Unveiling protoplanetary structure equations: Semi-analytical solutions via the homotopy analysis method.

This paper revisits the distribution of thermodynamic variables within initial protoplanets formed via gravitational instability (GI) across a broad mass spectrum ranging from (where denotes 1 Jupiter mass, equal to g), using the Homotopy Analysis Method (HAM), a novel approach in this context. Concerning heat transfer within the protoplanets, consideration is given to the convective mode. Our findings reveal a noteworthy alignment between the results obtained via the HAM, utilizing only the first four terms (third approximation), and numerical outcomes.

Estimating water levels caused by a tropical storm along the Bangladesh coast: A numerical approach.

In this study, water levels resulting from the dynamic interaction of tide and surge are estimated by solving a 2-D vertically integrated shallow water equations numerically. To solve the equations on the specific 2-D grid, the explicit Leapfrog scheme is implemented, adopting a staggered Arakawa C-grid. The domain's complex land-sea interface is approximated through the stair-step method in order to employ the finite difference technique.

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.

An effective method in investigating structures of polytropic protoplanets formed via gravitational instability.

In this article, we have reinvestigated the initial distribution of thermodynamic variables inside the protoplanets formed via gravitational instability having mass range ( ) by an embedded RKACeM(4,4) method assuming that the polytropic gas law holds in the protoplanets. The findings attained by our numerical experiments are recognized to be consistent with the results acquired through other notable investigations in this regard. Furthermore, the model is easily computable.

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 ( Jupiter mass = 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 giving the solution to the mentioned equation, respectively. On the other hand, only an exponential function is considered for one-stripe solutions.

On the implementation of novel RKARMS(4,4) algorithm to study the structures of initial extrasolar giant protoplanets.

The main motivation and novel notion of this present communication is to implement the recently suggested fourth order with four stages embedded RKARMS(4,4) algorithm to examine its efficiency in reinvesting the structures of extrasolar protoplanets formed via disk instability which being presented in Paul et al. [1] (G.C.