In this paper, we proposed a novel analytical technique for one-dimensional fractional heat equations with time fractional derivatives subjected to the appropriate initial condition. This new analytical technique, namely multistep reduced differential transformation method (MRDTM), is a simple amendment of the reduced differential transformation method, in which it is treated as an algorithm in a sequence of small intervals, in order to hold out accurate approximate solutions over a longer time frame compared to the traditional RDTM. The fractional derivatives are described in the Caputo sense, while the behavior of solutions for different values of fractional order [Formula: see text] compared with exact solutions is shown graphically. The analysis is accompanied by four test examples to demonstrate that the proposed approach is reliable, fully compatible with the complexity of these equations, and can be strongly employed for many other nonlinear problems in fractional calculus.

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We study some qualitative behaviour of a modified discrete-time host–parasitoid model. Modification of classical Nicholson–Bailey model is considered by introducing Pennycuick growth function for the host population. Furthermore, the existence and uniqueness of positive equilibrium point of proposed system is investigated. We prove that the positive solutions of modified system are uniformly bounded and the unique positive equilibrium point is locally asymptotically stable under certain parametric conditions. Moreover, it is also investigated that system undergoes Neimark–Sacker bifurcation by using standard mathematical techniques of bifurcation theory. Complexity and chaotic behaviour are confirmed through the plots of maximum Lyapunov exponents. In order to stabilise the unstable steady state, the feedback control strategy is introduced. Finally, in order to support theoretical discussions, numerical simulations are provided.

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This paper is devoted to establishing the existence theory for at least one solution to a coupled system of fractional order differential equations (FDEs). The problem under consideration is subjected to movable type integral boundary conditions over a finite time interval. Furthermore, we investigate the approximate solutions to the considered problem with the help of the differential transform. Moreover, some necessary conditions for the Hyers-Ulam type stability to the solution of the proposed problem are developed. The whole investigation has been illustrated by providing some suitable examples.

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