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Abstract: Sufficient conditions are given for the existence of solutions of impulsive boundary value problems for singular nonlinear fractional differential systems. We allow the nonlinearities p(t)f(t,y)$$p(t)f(t,y)$$ and q(t)g(t,x)$$q(t)g(t,x)$$ in fractional differential equations to be singular at t=0$$t\!=\!0$$. Both f$$f$$ and g$$g$$ may be super-linear and sub-linear. The analysis relies on some well-known fixed point theorems. The initial value problem discussed may be seen as a generalization of some ecological models. An example is given to illustrate the efficiency of the main theorems. A conclusion section is given at the end of the paper.

Abstract: Taking advantage of the Hirota bilinear form, four classes of lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation are presented through symbolic computation with Maple. Special choices of the involved parameters guaranteeing analyticity of the fourth solution are given, together with two particular lump-type solutions.

Side weirs with triangular channel are used as flow controlling devices in draining and irrigation networks. By installing a side weir on the main conduits side walls, the runoff overflows from the weir and are conducted toward the diversion channel. In this study, changing of the flow free surface and the turbulence of the flow field in triangular channels with side weir are numerically simulated using volume of fluid (VOF) scheme and RNG k–ε turbulence model. In the present paper, the pattern of the spatially varied flow with decreasing discharge in both subcritical and supercritical flow regimes for triangular channels with side weirs was simulated. The present numerical model has precisely predicted the changes of the water surface and the specific energy. In subcritical regime, the flow depth is from the beginning of the weir toward its end is followed by an increase and in supercritical conditions is followed by a reduction in depth. For both subcritical and supercritical regimes, a drop in the surface in the first third of the weir’s opening and a surface jump in the final third of its length has occurred. Along the mentioned surface jump the amount of the kinetic energy increases and the potential energy reduces. According to results of the simulation, the maximum longitudinal velocity for subcritical flow regime occurs in the first third of the length of the side weir and for supercritical flow regime, almost in the middle of the weir opening happens. In both subcritical and supercritical regimes, the maximum transverse velocity has occurred in the final third of the length of the side weir.

Abstract: The aim of this paper is to establish the existence of nonnegative solutions for a class of Schrödinger–Kirchhoff type problems driven by nonlocal integro-differential operators, that is, M∬R2N|u(x)−u(y)|pK(x−y)dxdy,∫RNV(x)|u|pdxLKu+V(x)|u|p−2u=G∬R2N|u(x)−u(y)|pK(x−y)dxdy,∫RNV(x)|u|pdxf(x,u)+h(x)    in RN,$$\begin{align*}&M\left(\mathop{\iint_{\mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,\int_{\mathbb{R}^N}V(x)|u|^pdx\right)\\&\kern10pt \left(\mathcal{L}_Ku+V(x)|u|^{p-2}u\right)\\&=G\left(\mathop{\iint_{\mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,\int_{\mathbb{R}^N}V(x)|u|^pdx\right)\nonumber\\&\kern11pt f(x,u)+h(x)\ \ \ \ {\rm in}\ \mathbb{R}^N,\end{align*}$$ where LK$\mathcal{L}_K$ is a nonlocal integro-differential operator with singular kernel K:RN∖{0}→(0,∞)$K:\mathbb{R}^N\,\backslash\,\{0\}\rightarrow(0,\infty)$, M,G$M,G$ are two nonnegative continuous functions on (0,∞)×(0,∞)$(0,\infty)\times(0,\infty)$, V∈C(RN,R+)$V\in C(\mathbb{R}^N,\mathbb{R}^+)$, h:RN→(0,∞)$h:\mathbb{R}^N\rightarrow (0,\infty)$ is a measurable function and f:RN×R→R$f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function. Employing several nonvariational techniques, we prove various results of existence of nonnegative solutions. The main feature of this paper is that the Kirchhoff function M$M$ can be zero at zero and the problem is not variational in nature.

Abstract: In this paper we study the global behavior of forest fires (FFs) in the Continental United States for the period 1984–2013. The data are obtained from a public domain catalog maintained by the Monitoring Trends in Burn Severity project. First we adopt clustering analysis to reduce the information dimensionality. Then we adopt mathematical tools commonly used in the analysis of dynamical systems, namely fractal dimension, entropy and fractional Fourier transform. The fractional techniques unveil FF patterns embedded in the data.

In this study, the flow turbulence and variations of the supercritical free surface flow in a circular channel along a side weir are simulated as three dimensional using the RNG k-ε turbulence model and volume of fluid (VOF) scheme. Comparison between the numerical model and experimental measurements shows that the numerical model simulates the free surface flow with good accuracy. According to the numerical model results, the specific energy variations along the side weir for the supercritical flow regime are almost constant and the energy drop is not significant but by increasing the side weir length the energy difference between the side weir upstream and downstream increases. Next, using the nonlinear regression (NLR) and analysis of the simulation results, some relationships for calculating the discharge coefficient of side weir on circular channels in supercritical flow regime are provided.

Abstract: In this work we study a fractional-order susceptible-infective-recovered-susceptible (SIRS) epidemic model with a nonlinear incidence rate. The incidence is assumed to be a convex function with respect to the infective class of a host population. Local and uniform stability analysis of the disease-free equilibrium is investigated. The conditions for the existence of endemic equilibria (EE) are given. Local stability of the EE is discussed. Conditions for the existence of Hopf bifurcation at the EE are given. Most importantly, conditions ensuring that the system exhibits backward bifurcation are provided. Numerical simulations are performed to verify the correctness of results obtained analytically.

Abstract: In this work, the auxiliary equation method is applied to derive exact solutions of nonlinear fractional Klein–Gordon equation and space-time fractional Symmetric Regularized Long Wave equation. Consequently, some exact solutions of these equations are successfully obtained. These solutions are formed in fractional complex transform to convert fractional differential equations into ordinary differential equations. The fractional derivatives are described in Jumarie’s modified Riemann–Liouville sense. The exact solutions founded by the suggested method indicate that the approach is easy to implement and powerful.