In physics, the majority of natural events have been researched and described using differential equations, each having its own initial and boundary conditions. These differential equations contain a large number of fundamental constants as well as other model parameters. They add to the equation's complexity and rounding errors, making the problem more difficult to solve. In this work, we provide a method for transforming these physics differential equations into dimensionless equations, which are significantly simpler. Nondimensionalization, by suitably substituting variables, is the process of removing some or all of the physical dimensions from an equation that contains physical quantities. Some benefits of these dimensionless equations include that they are simpler to identify when using well-known mathematical methods, need less time to compute, and do not round off errors. Through several examples we discuss, this method is useful not just in quantum mechanics but also in classical physics.

In this note, we prove a unique result for a nonlinear differential equation concerning a Nagumo-type source. An example is given to illustrate the theoretical result.

This paper is devoted to the study of a nonlinear fractional differential equation with a weakly singular source in Banach space. Using Bielecki type norm, we show that the problem has a unique solution. Furthermore, we introduce a result of the new Ulam-Hyers type stability for the main equation.

This paper is devoted to study a fractional equation involving Caputo-Katugampola derivative with nonlocal initial condition. Unlike previous papers, in this paper, the source function of problem is assumed having a singularity. We propose some reasonable conditions such that the problem has at least one mild solution or has a unique mild solution. The desired results are proved by using the Banach, Leray-Schauder and Krasnoselskii fixed point theorems. Some examples are given to confirm our theoretical findings. Keywords: Caputo-Katugampola fractional derivative; Nonlinear integral equations; existence 2010 MSC: 26A33; 35A01; 35A02; 35R11

In this paper we discus on a Lyapunov-type inequality for a fractional differential equation involving sequential generalized Caputo fractional derivatives with boundary conditions. The results presented in this paper is new to the corresponding results in the literature.