Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521893 unavailable APPLICATION OF DIFFERENTIAL TRANSFORM METHOD TO SOLVE HYBRID FUZZY DIFFERENTIAL EQUATIONS Paripour Mahmoud Department of Mathematics, Hamedan University of Technology, Hamedan, 65156-579, Iran Iran, Islamic Republic of Heidari Homa Hajilou Elahe 21 03 2015 5 3 (SUMMER) 203 217 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521893.html

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.

Hybrid systems Fuzzy Differential Equations Differential transformation method
Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521894 unavailable SOLVING NONLINEAR TWO-DIMENSIONAL VOLTERRA INTEGRAL EQUATIONS OF THE FIRST-KIND USING BIVARIATE SHIFTED LEGENDRE FUNCTIONS Nemati Somayeh Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran Ordokhani Y. 21 03 2015 5 3 (SUMMER) 219 230 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521894.html

In this paper, a method for ﬁnding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations of the ﬁrst-kind is proposed. This problem is transformedto a nonlinear two-dimensional Volterra integral equation of the second-kind. 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 second-kind 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.

Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521895 unavailable AN M/G/1 QUEUE WITH REGULAR AND OPTIONAL PHASE VACATION AND WITH STATE DEPENDENT ARRIVAL RATE Kalyanaraman Rathinasabapathy Professor of Mathematics, Annamalai University India Professor of Mathematics 2nd R Shanthi Assistant professor Annamalai University India Research Scholar 21 03 2015 5 3 (SUMMER) 231 244 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521895.html

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.

Vacation queue Supplementary variable probability generating function Performance Measures
Chen, H. Y., Ji-Hong Li and Nai-Shuo Tian, The GI/M/1 queue with phase-type working vacations and vacation interruptions, J. Appl. Math. Comput., 30, 121-141, 2009. Doshi, B. T., Queueing systems with vacations - a survey, Queueing System, 1, 29-66, 1986. Doshi, B. T., Single server queues with vacations, In: Takagi, H.(ed.), Stochastic Analysis of the computer and communication systems, 217-264, 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, 135-16, 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, 567-579, 2003. Ke, J. C., and Wang, K. H., Analysis of operating characteristics for the heterogeneous batch arrival queue with server startup and breakdowns, RAIRO-Oper. Res., Vol.37, 157-177, 2003. Keilson, J. and Servi, L. D., Dynamic of the M/G/1 vacation model, Operations Research, Vol.35 (4),575-582, 1987. Takagi, H., Vacation and Priority Systems Part1. Queueing Analysis: A Foundation of Performance Evaluation, Vol.1., North-Holland/Elsevier, Amsterdam, 1991. Tian, N., and Zhang, Z. G., The discrete time GI/Geo/1 queue with multiple vacations, Queueing Syst., 40, 283-294, 2002. Tian, N., and Zhang, Z. G., A note on GI/M/1 queues with phase-type setup times or server vacations, INFOR, 41,341-351, 2003. Tian, N., and Zhang, Z. G., Vacation queueing models: Theory and Applications, Springer, New York, 2006.
Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521896 unavailable A NOTE ON "A SIXTH ORDER METHOD FOR SOLVING NONLINEAR EQUATIONS" Assari Paria ORCID iD Islamic Azad University, Hamedan Branch Iran, Islamic Republic of Lotfi Taher Islamic Azad University, Hamedan Branch Iran, Islamic Republic of 21 03 2015 5 3 (SUMMER) 245 249 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521896.html

In this study, we modify an iterative non-optimal 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.

Nonlinear equation Multi-point method Convergence order optimal method
Cordero. A, Lotfi. T, Bakhtiari. P and Torregrosa. J. R, An efficient two-parametric family with memory for nonlinear equations, Numer Algor. DOI 10.1007/s11075-014-9846-8. Cordero. A, Lotfi. T, Mahdiani. K and Torregrosa. J. R, Two optimal general classes of iterative methods with eighth-Order, Acta Appl Math. DOI 10.1007/s10440-014-9869-0. Cordero. A, Lotfi. T, Torregrosa. J. R, Assari. P and Mahdiani. K, Some new bi-accelerator two-point methods for solving nonlinear equations, Comp. Appl. Math. DOI 10.1007/s40314-014-0192-1. Kung. H.T and Traub. J. F, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651. 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) 277-288. Lotfi. T, Magrenan. A. A, Mahdiani. K and Rainer. J. J, A variant of Steffensen–King’s type family with accelerated sixth-order 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 Steffensen-like iterative class by applying a suitable self-accelerator parameter. Mirzaee. F and Hamzeh. A, A sixth order method for solving nonlinear equations, International Journal of Mathematical Modelling and Computations. 4 (2014) 55-60. 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 third-order convergence, J. Appl. Math. Lett. 13 (8) (2000) 87-93.
Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521897 unavailable A STRONG COMPUTATIONAL METHOD FOR SOLVING OF SYSTEM OF INFINITE BOUNDARY INTEGRO-DIFFERENTIAL EQUATIONS Matinfar M. University of Mazandaran Iran, Islamic Republic of Riahifar Abbas University of Mazandaran Iran, Islamic Republic of Abdollahi H. University of Mazandaran Iran, Islamic Republic of 21 03 2015 5 3 (SUMMER) 251 258 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521897.html

The introduced method in this study consists of reducing a system of infinite boundary integro-differential equations (IBI-DE) 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.

Systems of infinite boundary integro-differential 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) 102-104. 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) 334-337. D. G. Sanikidze, On the numerical solution of a class of singular integral equations on an infinite interval, Differential Equations., 41(9) (2005) 1353-1358. M. Gulsu, B. Gurbuz, Y. Ozturk, M. Sezer, Laguerre polynomials approach for solving linear delay difference equations, Applied Mathematics Computation., (2011) 6765-6776. J. Pour-Mahmoud, M. Y. Rahimi-Ardabili, S. Shahmorad, Numerical solution of the system of Fredholm integro-differential equations by the Tau method., Applied Mathematics and Computation., 168 (2005) 465-478. J. Biazar, H. Ghazvini, M. Eslami, He's homotopy perturbation method for systems of integro-differential equations, Chaos, Solitions and Fractals., 39(3) (2009) 1253-1258. K. Maleknejad, F. Mirzaee, S. Abbasbandy, Solving linear integro-differential equations system by using rationalized Haar functions method, Applied Mathematics and Computation., 155(2) (2004) 317-328.
Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521898 unavailable NON-STANDARD FINITE DIFFERENCE METHOD FOR NUMERICAL SOLUTION OF SECOND ORDER LINEAR FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS Pandey Pramod Kumar Dyal Singh College (University of Delhi) India Department of Mathematics 21 03 2015 5 3 (SUMMER) 259 266 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521898.html

In this article we have considered a non-standard finite difference method for the solution of second order  Fredholm integro differential equation type initial value problems. The non-standard finite difference method and the composite trapezoidal quadrature method is used to transform the Fredholm integro-differential 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.

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 integro-differential equations of Fredholm type. J. Comput. Appl. Math. 72, 215-225 (1996). Zhao, J. and Corless, R.M. Corless, Compact finite difference method has been used for integro-differential equations. Appl. Math. Comput., : 271-288 (2006). Chang, S.H., On certain extrapolation methods for the numerical solution of integro-differential equations. J. Math. Comp., : 165-171 (1982). Yalcinbas, S., Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations.Appl. Math. Comput., 127: 195-206 (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 Integro-Differential Equations. World Applied Sciences Journal, 4: 321-325 (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). Springer-Verlag 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, 1035-1039 (1988). Pandey, P. K., Nonlinear Explicit Method for first Order Initial Value Problems. Acta Technica Jaurinensis, Vol. 6, No. 2, 118-125 (2013). Ramos, H., A non-standard explicit integration scheme for initial value problems. Applied Mathematics and Computation. 189, no.1,710-718 (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.233-239 (2012). Saadatmandia, A. and Dehghanb, M., Numerical solution of the higher-order linear Fredholm integro-differential-difference 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 High-order Integro-differential Equations. Analele Universitatii Oradea Fasc. Matematica, Tom XVI, pp. 247-257 (2009).
Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521899 unavailable OPTIMUM GENERALIZED COMPOUND LINEAR PLAN FOR MULTIPLE-STEP STEP-STRESS ACCELERATED LIFE TESTS Chandra Navin Pondicherry University India Department of Statistics Ahmad Kha Mashroor 21 03 2015 5 3 (SUMMER) 267 275 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521899.html

In this paper, we consider an  i.e., multiple step-stress 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 log-linear relationship between stress and lifetime and also we assume a generalized Khamis-Higgins model for the effect of changing stress levels. Taking into account that the problem of choosing the optimal time for 3-step step-stress tests under compound linear plan was initially attempted by Khamis and Higgins . We ever first have developed a generalized compound linear plan for multiple-step step-stress setting using variance-optimality criteria. Some numerical examples are discussed to illustrate the proposed procedures.

accelerated life testing Rayleigh distribution cumulative exposure model maximum likelihood estimate generalized compound linear plan
Al-Haj Ebrahem M., Al-Masri A., Optimum simple step-stress plan for log logistic cumulative exposure model. Metron LXV(1) (2007) 23-34. Bai D.S., Kim M.S., Lee S.H., Optimum Simple Step-Stress Accelerated Life Tests with Censoring. IEEE transactions on reliability 38(5) (1989) 528–532. Balakrishnan N., A synthesis of exact inferential results for exponential step-stress models and associated optimal accelerated life-tests. Metrika 69(2009) 351–396. Balakrishnan N., Han D., Optimal step-stress testing for progressively Type-I censored data from exponential distribution. Journal of statistical planning and inference 139(2009) 1782–1798. Balakrishnan N., Xie Q., Exact inference for a simple step-stress model with Type-I 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 step-stress model with Type-II 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 Step-Stress Accelerated Life Testing, in Applications of Reliability Theory and Survival Analysis. N. Chandra and G. Gopal, eds., Bonfring Publication, Coimbatore, India (2012) 57-65. Chandra N., Khan M.A., Optimum plan for step-stress accelerated life testing model under type-I censored samples. Journal of modern mathematics and statistics 7(5) (2013) 58-62. Chandra N., Khan M.A., Pandey M., Optimum test plan for 3-step step-stress accelerated life tests. International Journal of Performability Engineering 10(1) (2014) 03-14. Fan T.H., Wang W.L., Balakrishnan N., Exponential progressive step-stress life-testing 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) 1799-1808. Gouno E., Sen A., Balakrishnan N., Optimal step-stress test under progressive Type I censoring. IEEE transactions on reliability 53(2004) 83–393 Guan Q., Tang Y., Optimal step-stress test under Type-I censoring for multivariate exponential distribution. Journal of statistical planning and inference 142(7) (2012) 1908-1923. Hassan A.S., Al-Ghamdi A.S., Optimum step-stress accelerated life testing for Lomax distribution. Journal of Applied Sciences Research 5(12) (2009) 2153-2164. Khamis I.H., Optimum M-step step-stress design with k stress variables. Communications in Statistics - Simulation and Computation 26(4) (1997) 1301-1313. Khamis I.H., Higgins J.J., Optimum 3-step step-stress tests. IEEE transactions on reliability 45(2) (1996) 341-345. Lin C.T., Chou C.C., Balakrishnan N., Planning step-stress test plans under Type-I censoring for the log-location-scale case. Journal of statistical computation and simulations 83(10) (2013) 1852-1867. Meeker W.Q., Escobar L.A., Statistical Methods for Reliability Data. Wiley, New York, (1998). Miller R., Nelson W.B., Optimum simple step-stress plans for accelerated life testing. IEEE transactions on reliability R-32(1) (1983) 59–65. Nelson W.B., Accelerated life testing step-stress 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 type-I censoring with random removals. Quality & Quantity 45(3) (2011) 587-597. Srivastava P.W., Shukla R., Optimum Log-Logistic Step-Stress Model with Censoring. International Journal of Quality & Reliability Management 25(9) (2008) 968-976. Wang B.X., Interval estimation for exponential progressive Type-II censored step-stress accelerated life-testing. Journal of statistical planning and inference 140(2010) 2706–2718. Wu S.J., Lin Y.P., Chen Y.J., Planning step-stress life test with progressively Type-I group-censored exponential data. Statist Neerlandica 60(2006) 46–56. Wu S.J., Lin Y.P., Chen Y.J., Optimal step-stress test under Type-I progressive group-censoring with random removals. Journal of statistical planning and inference 138(2008) 817–826. Xiong C., Inference on a simple step-stress model with Type-II censored exponential data. IEEE transactions on reliability 47(1998) 142–146.
Islamic Azad University, Central Tehran Branch International Journal of Mathematical Modelling & Computations 2228-6225 Islamic Azad University, Central Tehran Branch 521900 unavailable EFFECT OF COUNTERPROPAGATING CAPILLARY GRAVITY WAVE PACKETS ON THIRD ORDER NONLINEAR ‎‎E‎VOLUTION EQUATIONS IN THE PRESENCE OF WIND FLOWING OVER WATER Dhar A. K. Mondal Joydev IIEST,WESTBENGAL,INDIA India IIEST,MATHEMATICS,SHIBPUR,WESTBENGAL, INDIA 21 03 2015 5 3 (SUMMER) 277 289 15 04 2016 15 04 2016 Copyright © 2015, Islamic Azad University, Central Tehran Branch. 2015 http://ijm2c.iauctb.ac.ir/article_521900.html

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.