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Über dieses Buch

This monograph is the first published book devoted to the theory of differential equations with non-instantaneous impulses. It aims to equip the reader with mathematical models and theory behind real life processes in physics, biology, population dynamics, ecology and pharmacokinetics. The authors examine a wide scope of differential equations with non-instantaneous impulses through three comprehensive chapters, providing an all-rounded and unique presentation on the topic, including:
- Ordinary differential equations with non-instantaneous impulses (scalar and n-dimensional case)- Fractional differential equations with non-instantaneous impulses (with Caputo fractional derivatives of order q ϵ (0, 1))- Ordinary differential equations with non-instantaneous impulses occurring at random moments (with exponential, Erlang, or Gamma distribution)
Each chapter focuses on theory, proofs and examples, and contains numerous graphs to enrich the reader’s understanding. Additionally, a carefully selected bibliography is included. Graduate students at various levels as well as researchers in differential equations and related fields will find this a valuable resource of both introductory and advanced material.



Chapter 1. Non-instantaneous Impulses in Differential Equations

The case of differential equations with instantaneous impulses is studied in the literature; so we begin with a brief overview of its statements and later we will compare it with the case of non-instantaneous impulses.

Ravi Agarwal, Snezhana Hristova, Donal O’Regan

Chapter 2. Non-instantaneous Impulses in Differential Equations with Caputo Fractional Derivatives

Fractional calculus is the theory of integrals and derivatives of arbitrary non-integer order, which unifies and generalizes the concepts of ordinary differentiation and integration. For more details on geometric and physical interpretations of fractional derivatives and for a general historical perspective we refer the reader to the monographs [42, 45, 101] and the cited references therein.

Ravi Agarwal, Snezhana Hristova, Donal O’Regan

Chapter 3. Non-instantaneous Impulses on Random Time in Differential Equations with Ordinary/Fractional Derivatives

In some real world phenomena a process may change instantaneously at uncertain moments and act non instantaneously on finite intervals. In modeling such processes it is necessarily to combine deterministic differential equations with random variables at the moments of impulses. The presence of randomness in the jump condition changes the solutions of differential equations significantly. The study combines methods of deterministic differential equations and probability theory. Note differential equations with random impulsive moments differs from the study of stochastic differential equations with jumps (see, for example, [105, 127–131, 134]). We will define and study nonlinear differential equations subject to impulses starting abruptly at some random points and their action continue on intervals with a given finite length. Inspired by queuing theory and the distribution for the waiting time, we study the cases of exponentially distributed random variables, Erlang distributed random variables and Gamma distributed random variables between two consecutive moments of impulses and the intervals where the impulses act are with a constant length.

Ravi Agarwal, Snezhana Hristova, Donal O’Regan


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