differential equation
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2022 ◽  
Vol 143 ◽  
pp. 104598
Yota Togashi ◽  
Kazuki Mizuo ◽  
Masahiko Osada ◽  
Tadashi Yamabe ◽  
Hiroshi Kameya

Sneha Dey ◽  
A. Ghorai ◽  

Exploration of dynamics of raindrops is one of the simple yet most complicated mechanical problems. Mass accretion from moist air during the motion of raindrop through resistive medium holds an arbitrary power law equation. Its integral part is the change of shape, terminal motions and terminal solutions, etc. Classical Newtonian formalism is used to formulate a mathematical model of generalized first order differential equation. We have discussed about the terminal velocity of raindrop and its variation with the extensive use of python program and library. It is found that terminal velocity 𝐯𝐓𝐜𝛂𝛃 is achieved within 20 seconds where 𝛂=, 𝛃=(𝟎,𝟏) and 𝐧=𝟎,𝟏,𝟐,𝟑,𝟒,…. Its variations due to mass accretion roughly follows the earlier predicted range 𝐠/𝟕 to 𝐠/𝟑.

Sneha Dey ◽  
Dr. A. Ghorai ◽  

The mass accretion of a raindrop in different layers of the atmosphere is not dealt with so far. A comprehensive brief study of the motion of raindrops through the atmosphere (i) without mass accretion, (ii) with mass accretion and (iii) finally pressure variation in the atmosphere with altitude using Bernoulli’s equation is illustrated. Acquirement of mass from moist air is mass accretion and mass accretion during the motion of raindrop through resistive medium holds an arbitrary power-law equation. Bernoulli’s equation when applied to it, the generalized first-order differential equation is reduced to a polynomial equation. Results show a single intersecting point of approximate terminal velocity 1 m/s and mass 10-06 mg as illustrated. Terminal velocity is achieved within 25 sec. There is the approximate exponential growth of terminal velocity. An increase in momentum is due to mass accretion during motion. Various conditions of no mass accretion and mass accretion show the same result while for atmospheric effect using Bernoulli’s equation the first-order differential equation reduces to a polynomial equation.

2022 ◽  
Vol 13 (2) ◽  
pp. 0-0

Nowadays, COVID-19 is considered to be the biggest disaster that the world is facing. It has created a lot of destruction in the whole world. Due to this COVID-19, analysis has been done to predict the death rate and infected rate from the total population. To perform the analysis on COVID-19, regression analysis has been implemented by applying the differential equation and ordinary differential equation (ODE) on the parameters. The parameters taken for analysis are the number of susceptible individuals, the number of Infected Individuals, and the number of Recovered Individuals. This work will predict the total cases, death cases, and infected cases in the near future based on different reproductive rate values. This work has shown the comparison based on 4 different productive rates i.e. 2.45, 2.55, 2.65, and 2.75. The analysis is done on two different datasets; the first dataset is related to China, and the second dataset is associated with the world's data. The work has predicted that by 2020-08-12: 59,450,123 new cases and 432,499,003 total cases and 10,928,383 deaths.

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 273
Mujahid Abbas ◽  
Muhammad Waseem Asghar ◽  
Manuel De la Sen

The aim of this paper is to propose a new faster iterative scheme (called AA-iteration) to approximate the fixed point of (b,η)-enriched contraction mapping in the framework of Banach spaces. It is also proved that our iteration is stable and converges faster than many iterations existing in the literature. For validity of our proposed scheme, we presented some numerical examples. Further, we proved some strong and weak convergence results for b-enriched nonexpansive mapping in the uniformly convex Banach space. Finally, we approximate the solution of delay fractional differential equations using AA-iterative scheme.

David P. Bourne ◽  
Charlie P. Egan ◽  
Beatrice Pelloni ◽  
Mark Wilkinson

AbstractWe give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

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