Piecewise differential equations: Theory, methods and applications
Résumé
Across many real-world issues, crossover tendencies are seen. As these operators are built on kernels that exhibit some qualities that arise in nature, some of these could be captured using the idea of piecewise differentiation and integration. Power-law processes, fading memory processes, and processes that mimic the generalized Mittag-Leer function are a few examples. The use of piecewise differential and integral operators, however, cannot be applied to all processes involving crossovers. For instance, when groundwater overabstraction causes it to ‡ow from confined to unconfined, a considerable alteration eventually manifests. The idea of piecewise differential equations, which can be thought of as an extension of the piecewise function to the framework of differential equations, is introduced in this work. While we concentrated on ordinary, it is important to note that partial differential equations can also be solved using the same technique. For both integer and noninteger instances, piecewise differential equations were introduced. We explained the usage of the Laplace transform for the linear case and demonstrated how a new class of Bode diagrams could be produced. We 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 straightforward ordinary differential equations to produce novel varieties of chaos. We think that this idea could lead to some significant conclusions in the future.
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