Journal Vol – 19 No – 4, April 2024



Arun Dubey, Dilip Kumar Jaiswal, Gulrana, A. K. Thakur



The stabilization of groundwater resources in excellent quality is crucial for both the environment and human societies. To examine the contaminant concentration pattern of infinite and semi-infinite aquifers, mathematical models provide accurate descriptions. The two-dimensional model for a semi-infinite heterogeneous porous medium with temporally dependent and space-dependent (degenerate form) dispersion coefficients for longitudinal and transverse directions is derived in this study. The Laplace Integral Transform Techniques (LITT) is used to find analytical solutions. The dispersion coefficient is considered the square of the velocity which represents the seasonal variation of the year in coastal/tropical regions. To demonstrate the solutions, the findings are presented graphically. Figures are drawn for different times for a function and discussed in the result and discussion section. It is also concluded that a two-dimensional model is more useful than a one-dimensional model for assessing aquifer contamination.


2-D Advection-dispersion equation,Aquifer,Heterogeneity,Pollution,Laplace transform,


I. A. Kumar, D. K. Jaiswal, N. Kumar,: ‘Analytical solutions to one dimensional advection-diffusion equation with variable coefficients in semi—infinite media’. J Hydrol., 380:330–337.(2010)

II. A. Sanskrityayn, N. Kumar,: ‘Analytical solution of ADE with temporal coefficients for continuous source in infinite and semi-infinite media’J. Hydrol. Eng. 23 (3). 06017008.(2018). 10.1061/(ASCE)HE.1943-5584.0001599.

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IV. C. K. Thakur, M. Chaudhary, van der Zee S.E.A.T.M., and M. K.Singh, : ‘Two-dimensional solute transport with exponential initial concentration distribution and varying flow velocity’, Pollution, 5(4), 721-737. (2019) 10.22059/poll.2019.275005.574

V. D. K. Jaiswal, A. Kumar and N. Kumar, : ‘Discussion on ‘Analytical solutions for advection-dispersion equations with time-dependent coefficients by Baoqing Deng, Fie Long, and Jing Gao.’ J. Hydrol. Eng. 25 (8): 07020012 (1-2).(2020).

VI. D. K. Jaiswal, A. Kumar, N. Kumar and M. K. Singh, : ‘Solute transport along temporally and spatially dependent flows through horizontal semi-infinite media: dispersion being proportional to square of velocity.’ J Hydrol. Eng. 16(3) : 228–238. (2011)

VII. D. K. Jaiswal, A. Kumar, N. Kumar and R. R. Yadav, : ‘Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media.’ J Hydro-environ Res 2: 254-263.(2009).

VIII. D. K. Jaiswal and Gulrana, ; ‘Study of Specially and Temporally Dependent Adsorption Coefficient in Heterogeneous Porous Medium.’ Appli and Applied Mathe: An Int Journal (AMM) 14 (1):485-496.(2019).

IX. D. K. Jaiswal, N. Kumar and R. R. Yadav, : ‘Analytical solution for transport of pollutant from time-dependent locations along groundwater.’ J. Hydro., 610.(2022).

X. J. Crank, : ‘The mathematics of diffusion.’ Oxford University Press, UK. (1975).

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XIX. D. K. Jaiswal, A. Dubey, V. Singh and P. Singh, : ‘Temporally Dependent Solute Transport in One-Dimensional Porous Medium: Analytical and Fuzzy Form Solutions.’ Mathematics in Engineering Science and Aerospace, 14(3), 711-719.(2023).

XX. P. Singh, P. Kumari and D. K Jaiswal,:‘An Analytical model with off diagonal impact on Solute Transport in Two-dimensional Homogeneous Porous Media with Dirichlet and Cauchy type boundary conditions.’GANITA, Vol.72(1), 299-309.(2022).

XXI. P. Singh, S. K. Yadav and N. Kumar, : ‘One-Dimensional Pollutant’s Advective-Diffusive Transport from a Varying Pulse-Type Point Source through a Medium of Linear Heterogeneity.’ J. Hydrol. Eng, 17(9): 1047–1052. (2012).

XXII. P. Singh, S. K. Yadav, O. V. Perig, : ‘Two-dimensional solute transport from a varying pulse type point source Modelling and simulation of diffusive processes.’ 211-232, Springer.(2014).

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XXIV. R. Kumar, A. Chatterjee, M. K.Singhand V. P. Singh, : ‘Study of solute dispersion with source/sink impact in semi-infinite porous medium.’ Pollution, 6(1),87-98,(2020). 10.22059/poll.2019.286098.656

XXV. R. R. Rumer, : ’Longitudinal dispersion in steady and unsteady flow.’ J Hydraul. Div. 88:147–173.(1962).

XXVI. R. R. Yadav and L. Kumar, : ‘Solute Transport for Pulse Type Input Point Source along Temporally and Spatially Dependent Flow.’ Pollution, 5(1): 53-70.(2019).

XXVII. R. R. Yadav, D. K. Jaiswal and Gulrana, : ‘Two-Dimensional Solute Transport for Periodic Flow in Isotropic Porous Media: An Analytical Solution.’ Hydrol Process. 26 (12):3425-3433.(2011). DOI: 10.1002/hyp.8398.

XXVIII. R. R. Yadav, D. K. Jaiswal, H. K. Yadav and Gulrana, : ‘Analytical solutions for temporally dependent dispersion through homogeneous porous media.’ Int. J. Hydrology Science and Technology, Vol. 2, No. 1, pp.101–115.(2012).

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XXXI. Y. Sun, A. S. Jayaraman, and G. S. Chirikjian, : ‘Lie group solutions of advection-diffusion equations.’ Phys. Fluids 33, 046604 (2021); 10.1063/5.0048467.

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A. P. Pushpalatha, S. Suganthi



A simple, finite and connected graph is denoted by G=(V,E). The primary Zagreb index, denoted as M1(G), characterizes the graph topologically by representing a squared degree sum of their vertices. Similarly, M2(G) denotes a second Zagreb index, that offers a topological measure of summing the degree of the product for adjacent vertices of graph G. We investigate a study of this topological indices M1(G)&M2(G) and got some interesting results also.


Zagreb indices,first Zagreb index,second Zagreb index,Fan graph,Barbell graph,Thorn graph,


I. Akhtar, S., Imran, M., Gao, W. and Farahani, M. R., : ‘On topological indices of honeycomb networks of graphene networks. Hacet.’ J. Math. Stat. 2018, 47(1) 19-35. 10.15672/HJMS.2017.464

II. Balaban A. T., Motoc I., Bonchev D., Mekenyan O., : ‘Topological indices for structure activity correlations.’ Topics Curr Chem,1983, 114:21-55
III. Das K. C., : ‘On comparing Zagreb indices of graphs.’ MATCH commun math comput Chem, 2010, 63: 433-440.

IV. Das, K. C, : ‘Maximizing the sum of the squares of the degrees of a graph.’ Discrete Math., 285, (2004), 57–66.

V. Das, K. C., Gutman, I. and Zhou, B., : ‘New upper bounds on Zagreb indices.’ J. Math. Chem., 46, (2009), 514–521.

VI. De, N., : ‘The vertex Zagreb index of some graph operations.’ Carpathian Math. Publ. 2016, 8(2), 215-223.

VII. Eliasi M, Iranmanesh A, Gutman I., : ‘Multiplicative versions of first Zagreb index., MATCH Commun Math comput chem, 2012, 68:217-230.

VIII. Farahani, M. R. and Kanna, M. R. (2015), : ‘Generalized Zagreb Index of V-Phenylenic Nanotubes and Nanotori.’ Journal of Chemical and Pharmaceutical Research, 7(11), 241-245.

IX. Gutman I., : ‘Multiplicative Zagreb indices of trees. Bull Soc Math Banja luka, 2011, 18:17-23.

X. Gutman I., Das K. C., : ‘The first Zagreb index 30 years after.’ MATCH Commun Math Comput Chem, 2004, 50: 83-92.

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XIV. Khalifcha, M. H., : ‘Yousefi-Azaria, H., Ashrafi, A. R., : ‘The first and second Zagreb indices of some graph operations.’ Discret. Appl. Math. 2009, 157, 804-811.

XV. Kexiang XU, : ‘The Zagreb indices of graphs with a given clique number.’ Applied Mathematics Letters , Volume 6, Issue 11 (2011), Pages1026-

XVI. Kinkar Ch. Das, Kexiang XU, Junki Nam., : ‘Zagreb indices of graphs’ Frontiers of Mathematics 2015.
XVII. K. C. Das, I. Gutman and B. Horoldagva (2012). : ‘Comparison between Zagreb indices and Zagreb coindices.’ MATCH Commun. Math. Comput. Chem., 68, pp.189 – 198
XVIII. K. C. Das, I. Gutman and B. Zhou (2009). : ‘New Upper Bounds on Zagreb Indices.’ J. Math. Chem.,46, pp. 514 – 521.

XIX. Lokesha, V., Deepika, T., : ‘Symmetric division deg index of tricyclic tetracyclic graphs.’ Int. J. Sci. Eng. Res.2016 ,7(5), 53-55.

XX. R. Pradeep Kumar, Soner Nandappa D., M. R. Rajesh Kanna., : ‘Redefined Zagreb, Randic, harmonic and GA indices of graphene.’ International Journal of Mathematical Analysis Vol.11, (2017), no.10, 493- 502. 10.12988/ijma.2017.7454

XXI. Sridhara G., Kanna , M. R. R. and Indumathi, R. S. : ‘Computation of topological indices of graphene.’ J. Nanometrial (2015) ID 969348.

XXII. K. Thilagavathi and A. Sangeetha Devi, : ‘Harmonious coloring and Proceedings of International Conference on Mathematical and Computer Science.’ Department of Mathematics Loyola College Chennai. (ICMCS 2009) Page no 50-52.

XXIII. F. Harary, : ‘Graph Theory.’ Addision Wesley, Reading Mass (1972).

XXIV. Yan, Z., Liu, H. and Liu, H., : ‘Sharp bounds for the second Zagreb index of unicyclic graphs.’ J. Math. Chem., 42, (2007), 565–574.

XXV. Zhou, B. and Gutman, I., : ‘Further properties of Zagreb indices.’ MATCH Commun. Math. Comput. Chem., 4, (2005),233–239.

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Nasir Uddin, Md. Eaqub Ali, Anish Kumar Adhikary, Shuvo Sarker, M. Ali Akbar, Pinakee Dey6



The existence of over-damped nonlinear differential equations results from a variety of engineering conundrums and physical natural occurrences. Non-oscillatory dynamics with forced over-damping are used in the simulation of nonlinear differential systems. For non-oscillatory nonlinear differential systems, it is possible to derive approximations of solutions using a variety of analytical methods, both with and without external forcing. This paper introduces a novel method for estimating solutions for highly nonlinear damped vibration systems subject to parameterized external forcing. The extended Krylov-Bogoliubov-Mitropolsky (KBM) technique and harmonic equilibrium (HM), which have both been previously developed in the literature, are the foundation of the suggested method. This method was initially created by Krylov-Bogoliubov to discover periodic details in second-order nonlinear differential equations. Several examples are provided to show how the suggested technique is applied. The process is fairly simple and straightforward, and using this formula, the result can be found with very marginal errors from the previous citations. The primary significance of this approach is in its ability to provide approximate analytical solutions of the first order that closely align with the findings obtained by numerical methods. These solutions are applicable to a variety of beginning scenarios and are distinct from those presented in earlier literature. Also, we illustrated the two-dimensional graph of all the solutions that we got in this article by using the data from the mentioned table. The results that we obtained from this method are effective and reliable for better measurements of strong nonlinearities.


Nonlinear non-autonomous system,Damped nonlinear system,External force Vary with time,Perturbation equation,


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III. G. M. Ismail, M. Abul-Ez, N. M. Farea, and N. Saad, : ‘Analytical approximations to nonlin ear oscillation of nanoelectro-mechanical resonators.’ Eur. Phys. J. Plus., vol. 134, no. 47, 2019.

IV. H. M. A., Chowdhury. M. S. H., Ismail G. M., and Yildirim A., : ‘A modified harmonic balance method to obtain higher-order approximations to strongly nonlinear oscillators.’ J. Interdiscip Math, vol. 23, no. 7, pp. 1325–1345, 2020. 10.1080/09720502.2020.1745385

V. H.-O.- Roshid, M. Z. Ali, P. Dey, and M. A. Akbar, : ‘Perturbation Solutions to Fifth Order Over-damped Nonlinear Systems.’ J. Adv. Math. Comput. Sci., vol. 32, no. 4, pp. 1–11, 2019, 10.9734/jamcs/2019/v32i430151.

VI. I. S. N. Murty, B. L. Deekshatulu, and G. Krisna. : ‘On an asymptotic method of Krylov-Bogoliubov for overdamped nonlinear systems.’ J. Frank Inst., vol. 288 (1), pp. 49–65, 1969. 10.1016/0016-0032(69)00203-1

VII. K. N. N. and N.N., Bogoliubov, : ‘Introduction to Nonlinear Mechanics.’ Princet. Univ. Press. New Jersey., 1947.

VIII. L. Cveticanin and G.M. Ismail, : ‘Higher-order approximate periodic solution for the os cillator with strong nonlinearity of polynomial type.’ Eur. Phys. J. Plus., vol. 134, 2019.

IX. L. M. J. Lu. ‘The VIM-Pad´e technique for strongly nonlinear oscillators with cubic and harmonic restoring force.’ J. Low Freq. Noise, Vib. Act. Control. 2018. 10.1177/1461348418813612

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XV. M. W. Ullah, M. S. Rahman, and M. A. Uddin. : ‘A modified harmonic balance method for solving forced vibration problems with strong nonlinearity.’ J. Low Freq. Noise, Vib. Act. Control., vol. 40, no. 2, pp. 1096 – 1104, 2021. 10.1177/1461348420923433

XVI. M. Yu., : ‘Problems on Asymptotic Method of non-stationary Oscillations’ (in Russian). 1964.

XVII. M. Shamsul Alam, : ‘Unified Krylov-Bogoliubov-Mitropolskii method for solving n-th order nonlinear system with slowly varying coefficients.’ J. Sound Vib., vol. 265 (5), pp. 987–1002, 2003.

XVIII. N. A. H, : ‘Perturbation Methods.’ J. Wiley, New York, 1973.

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XX. N. Sharif, Abdur Razzak, and M. Z. Alam, : ‘Modified harmonic balance method for solving strongly nonlinear oscillators.’ J. Interdiscip., vol. 22, no. 3, p. 353-375, 2019. 10.1080/09720502.2019.1624304

XXI. P. Dey, H. or Rashid, A. A. M, and U. M, S., : ‘Approximate Solution of Second Order Time Dependent Nonlinear Vibrating Systems with Slowly Varying Coefficients.’ Bull. Cal. Math. Soc, vol. 103, no. 5, pp. 371–38, 2011.
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XXIII. P. Dey, N. Uddin, and M. Alam. ‘An Asymptotic Method for Over-damped Forced Nonlinear Vibration Systems with Slowly Varying Coefficients.’ Br. J. Math. Comput. Sci., vol. 15, no. 3, pp. 1–8, 2016, 10.9734/bjmcs/2016/24531.

XXIV. P. Dey, S. M. A., and Z. A. M. : ‘Perturbation Theory for Damped Forced Vibrations with Slowly Varying Coefficients.’ J. Adv. Vib. Eng., vol. 9, no. 4, pp. 375–382, 2010.

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XXVIII. U. MW, R. MS, and U. MA., : ‘A modified harmonic balance method for solving forced vibration problems with strong nonlinearity.’ Vib. Act. Control, vol. 40, no. 2, p. 146134842092343, 2020.

XXIX. W. UV and L. L., : ‘On the detection of artifacts in harmonic balance solutions of nonlinear oscillators.’ Appl Math Model., vol. 65, pp. 408–414, 2019.

XXX. Z.L. Tao, G. H. Chen, and K.X. Bai, : ‘Approximate frequency-amplitude relationship for a singular oscillator.’ J. Low Freq. Noise, Vib. Act. Control. 2019. 10.1177/1461348419828880

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Arijit Mishra, Pinku Chandra Dey, Kamal Jyoti Barman



Let G be any graph. Then a one-one function f:V→ N is said to be a k-Zumkeller labeling of G if the induced function f^*: E→N defined by f^* (xy) =f(x)f(y) satisfies the following conditions: (i) For every xy∈E, f^* (xy) is a Zumkeller number. (ii) |f^* (E)|=k, where |f^* (E)| denotes the number of distinct Zumkeller numbers on the edges of G. In this paper, we prove the existence of k-Zumkeller labeling for certain graphs like tadpole, banana, friendship, and firecracker graphs.


Zumkeller number,banana graph,friendship graph,firecracker graph,tadpole graph,graph labeling.,


I. B. J. Balamurugan, K. Thirusangu, DG Thomas (2013), : ‘Strongly multiplicative Zumkeller labeling of graphs’. International Conference on Information and Mathematical Sciences, Elsevier, 349-354.
II. B. J. Balamurugan, K. Thirusangu, DG Thomas (2014), : ‘Zumkeller labeling of some cycle related graphs’. Proceedings of International Conference on Mathematical Sciences (ICMS – 2014), Elsevier, 549-553.
III. B. J. Balamurugan, K. Thirusangu and D.G. Thomas, : ‘Zumkeller labeling algorithms for complete bipartite graphs and wheel graphs.’ Advances in Intelligent Systems and Computing, Springer, 324 (2014), 405-413. 10.1007/978-81-322-2126-5_45
IV. B. J. Murali, K. Thirusangu, R. Madura Meenakshi, : ‘Zumkeller cordial labeling of graphs’. Advances in Intelligent Systems and Computing, Springer, 412 (2015), 533-541.
V. B. J. Balamurugan, K. Thirusangu and D.G. Thomas, : ‘Algorithms for Zumkeller labeling of full binary trees and square grids’. Advances in Intelligent Systems and Computing, Springer, 325 (2015), 183192.
VI. B. J. Balamurugan, K. Thirusangu and D.G. Thomas, : ‘k-Zumkeller Labeling for Twig Graphs’. Electronic Notes in Discrete Mathematics 48 (2015) 119126.
VII. F. Harary, : in Graph theory, Addison-Wesley, Reading Mass (1972).
VIII. I. Cahit, : ‘On cordial and 3-equitable labeling of graph’. Utilitas Math., 370 (1990), 189-198.
IX. J.A. Gallian, : ‘A dynamic survey of graph labeling’. Electronic J. Combin., 17 (2014), DS6.
X. Rosa, : ‘On certain valuations of the vertices of a graph’. N. B. Gordan and Dunad, editors, Theory of graphs, International Symposium, Paris (1966) 349359.
XI. S. Clark, J. Dalzell, J. Holliday, D. Leach, M. Liatti and M. Walsh, : ‘Zumkeller numbers’. Mathematical Abundance conference at Illinois State University, 18.04.2018.
XII. Y. Peng and K. P. S. Bhaskara Rao, : ‘On Zumkeller numbers’. J. Number Theory, 133(4) (2013), 1135-1155. 10.1016/j.jnt.2012.09.020

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M. Sathish Kumar, G. Veeramalai, S. Janaki, V. Ganesan



In this article, we examine the oscillation of a class of third-order damped nonlinear differential equations with multiple delays. Using the integral average and generalized Riccati techniques, new necessary criteria for the oscillation of equation solutions are established. The major effect is exemplified by an example.


Oscillation,nonlinear differential equations,third-order,delay arguments,damping,


I. A. Tiryaki, M. F. Aktas,: ‘Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping.’ Journal of Mathematical Analysis and Applications, 325 (2007), 54-68. 10.1016/j.jmaa.2006.01.001
II. C. S. Bose, R. Udhayakumar, A. M. Elshenhab, M. S. Kumar, J. S. Ro,: ‘Discussion on the Approximate Controllability of Hilfer Fractional Neutral Integro-Differential Inclusions via Almost Sectorial Operators’. Fractal and Fractional, 6(10), p.607.
III. G. S. Ladde, V. Lakshmikantham, B. G. Zhang,: ‘Oscillation Theory of Differential Equations with Deviating Arguments’. Monographs and Textbooks in Pure and Applied Mathematics 110, Marcel Dekker, New York, 1987.
IV. J. K. Hale. : ‘Theory of Functional Differential Equations’. Springer: New York, NY, USA, 1977.
V. M. Bohner, S.R. Grace, I. Sager, E. Tunc,: ‘Oscillation of third-order nonlinear damped delay differential equations’. Applied Mathematics and Computation, 278 (2016) 21-32. 10.1016/j.amc.2015.12.036
VI. M. Bohner, S.R. Grace, I. Jadlovska,: ‘Oscillation Criteria for Third-Order Functional Differential Equations with Damping’. Electronic Journal of Differential Equations, 2016 (215), 1-15.
VII. M. H. Wei, M. L. Zhang, X. L. Liu, Y. H. Yu. : ‘Oscillation criteria for a class of third order neutral distributed delay differential equations with damping’. Journal of Mathematics and Computer Science, 19 (2019), 19–28. 10.22436/jmcs.019.01.03
VIII. M. S. Kumar, O. Bazighifan, A. Almutairi, D. N. Chalishajar. : ‘Philos-type oscillation results for third-order differential equation with mixed neutral terms’, Mathematics, 9 (2021), ID 1021. 10.3390/math9091021
IX. M. Sathish Kumar, O. Bazighifan, Al-Shaqsi, F. Wannalookkhee, K. Nonlaopon,: ‘Symmetry and its role in oscillation of solutions of third-order differential equations’, Symmetry, 13, No 8, ID 1485;
X. M. Sathish Kumar, S. Janaki, V. Ganesan. : ‘Some new oscillatory behavior of certain third-order nonlinear neutral differential equations of mixed type’. International Journal of Applied and Computational Mathematics, 78 (2018), 1-14. 10.1007/s40819-018-0508-8
XI. M. Sathish Kumar, V. Ganesan. : ‘Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay’. AIMS Mathematics, 5, No 4, (2020), 3851-3874; 10.3934/math.2020250
XII. M. Sathish Kumar, V. Ganesan. : ‘Oscillatory behavior of solutions of certain third-order neutral differential equation with continuously distributed delay’. Journal of Physics: Conference Series, 1850, No 1 (2021), ID 012091. 10.1088/1742-6596/1850/1/012091
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XIV. S. K. Marappan, A. Almutairi, L. F. Iambor, O. Bazighifan. : ‘Oscillation of Emden–Fowler-type differential equations with non-canonical operators and mixed neutral terms’. S ymmetry, 15(2) (2023), p.553. 10.3390/sym15020553
XV. S. R. Grace, J. R. Graef, E. Tunc. : ‘On the oscillation of certain third order nonlinear dynamic equations with a nonlinear damping term’. Mathematica Slovaca, vol. 67, no. 2, 2017, pp. 501-508.
XVI. S. R. Grace. : ‘Oscillation criteria for third order nonlinear delay differential equations with damping’. Opuscula Mathematica, 35, no. 4 (2015), 485–497. 10.7494/OpMath.2015.35.4.485
XVII. Y. Sun, Y. Zhao, Q. Xie. : ‘Oscillation and Asymptotic Behavior of the Third-Order Neutral Differential Equation with Damping and Distributed Deviating Arguments’. Qualitative Theory of Dynamical Systems, 22, 50 (2023). 10.1007/s12346-022-00733-4
XVIII. Y. Wang, F. Meng, J. Gu. : ‘Oscillation criteria of third-order neutral differential equations with damping and distributed deviating arguments’. Advances in Difference Equations, 2021, 515 (2021).

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Asish Mitra, Soumya Sonalika



In the present study, we introduce a simple stochastic differential equation based on the Susceptible-Infectious (SI) model to simulate the progression of COVID-19. For a detailed study, a cumulative number of individuals infected with COVID-19 in Norway from 26 Feb 2020 to 09 March 2023 is utilized. The Euler-Maruyama (EM) method is used to solve the problem. Computer codes are developed in Matlab for the solution process.


Brownian Motion,Covid-19,Epidemiology,Euler-Maruyama (EM) Method,Stochastic Differential Equation (SDE),


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Kamal Jyoti Barman, Arijit Mishra



Indian cities are extending and growing very rapidly with the increase in population. As a result, there is a need to implement mass transit systems such as metro rail to meet their day-to-day mobility requirements. In recent years metro rail has grown in many Indian cities. Much like a graph that is made up of vertices and edges, a metro network is composed of stations and a metro route connecting them, where each station represents a vertex and any two vertices are adjacent whenever there is a link (metro route) between them. In this paper, we try to study the structure of a metro network via a graph theoretical approach.


Mass transit systems,Metro network,Metro network graph,


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II. S. K. Bisen. : ‘Graph theory use in transportation problems and railway networks’. International journal of science and research, 2017, Vol-6 (5), 1764-1768.

III. S. Stoilova and V. Stoev. : ‘An application of graph theory which examines the metro networks’. Transport Problems, 2015, vol-10 (2), 35-48.

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