Bounds on the critical times for the Fisher-KPP equation

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented. doi:10.1017/S1446181121000365

Positive solution of Hilfer fractional differential equations with integral boundary conditions

In this article, we have interested the study of the existence and uniqueness of positive solutions of the first-order nonlinear Hilfer fractional differential equation $$D_{0^{+}}^{\alpha ,\beta }y(t)=f(t,y(t)),\text{ }0<t\leq 1,$$ with the integral boundary condition $$I_{0^{+}}^{1-\gamma }y(0)=\lambda \int_{0}^{1}y(s)ds+d,$$ where $0<\alpha \leq 1,$ $0\leq \beta \leq 1,$ $\lambda \geq 0,$ $d\in \mathbb{R}^{+},$ and $D_{0^{+}}^{\alpha ,\beta }$, $I_{0^{+}}^{1-\gamma }$ are fractional ope\-rators in the Hilfer, Riemann-Liouville concepts, respectively. In this approach, we transform the given fractional differential equation into an equivalent integral equation. Then we establish sufficient conditions and employ the Schauder fixed point theorem and the method of upper and lower solutions to obtain the existence of a positive solution of a given problem. We also use the Banach contraction principle theorem to show the existence of a unique positive solution. The result of existence obtained by structure the upper and lower control functions of the nonlinear term is without any monotonous conditions. Finally, an example is presented to show the effectiveness of our main results.

Numerical Analysis of Cu + Al 2 O 3 / H 2 O Hybrid Nanofluid of Streamwise and Cross Flow with Thermal Radiation Effect: Duality and Stability

The motion of water conveying copper and aluminum nanoparticles on a heated moving sheet when thermal radiation and stretching/shrinking surface is significant and is investigated in this study to announce the increasing effects of volume fractions, thermal radiation, and moving parameters on this transport phenomenon. Furthermore, the flow of a Cu − Al 2 O 3 /water hybrid nanofluid across a heated moving sheet has been studied in both cross and streamwise directions. Thermal radiation effect is also considered, as this effect along with cross flow has not yet been investigated for the hybrid nanofluid in the published literature. Two distinct types of nanoparticles, namely, Al 2 O 3 (alumina) and Cu (copper), have been used to prepare hybrid nanofluid where water is considered as a base fluid. The system of nonlinear partial differential equations (PDEs) has been transferred to ordinary differential equations (ODEs) by compatible transformations before solving them by employing the III-stage Lobatto-IIIa method in bvp4c solver in MATLAB 2017 software. Temporal stability analysis has been carried out in order to verify stable branch between two branches by obtaining the smallest eigenvalue values. The branches obtained are addressed in depth against every applied parameter using figures and tables. The results show that there are three ranges of branches, no solution exists when λ > λ c , dual branches exist when 0.23 ≤ λ ≤ λ c , and a single solution exists when λ > 0.23 . Moreover, thermal layer thickness declines initially and then enhances in the upper and lower solutions for the higher values of the thermal radiation parameter.

Traveling waves and spreading properties for a reaction-diffusion competition model with seasonal succession*

In this paper, we investigate the propagation dynamics of a reaction–diffusion competition model with seasonal succession in the whole space. Under the weak competition condition, the corresponding kinetic system admits a globally stable positive periodic solution ( u ^ ( t ) , v ^ ( t ) ) . By the method of upper and lower solutions and the Schauder fixed point theorem, we first obtain the existence and nonexistence of traveling wave solutions connecting (0, 0) to ( u ^ ( t ) , v ^ ( t ) ) . Then we use the comparison arguments to establish the spreading properties for a large class of solutions.

Monotone Iterative and Upper–Lower Solution Techniques for Solving the Nonlinear ψ-Caputo Fractional Boundary Value Problem

The objective of this paper is to study the existence of extremal solutions for nonlinear boundary value problems of fractional differential equations involving the ψ−Caputo derivative CDa+σ;ψϱ(t)=V(t,ϱ(t)) under integral boundary conditions ϱ(a)=λIν;ψϱ(η)+δ. Our main results are obtained by applying the monotone iterative technique combined with the method of upper and lower solutions. Further, we consider three cases for ψ*(t) as t, Caputo, 2t, t, and Katugampola (for ρ=0.5) derivatives and examine the validity of the acquired outcomes with the help of two different particular examples.

BOUNDS ON THE CRITICAL TIMES FOR THE GENERAL FISHER–KPP EQUATION

The Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation is one of the prototypical reaction–diffusion equations and is encountered in many areas, primarily in population dynamics. An important consideration for the phenomena modelled by diffusion equations is the length of the diffusive process. In this paper, three definitions of the critical time are given, and bounds are obtained by a careful construction of the upper and lower solutions. The comparison functions satisfy the nonlinear, but linearizable, partial differential equations of Fisher–KPP type. Results of the numerical simulations are displayed. Extensions to some classes of reaction–diffusion systems and an application to a spatially heterogeneous harvesting model are also presented.

Existence of Solutions for a Class of Nonlinear Neumann BVPs in the Presence of Upper and Lower Solutions

In this article, we explore the monotone iterative technique (MI-technique) to study the existence of solutions for a class of nonlinear Neumann 4-point, boundary value problems (BVPs) defined as, \begin{eqnarray*} \begin{split} -\z^{(2)}(\y)=\x(\y,\z,\z^{(1)}),\quad 0<\y<1,\\ \z^{(1)}(0)=\lambda \z^{(1)}(\beta_1 ),\quad \z^{(1)}(1)=\delta \z^{(1)}(\beta_2), \end{split} \end{eqnarray*} where $ 0<\beta_1 \leq \beta_2 <1$ and $\lambda$, $\delta\in (0,1)$. The nonlinear term $ \x(\y,\z,\z^{(1)}): \Omega\rightarrow \mathbb{R} $, where $\Omega =[0,1]\times \mathbb{R}^2 $, is Lipschitz in $ \z^{(1)}(\y)$ and one sided Lipschitz in $ \z(\y)$. Using lower solution $l(\y)$ and upper solutions $u(\y)$, we develop MI-technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti maximum principle. Then under certain assumptions, we prove that these sequence converges uniformly to the solution $ \z(\y)$ in the specific region, where $ \frac{\partial\x}{\partial\z}<0 $ or $ \frac{\partial\x}{\partial\z}>0 $. To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi-point BVPs. We don’t require sign restriction which is very common and very strict condition.