iosr-Jm   Volume-17 ~ Issue-4 ~ Jul. – Aug. 2021

Abstract: In this paper we have solved initial value problem (IVP) for ordinary differential equations (ODE) by using Runge-Kutta fourth order method only. Despite various analytical methods for finding the solution of initial value problems, there are large number of ordinary differential equations which have no analytical solutions. In that case we have to solve ordinary differential equations by using numerical methods. There are numerous numerical methods such as Euler's, Heun's, Runge-Kutta third order and Runge-Kutta fourth order methods. Runge-Kutta fourth order method is the powerful numerical technique to solve initial value problems. In this paper different four examples have been solved by using MATLAB programming and numerical results have been shown in tables and graphs.

Key words: Initial value problem, ordinary differential equation, Runge-Kutta fourth order method and MATLAB programming.

[1]. Arefin, M.A. Gain, B. Karim R. and Hossain. S. (2020). A Comparative Exploration on Different Numerical Methods for Solving Ordinary Differential Equations, Journal of Mechanics of Continua and Mathematical Sciences, Vol. 15, pp. 1-11.
[2]. Gowri, P. Priyadharsini S. and Maheswari. T. (2017). A case study on Runge-Kutta 4th order differential equations and Its application, Impeial Journal of Interdisciplinary Research, 3
[3]. Hossen, M. Ahemed, Z. Kabir R. and Hossain. Z. (2019). A Comparative Investigation on Numerical Solution Of Initial Value Problem by Using Modified Euler's Method and Runge-Kutta Method, IOSR Journal of Mathematics, Vol. 15, pp. 40-45.
[4]. Islam. M.A. (2015). Accuracy Analysis of Numerical solutions of initial value problems (IVP) for ordinary differential equations (ODE), IOSR Journal of Mathematics, Vol. 11(3), 18-23.
[5]. Islam. M.A. (2015). Accurate solutions of initial value problems for ordinary differential equations with the fourth order Runge-Kutta method, Journal of Mathematics Research, Vol. 7(3), pp. 41..

Abstract: Cholera is an acute diarrheal disease caused by vibro-cholerae bacteria and the outbreak can occur in a situation where water supply, sanitation, food safety and hygiene are insufficient. We developed an epidemic model of SIQR-B, or Susceptible- Infectious-Quarantined-Recovered and Bacteria, type model for cholera infection. We incorporate control measures of treatment in quarantine and vaccination. The effective reproduction number is computed in terms of model parameters. The existence and stability of disease free and endemic steady statesare recognized and the stead states indicated to be locally and globally asymptotically stable whenever effective reproduction.......

Keywords: Cholera, Dynamical systems, Reproduction number, stability,Vaccination

[1]. Alexander Kramer et al. (2010).Modern Infectious Disease Epidemiology. Springer Science+Business Media, LLC 2010
[2]. C. E. Madubueze, S. C. Madubueze, S. Ajama. (2015). Bifurcation and Stability Analysis of the Dynamics of Cholera Model with Controls. International Scholarly and Scientific Research & Innovation 9(11)
[3]. E. Bertuzzo1, R. Casagrandi, M. Gatto, I. Rodriguez-Iturbe and A. Rinaldo. (2009). On spatially explicit models of cholera epidemics J. R. Soc. Interface (2010) 7, 321–333.
[4]. Ezekiel DangbemDamakoaIrépran, Antoine Perassoc, David Bekolle. (2017). Mathematical modelling and numerical simulations of the influence of hygiene and seasons on the spread of cholera. Mathematical Biosciences 296 (2018) 60–70
[5]. Jing'an Cui, ZhanminWu, and Xueyong Zhou. (2014). Mathematical Analysis of a Cholera Model with Vaccination. Journal of Applied Mathematics. 2014(324767), 16 pages.

Abstract: The use of neural network architecture in deep learning models is called as Artificial Neural Network (ANN). It is one of the most powerful machine learning algorithms applied to tasks across many domains. (finance, humanities, science. research and academics etc.). An ANN is a form of computation inspired by the structure and function of brain. [ Padhy, 2005] In this paper, we concentrate on the fundamentals of human neurons and how they are applied to artificial neurons to understand the principles of ANNs.

Keywords: Artificial Neural Network (ANN), Neural Network (NN) , artificial neuron

[1]. Padhy, N.P. , "Artificial Intelligence and Intelligent Systems", ISBN-13: 978-0-19-567154-4,400-437,2005
[2]. Han, J., Kamber, M., "Data Mining: Concepts and Techniques" ISBN:1-55860-489-8, 310-311 , 2001
[3]. Tsoukalas, L.H. and Uhring , R.E., " Fuzzy and Neural Approaches in Engineering", John Wiley , New York1996..

Abstract: The research sought to offer from Basic Secondary a basic geometric knowledge from the key informants to form principles for the teaching of quadrilaterals and their relationship with the real environment of sixth grade students that allows them to function in everyday life to guide you in the resolution of problems that includes knowledge about quadrilaterals. The methodology used was qualitative, based on the hermeneutical phenomenological method. The key informants were five (5) mathematics teachers who work at the "Francisco José Caldas" Educational Institution, in Colombia. Dialogues were used as information gathering techniques and instruments. Some didactic principles are reflected for the teaching of the quadrilaterals extracted from the information given by the informants.

Keywords: Teaching of quadrilaterals, Geometric knowledge, Daily life, Resolution of problems, Real environment.

[1]. Arrieche, B. (2019). Formación inicial del profesor de Educación Primaria en el área de Geometría y su Didáctica. Tesis doctoral no publicada, Universidad Pedagógica Experimental Libertador, Instituto Pedagógico "Rafael Alberto Escobar Lara" de Maracay.
[2]. Camargo, L y Acosta. M. (2012). La Geometría, su Enseñanza y su Aprendizaje. Rev. Fac. Cienc. Tecno (on line). 1-8.
[3]. Cantoral, R., Reyes, D., y Montiel, G. (2014). Socioepistemología, Matemáticas y Realidad. Revista Latinoamericana de Etnomatemática, 7, (3), 91-116.
[4]. Ferrés, J. (2001). Pedagogía de los medios audiovisuales y pedagogía con los medios audiovisuales. En J. Sancho (Coord.), Cuadernos para el 269: Análisis Para una Tecnología Educativa (pp. 115-142). Barcelona: Horsori Editorial.
[5]. Graterol, J. (2012). Hablando sobre enseñanza de la matemática con estudiantes futuros profesores de matemática. Números. 80, 119-134..

Abstract:We solve an open question proposed by Chang, Chuang, and Chen that was published in the International Journal of Production Economics, entitled "Short comments on technical note – The EOQ and
EPQ models with shortages derived without derivatives". Our solution approach has a unique character in that we find the minimum value before we derive the minimum point. Our results will be useful for researchers to develop their solution process from an algebraic point of view.

Key words: Economic Production Quantity; Economic Order Quantity; Shortage; Inventory Model

[1]. Càrdenas-Barrón, L.E., The economic production quantity (EPQ) with shortage derived algebraically, International Journal of Production Economics, Vol. 70, (2001), pp. 289–292.
[2]. Chang, S.K.J., Chuang, J.P.C., Chen, H.J., Short comments on technical note – The EOQ and EPQ models with shortages derived without derivatives, International Journal of Production Economics, Vol. 97, (2005), pp. 241-243.
[3]. Deng, P.S., Yen, C.P., Tung, C.T., Yu, Y.C., Chu, P., A technical note for the deteriorating inventory model with exponential timevarying demand and partial backlogging, International Journal of Information and Management Sciences, 17 (2), (2006), pp. 101- 108.
[4]. Deng, P.S., Improved inventory models with ramp type demand and Weibull deterioration, International Journal of Information and Management Sciences, 16 (4), (2005), pp. 79-86.
[5]. Deng, P.S., Yang, G.K.L., Chen, H.J., Chu, P., Huang, D., The criterion for the optimal solution of inventory model with stockdependent consumption rate, International Journal of Information and Management Sciences, 16 (2), (2005), pp. 97-109.

Abstract: The transportation problem deals with the concept of optimization of resources to minimize the cost that is incurred in transporting goods from the place of supply to the place of demand. Various traditional methods are used to solve this problem and allocate optimal resources to minimize the transportation cost. In this paper, we have proposed a new method, which uses NVM Algorithm, to solve a well-established transportation problem efficiently. NVM algorithm was tested and compared with various traditional methods and recently developed methods discussed in this paper, with 20 problems chosen from various research papers (from 1974-2019) and 2 problems with larger instances were randomly generated and the results came out to be optimal for most of the cases. The experimental....

Key Word: Transportation problem; Linear programming; NVM algorithm; Optimal solution; Initial basic feasible solution (IBFS).

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