Piecewise differential equations: theory, methods and applications

Across many real-world problems, crossover tendencies are seen. Piecewise differential operators are constructed by using different kernels that exhibit behaviors arising in several real -world problems; thus, crossover behaviors could be well modeled using these differential and integral operators. Power-law processes, fading memory processes and processes that mimic the generalized Mittag-Leffler function are a few examples. However, the use of piecewise differential and integral operators cannot be applied to all processes involving crossovers. For instance, a considerable alteration eventually manifests when groundwater over-abstraction causes it to flow from confined to unconfined aquifers. The idea of piecewise differential equations, which can be thought of as an extension of piecewise functions to the framework of differential equations, is introduced in this work. While we concentrate on ordinary differential equations, it is important to note that partial differential equations can also be constructed with the same technique. For both integer and non-integer instances, piecewise differential equations have been introduced. We have explained the usage of the Laplace transform for the linear case and demonstrated how a new class of Bode diagrams could be produced. We have provided some examples of numerical solutions as well as conditions for the existence and uniqueness of their solutions. We discussed a few scenarios in which we used chaos and non-linear ordinary differential equations to produce novel varieties of chaos. We believe that this idea could lead to some significant conclusions in the future.