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• Open Access   Article

  A Hybrid Semi-Analytical Method with Natural Transform for Solving Integro-Differential Equations

  George. A. Toma

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.1-7, Jun-2020

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  Abstract
  In this paper article, we proposed a new hybrid method to find analytical approximate or exact solutions for various many linear and nonlinear integro ? differential equations. This proposed hybrid method combines a semi-analytical method which is the homotopy perturbation method with the natural transform for solving initial value problems represented by linear and non-linear integro - differential equations of the first order. The basic steps of the proposed hybrid scheme are deriving the integro - differential equation twice so we got two additional initial conditions, then we applied the natural transform to the obtained equation that we got after the derivation, thus this equation has been converted to a new equation containing Natural transform variables and ,based on it, we constructed the Homotopy. We presented three problems to test the accuracy and effectiveness of the proposed hybrid technique, our numerical results were also compared with the exact solutions. The new proposed technique method is simple, effective and fast power, simplicity and efficiency.

  Key-Words / Index Term
  Homotopy Perturbation Method, Initial Value Problems , Linear and Nonlinear Integro-Differential Equations, Natural Transform.

  References
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  [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] Z. H. Khan, W. A. Khan, ?N-transform properties and applications,? NUST Journal of Engineering Sciences, Vol. 1, pp.127-133, 2008.
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  [6] 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.
  [7] 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.
  [8] E. E. Eladdad and E. A. Tarif, ?On the Coupling of the Homotopy Perturbation Method and New Integral Transform for Solving Systems of Partial Differential Equations, ? Advances in Mathematical Physics, 2019.
  [9] M. Tahmina Akter1, M. A. Mansur Chowdhury, ? Homotopy Perturbation Method for Solving Highly Nonlinear Reaction-Diffusion-Convection Problem, ? American Journal of Mathematics and Statistics 2019.
  [10] J. Biazar a, H. Ghazvini, ? Convergence of the homotopy perturbation method for partial differential equations, ? Nonlinear Analysis: Real World Applications, Elsevier, 2009.
  [11] J. Rashidinia, A. Tahmasebi, ? Approximate solution of linear integro-differential equations by using modified Taylor expansion method, ? World Journal of Modelling and Simulation, Vol. 9, No. 4, pp. 289-301, 2013.
  [12] B. Batiha, M.S.M. Noorani and I. Hashim, ? Numerical Solutions Of The Nonlinear Integro-Differential Equations , ? Int. J. Open Problems Compt. Math., Vol. 1, No. 1, 2008.
  [13] R. G. Shirvan, J. S. Nadjafi, M. Gachpazan, ? An Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Method, ? International Journal of Differential Equations, 2018.
  [14] L. Su, T. Yan, Y. Zhao, F. L, R. Liu, ? Numerical Solution of Integro-Differential Equations with Local Polynomial Regression, ? Open Journal of Statistics, 2012.
  [15] A.D. Chindhe, S.B. Kiwne, ? Application of Combine Natural Transform and Adomian Decomposition Method in Volterra
  Integro-Differential Equations, ? Mathematical Journal of
  Interdisciplinary Sciences, Vol. 5, No. 1, 2016.
  [16] M. Valizadeh, Y. Mahmoudi , and F. Dastmalchi Saei,
  ? Application of Natural Transform Method to Fractional
  Pantograph Delay Differential Equations, ? Mathematical Journal of Mathematics, 2019.
  

  Citation
  George. A. Toma , "A Hybrid Semi-Analytical Method with Natural Transform for Solving Integro-Differential Equations," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.1-7, 2020

• Open Access   Article

  A New Hybrid D-Numbers extended Consistent Linguistic Fuzzy Preference Relation with Enhanced DNIT model for selection of bowlers in IPL-2019

  Udaykumar S. Kashid, Archana U. Kashid, Saurabh N. Mehta, Vineeta Basotia

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.8-22, Jun-2020

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  Abstract
  The selection of the best alternatives totally depends upon the weight of criteria. The precise values of the criteria weights always have an influence on selecting the best alternatives and obtaining the optimised solution. But in real life to get more accuracy in a weight of criteria is always an open issue. In this research work to obtain precise weight of criteria, we have proposed new hybrid D-Numbers extended Consistent Linguistic Fuzzy Preference Relation with Enhanced D-Numbers Integral Theorem (HD-CLFPR with E-DNIT) as a mathematical model by removing the major drawback of existing D-Numbers Integral Theorem (DNIT) and enhanced it as E-DNIT. Thus, we have combined three different mathematical concepts (MCDM + Fuzzy set theory + D Numbers theory) to strengthen the capability of each other?s under fuzzy situation to avoid vagueness and abandon of incomplete information data. The main objective of this paper is in three phases: In first phase, we have developed new E-DNIT formula to allocate incomplete information data to complete information data under uncertain situation. In second phase, we have developed ?HD-CLFPR with E-DNIT? mathematical model to obtain the precise weight of selected four criteria. In third phase, for the validation, the proposed model has been implemented by using ?TOPSIS? method to select the best alternatives strategy for the ranking of bowlers in Indian premier league (IPL) 2019.

  Key-Words / Index Term
  Fuzzy Preference Relations; MCDM; Decision Theory; D-Numbers Theory; ?HD-CLFPR with E-DNIT

  References
  [1] Bernard ROY, ?Decision-aid and decision making,? Elsevier, European Journal of Operational Research, Vol.45, pp.324-331.,1990. https://doi.org/10.1016/0377-2217(90)90196-I.
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  [3] Carlsson, C., Fuller, R., ?Fuzzy multiple criteria decision making: Recent developments,? Elsevier,Fuzzy Sets and Systems, Vol.78, Issue.2, pp.139-153 ,1996.
  https://doi.org/10.1016/0165-0114(95)00165-4.
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  [5] Glenn Shafer, ?A Mathematical Theory of Evidence turns 40,? International Journal of Approximate Reasoning, Vol.79, pp.7?25, 2016. doi:10.1016/j.ijar.2016.07.009.
  [6] Udaykumar S. Kashid, Archana Kashid, Saurabh Mehta, ?A Review of Mathematical Multi-Criteria Decision Models with A case study,? International Journal of Research and Analytical Reviews (IJRAR) , Vol. 6, special issue, pp. 110-124, 2019. HTTP://DOI.ONE/10.1729/JOURNAL.20927
  [7] Huan-Jyh Shyurb, E. Stanley Lee, ?An extension of TOPSIS for group decision making,? Elsevier, Mathematical and Computer Modelling, Vol.45 pp. 801?813, 2007.
  doi:10.1016/j.mcm.2006.03.023.
  [8] T.C.Wang, Y.-H. Chen, ?Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP,? Elsevier, Information Sciences,Vol. 178, Issue.19, pp. 3755?3765, 2008. doi:10.1016/j.ins.2008.05.028
  [9] Lotfi A Zadeh, ?A simple view of the Dempster-Shafer theory of evidence and its implication for the rule of combination,? Journal AI Magazine archive,Vol. 7, Issue. 2, pp. 85 ? 90, 1986. DOI: https://doi.org/10.1609/aimag.v7i2.542.
  [10] Yong Deng, Felix T.S. Chan. ?A new fuzzy dempster MCDM method and its application in supplier selection,? Elsevier,Expert Systems with Applications,Vol. 38, Issue.8, pp. 9854?9861, 2011. https://doi.org/10.1016/j.eswa.2011.02.017 .
  [11] Deng Yong.,?D Numbers: Theory and Applications,? Journal of Information and Computational Science, Vol. 9, pp. 2421-2428, 2012.
  [12] Xinyang Denga, Yong Deng, ?D-Numbers theory- a generalization of Dempster-Shafer theory,? The Scientific World Journal, Vol.1402, pp. 1-37, 2014.
  http://dx.doi.org/10.1155/2014/358057.
  [13] Xinyang Deng, Xi Lua, Felix T.S. Chan, Rehan Sadiq, Sankaran Mahadevan , ?Yong Deng, D-CFPR: D numbers extended consistent fuzzy preference relations,? Elsevier, Knowledge-Based Systems, Vol. 73, pp. 61?68, 2014.
  [14] Y. Dong, Yinfeng Xu, Hongyi Li., ?On consistency measures of linguistic preference relations,? Elsevier, European Journal of Operational Research,? Vol. 189, Issue. 2, pp. 430-444, 2008. doi:10.1016/j.ejor.2007.06.013.
  [15] Dibyojyoti Bhattacharjee, Hemanta Saikia,?On Performance Measurement of Cricketers and Selecting an Optimum Balanced Team,? Taylor & Francis Journals- International Journal of Performance Analysis in Sport,Vol.14, pp. 262-275, 2014. DOI: 10.1080/24748668.2014.11868720.
  [16] Hermanus H. LEMMER, ?An analysis of players? performances in the first cricket twenty20 world cup series,? South African Journal for Research in Sport, Physical Education and Recreation, Vol. 30, Issue. 2, pp. 71-77, 2008.
  DOI:10.4314/sajrs.v30i2.25990 .
  [17] Udaykumar S. Kashid, Saurabh N. Mehta, Vineeta Basotia, ?A Hybrid ?AHP-TOPSIS? Mathematical Multi Criteria Decision-Making model for players performance evaluation and selection in IPL: A case study?, International Journal AU-FAIT, Vol.5, Issue.2, pp. 922-933, 2019.
  

  Citation
  Udaykumar S. Kashid, Archana U. Kashid, Saurabh N. Mehta, Vineeta Basotia, "A New Hybrid D-Numbers extended Consistent Linguistic Fuzzy Preference Relation with Enhanced DNIT model for selection of bowlers in IPL-2019," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.8-22, 2020

• Open Access   Article

  Possibility and Probability of Untested Positive Cases - A Reliability Check on The Efficacy of the Quality and Quantity of COVID-19 Testing.

  Raghuram K.S., Sathya Krishna K.R.

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.23-27, Jun-2020

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  Abstract
  With an exponential scaling up of COVID-19 testing, healthcare systems and lawmakers globally are overwhelmed with the unprecedented COVID-19 positive cases. Innovative strategies of testing like random testing and drive-thru testing are being implemented to identify the suspected cases, isolate, quarantine and treat them in isolation until an antidote or vaccine is developed for large scale administration. This paper dwells into the repercussions of using low-reliability testing kits and the possible damage caused by undetected positive cases due to the low accuracy of this testing. Bayes law is used to develop country-specific logarithmic functions to fit a model that identifies the probability of infected but undetected cases in any given population. Sixty-eight percent of total confirmed cases globally and the ?tests per million? data from thirteen countries are considered to develop a model. This predictive model identifies the probability of infected but undetected cases in any given population.

  Key-Words / Index Term
  COVID-19, Bayes law, False Positives, Sensitivity, Specificity, Covid-19 testing, Coronavirus, Wuhan, Testing Kit

  References
  [1] Eubank, S., Eckstrand, I., Lewis, B., Venkatramanan, S., Marathe, M., and Barrett, C., 2020. Commentary on Ferguson, et al., ?Impact of Non-pharmaceutical Interventions (NPIs) to Reduce COVID-19 Mortality and Healthcare Demand?. Bulletin of Mathematical Biology, 82(4). Pp 52.
  [2] S. Bullis, J. Crothers, S. Wayne and A. Hale, "A cautionary tale of false-negative nasopharyngeal COVID-19 testing", 2020, Science Direct ID Cases, vol. 20, p. e00791.
  [3] Gu?bjartsson, Dan?el F., et al. (2020). ?Early Spread of SARS-Cov-2 in the Icelandic Population.? Medrxiv Preprint. The New England of Medicine, Med 2020; 382:2302-2315, pp 01
  [4] Allam, M., Cai, S., Ganesh, S., Venkatesan, M., Doodhwala, S., Song, Z., Hu, T., Kumar, A., Heit, J., Study Group, C. and Coskun, A., 2020. COVID-19 Diagnostics, Tools, and Prevention. Diagnostics, 10(6), pp.409.
  [5] Ai, T., Yang, Z., Hou, H., Zhan, C., Chen, C., Lv, W., Tao, Q., Sun, Z. and Xia, L., 2020. Correlation of Chest CT and RT-PCR Testing in Coronavirus Disease 2019 (COVID-19) in China: A Report of 1014 Cases. Radiology, p.200642.
  [6] Stock, j. and Aspelun, K., 2020. Estimates of The Undetected Rate Among The SARS-Cov-2 Infected Using Testing Data From Iceland. Cold Spring Harbour Laboratory, BMJ, Yale, doi.org/10.1101/2020.04.06.20055582
  [7] Burkhardt, M., 2020. Estimating Possibly Undetected COVID-19 Infection Cases, Towards Data Science, pp01
  

  Citation
  Raghuram K.S., Sathya Krishna K.R., "Possibility and Probability of Untested Positive Cases - A Reliability Check on The Efficacy of the Quality and Quantity of COVID-19 Testing.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.23-27, 2020

• Open Access   Article

  An Examination of the Accuracy and Zero Stability of the Explicit Linear Two-Step Method for Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs)

  S.E. Fadugba

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.28-32, Jun-2020

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  In this paper, an explicit linear two-step method of maximal order containing one free parameter for the solution of IVPs in ODEs is presented. The performance measure of the method in terms of the accuracy and zero stability is examined. The bound of the local truncation error for the explicit linear one-step method has been investigated. Numerical example has been solved successfully via the explicit linear two-step method by varying the free parameter. The results obtained show that the explicit linear two-step method is zero stable and agrees with the exact solution. In the case of b = -5, the method is zero unstable. It can also be concluded that one order decrease in the values of the step length leads to third order decrease in the magnitude of the error bound of the method. The methodology can be applied to the solution of higher order ODEs emanated from real life situations with points of catastrophe.

  Key-Words / Index Term
  Bound, Explicit linear two-step method, Initial value problem, Local truncation error, Exact solution

  References
  [1] Fadugba, S., Ogunrinde, B. and Okunlola, T., ?Euler?s method for solving initial value problems in ordinary differential equations?, Pacific Journal of Science and Technology, Vol. 13, Issue 2, pp. 152-158, 2012.
  [2] Fadugba, S.E. and Ajayi, A.O., ?Comparative study of a new scheme and some existing methods for the solution of initial value problems in ordinary differential equations?, International Journal of Engineering and Future Technology, Vol. 14, Issue 3, pp. 47-56, 2017.
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  [8] Butcher, J.C., ?Numerical methods for ordinary differential equations?, John Wiley & Sons, Third Edition, 2016.
  [9] Fadugba S.E. and Falodun B.O., ?Development of a new one-step scheme for the solution of initial value problem (IVP) in ordinary differential equations?, International Journal of Theoretical and Applied Mathematics, Vol. 3, Issue 2, pp. 58-63, 2017..
  [10] Fadugba S.E. and Olaosebikan, T.E., ?Comparative study of a class of one-step methods for the numerical solution of some initial value problems in ordinary differential equations?, Research Journal of Mathematics and Computer Science, Vol. 2, Issue 9, DOI: 10.28933/rjmcs-2017-12-1801, 2018.
  [11] Qureshi, S. and Fadugba, S.E., ?Convergence of a numerical technique via interpolating function to approximate physical dynamical systems?, Journal of Advanced Physics, Vol. 7, pp. 446-450, 2018..
  [12] Fadugba, S.E. and Qureshi, S., ?Convergent numerical method using transcendental function of exponential type to solve continuous dynamical systems?, Punjab University Journal of Mathematics, Vol. 51, pp. 45-56, 2019.
  [13] Fadugba, S.E. and Idowu, J.O., ?Analysis of the properties of a third order convergence numerical method derived via transcendental function of exponential form?, International Journal of Applied Mathematics and Theoretical Physics, Vol. 5, 97-103, 2019.
  [14] Kayode S.J., ?A zero stable method for direct solution of fourth order ordinary differential equations?, American Journal of Applied Sciences, Vol. 5, Issue 11, pp. 1461-1466, 2009.
  [15] Kayode S.J., ?A zero-stable optimal order method for direct solution of second order differential equations?, Journal of Mathematics and Statistics, Vol. 6, Issue 3, pp. 367-371, 2010.
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  [18] Fadugba, S.E., ?Construction of an explicit linear two-step method of maximal order?, Advanced Science, Engineering and Medicine, Vol. 11, pp. 532-536, 2019.

  Citation
  S.E. Fadugba, "An Examination of the Accuracy and Zero Stability of the Explicit Linear Two-Step Method for Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs)," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.28-32, 2020

• Open Access   Article

  Mathematical Model of Population Interactions with Functional Responses and Harvesting Function

  S. T. Motuma

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.33-38, Jun-2020

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  Abstract
  In this paper, ecological interaction between two population species has been studied. Mutuality interaction has been considered between two populations. Without interactions both population grow logistically. The interaction between populations are proceeding following different types of functional responses and harvesting function. In these interactions, the population species are harvested proportionally. The proportional harvesting is introduced to the interaction in order to describe the removal of population from their habitat based on their density. According to this, both populations are harvested in this model. In order to study these interactions mathematical equation is constructed. It is shown that the model equation have both positive and bounded solutions. The dynamics of these populations have been studied. Local and global stability analysis is carried out based on the positive equilibrium point. Numerical simulations supporting theoretical results are also included here.

  Key-Words / Index Term
  Mutualism, Lyapunov function, Phase plane, Functional Response, Positivity, Boundedness and proportional Harvesting.

  References
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  [15] R. Ouncharoen Pinjai S, T. Dumrongpokaphan, and Y. Lenbury (2012). Global stability analysis of predator-prey model with harvesting and delay. Thai Journal of Mathematics, 8(3): 589- 605.
  [16] CVD. Le?n (2012). Lyapunov function for two-species cooperative systems. Applied Mathematics and Computation, 219(5): 2493-2497.
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  [20] R. Ahmad (2017). Global stability of two-species mutualism model with proportional harvesting International Journal of Advanced and Applied Sciences, 4(7), Pages: 74-79

  Citation
  S. T. Motuma, "Mathematical Model of Population Interactions with Functional Responses and Harvesting Function," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.33-38, 2020

• Open Access   Article

  Combined Effect of Chemical Reaction and Transverse Magnetic Field on Free Convection Flow past a Vertical Cone

  A. Kaushik

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.39-44, Jun-2020

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  Abstract
  The simultaneous effect of chemical reaction and magnetic field on an unsteady laminar free convection flow past an isothermal vertical cone is studied in this paper. The non linear governing equations are first written into a dimensionless form, and then Alternating Direction Implicit (ADI) Technique is applied to obtain numerical solutions of the governing equations. Flow variables are obtained and are presented graphically for different values of Prandtl number, Schmidt number. From numerical results, it is noticed that higher values of Prandtl number as well as Schmidt number results in decrease in velocity

  Key-Words / Index Term
  Chemical reaction, free convection flow, Magnetic field, vertical cone

  References
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  [10] P. Chandrakala and P. Narayana Bhaskar , Chemical Reaction Effects on MHD Flow Past an Impulsively Started Isothermal Vertical Plate with Uniform Mass Diffusion, International Journal of Applied Mechanics and Engineering , Vol. 19: Issue 2, pp 419-426, 2014.
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  [12] Santhosha, B., Younus, S., Kamala, G. and Ramana, M. M. V., Radiation and Chemical Reaction Effects on MHD Free Convective Heat and Mass Transfer Flow of a Viscoelastic Fluid Past a Porous Plate with Heat Generation/Absorption, International Journal of Chemical Sciences, Vol. 15(3), pp 170, 2017.
  [13] B. P. Garg and Shipra, Exact solution of MHD free convective and mass transfer flow near a moving vertical plate in the presence of heat source/sink, Journal of Rajasthan Academy of Physical Sciences, Vol.18, No.1&2, pp 25-44, 2019.
  [14] G. Palani and A. R. Ragavan, Free convection flow past a vertical cone with a chemical reaction in the presence of transverse magnetic field ,Journal of engineering physics and thermophysics, Vol. 88, no. 5, pp 1256-1263, 2015.
  [15] A. Kaushik, Numerical Solutions for unsteady flow past a semi- infinite plate using Alternating ?Direction-Implicit (ADI) Technique, International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.1, pp.9-14, 2020.
  [16] S. Mohana Murali, A.R. Vijayalakshmi, Effect of Thermal Radiation on a Linearly Accelerated Vertical Plate with Variable Mass Diffusion and Uniform Temperature Distribution in a Rotating Medium, International Journal of Computer Sciences and Engineering, Vol. 6, Issue 12, pp. 563-569, 2018.
  

  Citation
  A. Kaushik, "Combined Effect of Chemical Reaction and Transverse Magnetic Field on Free Convection Flow past a Vertical Cone," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.39-44, 2020

• Open Access   Article

  Impact of Numerical Technique on The Solution of Boundary Value Problem

  Nisu Jain

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.45-53, Jun-2020

  Abstract

  Full-Text HTML

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  Citation

  Abstract
  A comparison of orthogonal collocation method (OCM) and orthogonal collocation on finite elements (OCFE) is done to solve boundary value problem numerically and analytically. Both numerical techniques are applied on dimensionless form of the model, then equations are discretized using numerical techniques. The results are obtained by MATLAB ODE 15s system solver software. Comparison is shown both in tabulated and graphical form. Relative error is used to check the efficiency of the technique. 3-D graphs are used to specify solution variation for different values of parameter.

  Key-Words / Index Term
  OCM; OCFE; Boundary Value Problems; Laplace Transformation

  References
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  Citation
  Nisu Jain, "Impact of Numerical Technique on The Solution of Boundary Value Problem," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.45-53, 2020

• Open Access   Article

  Dynamics of Predator-prey populations and The Influence of Allee effect

  S.T. Motuma, T.G. Beyene

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.54-60, Jun-2020

  Abstract

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  Abstract
  In this paper, the interactions among three population species are studied. In these interactions, two mutuality prey population and a predator population are considered. Here, the first prey population considered as it is influenced by Allee effect while the second prey population is harvested and the predator population has alternative food source for survive. The three population interaction can be described as a food chain in which two preys help each other but the predator attacks both the preys following types of functional responses. These population interactions in ecology are modeled mathematically using ordinary differential equations.The biologically interesting equilibrium points of the system are identified and analyzed. The local and global stability analysis is carried out. Numerical examples of the model are conducted to support the results of the mathematical analysis. Finally, the results of the analysis and the discussion of the population dynamics areclearly studied.

  Key-Words / Index Term
  predator, Assumptions, Normalization, Positivity, Allee effect, Functional response, Stability

  References
  [1] F.Courchamp, J.Berec, Gascoigne J (2008). Allee effects in ecology and conservation. Oxford, N ew York, U SA: Oxford University Press
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  [4] A.Deredec, F.Courchmp,: Extinction thresholds in host parasite dynamics. Ann. Zool. Fenn. 40, 115?130 (2003)
  [5] L.Berec, E.Angulo, , Courchamp, F.: Multiple allee effects and population management. Trends Ecol. Evol. 22, 185? 191 (2007)
  [6] C.W. Clark, Mathematical Bio economic: The Optimal Management of Renewable Resources, Wiley, New York, 1990.
  [7] C.W. Clark, The Worldwide Crisis in Fisheries: Economic Models and Human Be- havior, Cambridge University Press, Cambridge, 2007.
  [8] M. Liermann, and R. Hilborn, Dispensation: evidence, modelsandimplications,Fish Fish., 2 (2001), 33-58
  [9] G.A.K. van Voorn, L. Hemerik, M.P. Boer, and B.W. Kooi, Heteroclinic orbits in- dicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469.
  [10] M.H. Wang, and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97.
  [11] J. Wang, J. Shi, and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011)
  [12] . Groom, M.: Allee effects limit population viability of an annual plant. Am. Nat. 151, 487?496 (1998)
  [13] P.A. Stephens, W.J. Sutherland, and R.P. Freckle ton, What is the Allee effect?, Oikos, 87 (1999), 185-190.
  [14] G. Wang, X.G Liang, and F.Z Wang, The competitive dynamicsofpopulationssubject to an Allee effect, Ecol. Model., 124 (1999), 183-192.
  [15] B. Dennis. Allee effects: population growth, critical density, and the chance of extinction.
  Natural Resource Modeling, 3(4):481?538, 1989.
  [16] M. A. Lewis and P. Kareiva. Allee dynamics and the spread of invading organisms. Theoretical Population Biology, 43(2):141?158, 1993.
  [17] L. Berec, E. Angulo, and F. Courchamp. Multiple Allee effects and population management. Trends in Ecology & Evolution, 22(4):185?191, 2007.
  [18] F. Courchamp, T. Clutton-Brock, and B. Grenfell. Inverse density dependence and the Allee effect. Trends in Ecology & Evolution, 14(10):405?410, 1999
  [19] P. A. Stephens and W. J. Sutherland. Consequences of the allee effect for behavior, ecology and conservation. Trends in Ecology & Evolution, 14(10):401?405, 1999.
  [20] N. Ali. (2016). A Food Chain Model with Strong Allee Effect in Prey, International Journal of Science and Research (IJSR),ISSN (Online): 2319-7064.

  Citation
  S.T. Motuma, T.G. Beyene, "Dynamics of Predator-prey populations and The Influence of Allee effect," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.54-60, 2020

• Open Access   Article

  Proving Unproved Euclidean Propositions on a New Foundational Basis

  A. Leon

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.61-81, Jun-2020

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  Abstract
  This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid?s Postulate 1), and Axiom 8 (an extended version of Euclid?s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid`s First Postulate, Euclid`s Second Postulate, Hilbert`s Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid`s Postulate 4, Posidonius-Geminus` Axiom, Proclus` Axiom, Cataldi`s Axiom, Tacquet`s Axiom 11, Khayyam`s Axiom, Playfair`s Axiom, and an extended version of Euclid`s Fifth Postulate.

  Key-Words / Index Term
  foundation of Euclidean geometry, sidedness, straightness, orthogonality, parallelism, convergence

  References
  [1] T. Heath, ?The Thirteen Books of Euclid`s Elements,? Second ed., vol. I, Dover Publications Inc, New York, pp. 153-374, 1956.
  [2] J. Playfair, ?Elements of Geometry,? W.E. Dean Printer and Publisher, New York, pp. 5-11, 1846.
  [3] D. Hilbert, ?The Foundations of Geometry,? The Open Court Publishing Company, La Salle, pp. 2-21, 1950.
  [4] G. Birkhoff y R. Beatley, ?Basic Geometry,? American Mathematical Society, Providence, pp. 38-164, 2000.
  [5] G. Cantor, ?Contributions to the founding of the theory of transfinite numbers,? Dover Publications Inc., New York, pp. 85-208, 1955.
  [6] G. Cantor, ??ber verschiedene Theoreme asu der Theorie der Punktmengen in einem n-fach ausgedehnten stetigen Raume Gn,? Acta Mathematica, Vol. 7, pp. 105-124, 1885.

  Citation
  A. Leon, "Proving Unproved Euclidean Propositions on a New Foundational Basis," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.61-81, 2020

• Open Access   Article

  The Prime Counting Function and The nth Term of Prime Number Series

  H.V. Kumarkhaniya

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.7 , Issue.3 , pp.82-88, Jun-2020

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  Abstract
  earliest works on the prime numbers are as old as 1500 BC yet as of the present, no known simple formula can distinguish the prime numbers from the composite numbers. The number of primes not greater than any natural number z is given by a function denoted by π(z), known as prime counting function. The value of which is approximated using the prime number theorem. It can be also computed using a function based on the work of B. Riemann. This article includes, derivation of simple formula of function that counts prime upto any given number z with essential calculations and an example to understand. The presented theoretical work is also aimed to derive the general formula for the nth term of the prime number series which is an exact formula for the value of nth prime. The nth prime can be computed using the derived formula and the algorithm presented in this article.

  Key-Words / Index Term
  prime numbers, prime counting function, nth prime number, composite numbers, number theory

  References
  [1] J. Williamson (translator and commentator), ?The Elements of Euclid,? Clarendon Press, Oxford, pp. 63, 1782.
  [2] R.S. Horsley, ?The Sieve of Eratosthenes. Being an account of his method of finding all the prime numbers,? Philosophical Transactions, Vol.62, pp. 327-347, 1772.
  [3] C.F. Gauss, ?K?nigliche Gesellschaft der Wissenschaften zu G?ttingen,? Werke, Vol.2, pp. 444-447, 1863.
  [4] A.M. Legendre, ?Essai sur la th?orie des nombres,? 2nd edition, Courcier, Paris, pp. 394, 1808.
  [5] B. Riemann, ??ber die Anzahl der Primzahlen unter einer gegebenen Gr?sse,? Monatsberichte der Berliner Akademie, pp. 671-680, Nov. 1859.

  Citation
  H.V. Kumarkhaniya, "The Prime Counting Function and The nth Term of Prime Number Series," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.7, Issue.3, pp.82-88, 2020