In this paper, we continue to study the (N, Q)-periodicity for a class of Weyl-like fractional difference equations of the form ^αv(n)=Lv(n+1)+p(n, v(n)), n∈ℤ. Some new existence theorems of (N, Q)-periodic mild solutions for the equations are established under weighted-sum conditions or the H ö lder continuity condition of order ϵ∈ (0,1) , which relax the Lipschitz-type conditions of [13]. The results are essentially a continuation and generalization of some recent results on this issue.
Fractional calculus has proven to be useful for modeling physical phenomena, as well being used for control system design. Non-integer models allow to capture complex dynamics without a significant increase in the number of parameters. Dielectric relaxation was studied in the 50s and 60s, which led to the proposal of two non-integer models in order to fit the frequency-domain data of the experiments: Cole-Cole and Davidson-Cole functions. Some years later, a more general function called the Havriliak-Negami function appeared. However, time-domain simulation of Havriliak-Negami function is a challenging task. Some approaches were proposed by using diffusive symbols in the 90s and early 2000s, but this still limited the range of the parameters. Sommacal’s works [43, 44] introduced new synthesis methods, one of them being based upon Oustaloup’s recursive approximation [35, 36]. The present work introduces two new additional approximations which will be compared to that of Sommacal’s works, as well as recursive identification of such type of models with different synthesis methods. The performance of real-time identification algorithm by using these synthesis methods will be tested by means of Monte Carlo simulation.
In this paper, we developed Fujita's critical exponent to determine whether or not the following fractional reaction-diffusion system admits global solutions or only blow-up solutions: (S) { u(t) = Delta(alpha )u + v(p), in R-N & times; (0, infinity), vt = Delta(alpha )v + u(q), in R-N & times; (0, infinity), u(& centerdot;, 0) = u(0) >= 0, v(& centerdot;, 0) = v(0) >= 0, in R-N, where p >= q >= 1 with pq > 1 and Delta alpha := - (-Delta)(alpha/2) denotes the fractional Laplace operator with 0 < alpha <= 2 on RN, N >= 2. More precisely, for the number (pq)(& lowast;) := alpha(p+1)/N + 1, we proved the following: (i) If (pq)(& lowast;) < pq, then (S) has a global solution for some initial data (u(0), v(0)). (ii) If (pq)(& lowast;) >= pq, then every nontrivial and non-negative solution to (S) blows up in finite time.
We aim to give closed integral form expressions for the Mathieu-type (a, A; α) -series and their alternating counterparts, which contain fractional powers special functions as generalized Fox-Wright hypergeometric function a_pΨ_q in their terms, together with their contiguous representations with respect to the additional parameters. Particular cases of a_pΨ _q are specified, including the Le Roy–type functions and their multi–index analogues and other special functions. The corresponding results are illustrated with important special cases.
