniform standing surface capillary gravity wave trains for longitudinal perturbation. Instability condition is obtained and graphs are plotted for maximum growth rateof instability and for wave number at marginal stability against wave steepness for some different values of dimensionless wind velocity.
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277
289


A. K.
Dhar
Iran


Joydev
Mondal
IIEST,WESTBENGAL,INDIA
India
IIEST,MATHEMATICS,SHIBPUR,WESTBENGAL, INDIA
IIEST,WESTBENGAL,INDIA
India
IIEST,MATHEMATICS,SH
Iran