iosr-Jm   Volume-18 ~ Issue-4 ~ Jul. – Aug. 2022

Abstract:In this paper, we developed a continuous third derivative block method using polynomial approximate solution for the solution of stiff first order initial value problems of ordinary differential equations. The development of the technique involved the interpolation and collocation of the polynomial approximate solution which give a Continuous Linear Multistep Method (CLMM). The CLMM is evaluated at some selected grid points to give discrete methods which are implemented in block form. Two cases among others are implemented, the methods are convergent and L-stable. Numerical results show that the methods are effective and computationally reliable for stiff problems.

Key Word: Stiff ODEs, Block Method, Third Derivative, interpolation and collocation, Stability

[1]. Yakubu, D. G. & Markus, S. (2016). Second derivatives of higher-order accuracy methods for the numerical integration of stiff IVPs Springers. 27, 963-977.
[2]. Yakubu, D. G. & Markus, S. (2016). Efficiency of second derivative multistep methods for the numerical integration of stiff IVPs. Journal of the Nigerian Mathematical Society. 12(5), 20-27.
[3]. Kumleng, G. M. Adee, S. O. Skwame, Y. (2013). Implicit Two Step Adam Moulton Hybrid Block Method with Two Off-Step Points for Solving Stiff Ordinary Differential Equations. Journal of Natural Sciences Research. 3(9), 77-81.
[4]. Eghige, J. O. &Okunuga, S. A. (2014). L (α)-Stable Second Derivative Block Multistep Formula for Stiff Initial Value Problems. International Journal of Applied Mathematics. 44(3),6-7.
[5]. Sharaban, T. & Azad, R. (2013). Numerical Approach for Solving Stiff Differential Equations. Global Journal of Science Frontier Research Mathematics and Decision Sciences.13 (6),7-18.

Abstract: Nirmala index of a graph is recently introduced degree based topological indexwhich is defined as N(.....

Keywords: Carbon nanocone C4[2], degree,multiplication degree, Nirmala index, reduced inverse Nirmala index,sum connectivity index. vertex degree sum

[1]. I.Gutman, J.Tosovic, Testing quality of molecular structure descriptors, vertex degree based topological indices,J.Serb.Chem.Soc.,78(6) (2013) 805-810.
[2]. M.R.R.Kanna, S.Roopa and H.L.Parashivmurthy, Topological indices of Vitamin D3, International Journal of Engineering and Technology, 7(4) (2018) 6276-6284.
[3]. E.Aslan, On the fourth atom bond connectivity index of carbon nanocones, Optoelectronics and Advanced Materials-Rapid Communications, 9(3-4) (2015) 525-527.
[4]. K.G.Mirajkar, B.R.Doddamani, Some topological indices of carbon nanocones [CNCk(n)] and Nanotori [C4C6C8(p,q)],International Journal of Scientific Research in Mathematical and Statistical Sciences,5(2) (2018) 35-39.
[5]. V.R.Kull,Nirmala index, International Journal of Mathematics Trends and Technology,63(3) (2021) 8-12.

Abstract: In-depth study of the parameters involved in every decision taken by decision-makers is very essential. Sensitivity generally concerns with inquiring the relationships between the independent and dependent variables in mathematical modeling problems. The motivation towards searching for such information can be adverse, depending on the situations. Sensitivity Analysis in transportation problems study when the predictions to be are far more reliable and in turn it allows the decision-makers to identify and take optimal decisions where the performance can be improved in the future. In this paper, the concept of sensitivity analysis on Fuzzy Transportation Problems (FTP) is carried out by applying transportation algorithms and using the ranking techniques on fuzzy numbers , optimum solutions is obtained and is related with the original fuzzy transportation problem. This study is used to identify how much variations in the input values for a given variable impact the results for a mathematical model..

Keywords: Sensitivity Analysis, Optimal Decisions, Fuzzy Transportation Problems, Trapezoidal fuzzy numbers.

[1]. Zadeh, L. A. (1996): "Fuzzy Sets, Information and Control", 8, pp. 338 -353.
[2]. Charnes. A., W, WCooper and A. Henderson, An introduction to Linear Programming, Wiley, New Work, 1953.
[3]. Chanas, D.Kuchta (1996), a concept of the optimal solution of the transportation problem with fuzzy cost coefficient, Fuzzy sets and Systems 82 pp299-305.
[4]. Sugunai Poonam, Abbas S.H., Guptha V. K. "Fuzzy Transportation Problem of Trapezoidal  -cut and Ranking Technique,
OPSEARCH, 39, no.5 and 6, pp251-266.
[5]. B. Thangaraj and M. Priyadharshini, Fuzzy transportation problem with the value and ambiguity indices of trapezoidal intutionistic fuzzy number, International Journal Mathematical Science with Computer Applicati ons, vol3, pp. 427-437, 2015.

Abstract: Adomian's decomposition method (ADM) and the variational iteration method (VIM) are well documented in the literature and have been found to be effective tools for obtaining an exact solution for initial boundary partial differential equations or, if this is not possible, for obtaining a highly accurate numerical solution. Various authors have compared the two approaches and concluded that, in general, both approaches accomplish the same result. In this study, both approaches are applied to Burgers' equation with three distinct initial boundary conditions. Adomian's decomposition method is typically computationally more challenging than the variational iteration method. VIM is typically applied to situations in which the initial condition is set to zero, and the elimination of the so-called "small terms or noisy terms" is a vital step.

Keywords: Adomian's decomposition method; Variational iteration method; Burger's equation

[1]. Adomian G. Solving Frontier Problems of Physics: The Decomposition Method. Boston, 1994.
[2]. Hetmanoik E, Slota D, Witula R, Zielonka A. Comparison of the adomian decomposition method and the variational iteration method in solving the moving boundary problem. Computers and Mathematics with Applications 2011; 61(8):1931–1934.
[3]. Abbaoui K, Cherruault Y. Convergence of adomian's method applied to differential equations. Computers and Mathematics with Applications 1994; 28(5):103–109.
[4]. AdomianG.Explicitsolutionsofnonlinearpartialdifferentialequations.Applied Mathematics and Computation1997; 88:117–126.
[5]. Basto M, Semiao V, Calheiros FL. Numerical study of modified adomian's method applied to burgers equation. Journal of Computational and Applied Mathematics 2007; 206(2):927–949.

Abstract: An S; Ic; I; R epidemic model is developed in this study to aid in the prevention and spread of diseases. The existence of a plausible region in which the disease can spread is demonstrated in this research by a careful analysis and study of the available knowledge. Quantitative study and the application of mathematical modeling techniques demonstrate that the disease has an endemic equilibrium point (EEP) and a disease free equilibrium (DFE).When the basic reproduction number R0 is less......

Keywords: Epidemic models, global stability, Lyapunov functions, Infectious diseases

[1]. Anderson Luiz Renal da Costa, Marcelo Amanajas Pires ,Rafail Lima Resque &SheyllaSusan Morena da Silva de Alemuda. Mathematical model of the infectious diseases: key concepts and Applications. Journal of Infectious Diseases and Epidemiology2021;7(5): 209.
[2]. S. Goldstein, F. Zhou, S. C. Hadler, B. P. Bell, E. E. Mast and H. S. Margolis, A mathematical model to estimate global hepatits B disease burden and vaccination impact, Int. J. Epidemiol., 34 (2005), 1329–1339.
[3]. Ghosh, M., Chandra, P., Sinha, P., and Shukla, J. B. Modelling the spread of carrier-dependent infectious diseases with environmental effect. Applied Mathematics and Computation, 2004;152(2) :385- 402.
[4]. E. O. Omondi,R.W.MbogoandL.S.Luboobi Mathematical analysis of sex-structured population model ofHIV infection in Kenya LETTERS IN BIOMATHEMATICS 2018; 5(1): 174–194
[5]. R. Naresh, S. Pandey and A. K. Misra, Analysis of a vaccination model for carrier dependent infectious diseases with environmental effects, Nonlinear Analysis: Modelling and Control,2008; 13:331–350.

Abstract: In this paper, the basic concepts of vague sets are reviewed and the concepts of vague regular alpha generalized closed sets in vague topological spaces are introduced. The basic properties of vague regular alpha generalized closed sets and their relation with other sets are discussed. Also some absorbing results are established with relevant examples.

Keywords: vague topology, vague regular α generalized closed set, vague regular α generalized open set.

[1]. V.Amarendra Babu, Ahmed Allam, T.Anitha, K.V.Rama Rao, vague topological sets and vague topological additive groups, international journal of science and innovative engineering and technology,(2017),issue volume2.
[2]. Atanssov.k, Intuitionistic fuzzy set, fuzzy set and systems,20(1986),87-96.
[3]. Borumandsaeid.A and Zarandi.A, vague set theory applied to BM- Algebras. International journal of algebra,5,5(2011),207-222.
[4]. Bustince.H, Burllio.P, vague sets are intuitionistic fuzzy sets, fuzzy sets and systems,(1996),79:403-405.
[5]. W.L.Gau and D.J.Buchrer, vague sets, IEEE transactions on systems, man and cybernetics,23,NO.20(1993),610-614..

Abstract: In this paper a mathematical model describing a within host cervical cancer infection with viral and cellular infection incorporating diffusion was formulated and analysed. The replenishment rate of the cells was represented by a logistic growth rate. The qualitative analysis of model showed that the infection dynamics can best be described by the thresh- old value R0w , in which for the value of R0w<1 the infection free equilibrium is globally asymptotically stable. This is theoretically in terpreted to means that cervical cancer is cleared from the body. On the other hand when R0w>1, the endemic equilibrium is globally asymptotically stable which implies viral persistence. The numerical results show that the movement of the virus makes the infection persist within the cells. This results in a more infected cells which implies that introduction the virus to purely uninfected cells results in propagation of the infection. Subject Classification: xxxxxx.....

Keywords: Diffusion, Human papilloma Virus, Reproduction Number, Stability Analysis

[1]. Castillo-Chavez C, Blower S, Van D. P, Krirschner D, Yakubu A. 2002.) Mathematical approaches for emerging and re-emerging infectious diseases: An introduction. The IMA Volumes in Mathematics and its Applications. Springer-Verlag, New York;
[2]. Castillo-Chavez, Feng Z. and Huang W. (2001). On the Computation of R0 and its Role on Global Stability, M-15553.
[3]. Clifford G. M., Smith, J. S., Plummer, M., Munoz, N., and Franceschi, S. (2003). Human papillomavirus types in invasive cervical cancer worldwide: a meta-analysis. British journal of cancer, 88(1), 63-73.
[4]. Feng Z., Velasco-Hernandez J., Tapia-Santos B. and Leite A.C M. (2011).A model for coupling within-host and between-host dynamics in an infectious disease. Springer Science+Business Media B.V. 2011
[5]. Mobisa B., Lawi G. O. and Nthiiri J. K. (2018). Modelling in Vivo HIV Dynamics under Combined Antiretroviral Treatment. Journal of Applied Mathematics, Volume 2018, Article ID 8276317, 11 pages
[6]. Sourisseau M., Sol-Foulon N., Porrot F., Blanchet F. and Schwartz O. (2007). Inefficient human immunodeficiency virus repli- cation in mobile lymphocytes. J Virol 81(2):1000–1012

Abstract: This study aims to identify the analysis of misconceptions in solving contextual problems of algebraic material using the Four-Tier Test for seventh grade students of SMP Negeri 1 Latambaga. The research method used is descriptive research. Sampling using a purposive sampling technique. The sample in this study consisted of 30 grade VII students of SMP Negeri 1 Latambaga. These misconceptions will be identified using diagnostic tests. The research instrument used is the Four-Tier Test. Based on the results of the study, it was found that students who identified misconceptions on all items in solving contextual problems in algebraic material had the highest percentage of 43.33%, false positive misconceptions 19.23%, and false negative misconceptions had the lowest percentage of 18.97% with 81.54 %........

Keywords: Contextual Problems; Algebraic Forms; Misconceptions;Four-Tier t test.

[1]. Dantes, N. (2014). Research Methods. Yogyakarta: CV Andi Offset.
[2]. Dzulfikar, A., &Vitantri, CA (2017). Mathematics Misconceptions in Elementary School Teachers. Suska Journal of Mathematics Education, 3 (1), 41–48. https://doi.org/10.24014/sjme.v3i1.3409
[3]. Fariyani, Q., Rusilowati, A., &Sugianto. (2015). Development of a Four-Tier Diagnostic Test to Reveal the Physics Misconceptions of Class X High School Students. Journal of Innovative Science Education, 4 (2), 41–49. https://journal.unnes.ac.id/sju/index.php/jise/article/view/9903
[4]. Fratiwi, NJ, Kaniawati, I., Suhendi, E., Suyana, I., &Samsudin, A. (2017). The Transformation of Two-Tier Test into Four-Tier Test on Newton's Laws Concepts. AIP Conference Proceedings, May, 1–5. https://doi.org/10.1063/1.4983967
[5]. Gurel, DK, Eryilmaz, A., & McDermott, LC (2015). A review and comparison of diagnostic instruments to identify students' misconceptions in science. Eurasia Journal of Mathematics, Science and Technology Education, 11 (5), 989–1008. https://doi.org/10.12973/eurasia.2015. 1369a

Abstract: Popa[6] explored the common fixed point forsemi-compatible maps choosing the family F4 of implicit real functionsin d-complete topological space. A class of implicit relationis used by Singh and Jain[8] to prove a common fixed point theorem in fuzzy metric space.In fuzzy normed spaces, Singh et al.[7] proved fixed point theoremfor two self-maps and Chauhan etal.[1] proved a result for common fixed point of four self-maps using the concept of compatibility. Main result of this paper presents a common fixed point theorem for six self-maps in fuzzy normed space employing a class of implicit relation.The result of Singh et al.[7], Popa[6], Singh and Jain[8] and Chauhan etal.[1] are generalized in this paper......

Keywords: Common fixed point, F- normed spaces or fuzzy normed space, weakly compatible mappings and semi-compatible, compatible mappings.

[1]. Chauhan, M. S., Badshah, V.H. and Sharma, D., Common fixed point theorems in Fuzzy normed space, Intl. Jour. Of Theoretical and Appl. Sci. 3(1),(2011),28-30.
[2]. George, A., Studies in fuzzy metric spaces, Ph.D. Thesis, Indian Institute of Technology, Madras.
[3]. Jose, M. and Santiago, M. L., Closed graph theorem in fuzzy normed spaces, Bull. Cal. Math. Soc. , 91(5), (1999), 375-378.
[4]. Jungck, G., Common fixed points for noncontinuous nonself maps on non-metric spaces, Far East J. Math. Sci., 4 (2) (1996), 199- 215.
[5]. Mishra, S.N., Sharma, S.N. and Singh, S.L., Common fixed point of maps in fuzzy metric spaces, Intl. Jour. Of Math. and Math. Sci. 17 (1994), 253-258.