The study of variable order differential equations is important in science and engineering for a better representation and analysis of dynamical problems. In the literature, there are several fractional order operators involving variable orders. In this article, we construct a nonlinear variable order fractional differential system with a p-Laplacian operator. The presumed problem is a general class of the nonlinear equations of variable orders in the ABC sense of derivatives in combination with Caputo’s fractional derivative. We investigate the existence of solutions and the Hyers–Ulam stability of the considered equation. The presumed problem is a hybrid in nature and has a lot of applications. We have given its particular example as a waterborne disease model of variable order which is analysed for the numerical computations for different variable orders. The results obtained for the variable orders have an advantage over the constant orders in that the variable order simulations present the fluctuation of the real dynamics throughout our observations of the simulations.

In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. Some examples are also presented to test the strength and applicability of the results obtained.

A recent addition to the class of integral transforms is the quaternion quadratic-phase Fourier transform (Q-QPFT), which generalizes various signal and image processing tools. However, this transform is insufficient for addressing the quadratic-phase spectrum of non-stationary signals in the quaternion domain. To address this problem, we, in this paper, study the (two sided) quaternion windowed quadratic-phase Fourier transform (QWQPFT) and investigate the uncertainty principles associated with the QWQPFT. We first propose the definition of QWQPFT and establish its relation with quaternion Fourier transform (QFT); then, we investigate several properties of QWQPFT which includes inversion and the Plancherel theorem. Moreover, we study different kinds of uncertainty principles for QWQPFT such as Hardy’s uncertainty principle, Beurling’s uncertainty principle, Donoho–Stark’s uncertainty principle, the logarithmic uncertainty principle, the local uncertainty principle, and Pitt’s inequality.