A comparative analysis for the solution of nonlinear Burgers' equation.

In this work, a comparative study of seven well-known mathematical techniques for the coupled Burgers' equations is reported. The techniques involve in this comparison are as follows: Laplace transform Adomian decomposition method, Laplace transform homotopy perturbation method, Variational iteration method, Variational iteration decomposition method, Variational iteration homotopy perturbation method, the optimal homotopy asymptotic method, and OHAM with Daftardar-Jafari polynomial. Here we considered a practical example which consists of coupled Burgers' equations with the kinematic viscosity ε=1. Convergence and stability analysis is a major part of this analysis. After a careful observation, it is found that the variational iteration method has faster convergence than all the remaining methods. Adomian decomposition method and Homotopy perturbation method show weaker stability in comparison with other involved techniques.

A fuzzy fractional model of coronavirus (COVID-19) and its study with Legendre spectral method.

The virus which belongs to the family of the coronavirus was seen first in Wuhan city of China. As it spreads so quickly and fastly, now all over countries in the world are suffering from this. The world health organization has considered and declared it a pandemic. In this presented research, we have picked up the existing mathematical model of corona virus which has six ordinary differential equations involving fractional derivative with non-singular kernel and Mittag-Leffler law. Another new thing is that we study this model in a fuzzy environment. We will discuss why we need a fuzzy environment for this model. First of all, we find out the approximate value of ABC fractional derivative of simple polynomial function ( t - a ) n . By using this approximation we will derive and developed the Legendre operational matrix of fractional differentiation for the Mittag-Leffler kernel fractional derivative on a larger interval [ 0 , b ] , b ⩾ 1 , b ∈ N . For the numerical investigation of the fuzzy mathematical model, we use the collocation method with the addition of this newly developed operational matrix. For the feasibility and validity of our method we pick up a particular case of our model and plot the graph between the exact and numerical solutions. We see that our results have good accuracy and our method is valid for the fuzzy system of fractional ODEs. We depict the dynamics of infected, susceptible, exposed, and asymptotically infected people for the different integer and fractional orders in a fuzzy environment. We show the effect of fractional order on the suspected, exposed, infected, and asymptotic carrier by plotting graphs.