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Register a new userRational Solutions of Abel Differential Equations
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Luis \\'Angel Calder\\'on P\\'erez
Publication date
2021
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English
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We study the rational solutions of the Abel equation $x=A(t)x^3+B(t)x^2$ where $A,Bin C[t]$. We prove that if $deg(A)$ is even or $deg(B)>(deg(A)-1)/2$ then the equation has at most two rational solutions. For any other case, an upper bound on the number of rational solutions is obtained. Moreover, we prove that if there are more than $(deg(A)+1)/2$ rational solutions then the equation admits a Darboux first integral.
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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.
This paper concerns with a mathematical modelling of biological experiments, and its influence on our lives. Fractional hybrid iterative differential equations are equations that interested in mathematical model of biology. Our technique is based on the Dhage fixed point theorem. This tool describes mixed solutions by monotone iterative technique in the nonlinear analysis. This method is used to combine two solutions: lower and upper. It is shown an approximate result for the hybrid fractional differential equations iterative in the closed assembly formed by the lower and upper solutions.
We investigate local fractional nonlinear Riccati differential equations (LFNRDE) by transforming them into local fractional linear ordinary differential equations. The case of LFNRDE with constant coefficients is considered and non-differentiable solutions for special cases obtained.
In the study of ordinary differential equations (ODEs) of the form $hat{L}[y(x)]=f(x)$, where $hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)propto u(x)$ and $hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $hat{L}=hat{D}^n$ for $ngeq 2$. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution $u(x)$, introducing a parameter $epsilon$ such that $u(x)rightarrow u(x;epsilon)$, and Taylor expanding $u(x;epsilon)$ about $epsilon = 0$. The coefficients of this expansion $frac{partial^k u}{partialepsilon^k}big{|}_{epsilon=0}$ yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. While the ideas can be introduced at the undergraduate level, we could not find any elementary or advanced text that illustrates these ideas with appropriate generality.
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