Revisiting Fisher-KPP model to interpret the spatial spreading of invasive cell population in biology.

In this paper, the homotopy analysis method, a powerful analytical technique, is applied to obtain analytical solutions to the Fisher-KPP equation in studying the spatial spreading of invasive species in ecology and to extract the nature of the spatial spreading of invasive cell populations in biology. The effect of the proliferation rate of the model of interest on the entire population is studied. It is observed that the invasive cell or the invasive population is decreased within a short time with the minimum proliferation rate. The homotopy analysis method is found to be superior to other analytical methods, namely the Adomian decomposition method, the homotopy perturbation method, etc. because of containing an auxiliary parameter, which provides us with a convenient way to adjust and control the region of convergence of the series solution. Graphical representation of the approximate series solutions obtained by the homotopy analysis method, the Adomian decomposition method, and the Homotopy perturbation method is illustrated, which shows the superiority of the homotopy analysis method. The method is examined on several examples, which reveal the ingenuousness and the effectiveness of the method of interest.

Solving protoplanetary structure equations using Adomian decomposition method.

In this paper, the distribution of thermodynamic variables in the protoplanets formed by gravitational instability in the mass range 0.3 - 10 M J ( 1 M J = 1 Jupiter mass = 1.8986 × 10 30 gm) is investigated in their initial state by solving the structure equations via the Adomian decomposition method. Concerning the heat transfer in the protoplanets, the mode of convection is taken into account. The outcomes indicate that there is a reasonably good agreement between the Adomian semi-analytical solution containing only first 8 terms and the numerical results.

Lump, lump-stripe, and breather wave solutions to the (2 + 1)-dimensional Sawada-Kotera equation in fluid mechanics.

The present study investigates the lump, one-stripe, lump-stripe, and breather wave solutions to the (2+1)-dimensional Sawada-Kotera equation using the Hirota bilinear method. For lump and lump-stripe solutions, a quadratic polynomial function, and a quadratic polynomial function in conjunction with an exponential term are assumed for the unknown function f giving the solution to the mentioned equation, respectively. On the other hand, only an exponential function is considered for one-stripe solutions. Besides, the homoclinic test approach is adopted for constructing breather wave solutions. The propagations of the attained lump, lump-stripe, and breather wave solutions are shown through some graphical illustrations. The graphical outputs demonstrate that the lump wave moves along the straight line y = - x and exponentially decreases away from the origin of the spatial domain. On the other hand, lump-kink solutions illustrate the fission and fusion interaction behaviors upon the selection of the free parameters. Fission and fusion processes show that the stripe soliton splits into a stripe soliton and a lump soliton, and then the lump soliton merges into one stripe soliton. In addition, the achieved breather waves illustrate the periodic behaviors in the xy-plane. The outcomes of the study can be useful for a better understanding of the physical nature of long waves in shallow water under gravity.