Volume-7 , Issue-2 , Apr 2020, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Magnetic Field a Heat Generation Effects on Second Grade Fluid Flow past an Oscillating Vertical Plate in Porous Medium

  Rahul Mehta, H. R. Kataria

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.1-8, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  The present paper is concerned with the study of heat generation/absorption effect on unsteady natural convective MHD Second grade fluid flow past an oscillating vertical plate in presence of thermal radiation and chemical reaction. It is assumed that the bounding plate has ramped temperature with ramped surface concentration and isothermal temperature with ramped surface concentration through porous medium. Governing non-dimensional equations are solved using Laplace transform technique and analytic expressions are obtained of velocity, temperature and concentration profiles. For both thermal plates, analytic expressions of Nusselt Number and Sherwood Number are derived and presented in tabular form. The effects of Magnetic parameter M, second grade fluid α, Heat generation/absorption parameter H, thermal radiation parameter Nr, chemical reaction parameter Kr in time variable t on velocity, temperature and concentration profiles are discussed through several graphs.

  Key-Words / Index Term
  MHD; Second grade fluid; Porous medium; Nusselt Number; Sherwood Number

  References
  [1] W. Tan, T. Masuoka, Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary, Int J Non-Linear, 40, 515–522 (2005).
  [2] M. M. Rashidi, A. M. Siddiqui, M. Asadi, Application of homotopy analysis method to the unsteady squeezing flow of a second-grade fluid between circular plates, Math Probl Eng, 18, 706840 (2010).
  [3] M. Sheikholeslami, Magnetohydrodynamic nanofluid forced convection in a porous lid driven cubic cavity using Lattice Boltzmann Method, Journal of Molecular Liquids,(2017). 10.1016/j.molliq.2017.02.020
  [4] M. Sheikholeslami, M. M. Bhatti, Active method for nanofluid heat transfer enhancement by means of EHD, International Journal of Heat and Mass Transfer 109 (2017) 115–122.
  [5] M. Sheikholeslami, S.A. Shehzad, Thermal radiation of ferrofluid in existence of Lorentz forces considering variable viscosity, International Journal of Heat and Mass Transfer 109 (2017) 82–92.
  [6] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using Lattice Boltzmann Method, International Journal of Heat and Mass Transfer, 108 (2017) 1870-1883.
  [7] M. Sheikholeslami, M. Shamlooei, Fe3O4- H2O nanofluid natural convection in presence of thermal radiation, International Journal of Hydrogen Energy(2016), http://dx.doi.org/10.1016/j.ijhydene.2017.02.031
  [8] M. Sheikholeslami, CuO-water nanofluid free convection in a porous cavity considering darcy law, The European Physical Journal Plus, (2017) 132: 55 DOI 10.1140/epjp/i2017-11330- 3
  [9] M. Sheikholeslami, T. Hayat, A. Alsaedi, Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using Lattice Boltzmann Method, International Journal of Heat and Mass Transfer, (2017) 108 : 1870-1883
  [10] H. R. Kataria, A. Mittal, Mathematical model for velocity and temperature of gravity-driven convective optically thick nanofluid flow past an oscillating vertical plate in presence of magnetic field and radiation. Journal of Nigerian Mathematical Society, (2015) 34, 303– 317.
  [11] H. R. Kataria, A. S. Mittal, Velocity, mass and temperature analysis of gravitydriven convection nanofluid flow past an oscillating vertical plate in presence of magnetic field in a porous medium, Applied Thermal Engineering, (2017) 110, 864-874.
  [12] T. Hayat, N. Ahmed, M. Sajid, S. Asghar, On the MHD flow of a second grade fluid in a porous channel, Computers & Mathematics with Applications, 54, 407–414 (2007).
  [13] T. Hayat, C. Fetecau, M. Sajid, Analytic solution for MHD Transient rotating flow of a second grade fluid in a porous space, Nonlinear Analysis: Real World Applications, 9, 1619–1627 (2008).
  [14] T. Hayat, S. Saif, Z. Abbas, The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet, Physics Letters A, 372, 5037–5045 (2008).
  [15] Samiulhaq, S. Ahmad, D. Vieru, I. Khan, S. Shafie, Unsteady Magnetohydrodynamic Free Convection Flow of a Second Grade Fluid in a Porous Medium with Ramped Wall Temperature PLOS ONE 9(5): e88766. doi:10.1371/journal.pone.0088766 (2015)
  [16] M. H. Abolbashari, N. Freidoonimehr, F. Nazari, M.M. Rashidi, Entropy Analysis for an Unsteady MHD Flow past a Stretching Permeable Surface in Nano-Fluid, Powder Technology 267 (2014) 256-267.
  [17] M. M. Rashidi, M. Ali, N. Freidoonimehr, B. Rostami, M. Anwar Hossain, Mixed convective heat transfer for MHD viscoelastic fluid flow over a porous wedge with thermal radiation, Advances in Mechanical Engineering, Volume 2014 (2014) Article number 735939.
  [18] M.M. Rashidi, E. Erfani, Analytical Method for Solving Steady MHD Convective and Slip Flow due to a Rotating Disk with Viscous Dissipation and Ohmic Heating, Engineering Computations 29 (6) (2012) 562–579.
  [19] N. Freidoonimehr, M.M. Rashidi, S. Mahmud, Unsteady MHD free convective flow past a permeable stretching vertical surface in a nano-fluid, International Journal of Thermal Sciences 87 (2015) 136-145.
  [20] M.M. Rashidi, E. Momoniat, B. Rostami, Analytic approximate solutions for MHD boundary-layer viscoelastic fluid flow over continuously moving stretching surface by homotopy analysis method with two auxiliary parameters, Journal of Applied Mathematics, Volume 2012.
  [21] S. Abbasbandy, T. Hayat, A. Alsaedi, M.M. Rashidi, Numerical and Analytical Solutions for Falkner-Skan Flow of MHD Oldroyd-B fluid, International Journal of Numerical Methods for Heat and Fluid Flow 24 (2) (2014) 390-401.
  [22] J Zhou, M Hatami, D Song, & D Jing, Design of microchannel heat sink with wavy channel and its time-efficient optimization with combined RSM and FVM methods. International Journal of Heat and Mass Transfer, (2016) 103, 715-724.
  [23] D Song, M Hatami, Y Wang, D Jing, & Y Yang, Prediction of hydrodynamic and optical properties of TiO 2/water suspension considering particle size distribution. International Journal of Heat and Mass Transfer, (2016) 92, 864-876.
  [24] S Ghasemi, M Hatami, A K Sarokolaie, & D D Ganji Study on blood flow containing nanoparticles through porous arteries in presence of magnetic field using analytical methods. Physica E: Low-dimensional Systems and Nanostructures, (2015) 70, 146-156.
  [25] M Fakour, A Vahabzadeh, D D Ganji, & M Hatami, M. Analytical study of micropolar fluid flow and heat transfer in a channel with permeable walls. Journal of Molecular Liquids, (2015) 204, 198-204.
  [26] M Hatami, M Sheikholeslami, M Hosseini, D D Ganji, Analytical investigation of MHD nanofluid flow in non-parallel walls, Journal of Molecular Liquids 194 (2014) 251–259.
  [27] H. R. Kataria, H. R. Patel, (2015): Effect of magnetic field on unsteady natural convective flow of a micropolar fluid between two vertical walls. Ain Shams Engineering Journal, doi. 10.1016/j.asej.2015.08.013.
  [28] H. R. Kataria, H. R. Patel, (2016): Radiation and chemical reaction effects on MHD Casson fluid flow past an oscillating vertical plate embedded in porous medium, Alexandria Engineering Journal , 55, 583-595
  [29] H. R. Kataria, H. R. Patel, (2016): Soret and heat generation effects on MHD Casson fluid flow past an oscillating vertical plate embedded through porous medium, Alexandria Engineering Journal 55, 2125–2137
  [30] H. R. Kataria, H. R. Patel, (2016): Effect of thermo-diffusion and parabolic motion on MHD Second grade fluid flow with ramped wall temperature and ramped surface concentration, Alexandria Engineering Journal, 10.1016/j.aej.2016.1
  [31] H. R. Kataria, H. R. Patel, Heat and Mass Transfer in MHD Second Grade Fluid Flow with Ramped Wall Temperature through Porous Medium, Mathematics Today Vol.32 (2016) 67-83.
  [32] S M Rassoulinejad-Mousavi, H R Seyf, S Abbasbandy, "Heat transfer through a porous saturated channel with permeable walls using two-equation energy model", Journal of Porous Media, 2013, 16 (3), 241-254.
  [33] S M Rassoulinejad-Mousavi, S Abbasbandy, "Analysis of Forced Convection in a Circular Tube Filled With a Darcy–Brinkman–Forchheimer Porous Medium Using Spectral Homotopy Analysis Method". ASME. J. Fluids Eng., 2011,133(10),101207.
  [34] H R Seyf, S M Rassoulinejad-Mousavi, "An Analytical Study for Fluid Flow in Porous Media Imbedded Inside a Channel With Moving or Stationary Walls Subjected to Injection/Suction", ASME. J. Fluids Eng., 2011, 133(9), 091203.
  [35] S K Mohammadian, S M Rassoulinejad-Mousavi, Y Zhang, "Thermal management improvement of an air-cooled high-power lithium-ion battery by embedding metal foam", Journal of Power Sources, 2015, 296, 305-313.
  [36] S M Rassoulinejad-Mousavi, S Abbasbandy, H H Alsulami, "Analytical flow study of a conducting Maxwell fluid through a porous saturated channel at various wall boundary conditions", Eur. Phys. J. Plus, 2014, 129: 181.
  [37] S M Rassoulinejad-Mousavi, H Yaghoobi, "Effect of Non-linear Drag Term on Viscous Dissipation in a Fluid Saturated Porous Medium Channel with Various Boundary Conditions at Walls", Arab. J. Sci. Eng. , 2014, 39 (2), 1231–1240.
  [38] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336.
  [39] S. Rosseland, Astrophysik und atom-theoretischeGrundlagen, Springer-Verlag, Berlin, 1931.
  [40] G. S. Seth, A K Singha, R Sharma, MHD natural convection flow with hall effects, radiation and Heat absorption over an exponentially accelerated vertical Plate with ramped temperature, Ind. J. Sci. Res. and Tech, 5 (2015) 10-22.
  [41] G. S. Seth, S M Hussain, S Sarkar, Hydromagnetic natural convection flow with heat And mass transfer of a chemically reacting and heat Absorbing fluid past an accelerated moving vertical plate with ramped temperature and ramped surface Concentration through a porous medium, Journal of the Egyptian Mathematical Society, 23 (2015) 197–207.
  

  Citation
  Rahul Mehta, H. R. Kataria, "Magnetic Field a Heat Generation Effects on Second Grade Fluid Flow past an Oscillating Vertical Plate in Porous Medium," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.1-8, 2020

• Open Access   Article

  Numerical analysis of nonlinear radiation, viscous and Ohmic dissipation effects on steady Magnetohydrodynamic forced convection flow over a shrinking surface with internal heat generation/absorption

  J. Wilfred Samuel Raj, S.P. Anjali Devi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.9-6, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  An analysis has been carried out to investigate nonlinear radiation effects on MHD boundary layer flow over a shrinking surface with internal heat generation/absorption, viscous and Ohmic dissipation. The governing nonlinear partial differential equations are reduced to nonlinear ordinary differential equations using similarity transformations. In this analysis, Nachtsheim Swigert shooting iteration scheme coupled with a fourth order Runge Kutta integration algorithm is adopted to solve the nonlinear boundary value problem. The effects of Magnetic parameter, Suction parameter, Prandtl number, Heat generation/absorption parameter, Eckert number, Radiation parameter and Temperature ratio parameter on dimensionless velocity, skin friction coefficient, dimensionless temperature and dimensionless rate of heat transfer are discussed. It is apparent from the investigation that the increasing effect of Temperature ratio parameter is to enhance the thermal boundary layer thickness and reduce the dimensionless rate of heat transfer.

  Key-Words / Index Term
  Boundary layer flow, Kummer’s function, Magnetic field, Thermal radiation

  References
  [1] R. Viskanta, R.J. Grosh, “Boundary layer in thermal radiation absorbing and emitting media”, International Journal of Heat and Mass Transfer, 5, pp. 795–806, (1962).
  [2] M. Ali, T. Chen, B. Armaly, “Natural convection - Radiation interaction in boundary layer flow over horizontal surfaces”, AIAA / ASME 3rd Joint Thermophysics, Fluids, Plasma and Heat Transfer Conference, (1982).
  [3] M.A. Hossain, H.S. Takhar, “Radiation effect on mixed convection along a vertical plate with uniform surface temperature”, Heat Mass Transfer, 37, pp. 329-331, (2001).
  [4] E.M.A. Elbashbeshy, “Radiation effect on heat transfer over a stretching surface”, Canadian Journal of Physics, 78, pp. 1107-1112, (2000).
  [5] Ahmed Y. Ghaly, M.E. Elsayed, Elbarbary, “Radiation effect on MHD free-convection flow of a gas at a stretching surface with a uniform free stream”, Journal of Applied Mathematics, 2, pp.93-103, (2002).
  [6] Mahmoud E.M. Ouaf, “Exact solution of thermal radiation on MHD flow over a stretching porous sheet”, Applied Mathematics and Computation, 170, pp. 1117–1125, (2005).
  [7] T. Hayat, Z. Abbas, S.M.M. EL-Kabeir, M.A. EL-Hakiem, A.M. Rashad, “Heat transfer analysis on the MHD flow of a second grade fluid in a channel with porous medium”, Chaos,Solitons & Fractals, 38, pp. 556-567, (2008).
  [8] T. Hayat, R. Sajjd, Z. Abbas, M. Sajid, Awatif A. Hendi, “Radiation effects on MHD flow of Maxwell fluid in a channel with porous medium”, International Journal of Heat and Mass Transfer, 54, pp. 854-862, (2011).
  [9] S. Mukhopadhyay, K. Bhattacharyya, G.C. Layek, “Steady boundary layer flow and heat transfer over a porous moving plate in presence of thermal radiation”, International Journal of Heat and Mass Transfer,54, pp. 2751-2757, (2011).
  [10] V.M. Soundalgekar, Ioan Pop, “Viscous dissipation effects on unsteady free convective flow past an infinite vertical porous plate with variable suction”, International Journal of Heat and Mass Transfer, 17, pp. 85-92, (1974).
  [11] M.A. Hossain, “Viscous and Joules heating effects on MHD free convection flow with variable plate temperature”, International Journal of Heat and Mass Transfer, 35, pp. 3485-3487, (1992).
  [12] S.K. Khan, M.S. Abel, R.M. Sonath, “Viscoelastic MHD flow heat and mass transfer over a porous stretching sheet dissipation of energy and stress work”, Heat Mass Transfer, 40, pp.47-57, (2003).
  [13] R. Cortell, “Suction, viscous dissipation and thermal radiation effects on the flow and heat transfer of a power-law fluid past an infinite porous plate”, Chemical Engineering Research and Design, 89, pp.85- 93, (2011).
  [14] M. Fathizadeh, M. Madani, Y. Khan, N., Faraz, A. Yildrun, S. Tutkum, “An effective modification of the homotopy perturbation method for MHD viscous flow over a stretching sheet”, Journal of King Saud University, 25, pp.107-113, (2013).
  [15] M. Miklavcic, C.Y. Wang, “Viscous flow due to a shrinking sheet”, Quarterly of Applied Mathematics, 64, pp.283-290, (2006).
  [16] T. Fang, “Boundary layer flow over a shrinking sheet with power law velocity”. International Journal of Heat Mass Transfer, 51, pp.5838-5843, (2008).
  [17] T. Fang, J. Zhang, “Closed-form exact solutions of MHD viscous flow over a shrinking sheet”, Communication in Nonlinear Science and Numerical Simulation, 14, pp.2853-2857, (2009).
  [18] T. Fang, J. Zhang, “Thermal boundary layers over a shrinking sheet: an analytical solution”, Acta Mechanica, 209, pp.325-343, (2010).
  [19] K. Bhattacharya, “Effects of heat source/sink on MHD flow and heat transfer over a shrinking sheet with mass suction”, Chemical Engineering Research Bulletin, 15, pp.12-17, (2011).
  [20] K. Bhattachaarya, “Boundary layer flow and heat transfer over an exponentially shrinking sheet”, Chinese Physics Letter 28, pp.1-4, (2011).
  [21] M. Ashraf, S. Ahmad, “Radiation effects on MHD boundary layer stagnation point flow towards a heated shrinking sheet”, World Applied Science Journal, 13, pp. 1748-1756, (2011).
  [22] K. Bhattachaarya, G.C. Layek, “Effects of suction/blowing on steady boundary layer stagnation point flow and heat transfer towards a shrinking sheet with thermal radiation”, International Journal of Heat and Mass Transfer, 54, pp. 302-307, (2011).
  [23] K. Bhattacharyya, S. Mukhopadhyay, G.C. Layek, I. Pop, “Effect of thermal radiation on micropolar fluid flow and heat transfer over a porous shrinking sheet”, International Journal of Heat and Mass Transfer, 55, pp. 2945-2952, (2012).
  [24] C. Midya, “Study of boundary layer flow and heat transfer over an axisymmetric shrinking sheet with suction”, Journal of Global Research in Mathematical Archives, 1, pp. 70-77, (2013).
  [25] A.M. Rohini, S. Ahmad, I. Pop, “Flow and heat transfer at a stagnation point over an exponentially shrinking vertical sheet with suction”, International Journal of Thermal Sciences, 75, pp. 164-170, (2014).
  [26] S.P. Anjali Devi, J. Wilfred Samuel Raj, “Nonlinear radiation effects on hydromagnetic boundary layer flow and heat transfer over a shrinking surface”, Journal of Applied Fluid Mechanics, 8, pp. 613-621, (2015).
  [27] Md. Sharif Uddina, K. Bhattacharyya, “Thermal boundary layer in stagnation-point flow past a permeable shrinking sheet with variable surface temperature”, Propulsion and Power Research, 6, pp. 186–194, (2017).
  [28] N.S. Ismail, N.M. Arifin, R. Nazar, N. Bachok, “Stability analysis of unsteady MHD stagnation point flow and heat transfer over a shrinking sheet in the presence of viscous dissipation”, Chinese Journal of Physics, 57, pp. 116-126, (2019).
  [29] Chakrabarti, A.S. Gupta, “Hydromagnetic flow and heat transfer over a stretching sheet”, Quart. Appl. Math., 37, pp. 73-78, (1979).
  [30] M.Q. Brewster, Thermal Radiative Transfer Properties, Wiley, Newyork, (1972).
  

  Citation
  J. Wilfred Samuel Raj, S.P. Anjali Devi, "Numerical analysis of nonlinear radiation, viscous and Ohmic dissipation effects on steady Magnetohydrodynamic forced convection flow over a shrinking surface with internal heat generation/absorption," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.9-6, 2020

• Open Access   Article

  Covariate Measurement Error in a Non-Linear Model for Longitudinal Data

  Rosa C. S. Oliveira

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.17-33, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this paper, we study nonlinear generalized regression models for the analysis of longitudinal data. We indicate a two-stage approach that, in the first stage models a linear model to the longitudinal data to estimate the random effect that characterizes the trajectory for each individual and in the second stage uses the estimated intercept and slope as predictors for the covariates used in a probit model. We show that the straight use of the estimates in the probit model produces biased estimates of their outcome. We show how to adapt a Regression Calibration and a Pseudo-Likelihood approach to this context and compare these approaches with a naive analysis where the estimation error is ignored. Regression Calibration and Pseudo-Likelihood methods seem to be the best choice as they perform well, even for small sample size, if data is not very noisy. It is fair to say, mainly, that the Regression Calibration and Pseudo-Likelihood method perform equally well, nonetheless, Regression Calibration seems to perform a bit better. Nevertheless, naive approach has smaller Mean Squared Error, has a great absolute bias. Regression Calibration seems to be the best. Our study indicates that correcting for measurement errors instead of falsely assume that errors are not present will produce less bias than ignoring exposure measurement error in the analysis.

  Key-Words / Index Term
  measurement error, errors-in-variables, general nonlinear model, longitudinal data

  References
  [1]. Póvoa P. C-reactive protein: a valuable marker of sepsis.. Intensive Care Medicine 2002; 28 (3): 235–243.
  [2]. Tosteson JP, Demidenko E. Covariate measurement error and the estimation of random effect parameters in a mixed modelfor longitudinal data. Statistics in Medicina 1998; 17: 1959–1971.
  [3]. Carroll R, Ruppert D, Stefanski L, Crainiceanu C. Measurement Error in Nonlinear Models: A Modern Perspective. Chapman and Hall . 2006.
  [4]. Rosner B, Spiegelman D, Willett WC. Correction of logistic regression relative risk estimates and confidence intervals formeasurement error: the case of multiple covariates measured with error. American Journal of Epidemiology 1990; 132:734–745.
  [5]. W. SD, . PK. Likelihood analysis for errors-in-variables regression with replicate measurements.. Biometrika 1996; 83:813–824.
  [6]. Carroll RJ, Küchenhoff H, Lombard F, Stefanski LA. Asymptotics for the SIMEX estimator in nonlinear measurement errormodels. Journal of the American Statistical Association, 1996; 91: 242–250.
  [7]. R.C. B, R.A. B, F.B. C, R.P. D, A.M. F, W.J. KWSRS. Definitions for sepsis and organ failure and guidelines for the use ofinnovative therapies in sepsis.. In: . 101. ; 1992: 1644-1655.
  [8]. Póvoa P, M. TPA, Carneiro AH, SACiUCI. PCASSG. C-reactive protein, an early marker of community-acquired sepsis resolution: a multi-center prospective observational study. Critical Care 2011; 15 (4).
  [9]. Lawrence Gould CMIJQGMSBF. Joint modeling of survival and longitudinal non-survival data: current methods and issues.Report of the DIA Bayesian joint modeling working group.. Stat Med. 2015; 34(14): 2181-95.
  [10]. McCulloch CE, Searle SR. Generalized, Linear and Mixed Models. Wiley InterScience . 2001.
  [11]. Akaike H. Information Theory and an Extension of The Maximum Likelihood Principle. International Symposium on Information Theory 1973; 2: 267–281.
  [12]. Erling BA. Asymptotic Properties of Conditional Maximum-Likelihood Estimators. Journal of the Royal Statistical Society, Series B 1971; 32: 283–301.
  [13]. Breslow NE, Holubkov R.Weighted likelihood, pseudo-likelihood and maximum likelihood methods for logistic regression analysis of two-stage data. Statistics in Medicine 1997; 16: 103–116.
  [14]. Carroll RJ, Spiegelman CH, Gordon Lan KK, Bailey KT, Abbott RD. On errors-in-variables for binary regression models. Biometrika 1984; 71: 19–25.
  [15]. Carroll RJ, Küchenhoff H, Lombard F, Stefanski LA. Asymptotics for the SIMEX estimator in nonlinear measurement error models. Journal of the American Statistical Association, 1996; 91: 242–250.
  [16]. Chen B, Yi GY, Cook RJ.Weighted Generalized Estimating Functions for Longitudinal Response and Covariate Data That Are Missing at Random. Journal of the American Statistical Association 2010; 105: 336–353.
  [17]. J. G. C-reactive protein: risk factor, biomarker and/or therapeutic target–. Canadian Journal of Cardiology 2010; 26, Suppl A: 41A–44A.
  [18]. Huang YJ,WangCY. Consistent functional methods for logistic regression with errors in covariates. Journal of the American Statistical Association 2001; 96: 1469–1482.
  [19]. Ma Y, Tsiatis AA. Closed form semiparametric estimators for measurement error models. Stat. Sin. 2006; 16: 183–193.
  [20]. Chen MH, Ibrahim JG, Shao QM. Propriety of the Posterior Distribution and Existence of the MLE for Regression Models with Covariates Missing at Random. Journal of the American Statistical Association 2004; 99: 421–438.
  [21]. Nugent W, Graycheck L, Basham R. A Devil Hidden in the Details: The Effects of Measurement Error in Regression Analysis. Journal of Social Service Research 2000; 27: 53–75.
  [22]. Cox C. Nonlinear quasi-likelihood models: applications to continuous proportions. Computational Statistics and Data Analysis 1996; 21: 449–461.
  [23]. Cox DR. Partial likelihood. Biometrika 1975; 62: 269–276.
  [24]. Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data, 2nd edition. Oxford, U.K.: Oxford University Press. . 2002.
  [25]. Fitzmaurice GM, Laird NM, Ware JH. Applied longitudinal analysis. Wiley-Interscience . 2004.
  [26]. Fitzmaurice GM, Laird NM, Zahner GEP. Multivariate Logistic Models for Incomplete Binary Responses. Journal of the American Statistical Association 1996; 91: 99–108.
  [27]. Fitzmaurice GM, Lipsitz SR, Molenberghs G, Ibrahim JG. A Protective Estimator for Longitudinal Binary Data Subject to Non-Ignorable Non-Monotone Missingness. Journal of the Royal Statistical Society. Series A 2005; 168: 723–735.
  [28]. Fitzmaurice GM, Molenberghs G, Lipsitz SR. Regression Models for Longitudinal Binary Responses with Informative Drop-Outs. Journal of the Royal Statistical Society. Series B 1995; 57: 691–704.
  [29]. Fuller W. Measurement Error Models. John Wiley and Sons . 1987.
  [30]. Laird NM, Ware JH. Random effects models for longitudinal data. Biometrics 1982; 38: 963–974.
  [31]. Li Y, Lin X. Covariate Measurement Errors in Frailty Models for Clustered Survival Data. Biometrika 2000; 87 (4): 849–866.
  [32]. Chaganty NR, Joe H. Efficiency of Generalized Estimating Equations for Binary Responses. Journal of the Royal Statistical Society, Series B 2004; 66: 851–860.
  [33]. Cook JR, Stefanski L. Simulation-extrapolation estimation in parametric measurement error models. Journal of the American Statistical Association, 1994; 89: 1314–1328.
  [34]. W. SD. Semiparametric maximum likelihood for measurement error regression.. Biometrika 2001; 57: 53–61.
  [35]. Stefanski LA, Buzas JS. Instrumental variable estimation in binary regression measurement error models. Journal of the American Statistical Association 1995; 90: 541–550.
  [36]. Stefanski LA, Carroll R. Covariate measurement error in logistic regression. Annals of Statistics 1985; 13: 1335–1351.
  [37]. Stefanski LA, Carroll R. Conditional scores and optimal scores for generalized linear measurement-error models. Biometrika 1987; 74: 703–716.
  

  Citation
  Rosa C. S. Oliveira, "Covariate Measurement Error in a Non-Linear Model for Longitudinal Data," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.17-33, 2020

• Open Access   Article

  A Mathematical Model for Co-infection of HPV and HSV-II with Drug Resistance Compartment

  E.D. Gurmu, B.K. Bole, P.R. Koya

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.34-46, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this paper, we proposed and analysed a compartmental HPV-HSV-II coinfection model describing the transmission dynamics of HPV and HSV-II using the stability theory of differential equations. The well possedness of the formulated model equations was proved and the equilibrium points of the model have been identified. In addition, the basic reproductive number that governs the disease transmission was obtained from the largest eigenvalue of the next-generation matrix. Both local and global stability of the disease free equilibrium and endemic equilibrium point of the model equation was established using basic reproduction number. Our model revealed that the disease-free equilibrium of the HPV only model, HSV-II only model and HPV-HSV-II co-infection is locally asymptotically stable when the basic reproduction number is less than one. The endemic states are considered to exist when the basic reproduction number for each model is greater than one. We found from the analysis of the impact of HSV-II on HPV that HPV infection increases the risk of HSV-II similarly; HSV-II infection increases the risk for HPV. Sensitivity analysis was carried out on the model parameters in order to determine their impact on the disease dynamics. Numerical simulations of the model equations was carried out using the software DE Discover 2.6.4 and MATLAB R2015b with ODE45 solver and the results are displayed graphically and discussed. We simulate the HPV-HSV-II coinfection model by varying the force of infection to see its effects on infected HPV population, infected HSV-II population, HPV-HSV-II coinfection population and HSV-II- Cervical Cancer coinfected population. The result illustrated that, as force of infection increases or decreases, the infections increases or decreases.

  Key-Words / Index Term
  Co-infection, HPV, HSV-II, Cervical Cancer, Stability, Reproduction number.

  References
  [1] Naveed Shahzad, Roman Farooq, Bilal Aslam and Muhammad Umer, “Microbicides for the Prevention of HPV, HIV-1, and HSV-2: Sexually Transmitted Viral Infections”, Intech Open, and DOI: 10.5772/intechopen.68927(September 6th 2017).
  [2] Eshetu Dadi Gurmu, Purnachandra Rao Koya . “Sensitivity Analysis and Modeling the Impact of Screening on the Transmission Dynamics of Human Papilloma Virus (HPV)”. American Journal of Applied Mathematics. 7(3), 70-79, (2019)..
  [3] Rahman, SM Ashrafur, “Study of Infectious Diseases by Mathematical Models: Predictions and Controls”, Electronic Thesis and Dissertation Repository. 3487, (2016).
  [4] A.S. Bergot, A. Kassianos, I.A. Frazer, D Mittal, “New Approaches to immunotherapy for HPV associated cancers”, OPEN ACESS, Cancers, 346-3495, (2011).
  [5] D.R. Lowy and J.T. Schiller, “Prophylactic human papillomavirus vaccines”, J C Invest. 116(5), 1167-1173, (2006).
  [6] WHO list of priority medical devices for cancer management, Geneva: World Health Organization. Licence: CC BY-NC-SA 3.0 IGO, (2017).
  [7] Murtaza Mustafa, EM. Illzam, RK. Muniandy AM. Sharifah, MK. Nang, B. Ramesh, “Herpes simplex virus infections, Pathophysiology and Management”, Journal of Dental and Medical Sciences (IOSR-JDMS).15(7), 85-91, (2016).
  [8] Rajul Patel, Oliver J Kennedy, Emily Clarke, Anna Geretti, Arvid Nilsen, Stephan Lautenschlager, John Green, Gilbert Donders,Willem van der Meijden, Mikhail Gomberg, Harald Moi, and Elizabeth Foley, “European guidelines for the management of genital herpes”. International Journal of STD & AIDS. 0(0), 1–14, (2017).
  [9] World Health Organization, “WHO Guidelines for the Treatment of Genital Herpes Simplex Virus” (2016).
  [10] H.W. Hethcote, “Qualitative analyses of communicable disease models”, Mathematical Biosciences, vol. 28, no. 3-4, pp. 335–356, (1976).
  [11] N. C. Grassly and C. Fraser, “Mathematical models of infectious disease transmission”, Nature Reviews Microbiology, vol. 6, no.6, pp. 477–487, (2008).
  [12] Udoy S. Basak, Jannatun Nayeem, Chandra N. Podder, “Mathematical study of HIV and HSV-II co-infec tion”, American Journal of Mathematics and Statistics, 5(1), 15-23, (2011).
  [13] A. Mhlanga, C. P. Bhunu, and S. Mushayabasa, “A Computational Study of HSV-2 with Poor Treatment Adherence”, Hindawi Publishing Corporation Abstract and Applied Analysis, Article ID 850670, 15, (2015).
  [14] G. G. Sanga, O. D. Makinde, E.S. Massawe and L. Namkinga, “Modelling Co-dynamics of Cervical Cancer and HIV Disease”, Glob.J.Appl.Math. 13(6), 2057-2078, (2017).
  [15] Maja Hühns, Georg Simm, Andreas Erbersdobler, and Annette Zimpfer, “HPV Infection, but Not EBV or HHV-8 Infection, Is Associated with Salivary Gland Tumours”, BioMed Research International, Article ID 829349, 7, (2015).
  [16] P. V. D. Driessche, J.Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Math. Biosci, 180,29–48, (2002).
  [17] C. Castillo-Chavez, B. Song, “Dynamical models of tuberculosis and their applications”, Math. Biosci. Eng. 1 (2) ,361–404, (2004).
  [18] Eshetu Dadi Gurmu, Boka Kumsa Bole, Purnachandra Rao Koya, “Mathematical Modelling of HIV/AIDS Transmission Dynamics with Drug Resistance Compartment”, American Journal of Applied Mathematics. 8, (1), pp. 34-45, (2020).
  [19] Eshetu Dadi Gurmu and Purnachandra Rao Koya, “Impact of Chemotherapy treatment of SITR Compartmentalization and Modeling of Human Papilloma Virus (HPV)”, Journal of Mathematics (IOSR – JM). 15(3), 17 – 29, (2019).
  

  Citation
  E.D. Gurmu, B.K. Bole, P.R. Koya, "A Mathematical Model for Co-infection of HPV and HSV-II with Drug Resistance Compartment," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.34-46, 2020

• Open Access   Article

  Development of Homotopy Perturbation Method for Solving Nonlinear Algebraic Equations

  Bachir. N. Kharrat, George. A. Toma

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.47-50, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this paper, we proposed expanding the application of the He`s homotopy perturbation method to solve nonlinear algebraic equations using the first seven terms of Taylor`s series. The main objective of our research is to find an approximate solution with high accuracy for solving non-linear algebraic equations. In the proposed hybrid scheme, we combined the homotopy perturbation method with the Taylor`s series for an analytical function, by truncating the first seven terms of Taylor`s series and then apply the homotopy perturbation method. We demonstrated the efficacy of the proposed method by finding an approximate solution with high accuracy. This solution is the intersection of the graph with the horizontal axis. We also found a new iterative formula that gives, in every iteration, an approximate solution that converges from the exact solution faster and with small error. Numerical examples are given to show the efficiency and accuracy of the proposed iterative scheme.

  Key-Words / Index Term
  Homotopy Perturbation Method, Iterative Scheme, Nonlinear Algebraic Equations, Taylor`s Series

  References
  [1] M. Javidi, A. Golbabi, “Modified homotopy perturbation method for solving non-integral equation,” Choas, Solitons and Fractals,Vol. 40, pp. 1408-1412, 2009.
  [2] J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” Int. J. Non-Linear Mechanics,Vol. 35, No. 1, pp. 37-43, 2000.
  [3] B. N. Kharrat, G. Toma, “A New Hybrid Sumudu Transform With Homotopy Perturbation Method For Solving Boundary Value Problems,” Middle-East Journal of Scientific Research, Vol. 28, No. 2, pp.142-149, 2020.
  [4] B. N. Kharrat, G. Toma, “Modified Homotopy Perturbation Method by Using Sumudu Transform for Solving Initial Value Problems Represented By System of Nonlinear Partial Differential Equations,” World Applied Sciences Journal, Vol. 36, No. 7, pp.844-849, 2018.
  [5] T. Liu, X. Qin, Q. LiB, “An optimal fourth-order family of modified Cauchy methods for finding solutions of nonlinear equations and their dynamical,” Open Math, Vol. 17, pp.1567-1589, 2019.
  [6] S. Abbasband, “Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition,” Applied Mathematics and Computation, Vol. 145, pp.887-893, 2003.
  [7] P.K. Bera. S.K. Das, P. Bera, “Applications of the Aboodh Transform and the Homotopy Perturbation Method to the Nonlinear Oscillators,” International Journal of Computer Sciences and Engineering , Vol. 6, pp.2347-2693, 2018.
  [8] P.K. Bera, S.K. Das, P. Bera, “A Study of Nonlinear Vibration of Euler-Bernoulli Beams Using Coupling Between The Aboodh Transform And The Homotopy Perturbation Method,” International Journal of Computer Sciences and Engineering , Vol. 5, pp.2347-2693, 2017.
  

  Citation
  Bachir. N. Kharrat, George. A. Toma, "Development of Homotopy Perturbation Method for Solving Nonlinear Algebraic Equations," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.47-50, 2020

• Open Access   Article

  BTIB Designs through General Efficiency Balanced Row-Column Designs and their Optimalities

  M.S. Prajapati, J.V. Jayswal

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.51-54, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this article, we propose some methods of construction of rectangular-type and step-type balanced treatment incomplete block designs via general efficiency balanced row-column designs, and examine their A- and MV- optimalities.

  Key-Words / Index Term
  A-optimality, MV-optimality, General efficiency balanced row-column (GEBRC) design, Rectangular (R-) type, Step (S-)type, Balanced treatment incomplete block (BTIB)design.

  References
  [1]. BECHHOFER, R.E. and TAMHAANE , A.C. Incomplete block design for comparing treatments with a control : General theory, Technometrics,Vol.23, pp.45-57,1981,Corrigendum, Vol.24, 71, 1982.
  [2]. CALINSKI, T. On the notation of balance in block designs. Recent development in statistics (J.R.Barra et. al. ed.) North Holland, pp. 365-374, 1977.
  [3]. DAS, M.N. and GHOSH, D.K. Balance incomplete block designs. Sankhya ̅ B,Vol.47, Issue.1, pp.67-77, 1985.
  [4]. GUPTA, S. A note on the notion of balance in block designs. Calcutta Statist.Assoc.Bull. , Vol.39, pp.85-89, 1987.
  [5]. GUPTA, V.K. and PRASAD GANDHI, N.S. (1990) On general efficiency balanced row-column designs. Calcutta statist. Assoc. Bull. Vol.39, pp.173-183, 1990.
  [6]. HEDAYAT, A.S. and MAJUMDAR, D. A-optimal incomplete block designs for control-test treatment comparisons, Technometrics, Vol.26, pp.363-370, 1984.
  [7]. HEDAYAT, A.S. and MAJUMDAR, D. Model robust optimal designs for comparing test treatments with a control, J.statist.plann, inference, Vol.18, pp.25-33, 1988.
  [8]. KAGEYAMA, S. AND MUKHERJEE, R. General balanced designs through reinforcement. Sankhya ̅ B, Vol.48, Issue.3, pp.380-387, 1986.
  [9]. MAJUMDAR, D. and NOTZ, W. (1983). Optimal incomplete designs for comparing test treatments with a control, Ann. Statist., Vol.11, pp.258-266, 1983.
  [10]. NOTZ, W. AND TAMHANE, A.C. Incomplete block designs for comparing treatments with a control: Minimal complete sets of generator designs for k=3, p=3 (1), Comm.Statist. A-12, pp.1391-1412, 1983.
  

  Citation
  M.S. Prajapati, J.V. Jayswal, "BTIB Designs through General Efficiency Balanced Row-Column Designs and their Optimalities," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.51-54, 2020

• Open Access   Article

  Correction when using Estimates in Logistic Regression for Longitudinal Data: An example of sepsis and C-reactive protein

  Rosa C. S. Oliveira

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.55-64, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  A usual aim in longitudinal studies is to characterize the relationship between a dichotomous outcome and both time-independent and time-dependent covariates. In this paper, we study nonlinear generalized mixed-effects regression models for analysis of longitudinal data. We use a two-stage approach that first fits a linear model to the longitudinal data estimating the random effect that describes the trend for each method and then uses the estimated intercept and slope as predictors for the covariates used in a probit model. We show how to adapt a regression calibration and the best linear unbiased obtained from a linear mixed model approaches to this context and compare these approaches with a naive analysis where the estimation error is ignored. The methods are applied to health data collected during a study designed to evaluate the epidemiology of community-acquired sepsis in a larger cohort of infected intensive care unit patients. We show that the straight use of the estimates in the probit model produces biased estimates of their outcome. Nonetheless, regression calibration and linear mixed effect offer little or no advantage when sample sizes are small, they perform best when samples are reasonably large and especially when the error of prediction (measurement error) or the effects are not small. Our study indicates that the naive approach produces weak results and that regression calibration and linear mixed effect method provides a way to obtain unbiased estimators, especially when is correction advocated, practically indistinguishable.

  Key-Words / Index Term
  error of prediction, general nonlinear model, longitudinal data

  References
  [1]. Póvoa P, M. TPA, Carneiro AH, SACiUCI. PCASSG. C-reactive protein, an early marker of community-acquired sepsisresolution: a multi-center prospective observational study. Critical Care 2011; 15 (4).
  [2]. Oliveira R, Teixeira-Pinto A. in press. Lets see 2019; 0: 1959–1971.
  [3]. Wang N, Lin X, Gutierrez RG, Carroll RJ. Bias Analysis and SIMEX Approach in Generalized Linear Mixed Measurement Error Models. Journal of the American Statistical Association 1998; 93(441): 249-261. doi: 10.1080/01621459.1998.10474106
  [4]. Fuller W. Measurement Error Models. John Wiley and Sons . 1987.
  [5]. Póvoa P. C-reactive protein: a valuable marker of sepsis.. Intensive Care Medicine 2002; 28 (3): 235–243.
  [6]. R.C. B, R.A. B, F.B. C, R.P. D, A.M. F, W.J. KWSRS. Definitions for sepsis and organ failure and guidelines for the use of innovative therapies in sepsis.. In: . 101. ; 1992: 1644-1655.
  [7]. McCulloch CE, Searle SR. Generalized, Linear and Mixed Models. Wiley InterScience . 2001.
  [8]. Akaike H. Information Theory and an Extension of The Maximum Likelihood Principle. International Symposium on Information Theory 1973; 2: 267–281.
  [9]. Erling BA. Asymptotic Properties of Conditional Maximum-Likelihood Estimators. Journal of the Royal Statistical Society, Series B 1971; 32: 283–301.
  [10]. Armstrong B. Measurement error in the generalized linear model. Commun Stat B Simula Computa 1985; 14: 529–544.
  [11]. Armstrong B, Whittemore AS, Howe GR. Analysis of case-control data with covariate measurement error: application to diet and colon cancer. Stat Med 1989; 8: 1151–1163.
  [12]. Breslow NE, Holubkov R.Weighted likelihood, pseudo-likelihood and maximum likelihood methods for logistic regression analysis of two-stage data. Statistics in Medicine 1997; 16: 103–116.
  [13]. Carroll R, Ruppert D, Stefanski L, Crainiceanu C. Measurement Error in Nonlinear Models: A Modern Perspective. Chapman and Hall . 2006.
  [14]. Carroll RJ, Spiegelman CH, Gordon Lan KK, Bailey KT, Abbott RD. On errors-in-variables for binary regression models. Biometrika 1984; 71: 19–25.
  [15]. Carroll RJ, Küchenhoff H, Lombard F, Stefanski LA. Asymptotics for the SIMEX estimator in nonlinear measurement error models. Journal of the American Statistical Association, 1996; 91: 242–250.
  [16]. Carroll RJ, Küchenhoff H, Lombard F, Stefanski LA. Asymptotics for the SIMEX estimator in nonlinear measurement error models. Journal of the American Statistical Association, 1996; 91: 242–250.
  [17]. Chen B, Yi GY, Cook RJ.Weighted Generalized Estimating Functions for Longitudinal Response and Covariate Data That Are Missing at Random. Journal of the American Statistical Association 2010; 105: 336–353.
  [18]. Diggle PJ, Liang KY, Zeger SL. The Analysis of Longitudinal Data. Clarendon Press, Oxford . 1993.
  [19]. J. G. C-reactive protein: risk factor, biomarker and/or therapeutic target–. Canadian Journal of Cardiology 2010; 26, SupplA: 41A–44A.
  [20]. Huang YJ,WangCY. Consistent functional methods for logistic regression with errors in covariates. Journal of the American Statistical Association 2001; 96: 1469–1482.
  [21]. Ma Y, Tsiatis AA. Closed form semiparametric estimators for measurement error models. Stat. Sin. 2006; 16: 183–193.
  [22]. Chen MH, Ibrahim JG, Shao QM. Propriety of the Posterior Distribution and Existence of the MLE for Regression Models with Covariates Missing at Random. Journal of the American Statistical Association 2004; 99: 421–438.
  [23]. Nugent W, Graycheck L, Basham R. A Devil Hidden in the Details: The Effects of Measurement Error in Regression Analysis. Journal of Social Service Research 2000; 27: 53–75.
  [24]. Cox C. Nonlinear quasi-likelihood models: applications to continuous proportions. Computational Statistics and Data Analysis 1996; 21: 449–461.
  [25]. Cox DR. Partial likelihood. Biometrika 1975; 62: 269–276.
  [26]. Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data, 2nd edition. Oxford, U.K.: Oxford University Press. . 2002.
  [27]. Fitzmaurice GM, Laird NM, Ware JH. Applied longitudinal analysis. Wiley-Interscience . 2004.
  [28]. Fitzmaurice GM, Laird NM, Zahner GEP. Multivariate Logistic Models for Incomplete Binary Responses. Journal of the American Statistical Association 1996; 91: 99–108.
  [29]. Fitzmaurice GM, Lipsitz SR, Molenberghs G, Ibrahim JG. A Protective Estimator for Longitudinal Binary Data Subject toNon-Ignorable Non-Monotone Missingness. Journal of the Royal Statistical Society. Series A 2005; 168: 723–735.
  [30]. Fitzmaurice GM, Molenberghs G, Lipsitz SR. Regression Models for Longitudinal Binary Responses with Informative Drop-Outs. Journal of the Royal Statistical Society. Series B 1995; 57: 691–704.
  [31]. Laird NM, Ware JH. Random effects models for longitudinal data. Biometrics 1982; 38: 963–974.
  [32]. Li Y, Lin X. Covariate Measurement Errors in Frailty Models for Clustered Survival Data. Biometrika 2000; 87 (4): 849–866.
  [33]. Chaganty NR, Joe H. Efficiency of Generalized Estimating Equations for Binary Responses. Journal of the Royal Statistical Society, Series B 2004; 66: 851–860.
  [34]. Cook JR, Stefanski L. Simulation-extrapolation estimation in parametric measurement error models. Journal of the American Statistical Association, 1994; 89: 1314–1328.
  [35]. Rosner B, Spiegelman D, Willett WC. Correction of logistic regression relative risk estimates and confidence intervals for measurement error: the case of multiple covariates measured with error. American Journal of Epidemiology 1990; 132:
  [36]. W. SD. Semiparametric maximum likelihood for measurement error regression.. Biometrika 2001; 57: 53–61.
  [37]. Stefanski LA, Buzas JS. Instrumental variable estimation in binary regression measurement error models. Journal of the American Statistical Association 1995; 90: 541–550.
  [38]. Stefanski LA, Carroll R. Covariate measurement error in logistic regression. Annals of Statistics 1985; 13: 1335–1351.
  [39]. Stefanski LA, Carroll R. Conditional scores and optimal scores for generalized linear measurement-error models. Biometrika 1987; 74: 703–716.
  [40]. Tosteson JP, Demidenko E. Covariate measurement error and the estimation of random effect parameters in a mixed model for longitudinal data. Statistics in Medicina 1998; 17: 1959–1971.
  

  Citation
  Rosa C. S. Oliveira, "Correction when using Estimates in Logistic Regression for Longitudinal Data: An example of sepsis and C-reactive protein," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.55-64, 2020

• Open Access   Article

  On Control and Differential Game Problems in Hilbert Space

  F. Allahabi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.65-69, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this paper we investigate a control and pursuit game problems of an infinite system of first order differential equations. This work suggest a new approach to construct the control functions of pursuers in Hilbert space. First, in the control problem, we obtain the sufficient conditions for possibility to bring the control system from the initial point to the origin in finite time. Second, in the pursuit game problem with geometric and integral constraints on the controls of players, we established the sufficient conditions for which pursuit game is possible to complete the pursuit.

  Key-Words / Index Term
  Differential game, pursuit game, control, admissibility, pursuit time

  References
  [1] O. A. Ladyzhenskaya, “Boundary-Value Problems of Matahematical Physics,” [in Russian], Nauka, Moscow, 1973.
  [2] F.L. Chernous’ko, “Bounded Controls in Systems with Distributed Parameters,” Prikl. Mat. Mekh, Vol. 56, Issue.5, pp. 810-826, 1992.
  [3] F.L. Chernous’ko, “On the construction of a bounded control in oscillatory systems,” Journal of Applied Mathematics and Mechanics, Vol.52, Issue.4, pp. 426-433, 1988.
  [4] F.L. Chernous’ko, “Decomposition and suboptimal control in dynamical systems,” Journal of Applied Mathematics and Mechanics, Vol.54, Issue.6, pp. 727-734, 1990.
  [5] N. Yu. Satimov, M. Tukhtasinov, “On some game problems for rst-order controlled evolution equations,” Differential Equations, Vol.41. No.8, pp.1169-1177, 2005.
  [6] N. Yu. Satimov, M. Tukhtasinov, “Game Problems on a Fixed Interval in Controlled First-Order Evolution Equations,” Mathematical Notes, Vol.80, No.4, pp. 578-589, 2006.
  [7] G.I. Ibragimov, F. Allahabi, A. Kuchkarov “A pursuit problem in an infinite system of second-order differential equations,” Ukr. Math. J, Vol.65, No.8, pp.1203-1216, 2014
  [8] M. Tukhtasinov, M. Sh. Mamatov, “On Pursuit Problems in Controlled Distributed Systems,” Mathematical Notes, Vol.84, No.2, pp.256–262, 2008.
  [9] G.I. Ibragimov, A. Akhmedov, Puteri Nur Izzati, N. Abdul
  Manaf, “Pursuit Differential Game Described by infinite First Order 2-Systems of Differential Equations,” Malaysian Journal of Mathematical Sciences. Vol.11, Issue.2, pp.181-190, 2017.
  [10] F. Allahabi, , M. Abdulkawi Mahiub “A Problem of Pursuit Game with Various Constraints on Controls of Players,” International Journal of Partial Differential Equations and Applications, Vol.6, No.1, pp.13-17, 2019.
  [11] Arti Kaushik, “Some Problems of Non Homogeneous Differential Equation of First Order and First Degree, Solved By the Method of Homogeneous Differential Equation of First Order and First Degree,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, No.14, pp.40-43, 2019.
  [12] Arti Kaushik, “Numerical Study of a Steady State Two Dimensions Heat equation using TDMA Technique,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.4, No.1, pp.6-11, 2017.
  

  Citation
  F. Allahabi, "On Control and Differential Game Problems in Hilbert Space," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.65-69, 2020

• Open Access   Article

  Topological Indices of some Nanostructures

  M. Bhanumathi and M. Thusleem Furjana

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.70-75, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  Let G be a connected (p, q) graph and let T(G) be the total graph of a graph G. In this paper, we define three new topological indices, first eccentricity T-index, second eccentricity T-index and third eccentricity T-index using Total graph of a graph. Also, we evaluate the value these indices for some nanostructures and for cycles and complete graphs.

  Key-Words / Index Term
  Topological indices, Nanocones, Total graph

  References
  [1] R. Balakrishnan and K. Ranganathan, “A Text book of Graph Theory”, Springer verlag, Newyork, 2012.
  [2] M.Ghorhani, Mohammad A. Hosseinzadeh, “A new version of Zagreb indices ”, Filomat 26:1,pp. 93-100, 2012.
  [3] I.Gutman, N.Trinajstic, Graph Theory and Molecular orbitals. “Total II-electron energy of alternant hydrocarbons”, chems. phys. lett.,17, pp. 535-538, 1972.
  [4] F. Harary, “Graph Theory”, Narosa Publication House, 2001.
  [5] Keerthi G. Mirajkar, Bhagyashri R. Doddamani, “Some Topological indices of Carbon Nanocones [CNCk(n)] and Nanotori [C4C6C8(P,Q)] ”, International journal of Scientific research in mathematics and statistical sciences, volume-5, Issue -2, pp 35-39, April 2018.
  [6] Mohammad Reza Rostami, Benzenoid systems: “A computational study of two topological indices”, Optoelectronics and Advanced materials-Rapid communications, Vol. 7, No. 9-10, pp 776-780, 2013.
  [7] Rajesh Kanna M. R, Pradeep Kumar .R, Jagadeesh .R, “Computation of Topological Indices of Dutch wind mill graph”, Open journal of Discrete Mathematics, Volume 6, pp 74-81, 2016.
  

  Citation
  M. Bhanumathi and M. Thusleem Furjana, "Topological Indices of some Nanostructures," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.70-75, 2020

• Open Access   Article

  Application of TOPSIS Method for Decision Making

  R. M. Zulqarnain, M. Saeed, N. Ahmad, F. Dayan, B. Ahmad

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.2 , pp.76-81, Apr-2020

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this paper, we discuss the order preference by similarity ideal solution (TOPSIS) method with basic concepts and determine the TOPSIS algorithm. Secondly, we construct a graphical model for the TOPSIS method by using the TOPSIS algorithm. Finally, we use the developed method for decision making in our daily life. In this work, we use the TOPSIS method for the selection of a car by using hypothetical data and examined that the civic is the best automotive car according to given parameters.

  Key-Words / Index Term
  Multiple Criteria Decision Making (MCDM), TOPSIS, Positive Ideal Solution (PIS), Negative Ideal Solution (NIS)

  References
  [1] C. Hawng, K. Yoon, "Multiple Attribute Decision Making: Methods and Applications," A State of the Art Survey 1. Berlin Heidelberg Springer, 1981.
  [2] Y. Bi, D. Lai, H. Yan, “Synthetic evaluation of the effect of health promotion : Impact of a UNICEF project in 40 poor western counties of China,” R. Soc. Public Heal, vol. 124, pp. 376–391, 2010.
  [3] I. Chamodrakas, D. Martakos, “A utility-based fuzzy TOPSIS method for energy-efficient network selection in heterogeneous wireless networks,” Appl. Soft Comput. J, vol. 11, no. 4, pp. 3734–3743, 2011.
  [4] R. M. Zulqarnain, M. Saeed, B. Ali, N. Ahmad, L. Ali, S. Abdal, “Application of Interval Valued Fuzzy Soft Max-Min Decision Making Method,” International Journal of Mathematical Research, vol. 9, no. 1, pp. 11-19, 2020.
  [5] M. Behzadian, S. K. Otaghsara, M. Yazdani, J. Ignatius, “A state-of the-art survey of TOPSIS applications,” Expert Syst. Appl, Vol. 39, no. 17, pp. 13051–13069, 2012.
  [6] F. Dayan, M. Zulqarnain, H. Naseer, “A Ranking Method for Students of Different Socio Economic Backgrounds Based on Generalized Fuzzy Soft Sets,” Int. J. Sci. Res, Vol. 6, no. 9, pp. 691–694, 2017.
  [7] L. Yang, J. Deuse, “Multiple-attribute decision-making approach for an energy-efficient facility layout design,” Publ. by Elsevier B.V, vol. 3, pp. 149–154, 2012.
  [8] G. La, G. Aiello, C. Rastellini, R. Micale, L. Cicalese, “Multi-Criteria Decision Making support system for pancreatic islet transplantation,” Expert Syst. Appl, Vol. 38, no. 4, pp. 3091–3097, 2011.
  [9] T. Chen, “A signed-distance-based approach to importance assessment and multi-criteria group decision analysis based on interval type-2 fuzzy set,” Knowl Inf Syst, Vol. 35, pp. 193–231, 2013.
  [10] R. Kuo, Y. Wu, T. Hsu, “Integration of fuzzy set theory and TOPSIS into HFMEA to improve outpatient service for elderly patients in Taiwan,” J. Chinese Med. Assoc, Vol. 75, pp. 341–348, 2012.
  [11] K. Im, H. Cho, “A systematic approach for developing a new business model using morphological analysis and integrated fuzzy approach,” Expert Syst. Appl, Vol. 40, no. 11, pp. 4463–4477, 2013.
  [12] K. K. damghani, S. Sadi-nezhad, M. Tavana, “Solving multi-period project selection problems with fuzzy goal programming based on TOPSIS and a fuzzy preference relation,” Inf. Sci. (Ny), Vol. 1, 2013.
  [13] M. Nakhaeinejad, N. Nahavandi, “An interactive algorithm for multi-objective flow shop scheduling with fuzzy processing time through resolution method and TOPSIS,” Int J Adv Manuf Technol, no. July, 2012.
  [14] H. Hosseini, A. S. Milani, “An improvement of quantitative strategic planning matrix using multiple criteria decision making and fuzzy numbers,” Appl. Soft Comput. J, Vol. 12, no. 8, pp. 2246–2253, 2012.
  [15] A. Taleizadeh, S. Taghi, A. Niaki, “A hybrid method of Pareto TOPSIS and genetic algorithm to optimize multi- product multi-constraint inventory control systems with random fuzzy replenishments A Hybrid Method of Pareto , TOPSIS and Genetic Algorithm to Optimize Multi-Product Multi-Constr,” Math. Comput. Model, 2009.
  [16] L. I. Tong, C. H. Wang, H. C. Chen, “Optimization of multiple responses using principal component analysis and,” Int J Adv Manuf Techno, Vol. 27, pp. 407–414, 2005.
  [17] R. A. Krohling, V. C. Campanharo, “Fuzzy TOPSIS for group decision making: A case study for accidents with oil spill in the sea,” Expert Syst. Appl, Vol. 38, no. 4, pp. 4190–4197, 2011.
  [18] X. Wang, H. K. Chan, “A hierarchical fuzzy TOPSIS approach to assess improvement areas when implementing green supply chain initiatives,” Int. J. Prod. Res, Vol. 51, no. 10, pp. 3117–3130, 2013.
  [19] Y. Kim, E. S. Chung, S. M. Jun, S. U. Kim, “Prioritizing the best sites for treated wastewater instream use in an urban watershed using fuzzy TOPSIS,” Resour. Conserv. Recycl, Vol. 73, pp. 23–32, 2013.
  [20] P. Li, H. Qian, J. Wu, J. Chen, “Sensitivity analysis of TOPSIS method in water quality assessment: I. Sensitivity to the parameter weights,” Environ. Monit. Assess, Vol. 185, no. 3, pp. 2453–2461, 2013.
  [21] A. Awasthi, S. S. Chauhan, S. K. Goyal, “A multi-criteria decision making approach for location planning for urban distribution centers under uncertainty,” Math. Comput. Model, Vol. 53, no. 1–2, pp. 98–109, 2011.
  [22] M. J. Ostad-Ahmad-Ghorabi, M. Attari, “Advancing environmental evaluation in cement industry in Iran,” J. Clean. Prod, Vol. 41, pp. 23–30, 2013.
  [23] P. Li, J. Wu, H. Qian, “Groundwater quality assessment based on rough sets attribute reduction and TOPSIS method in a semi-arid area, China,” Env. Monit Assess, Vol. 184, no. Sep, pp. 4841–4854, 2012.
  [24] Y. Sun, Z. Liang, C. Shan, H. Viernstein, F. Unger, “Comprehensive evaluation of natural antioxidants and antioxidant potentials in Ziziphus jujuba Mill . var . spinosa ( Bunge ) Hu ex H . F . Chou fruits based on geographical origin by TOPSIS method,” Food Chem, Vol. 124, no. 4, pp. 1612–1619, 2011.
  [25] D. Yong, “Plant location selection based on fuzzy TOPSIS,” Int J Adv Manuf Technol, Vol. 28, no. Aug, pp. 839–844, 2006.
  [26] S. J. J. C, C. L. Hwang, "Fuzzy Multiple Attribute Decision Making Methods," Springer-Verlag Berlin Heidelberg, Chapter 5, 1992.
  [27] R. M. Zulqarnain, S. Abdal, B. Ali, L. Ali, F. Dayan, M. I. Ahamad, Z. Zafar, “Selection of Medical Clinic for Disease Diagnosis by Using TOPSIS Method,” Int. J. Pharm. Sci. Rev. Res, Vol. 61, no. 1, pp. 22-27, 2020.
  [28] C. T. Chen, “Extensions of the TOPSIS for group decision-making under fuzzy environment,” Fuzzy Sets Syst, Vol. 114, no. 1, pp. 1–9, 2000.
  [29] M. Anisseh, F. Piri, M. Reza, “Fuzzy extension of TOPSIS model for group decision,” Artif Intell Rev, Vol. 38, pp. 325–338, 2012.
  [30] L. Dymova, P. Sevastjanov, A. Tikhonenko, “An approach to generalization of fuzzy TOPSIS method,” Inf. Sci. (Ny), Vol. 238, no. February, pp. 149–162, 2013.
  [31] S. Rouhani, M. Ghazanfari, M. Jafari, “Evaluation model of business intelligence for enterprise systems using fuzzy TOPSIS,” Expert Syst. Appl, Vol. 39, no. 3, pp. 3764–3771, 2012.
  [32] M. Zulqarnain, F. Dayan, “Choose Best Criteria for Decision Making Via Fuzzy Topsis Method,” Math. Comput. Sci, Vol. 2, no. 6, p. 113, 2017.
  [33] M. Zulqarnain, F. Dayan, “Selection Of Best Alternative For An Automotive Company By Intuitionistic Fuzzy TOPSIS Method,” Int. J. Sci. Technol. Res, Vol. 6, no. 10, pp. 126–132, 2017.
  [34] T. Kaya, C. Kahraman, “Multicriteria decision making in energy planning using a modified fuzzy TOPSIS methodology,” Expert Syst. Appl, Vol. 38, no. 6, pp. 6577–6585, 2011.
  [35] F. Cavallaro, “Fuzzy TOPSIS approach for assessing thermal-energy storage in concentrated solar power ( CSP ) systems,” Appl. Energy, Vol. 87, no. 2, pp. 496–503, 2010.
  [36] F. Dayan, M. Zulqarnain, “On Generalized Interval Valued Fuzzy Soft Matrices,” Am. J. Math. Comput. Model, Vol. 3, no. 1, pp. 1–9, 2018.
  [37] R. M. Zulqarnain, X. L. Xin, B. Ali, S. Abdal, A. Maalik, L. Ali, M. I. Ahamad, Z. Zafar, “Disease identification using trapezoidal fuzzy numbers by Sanchez’s approach,” Int. J. Pharm. Sci. Rev. Res, Vol. 61, no 1, pp. 13-18, 2020.
  [38] G. R. Jahanshahloo, F. H. Lotfi, A. R. Davoodi, “Extension of TOPSIS for decision-making problems with interval data : Interval efficiency,” Math. Comput. Model, Vol. 49, no. 5–6, pp. 1137–1142, 2009.
  [39] L. Dymova, P. Sevastjanov, A. Tikhonenko, “A direct interval extension of TOPSIS method,” Expert Syst. Appl, no. March, 2013.
  [40] Z. Yue, “An extended TOPSIS for determining weights of decision makers with interval numbers,” Knowledge-Based Syst, Vol. 24, no. 1, pp. 146–153, 2011.
  [41] M. Zulqarnain, M. Saeed, “A New Decision Making Method on Interval Valued Fuzzy Soft Matrix (IVFSM),” Br. J. Math. Comput. Sci, Vol. 20, no. 5, pp. 1–17, 2017.
  [42] M. Zulqarnain, M. Saeed, “An Application of Interval Valued Fuzzy Soft Matrix in Decision Making Problem,” Sci. Int, Vol. 28, no. 3, pp. 2261–2264, 2016.
  [43] M. Zulqarnain, M. Saeed, M. F. Tbassum, “Comparison Between Fuzzy Soft Matrix (FSM) and Interval Valued Fuzzy Soft Matrix (IVFSM) in Decision Making,” Sci. Int, Vol. 28, no. 5, pp. 4277–4283, 2016.
  [44] M. S. Mahmoodzadeh, J. Shahrabi, M. Pariazar, “Project selection by using a fuzzy AHP and topsis technique,” World Acad. Sci. Eng. Technol, Vol. 1, no. 6, pp. 270–275, 2007.
  [45] M. Velasquez, P. T. Hester, “An Analysis of Multi-Criteria Decision Making Methods,” Int. J. Oper. Res, Vol. 10, no. 2, pp. 56–66, 2013.
  [46] D. Li, “Extension of the TOPSIS for multi-Attribute group Decision making under Atanassov IFS environments,” Int. J. Fuzzy Syst. Appl, Vol. 1, no. 4, pp. 47–61, 2011.
  [47] R. M. Zulqarnain, S. Abdal, A. Maalik, B. Ali, Z. Zafar, M. I. Ahamad, S. Younas, A. Mariam, F. Dayan., “Application of TOPSIS Method in Decision Making Via Soft Set,” Biomed. J. Sci. Tech. Res, Vol. 24, no. 3, pp. 18208–18215, 2020.
  [48] M. Sevkli, S. Zaim, A. Turkyılmaz, M. Satır, “An Application of Fuzzy Topsis Method for Supplier Selection,” IEEE Int. Conf. Fuzzy Syst, no. July, 2010.
  [49] M. Zulqarnain, F. Dayan, M. Saeed, “TOPSIS Analysis for The Prediction of Diabetes Based on General Characteristics of Humans,” Int. J. Pharm. Sci. Res, Vol. 9, no. 7, pp. 2932-2939, 2018.
  [50] S. Eraslan, “A Decision Making Method via TOPSIS on Soft Sets,” J. New Results Sci, Vol. 8, pp. 57–71, 2015.
  [51] D. Y. Cheng, K. M. Chao, C. C. Lo, C. F. Tsai, A user centric service-oriented modeling approach, Vol. 14, no. 4. 2011.
  [52] M. R. Fathi, H. Zarei Matin, M. Karimi Zarchi, S. Azizollahi, “The Application of Fuzzy TOPSIS Approach to Personnel Selection for Padir Company, Iran,” J. Manag. Res., Vol. 3, no. 2, pp. 1–14, 2011.
  [53] E. Triantaphyllou, B. Shu, S. N. Sanchez, T. Ray, “Multi-Criteria Decision Making : An Operations Research Approach,” Encycl. Electr. Electron. Eng, Vol. 15, no. February, pp. 175–186, 1998.
  [54] W. Sa, “The mean error estimation of TOPSIS method using a fuzzy reference models,” J. Theor. Appl. Comput. Sci, Vol. 7, no. 3, pp. 40–50, 2013.
  
  

  Citation
  R. M. Zulqarnain, M. Saeed, N. Ahmad, F. Dayan, B. Ahmad, "Application of TOPSIS Method for Decision Making," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.2, pp.76-81, 2020