Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic behavior of solutions of linear multi-order fractional differential equation systems
147
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we investigate some aspects of the qualitative theory for multi-order fractional differential equation systems. First, we obtain a fundamental result on the existence and uniqueness for multi-order fractional differential equation systems. Next, a representation of solutions of homogeneous linear multi-order fractional differential equation systems in series form is provided. Finally, we give characteristics regarding the asymptotic behavior of solutions to some classes of linear multi-order fractional differential equation systems.
rate research
Read More
We present some distinct asymptotic properties of solutions to Caputo fractional differential equations (FDEs). First, we show that the non-trivial solutions to a FDE can not converge to the fixed points faster than $t^{-alpha}$, where $alpha$ is the order of the FDE. Then, we introduce the notion of Mittag-Leffler stability which is suitable for systems of fractional-order. Next, we use this notion to describe the asymptotical behavior of solutions to FDEs by two approaches: Lyapunovs first method and Lyapunovs second method. Finally, we give a discussion on the relation between Lipschitz condition, stability and speed of decay, separation of trajectories to scalar FDEs.
Using a temporally weighted norm we first establish a result on the global existence and uniqueness of solutions for Caputo fractional stochastic differential equations of order $alphain(frac{1}{2},1)$ whose coefficients satisfy a standard Lipschitz condition. For this class of systems we then show that the asymptotic distance between two distinct solutions is greater than $t^{-frac{1-alpha}{2alpha}-eps}$ as $t to infty$ for any $eps>0$. As a consequence, the mean square Lyapunov exponent of an arbitrary non-trivial solution of a bounded linear Caputo fractional stochastic differential equation is always non-negative.
We give a necessary and sufficient condition for a system of linear inhomogeneous fractional differential equations to have at least one bounded solution. We also obtain an explicit description for the set of all bounded (or decay) solutions for these systems.
Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
We comprehensively study admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. For each class from the constructed chain of nested gauged classes of such systems, we single out its singular subclass, which appears to consist of systems being similar to the elementary (free particle) system whereas the regular subclass is the complement of the singular one. This allows us to exhaustively describe the equivalence groupoids of the above classes as well as of their singular and regular subclasses. Applying various algebraic techniques, we establish principal properties of Lie symmetries of the systems under consideration and outline ways for completely classifying these symmetries. In particular, we compute the sharp lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by systems from each of the above classes and subclasses. We also show how equivalence transformations and Lie symmetries can be used for reduction of order of such systems and their integration. As an illustrative example of using the theory developed, we solve the complete group classification problems for all these classes in the case of two dependent variables.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
