Comparing the Atangana–Baleanu and Caputo–Fabrizio derivative with fractional order: Allen Cahn model

In 2015 Caputo and Fabrizio suggested a new operator with fractional order, this derivative is based on the exponential kernel. Earlier this year 2016 Atangana and Baleanu developed another version which used the generalized Mittag-Leffler function as non-local and non-singular kernel. Both operators show some properties of filter. However the Atangana and Baleanu version has in addition to this, all properties of fractional derivative. In this work, we aimed to represent the model by Allen–Cahn with both derivatives in order to see their difference in a real world problem. Both modified models will be solved numerically via the Crank–Nicholson scheme and their numerical simulations are presented to check the effectiveness of the both kernels.     

Chaos control of fractional order Rabinovich–Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich–Fabrikant system

In this article the local stability of the Rabinovich–Fabrikant (R–F) chaotic system with fractional order time derivative is analyzed using fractional Routh–Hurwitz stability criterion. Feedback control method is used to control chaos in the considered fractional order system and after controlling the chaos the authors have introduced the synchronization between fractional order non-chaotic R–F system and the chaotic R–F system at various equilibrium points. The fractional derivative is described in the Caputo sense. Numerical simulation results which are carried out using Adams–Boshforth–Moulton method show that the method is effective and reliable for synchronizing the systems.     

Chaos in a topologically transitive system

The chaotic phenomena have been studied in a topologically transitive system and it has been shown that the erratic time dependence of orbits in such a topologically transitive system is more complicated than what described by the well-known technology "Li-Yorke chaos". The concept "sensitive dependency on initial conditions" has been generalized, and the chaotic phenomena has been discussed for transitive systems with the generalized sensitive dependency property.     

Dynamic analysis of a fractional order prey–predator interaction with harvesting

In recent years, prey–predator models appearing in various fields of mathematical biology have been proposed and studied extensively due to their universal existence and importance. In this paper, we introduce a fractional-order prey–predator model and deals with the mathematical behaviors of the model. The dynamical behavior of the system is investigated from the point of view of local stability. We also carry out a detailed analysis on the stability of equilibrium. Numerical simulations are presented to illustrate the results.     

Periodicity in a nonlinear discrete predator–prey system with state dependent delays

With the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear discrete state dependent delays predator–prey systemwhere $a_i,c_j,d_i:Z\to R^{+}$ are positive $\omega$-periodic, $\omega$ is a fixed positive integer. $b_1,b_2:Z\to R^{+}$ are $\omega$-periodic and ${\sum }_{k=0}^{\omega -1}{b}_i(k)>0$. ${\tau }_i,{\sigma }_j,{\rho }_i:Z\times R\times R\to R(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are $\omega$-periodic with respect to their first arguments, respectively. ${\alpha }_i,{\beta }_j,{\gamma }_i(i=1,2,\dots ,n,j=1,2,\dots ,m)$ are positive constants.     

The existence of an extremal solution to a nonlinear system with the right-handed Riemann–Liouville fractional derivative

A monotone iterative method is applied to show the existence of an extremal solution for a nonlinear system involving the right-handed Riemann–Liouville fractional derivative with nonlocal coupled integral boundary conditions. Two comparison results are established. As an application, an example is presented to demonstrate the efficacy of the main result.     

Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.     

Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces

In this paper, we shall be concerned with the existence result of the following problem,

$$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\left( a(x,u,\nabla u)\right) -\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(0.1)

with the second term f belongs to \(L^1(\Omega )\). The growth and the coercivity conditions on the monotone vector field a are prescribed by a generalized N-function M. We assume any restriction on M, therefore we work with Musielak–Orlicz spaces which are not necessarily reflexive. The lower order term \(\Phi \) is a Carathéodory function satisfying only a growth condition.

     

On a system of nonlinear wave equations with Balakrishnan–Taylor damping

In this paper, we study the initial-boundary value problem for a coupled system of nonlinear viscoelastic wave equations of Kirchhoff type with Balakrishnan–Taylor damping terms. For certain class of relaxation functions and certain initial data, we prove that the decay rate of the solution energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type. Also, we show that nonlinear source of polynomial type is able to force solutions to blow up in finite time even in presence of stronger damping.     

Analysis of the nonlinear vibration of a two-mass–spring system with linear and nonlinear stiffness

An analytical approach is developed for areas of nonlinear science such as the nonlinear free vibration of a conservative, two-degree-of-freedom mass–spring system having linear and nonlinear stiffnesses. The main contribution of this research is twofold. First, it introduces the transformation of two nonlinear differential equations for a two-mass system using suitable intermediate variables into a single nonlinear differential equation and, more significantly, the treatment of a nonlinear differential system by linearization coupled with Newton’s method. Secondly, the major section is the solving of the governing nonlinear differential equation where the displacement of the two-mass system can be obtained directly from the linear second-order differential equation using a first-order variational approach. The aforementioned approach proposed by J.H. He, who actually developed the method, is exactly He’s variational method. This approach is an explicit method with high validity for resolving strong nonlinear oscillation system problems. Two examples of nonlinear two-degree-of-freedom mass–spring systems are analyzed, and verified with published results and exact solutions. The method can be easily extended to other nonlinear oscillations and so could be widely applicable in engineering and science.     

Existence of positive solutions of Sturm–Liouville boundary value problems for a nonlinear singular second order differential system with a parameter

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of Sturm–Liouville boundary value problems for a nonlinear singular second order ordinary differential system with a parameter. Some well-known results in the literature are generalized and improved. An example is presented to illustrate the application of our main result.     

Inversion of the Kipriyanov–Radon transform via fractional derivatives in a one-dimensional parameter

This paper considers the Kipriyanov–Radon transform constructed as a special Radon transform adopted for dealing with singular Bessel differential operators of the corresponding indices acting on a part of the variables. The authors obtain inversion formulas generalizing the classical formulas for the Radon transform of axially-symmetric functions and relating to the integro-differentiation of fractional order in a one-dimensional parameter. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Note on a nonlocal Sturm–Liouville problem with both right and left fractional derivatives

In this paper, we research the geometric multiplicity of eigenvalues for a nonlocal Sturm–Liouville eigenvalue problem. To this end, we study the uniqueness of solutions for a nonlocal Sturm–Liouville problem under some initial value conditions.     

Boundedness in a three-dimensional chemotaxis–haptotaxis model with nonlinear diffusion

The quasilinear chemotaxis–haptotaxis system

$\{\begin{array}{cc}{u}_t=\mathrm{?}?(D(u)\mathrm{?}u)?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)\hfill & \hfill \\ \phantom{u_t=}+\mu u(1?u?w),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v?v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}$

is considered under homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }?{\mathbb{R}}^3$. Here $\chi >0$, $\xi >0$ and $\mu >0$, $D(u)\ge c_D{u}^{m?1}$ for all $u>0$ with some $c_D>0$ and $D(u)>0$ for all $u\ge 0$. It is shown that if the ratio $\frac{\chi }{\mu }$ is sufficiently small, then the system possesses a unique global classical solution that is uniformly bounded. Our result is independent of m.     

Optimal control of a nonlinear parabolic–elliptic system

In this paper, we shall study the problem of optimal control of the parabolic–elliptic system

_ut+_(f(t,x,u))x+g(t,x,u)+_Px−_(a(t,x)ux)x=_f0+B^∗ν$u_t+{\left(f(t,x,u)\right)}_x+g(t,x,u)+P_x-{\left(a(t,x)u_x\right)}_x=f_0+B^{\ast }\nu$

Global stability of a nonlinear stochastic predator–prey system with Beddington–DeAngelis functional response

Stochastically asymptotic stability in the large of a predator–prey system with Beddington–DeAngelis functional response with stochastic perturbation is considered. The result shows that if the positive equilibrium of the deterministic system is globally stable, then the stochastic model will preserve this nice property provided the noise is sufficiently small. Some simulation figures are introduced to support the analytical findings.