Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.
Exact nonlinear partial differential equation solutions are critical for
describing new complex characteristics in a variety of fields of applied
science. The aim of this research is to use the F-expansion method to
find the generalized solitary wave solution of the regularized long wave
(RLW) equation of fractional order. Fractional partial differential
equations can also be transformed into ordinary differential equations
using fractional complex transformation and the properties of the
modified Riemann–Liouville fractional-order operator. Because of the
chain rule and the derivative of composite functions, nonlinear
fractional differential equations (NLFDEs) can be converted to ordinary
differential equations. We have investigated various set of explicit
solutions with some free parameters using this approach. The solitary
wave solutions are derived from the moving wave solutions when the
parameters are set to special values. Our findings show that this
approach is a very active and straightforward way of formulating exact
solutions to nonlinear evolution equations that arise in mathematical
physics and engineering. It is anticipated that this research will
provide insight and knowledge into the implementation of novel methods
for solving wave equations.
In this paper, we work on a general comparison principle for Caputo fractional-order ordinary differential equations. A full result on maximal solutions to Caputo fractional-order systems is given by using continuation of solutions and a newly proven formula of Caputo fractional derivatives. Based on this result and the formula, we prove a general fractional comparison principle under very weak conditions, in which only the Caputo fractional derivative is involved. This work makes up deficiencies of existing results.
Book Review: Existence theorems for ordinary differential equations
