Narrow Results

In this paper, we have studied the concentration profile of Cytosolic calcium ion (Ca${}^{2+})$ with the aid of fractional calculus. A mathematical model has been considered to examine the influence of fractional advection diffusion equation (cross flow) for the calcium profile. A closed form solution of the fractional advection diffusion equation, arising in study of diffusion of cytosolic calcium in astocytes cell, has been obtained by using Sumudu transform techniques. Graphs for the calcium concentration profiles have been simulated for certain values of the parameters to examine the various effects on concentrations of Cytosolic calcium ion.

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time $t$ on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

Several significant questions of mathematics and mathematical physics have been effectively explained and answered through the use of fractional kinetic equations containing special functions. Due to the high importance of arbitrary-order kinetic equations, the aim of this work is to obtain the solution of a new arbitrary-order kinetic equation related to the Riemann xi function. The Sumudu transform technique is used to solve it. The findings are plotted with the help of MATLAB R2016a. The outcomes of the paper are in the form of an infinite series representation of the Riemann xi function and in terms of the Mittag-Leffler function.

In this paper, the one-dimensional nonlinear temporal fractional-order Fisher’s equation is solved by using the Sumudu residual power series method (SRPSM), a powerful computing technique. This method combines the RPSM and Sumudu transform to provide approximations of solutions using the notion of limit, in contrast to the standard residual power series method, which necessitates computing fractional derivatives. With reference to the equation under consideration, this work compares and validates the solution obtained by using the considered method. The applicability, effectiveness, and dependability of the provided strategy were confirmed through the analysis of this equation and numerical simulations.