Volume-6 , Issue-5 , Oct 2019, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Conversion of Ordinary Differential Equation with Initial Condition (Initial Value Problem) into Volterra Integral Equation by Using Two Methods, Both the Methods Give Different and Perfect Volterra Integral Equation for Some Problems

  Arun Joshi, Anuj Joshi, Ritu

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.6 , Issue.5 , pp.76-82, Oct-2019

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  This is based on some of the ordinary differential equation with initial boundary condition problem or initial value problem solved by the two methods and converted the ordinary differential equation with initial boundary condition or initial value problem into Volterra integral equation. The solution of these ordinary differential equation with initial condition or initial value problem by using the two methods are different solution but the solutions are correct. In Both the methods it always convert the ordinary differential equation with initial boundary condition or initial value problem into volterra integral equation. We are applying the two different methods in some of the ordinary differential equation with initial boundary condition or initial value problem .Then solutions of some problems give different volterra integral equation and some problems give same volterra integral equation but all the solutions are correct and perfect because all the solutions are satisfied by the condition of volterra integral equation. This shows that both the methods are correct to convert the ordinary differential equation with initial boundary condition problem or initial value problem into volterra integral equation

  Key-Words / Index Term
  Ordinary differential equation, Initial value problem, Initial boundary condition, Volterra integral equation, linear Volterra integral equation, linear Volterra integral equation of second kind

  References
  [1] A. J. Jerri, “Introduction to Integral Equations with Applications”, Wiley, New York, 1999.
  [2] A. M. Wazwaz, “A First Course in Integral Equations”, World Scientific, Singapore, 1997.
  [3] B. L. Moiseiwitsch, “Integral Equations, Longman”, London, New York, 1977.
  [4] D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind”, J. Assoc. Comput.,page 84–96, March 9, 1962.
  [5] L. M. Delves, J. L. Mohamed, “Computational Methods for Integral Equations”, Cambridge University Press, Cambridge, 1985.
  [6] K. Maleknejad, N. Aghazadeh, “Numerical Solution of Volterra Integral Equations of the Second Kind with Convolution Kernel by Using Taylor-Series Expansion Method”, Appl. Math. Comput., page 915–922, 2005.
  [7] P. Linz, “Analytical and Numerical Methods for Volterra Equations”, SIAM Stud. Appl.Math., 1985.
  [8] J. M. Bownds, B. Wood,“A Note on Solving Volterra Integral Equations with Convolution Kernels”, Appl. Math. Comput, page 307–315, 1977.
  [9] K. E. Atkinson, “The Numerical Solution of Integral Equations of the Second Kind”, Cambridge University Press, Cambridge, 1997.
  [10] M. Nadir, A. Rahmoune, Modi.ed,“Method for solving linear Volterra integral equations of second kind”,page 141-146, IJMM, 2007.
  [11] M. Nadir, S. Guechi, “Integral Equations and their Relationship to Differential Equations with Initial Conditions, in General Letters in Mathematics, page 1-9, 2016.
  [12] J. D. Logan, “A First Course in Differential Equations”, Springer Science, 2006.
  [13] K. Al-Khaled, M. Naim Anwar, “Numerical comparison of methods for solving second order ordinary initial value problems”, in Applied Mathematical Modelling, 2007.
  [14] P. Linz, “Linear Multistep Methods for Volterra Integrodifferential Equations”, Journal of the Association for Computing Machinery, Vol. 16, pp. 295–301, 1969.
  [15] L. Taverini, “One-Step Methods for the Numerical Solution of Volterra Functional Differential Equations”, SIAM Journal on Numerical Analysis, Vol. 8, pp. 786–795, 1975.
  [16] J. M. Bownds, B. Wood, “On Numerically Solving Non-linear Volterra Equations with Fewer Computations”, SIAM Journal on Numerical Analysis, Vol. 13, pp. 705–719, 1976.
  [17] F. Dehoog, R. Weiss, “High Order Methods for Volterra Integral Equations of the First Kind”, SIAM Journal on Numerical Analysis, Vol. 10, pp. 647–664, 1973.
  [18] M. D. Raisinghania, “Integral equation and boundary value problems”, S.chand and company ltd, New delhi, India, page 2.1-2.22, 2007.
  

  Citation
  Arun Joshi, Anuj Joshi, Ritu, "Conversion of Ordinary Differential Equation with Initial Condition (Initial Value Problem) into Volterra Integral Equation by Using Two Methods, Both the Methods Give Different and Perfect Volterra Integral Equation for Some Problems," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.5, pp.76-82, 2019

• Open Access   Article

  Time Series Analysis of Atmospheric Particulate Matter of Bengaluru City

  Shrivallabha S., Kumaresh P. Nelavigi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.6 , Issue.5 , pp.83-85, Oct-2019

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  Particulate matter (PM) is composed of inert carbonaceous cores with multiple layers of various absorbed molecules, including metals, organic pollutants, acid salts and biological elements, such as endotoxins, allergens and pollen fragments. PM is classified in the following types -“Total suspended particulates” (TSP) is a name given to particles of sizes up to about 50μm. The larger particles in this class are too big to pass through our noses or throats and so, they cannot enter our lungs. They are often from wind-blown dust and may cause soiling of buildings and clothes. However, TSP samples may also contain the small PM10 and PM2.5 particles that may enter into our lungs. Total suspended particulates (TSP) with additional subcategories of particles smaller than 10μm (PM10) and particles smaller than 2.5μm (PM2.5) are discussed. Size and chemical composition are among the most important parameters influencing the way in which airborne particles interact with the environment. This paper presents a time series analysis of particulate matter (PM10) in Bengaluru city, Karnataka, India from April 2018 to November 2018. An ARIMA(Auto-Regressive Integrated Moving Average ) model of time series analysis is used for analysis and forecasting of the future concentration of the air pollutant. The data set of daily average PM10 concentration collected from Karnataka State Pollution Control Board was good fitted with an ARIMA model as per Ljung –Box test. The cross validation of model is done using residual analysis

  Key-Words / Index Term
  Air pollution, Forecasting, Particulate matter, stationary, non-stationary, Time series analysis, ARIMA, ARMA, PM10

  References
  [1] Seetharam, A.L. & Simha, B.L.U., 2009. Urban Air Pollution – Trend and Forecasting of Major Pollutants by Timeseries Analysis. International Journal of Civil and Environmental Engineering, 1(2), pp.71–74
  [2] Burden of disease from the joint effects of household and ambient Air pollution for 2016. , Public Health, Social and Environmental Determinants of Health Department, World Health Organization, (May 2018).
  [3] Chadchan, J. & Shankar, R., 2012. An analysis of urban growth trends in the post-economic reforms period in India. International Journal of Sustainable Built Environment, 1(1), pp.36–49. Available at: http://dx.doi.org/10.1016/j.ijsbe.2012.05.001.
  [4] Dadhich, P.N. & Hanaoka, S., 2012. Spatial investigation of the temporal urban form to assess impact on transit services and public transportation access. Geo-Spatial Information Science, 15(3), pp.187–197.
  [5] Ravindra, K. et al., 2003. Variation in spatial pattern of criteria air pollutants before and during initial rain of monsoon. Environmental Monitoring and Assessment, 87(2), pp.145–153.
  [6] Khandelwal, S. et al., 2018. Assessment of land surface temperature variation due to change in elevation of area surrounding Jaipur, India. Egyptian Journal of Remote Sensing and Space Science, 21(1), pp.87–94.
  [7] Tandon, A., Yadav, S. & Attri, A.K., 2008. City-wide sweeping a source for respirable particulate matter in the atmosphere. Atmospheric Environment, 42(5), pp.1064–1069.
  [8] Cropper, M.L. et al., 1997. The Health Effects of Air Pollution in Delhi , India. The World Bank, (December 1997), p.40.
  [9] Ren, C. & Tong, S., 2008. Health effects of ambient air pollution – recent research development and contemporary methodological challenges. Environmental Health, 7(1), p.56. Available at: http://www.ehjournal.net/content/7/1/56.
  [10] Lovett, G.M. et al., 2009. Effects of air pollution on ecosystems and biological diversity in the eastern United States. Annals of the New York Academy of Sciences, 1162, pp.99–135.
  [11] Ilyas, S.Z. et al., 2010. Air pollution assessment in urban areas and its impact on human health in the city of Quetta, Pakistan. Clean Technologies and Environmental Policy, 12(3), pp.291–299. Available at: http://dx.doi.org/10.1007/s10098-009-0209-4.
  [12] Middleton, A.J.T. et al., 2013. Association of Schools of Public Health Air Pollution Research Seminar. , 72(4), pp.367–374.
  [13] Gupta, S., Kankaria, A. & Nongkynrih, B., 2014. Indoor air pollution in India: Implications on health and its control. Indian Journal of Community Medicine, 39(4), p.203. Available at: http://www.ijcm.org.in/text.asp?2014/39/4/203/143019.
  [14] Hirota, K. et al., 2017. A Methodology of Health Effects Estimation from Air Pollution in Large Asian Cities. Environments, 4(3), p.60. Available at: http://www.mdpi.com/2076-3298/4/3/60.
  [15] Orimoogunje, O.O.I. & Balogun, V.S., 2015. An Assessment of Seasonal Variation of Air Pollution in Benin City, Southern Nigeria. Atmospheric and Climate Sciences, 5(July), pp.209–2018.
  [16] Gardner, E.S., 2006. Exponential smoothing: The state of the art Part II, International Journal of Forecasting,22(4), 637-666.
  [17] Sahu, S.K., Gelfand, A.E. & Holland, D.M., 2007. High-resolution space-time ozone modeling for assessing trends. Journal of the American Statistical Association, 102(480), pp.1221–1234.
  [18] Siew, L.Y. et al., 2008. Arima and Integrated Arfima Models for Forecasting Air Pollution Index in Shah Alam , Selangor. The Malaysian Journal of Analytical Science, 12(1), pp.257–263.
  [19] Kumar, U. & Jain, V.K., 2010. ARIMA forecasting of ambient air pollutants (O3, NO, NO2 and CO). Stochastic Environmental Research and Risk Assessment, 24(5), pp.751–760. Available at: http://link.springer.com/10.1007/s00477-009-0361-8.
  [20] Kim, S.E., 2010. Tree-based threshold modeling for short-term forecast of daily maximum ozone level. Stochastic Environmental Research and Risk Assessment, pp.19–28.
  

  Citation
  Shrivallabha S., Kumaresh P. Nelavigi, "Time Series Analysis of Atmospheric Particulate Matter of Bengaluru City," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.5, pp.83-85, 2019

• Open Access   Article

  Probabilistic Metric Space, Menger Space and Some Common Fixed Point Theorem

  Krishnadhan Sarkar, Dinanath Barman, Mithun Paul, Kalishankar Tiwary

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.6 , Issue.5 , pp.86-90, Oct-2019

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  In this note we use the concept of semi compatible pair of reciprocal continuous maps in Probabilistic metric space and in Menger Space. We define existing definitions in this literature, and prove a common fixed point theorem in these spaces. Our results improve many well-known existing results in this literature

  Key-Words / Index Term
  Probabilistic metric space, Menger space, Semi compatible maps, Reciprocal continuous maps

  References
  [1]. S. Banach, “Sur les opérationsdans les ensembles abstraitsetleur application aux équations intégrales”. Fundam. Math., French, Vol. 3, pp.133–181, 1992.
  [2]. H. Bouhadjera and C. Godet–Thobie, “Common fixed point theorems for pairs of sub- compatible maps”, arXiv : 0906 .3159 v2, 2009.
  [3]. B. S .Choudhury, S. K. Bhandari, “Kannan-type cyclic contraction results in 2-Menger space,” Mathematica Bohemica, Vol. 141, No. 1, pp.37-58, 2016.
  [4]. B. S. Choudhury, K. Das, “A new contraction principle in Menger spaces”. Acta Math. Sin., Engl. Ser. Vol.24, pp.1379–1386, 2008.
  [5]. B. S. Choudhury, K. Das, “A coincidence point result in Menger spaces using a control function.” Chaos Solitons Fractals, Vol. 42, pp.3058–3063, 2009.
  [6]. B. S. Choudhury, K. Das, “Fixed points of generalized Kannan type mappings in gener- alized Menger spaces’. Commun. Korean Math. Soc. Vol. 24, pp.529–537, 2009.
  [7]. B. S. Choudhury, K. Das, S. K. Bhandari, “A generalized cyclic C-contraction principle in Menger spaces using a control function”. Int. J. Appl. Math. Vol.24, pp.663–673, 2011.
  [8]. B. S. Choudhury, K. Das, S. K. Bhandari, “Fixed point theorem for mappings with cyclic contraction in Menger spaces.” Int. J. Pure Appl. Sci. Technol. Vol. 4, pp.1–9, 2011.
  [9]. P. N. Dutta, B. S. Choudhury, “A generalized contraction principle in Menger spaces using a control function”. Anal. Theory Appl., Vol.26, 110–121, 2010.
  [10]. A .Jain and B. Singh,” Common fixed point theorems in Menger space through compatible maps of type (A)”, Chh. J. Sci.Tech.,Vol. 2, pp.1-12, 2005.
  [11]. S. Kumar and R. Chugh, “Common fixed point theorems using minimal commutativity and reciprocal continuity conditions in metric spaces”, Sci. Math. Japan,, Vol. 56, pp. 269-275,2002
  [12]. S. Kumar and B. D. Pant, “A common fixed point theorem in probabilistic metric space using implicit relatios,” Filomat, Vol. 22(2) ,pp. 43-52, 2008
  [13]. P. Malviya, V. Gupta, V.H. Badshah, “Common fixed point theorem for semi compatible pairs of reciprocal continuous maps in Menger spaces’, Annals of Pure and Applied Mathematics, Vol. 11, No. 2, pp. 139-144, 2016.
  [14]. K .Menger, “Statistical metrics,” Poc. Nat. Acad. Sci., vol. 28, pp. 535-537, 1942.
  [15]. S. N. Mishra, “Common fixed points of compatible mappings in PM-space,” Math. Japon., 36(2), pp. 283-289, 1991
  [16]. B. D. Pant and S. Chouhan, “Common fixed point theorems for semi-compatibility maps using implicit relation”, Int. J. of Math. Analysis, 3(28), pp. 1389-1398, 2009.
  [17]. K. Sarkar and K. Tiwary, “Common Fixed Point Theorems for Weakly Compatible Mappings on Cone Banach Space,” International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol: 5, Issue-2, pp. 75-79, 2018.
  [18]. B. Schweizer, A. Sklar, “Probabilistic Metric Spaces.” North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing, New York, 1983.
  [19]. V.M. Sehgal, “Some fixed point theorems in functions analysis and probability,” Ph.D. dissertation, Wayne State Univ. Michigan, 1966.
  [20]. V. M. Sehgal and A. T. Bharucha-Reid, “Fixed points of contraction mappings on probabilistic metric spaces”, Math. Systems Theory, Vol. 6 ,pp. 97-102, 1972
  [21]. Y. Shi, L. Ren, X. Wang, “The extension of fixed point theorems for set valued mapping”. J. Appl. Math. Comput., 13,pp. 277–286, 2003.
  [22]. S. L. Singh, B. D. Pant and R. Talwar, “Fixed points of weakly commuting mappings on Menger spaces,” Jnanabha, 23, pp.115-122, 1993.
  [23]. S. L. Singh and B. D. Pant, “Common fixed point theorems in probabilistic metric space and extension to uniform spaces,” Honam. Math. J., pp. 1-12, 1984.
  

  Citation
  Krishnadhan Sarkar, Dinanath Barman, Mithun Paul, Kalishankar Tiwary, "Probabilistic Metric Space, Menger Space and Some Common Fixed Point Theorem," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.5, pp.86-90, 2019

• Open Access   Article

  A Time Series Analysis of Nigeria All Share Index (1985-2018) Buys-Ballot Methodology

  D. Enegesele, O.A.P, Otaru

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.6 , Issue.5 , pp.91-98, Oct-2019

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  Classical time-series analysis is the decomposition of time-series into components due to seasonal variation, trend, cyclic variation and irregular fluctuations. A good knowledge of this method is necessary because most real life data are not normally distributed hence most analyst are faced with the problem of choice of appropriate transformation, choice of model for decomposition and difficulty to determine the presence of seasonal effect in a series to achieve unbiased decision. Thus, the study aims to identify the best transformation for modeling of Nigeria All Share Index (ASI), an appropriate choice of model for decomposition, the seasonal effect as well as trend pattern of Nigeria All Share Index (ASI). The study applied Buys-Ballot methodology to assess the characteristics of ASI. Evidence of the result showed that logarithm transformation is best suitable for ASI series and multiplication model is the appropriate model for decomposition. Also, result revealed that ASI contains seasonal effect and quadratic trend curve best fit ASI with p<0.05 and coefficient of determination of 0.974. We therefore recommend that appropriate transformation approach should be used to obtain stability of the variance of time series data and care should be taken in the choice of appropriate model for decomposition

  Key-Words / Index Term
  Normality, Buys-Ballot, Transformation, Model, Trend, Seasonal, Decomposition

  References
  [1] M.S. Bartlett, “The Uses of Transformation”, Biometrika, Vol3, pp. 39-52, 1947.
  [2] G.E.P Box, G.M,Jenkins, “Time Series Analysis: Forecasting and Control”, Holden-Day, San Francisco, 1976.
  [3] G.E.P Box, G. M Jenkins, G. C, Reinsel, “Time Series Analysis, Forecasting and Control”, 3rd Edition, Prentice-Hall, 1994.
  [4] C. Chatfield, “The Analysis of Time Series: An Introduction”, Chapman and Hall/CRC Press, Boca Raton, 2004.
  [5] I.S.Iwueze, A.C. Akpanta, “On Applying the Bartlett Transformation Method to Time Series Data”, Journal of Applied Mathematics, Vol 2, pp. 633-645, 2009.
  [6] I. S. Iwueze, E. C. Nwogu, “Buys-Ballot Estimates for Time Series Decomposition”, Global Journal of Mathematics, Vol. 3 No. 2, pp 83-98, 2004.
  [7] I.S., Iwueze, E.C. Nwogu, J. Ohakwe, J.C. Ajaraogu, “Uses of the Buys-Ballot Table in Time Series Analysis”, Applied Mathematics, pp 633-645, 2011.
  [8] E.F. Fama, “Efficient Capital Markets: A Review of Theory and Empirical Work”,Journal of Finance,Vol. 25,pp.383-417. 1970.
  [9] T. F. Gosnell,A. J. Keown, J. M. Pinkerton, “The Intraday Speed of Stock Price Adjustment to MajorDividend Changes: Bid-Ask Bounce and Order Flow Imbalances”,Journal of Banking and Finance, Vol.20, pp.247-266. 1996.
  [10] J. Patell, M. Wolfson, “The Intraday Speed of Adjustment of Stock Prices to Earnings and Dividend Announcements”, Journal of Financial Economics, Vol. 13,pp 223-252. 1984.
  [11] N. Seyhun,“Insiders’ Profits, Cost of Trading and market Efficient”, Journal of Financial Economics,Vol.16, pp.189-212. 1986
  [12] M. A. Adebiyi, “Capital Market Performance and the Nigerian Economic Growth Management in Nigeria Lagos”: University Press. 2005
  [13] O. Ajakaiye, T. Fakiyesi, “Global Financial Crisis Discussion Series Paper 8: Nigeria”, Overseas Development Institute London. 2009
  [14] N. G. Mankiw,“Macro Economics”, 4th Edition, , UK: Macmillan Press Ltd.pp.472 1999.
  [15] A.O. Ologunde, D.O. Elumilade, T. O. Asaolu, “Stock Market Capitalization and Interest Rate in Nigeria: A Time Series Analysis”, International Research Journal of Finance and Economics, Issue 4, pp.154-166. 2006
  [16] H. Alile, “Government must Divest”,The Business Concord of Nigeria 2nd December, pp.8.1997
  [17] C.O.Mgbame,O.J.Ikhatua, “Accounting Information and Stock Volatility in the Nigerian Capital Market: A Garch Analysis Approach”, International Review of Management and Business Research, Vol 2, No 1, pp. 265 – 281. 2013
  [18] D. Aurangzeb, “Factors Affecting Performance of Stock Market: Evidence from South Asian Countries”,International Journal of Academic Research in Business and Social Sciences, Vol 2, No. 9, pp.1-15.2012.
  [19] N. Okereke-Onyuike, “Strategies for Stock Market Recovery”, A Paper presented to the Ad-hoc Committee of the House of Representatives during the Capital Market Probe. Business World, pp.14-21. 2012.
  [20] CBN, Central Bank of Nigeria Statistical Bulletin, Various Issues. Abuja: Central bank of Nigeria Publication (CBN). 2017.
  [21] W.W.W Wei, “Time Series Analysis, Univariate and Multivariate Methods”, Addison- Wesley, Redwood City, 1989.
  [22] H.Wold, “A study in the Analysis of Stationary Time Series”. 2nd Edition, Stockholm: Almqrist and witsett, 1938.
  

  Citation
  D. Enegesele, O.A.P, Otaru, "A Time Series Analysis of Nigeria All Share Index (1985-2018) Buys-Ballot Methodology," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.6, Issue.5, pp.91-98, 2019