We define and study a tritrophic bioeconomic model of Lotka-Volterra with a prey, middle predator, and top predator populations. These fish populations are exploited by two fishermen. We study the existence and the stability of the equilibrium points by using eigenvalues analysis and Routh-Hurwitz criterion. We determine the equilibrium point that maximizes the profit of each fisherman by solving the Nash equilibrium problem. Finally, following some numerical simulations, we observe that if the price varies, then the profit behavior of each fisherman will be changed; also, we conclude that the price change mechanism improves the fishing effort of the fishermen.

In this article, the linearization problem of fifth-order ordinary differential equation is presented by using the generalized Sundman transformation. The necessary and sufficient conditions which allow the nonlinear fifth-order ordinary differential equation to be transformed to the simplest linear equation are found. There is only one case in the part of sufficient conditions which is surprisingly less than the number of cases in the same part for order 2, 3, and 4. Moreover, the derivations of the explicit forms for the linearizing transformation are exhibited. Examples for the main results are included.

We discuss the existence of global solutions to the magnetohydrodynamics (MHD) equations, where the effects of finite Larmor radius corrections are taken into account. Unlike the usual MHD, the pressure is a tensor and it depends on not only the density but also the magnetic field. We show the existence of global solutions by the energy methods. Our techniques of proof are based on the existence of local solution by semigroups theory and a priori estimate.

It is known that power series expansion of certain functions such as $\mathrm{sech}(x)$ diverges beyond a finite radius of convergence. We present here an iterative power series expansion (IPS) to obtain a power series representation of $\mathrm{sech}(x)$ that is convergent for all $x$. The convergent series is a sum of the Taylor series of $\mathrm{sech}(x)$ and a complementary series that cancels the divergence of the Taylor series for $x\ge \pi /\mathrm{2}$. The method is general and can be applied to other functions known to have finite radius of convergence, such as $\mathrm{1}/(\mathrm{1}+x^{\mathrm{2}})$. A straightforward application of this method is to solve analytically nonlinear differential equations, which we also illustrate here. The method provides also a robust and very efficient numerical algorithm for solving nonlinear differential equations numerically. A detailed comparison with the fourth-order Runge-Kutta method and extensive analysis of the behavior of the error and CPU time are performed.

We prove some uniqueness results for the solution of two kinds of Dirichlet boundary value problems for second- and fourth-order linear elliptic differential equations with discontinuous coefficients in polyhedral angles, in weighted Sobolev spaces.

We propose and analyse an human immunodeficiency virus (HIV) infection model with spatial diffusion and delay in the immune response activation. In the proposed model, the immune response is presented by the cytotoxic T lymphocytes (CTL) cells. We first prove that the model is well-posed by showing the global existence, positivity, and boundedness of solutions. The model has three equilibria, namely, the free-infection equilibrium, the immune-free infection equilibrium, and the chronic infection equilibrium. The global stability of the first two equilibria is fully characterized by two threshold parameters that are the basic reproduction number $R_{\mathrm{0}}$ and the CTL immune response reproduction number $R_{\mathrm{1}}$. The stability of the last equilibrium depends on $R_{\mathrm{0}}$ and $R_{\mathrm{1}}$ as well as time delay $\mathrm{\tau }$ in the CTL activation. We prove that the chronic infection equilibrium is locally asymptotically stable when the time delay is sufficiently small, while it loses its stability and a Hopf bifurcation occurs when $\mathrm{\tau }$ passes through a certain critical value.

An analytical-approximate method is proposed for a type of nonlinear Volterra partial integro-differential equations with a weakly singular kernel. This method is based on the fractional differential transform method (FDTM). The approximate solutions of these equations are calculated in the form of a finite series with easily computable terms. The analytic solution is represented by an infinite series. We state and prove a theorem regarding an integral equation with a weak kernel by using the fractional differential transform method. The result of the theorem will be used to solve a weakly singular Volterra integral equation later on.

Considering the phenomenon of the mean reversion and the different speeds of stock prices in the bull market and in the bear market, we propose four dynamic models each of which is represented by a parameterized ordinary differential equation in this study. Based on existing studies, the models are in the form of either the logistic growth or the law of Newton’s cooling. We solve the models by dynamic integration and apply them to the daily closing prices of the Taiwan stock index, Taiwan Stock Exchange Capitalization Weighted Stock Index. The empirical study shows that some of the models fit the prices well and the forecasting ability of the best model is acceptable even though the martingale forecasts the prices slightly better. To increase the forecasting ability and to broaden the scope of applications of the dynamic models, we will model the coefficients of the dynamic models in the future. Applying the models to the market without the price limit is also our future work.

We study the equations governing the high-speed transmission in long-haul electrical systems $i{\partial }_{t}u-\left(\mathrm{1}/\mathrm{3}\right){\left|{\partial }_x\right|}^{\mathrm{3}}u=i\lambda {\partial }_x\left({\left|u\right|}^{\mathrm{2}}u\right)$, $\left(t,x\right)\in {\mathbb{R}}^{+}\times \mathbb{R}$, $u\left(\mathrm{0},x\right)=u_{\mathrm{0}}\left(x\right)$, $x\in \mathbb{R},$ where $\lambda \in \mathbb{R},{\left|{\partial }_x\right|}^{\alpha }={\mathcal{F}}^{-\mathrm{1}}{\left|\xi \right|}^{\alpha }\mathcal{F}$, and $\mathcal{F}$ is the Fourier transformation. Our purpose in this paper is to obtain the large time asymptotics for the solutions under the nonzero mass condition $\int u_{\mathrm{0}}\left(x\right)dx\ne \mathrm{0}.$

The mathematical simulation of water contaminant measurement is often used to assess the water quality. The monitoring point placement for water quality measurement in an opened-closed reservoir can give accurate or inaccurate assessment. In this research, the mathematical model of the approximated water quality in an opened-closed reservoir with removal mechanism system is proposed. The water quality model consists of the hydrodynamic model and the dispersion model. The hydrodynamic model is used to describe the water current in the opened-closed reservoir. The transient advection-diffusion equation with removal mechanism provides the water pollutant concentration. The water velocity from the hydrodynamic model is plugged into the dispersion model. The finite difference techniques are used to approximate the solution of the water quality model. The proposed numerical simulations give a suitable area of zonal removal mechanism placement. The proposed simulations also give the overall and specified approximated water quality for each point and time when the exit gate is opened on the different periods of time. In addition, the proposed techniques can give a suitable period of time to open the exit gate to achieve a good agreement water quality by using contaminant removal mechanism.

This paper presents a computational study of the stability of the steady state solutions of a biological model with negative feedback and time delay. The motivation behind the construction of our system comes from biological gene networks and the model takes the form of an integro-delay differential equation (IDDE) coupled to a partial differential equation. Linear analysis shows the existence of a critical delay where the stable steady state becomes unstable. Closed form expressions for the critical delay and associated frequency are found and confirmed by approximating the IDDE model with a system of $N$ delay differential equations (DDEs) coupled to $N$ ordinary differential equations. An example is then given that shows how the critical delay for the DDE system approaches the results for the IDDE model as $N$ becomes large.

This article discusses the evolution of Benjamin-Bona-Mahony (BBM) wave packet’s envelope. The envelope equation is derived by applying the asymptotic series up to the third order and choosing appropriate fast-to-slow variable transformations which eliminate the resonance terms that occurred. It is obtained that the envelope evolves satisfying the Nonlinear Schrodinger (NLS) equation. The evolution of NLS envelope is investigated through its exact solution, Soliton on Finite Background, which undergoes modulational instability during its propagation. The resulting wave may experience phase singularity indicated by wave splitting and merging and causing amplification on its amplitude. Some parameter values take part in triggering this phenomenon. The amplitude amplification can be analyzed by employing Maximal Temporal Amplitude (MTA) which is a quantity measuring the maximum wave elevation at each spatial position during the observation time. Wavenumber value affects the extreme position of the wave but not the amplitude amplification. Meanwhile, modulational frequency value affects both terms. Comparison of the evolution of the BBM wave packet to the previous results obtained from KdV equation gives interesting outputs regarding the extreme position and the maximum wave peaking.

We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in $L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and the right-hand side belongs to $L^{\mathrm{1}}\left(\mathrm{\Omega }\right)$; we extend the results where the case of linear finite elements approximation is considered. We prove that the unique solution of the discrete problem converges in $W_{\mathrm{0}}^{\mathrm{1},q}\left(\mathrm{\Omega }\right)$ for every $q$ with ($d=\mathrm{2}$ or $d=\mathrm{3}$) to the unique renormalized solution of the problem. Statements and proofs remain valid in our case, which permits obtaining a weaker result when the right-hand side is a bounded Radon measure and, when the coefficients are smooth, an error estimate in $W_{\mathrm{0}}^{\mathrm{1},q}\left(\mathrm{\Omega }\right)$ when the right-hand side $f$ belongs to $L^r\left(\mathrm{\Omega }\right)$ verifying $T_k\left(f\right)\in H^{\mathrm{1}}\left(\mathrm{\Omega }\right)$ for every $k>\mathrm{0}$, for some $r>\mathrm{1}.$

The approximate analytical solution of the fractional Cahn-Hilliard and Gardner equations has been acquired successfully via residual power series method (RPSM). The approximate solutions obtained by RPSM are compared with the exact solutions as well as the solutions obtained by homotopy perturbation method (HPM) and q-homotopy analysis method (q-HAM). Numerical results are known through different graphs and tables. The fractional derivatives are described in the Caputo sense. The results light the power, efficiency, simplicity, and reliability of the proposed method.

A mathematical model of dispersed bioparticle-blood flow through the stenosed coronary artery under the pulsatile boundary conditions is proposed. Blood is assumed to be an incompressible non-Newtonian fluid and its flow is considered as turbulence described by the Reynolds-averaged Navier-Stokes equations. Bioparticles are assumed to be spherical shape with the same density as blood, and their translation and rotational motions are governed by Newtonian equations. Impact of particle movement on the blood velocity, the pressure distribution, and the wall shear stress distribution in three different severity degrees of stenosis including 25%, 50%, and 75% are investigated through the numerical simulation using ANSYS 18.2. Increasing degree of stenosis severity results in higher values of the pressure drop and wall shear stresses. The higher level of bioparticle motion directly varies with the pressure drop and wall shear stress. The area of coronary artery with higher density of bioparticles also presents the higher wall shear stress.

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations ${\mathrm{D}}_t^{\alpha }x\left(t\right)=Ax\left(t\right)+{\mathrm{D}}_t^{\alpha -\mathrm{1}}F\left(t,x\left(t\right),Bx\left(t\right)\right), t\in \mathbb{R},$ where , $A$ is a linear densely defined operator of sectorial type on a complex Banach space $X$ and $B$ is a bounded linear operator defined on $X$, $F$ is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity $F$ is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

By combining the techniques of fractional calculus with measure of weak noncompactness and fixed point theorem, we establish the existence of weak solutions of multipoint boundary value problem for fractional integrodifferential equations.

We consider the nonlinear eigenvalue problem ${\left[D\left(u\right)u^{\mathrm{\prime }}\right]}^{\mathrm{\prime }}+\lambda f\left(u\right)=\mathrm{0}$, $u(t)>\mathrm{0}$, $t\in I≔(\mathrm{0,1})$, $u(\mathrm{0})=u(\mathrm{1})=\mathrm{0}$, where $D(u)=u^k$, $f(u)=u^{\mathrm{2}n-k-\mathrm{1}}+\sin u$, and $\lambda >\mathrm{0}$ is a bifurcation parameter. Here, $n\in \mathbb{N}$ and $k$ () are constants. This equation is related to the mathematical model of animal dispersal and invasion, and $\lambda$ is parameterized by the maximum norm $\alpha ={‖u_{\lambda }‖}_{\mathrm{\infty }}$ of the solution $u_{\lambda }$ associated with $\lambda$ and is written as $\lambda =\lambda (\alpha )$. Since $f(u)$ contains both power nonlinear term $u^{\mathrm{2}n-k-\mathrm{1}}$ and oscillatory term $\sin u$, it seems interesting to investigate how the shape of $\lambda (\alpha )$ is affected by $f(u)$. The purpose of this paper is to characterize the total shape of $\lambda (\alpha )$ by $n$ and $k$. Precisely, we establish three types of shape of $\lambda (\alpha )$, which seem to be new.

In this paper, we will consider the existence of a strong solution for stochastic differential equations with discontinuous drift coefficients. More precisely, we study a class of stochastic differential equations when the drift coefficients are an increasing function instead of Lipschitz continuous or continuous. The main tools of this paper are the lower solutions and upper solutions of stochastic differential equations.

We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, in electrokinetics, and in the modeling of biomolecules’ electrostatics. The first method, considered for the problem, is the widely known Ghost-Fluid Method (GFM) and the second method is the recently introduced Voronoi Interface Method (VIM). The VIM method uses Voronoi partitions near the interface to construct local configurations that enable the use of the Ghost-Fluid philosophy in one dimension. Both methods lead to symmetric positive definite linear systems. The Ghost-Fluid Method is generally first-order accurate, except in the case of both a constant discontinuity in the solution and a constant diffusion coefficient, while the Voronoi Interface Method is second-order accurate in the $L^{\mathrm{\infty }}$-norm. Therefore, the Voronoi Interface Method generally outweighs the Ghost-Fluid Method except in special case of both a constant discontinuity in the solution and a constant diffusion coefficient, where the Ghost-Fluid Method performs better than the Voronoi Interface Method. The paper includes numerical examples displaying this fact clearly and its findings can be used to determine which approach to choose based on the properties of the real life problem in hand.

This paper describes a numerical solution for mathematical model of the transport equation in a simple rectangular box domain. The model of street tunnel pollution distribution using two-dimension advection and three-dimension diffusion is solved numerically. Because of the nature of the problem, the model is extended to become three-dimension advection and three-dimension diffusion to study the sea-sand mining pollution distribution. This model with various advection and diffusion parameters and the boundaries conditions is also solved numerically using a finite difference (FTCS) method.

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is $\{-\mathrm{d}\mathrm{i}\mathrm{v}[(\mathrm{1}+{\left|u\right|}^q)\nabla u]+u=\gamma \left({\left|\nabla u\right|}^{\mathrm{2}}/{\left(\mathrm{1}+\left|u\right|\right)}^{\mathrm{1}-q}\right)+f\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega },\hspace{0.17em}\hspace{0.17em}u=\mathrm{0}\hspace{0.17em}\hspace{0.17em}\mathrm{o}\mathrm{n}\hspace{0.17em}\hspace{0.17em}\partial \mathrm{\Omega },\}$, where , $\gamma >\mathrm{0}$ and $f$ belongs to some appropriate Lebesgue space. We give assumptions on $f$ with respect to $q$ and $\gamma$ to show the existence and regularity results for the solutions of previous equation.

In this paper we consider a numerical method for solving nonhomogeneous backward heat conduction problem. Coupled with the likewise Crank Nicolson scheme and an intermediate variable, the backward problem is transformed to a nonhomogeneous Helmholtz type problem; the unknown initial temperature can be obtained by solving this Helmholtz type problem. To illustrate the effectiveness and accuracy of the proposed method, we solve several problems in both two and three dimensions. The results show that this numerical method can solve nonhomogeneous backward heat conduction problem effectively and precisely, even though the final temperature is disturbed by significant noise.

In this paper, we consider a priori error estimates for the finite volume element schemes of optimal control problems, which are governed by linear elliptic partial differential equation. The variational discretization approach is used to deal with the control. The error estimation shows that the combination of variational discretization and finite volume element formulation allows optimal convergence. Numerical results are provided to support our theoretical analysis.

This paper presents theoretical results about control of the coefficient function in a hyperbolic problem with Dirichlet conditions. The existence and uniqueness of the optimal solution for optimal control problem are proved and adjoint problem is used to obtain gradient of the functional. However, a second adjoint problem is given to calculate the gradient on the space $W_{\mathrm{2}}^{\mathrm{1}}\left(\mathrm{0},l\right).$ After calculating gradient of the cost functional and proving the Lipschitz continuity of the gradient, necessary condition for optimal solution is constructed.

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem’s nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

In this paper, we study the dynamics of a viral infection model formulated by five fractional differential equations (FDEs) to describe the interactions between host cells, virus, and humoral immunity presented by antibodies. The infection transmission process is modeled by Hattaf-Yousfi functional response which covers several forms of incidence rate existing in the literature. We first show that the model is mathematically and biologically well-posed. By constructing suitable Lyapunov functionals, the global stability of equilibria is established and characterized by two threshold parameters. Finally, some numerical simulations are presented to illustrate our theoretical analysis.