International Journal of Differential Equations Articles (Project Euclid)
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The latest articles from International Journal of Differential Equations on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2017 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Tue, 18 Jul 2017 22:02 EDTTue, 18 Jul 2017 22:02 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Modelling the Potential Role of Media Campaigns in Ebola Transmission
Dynamics
http://projecteuclid.org/euclid.ijde/1487905251
<strong>Sylvie Diane Djiomba Njankou</strong>, <strong>Farai Nyabadza</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 13 pages.</p><p><strong>Abstract:</strong><br/>
A six-compartment mathematical model is formulated to investigate the
role of media campaigns in Ebola transmission dynamics. The model
includes tweets or messages sent by individuals in different
compartments. The media campaigns reproduction number is computed and
used to discuss the stability of the disease states. The presence of a
backward bifurcation as well as a forward bifurcation is shown
together with the existence and local stability of the endemic
equilibrium. Results show that messages sent through media have a more
significant beneficial effect on the reduction of Ebola cases if they
are more effective and spaced out.
</p>projecteuclid.org/euclid.ijde/1487905251_20170718220210Tue, 18 Jul 2017 22:02 EDTA Family of Boundary Value Methods for Systems of Second-Order Boundary
Value Problems
http://projecteuclid.org/euclid.ijde/1487905252
<strong>T. A. Biala</strong>, <strong>S. N. Jator</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 12 pages.</p><p><strong>Abstract:</strong><br/>
A family of boundary value methods (BVMs) with continuous coefficients is
derived and used to obtain methods which are applied via the block
unification approach. The methods obtained from these continuous BVMs
are weighted the same and are used to simultaneously generate
approximations to the exact solution of systems of second-order
boundary value problems (BVPs) on the entire interval of integration.
The convergence of the methods is analyzed. Numerical experiments were
performed to show efficiency and accuracy advantages.
</p>projecteuclid.org/euclid.ijde/1487905252_20170718220210Tue, 18 Jul 2017 22:02 EDTA Trigonometrically Fitted Block Method for Solving Oscillatory
Second-Order Initial Value Problems and Hamiltonian Systems
http://projecteuclid.org/euclid.ijde/1487905253
<strong>F. F. Ngwane</strong>, <strong>S. N. Jator</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we present a block hybrid trigonometrically fitted
Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are
functions of the frequency and the step-size for directly solving
general second-order initial value problems (IVPs), including
Hamiltonian systems such as the energy conserving equations and
systems arising from the semidiscretization of partial differential
equations (PDEs). Four discrete hybrid formulas used to formulate the
BHTRKNM are provided by a continuous one-step hybrid trigonometrically
fitted method with an off-grid point. We implement BHTRKNM in a
block-by-block fashion; in this way, the method does not suffer from
the disadvantages of requiring starting values and predictors which
are inherent in predictor-corrector methods. The stability property of
the BHTRKNM is discussed and the performance of the method is
demonstrated on some numerical examples to show accuracy and
efficiency advantages.
</p>projecteuclid.org/euclid.ijde/1487905253_20170718220210Tue, 18 Jul 2017 22:02 EDTAsymptotics for the Ostrovsky-Hunter Equation in the Critical Case
http://projecteuclid.org/euclid.ijde/1487905254
<strong>Fernando Bernal-Vílchis</strong>, <strong>Nakao Hayashi</strong>, <strong>Pavel I. Naumkin</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 21 pages.</p><p><strong>Abstract:</strong><br/>
We consider the Cauchy problem for the Ostrovsky-Hunter equation ${\partial }_{x}({\partial }_{t}u-(b/\mathrm{3}){\partial }_{x}^{\mathrm{3}}u-{\partial }_{x}\mathcal{K}{u}^{\mathrm{3}})=au$ ,
$(t,x)\in {\mathbb{R}}^{\mathrm{2}}$ , $u(\mathrm{0},x)={u}_{\mathrm{0}}(x)$ ,
$x\in \mathbb{R}$ , where $ab>\mathrm{0}$ . Define ${\xi }_{\mathrm{0}}={(\mathrm{27}a/b)}^{\mathrm{1}/\mathrm{4}}$ . Suppose that
$\mathcal{K}$ is a pseudodifferential operator
with a symbol $\stackrel{^}{K}(\xi )$ such that $\stackrel{^}{K}(\pm{\xi }_{\mathrm{0}})=\mathrm{0}$ , $\mathrm{I}\mathrm{m} \stackrel{^}{K}(\xi )=\mathrm{0}$ , and $|\stackrel{^}{K}(\xi )|\le C$ .
For example, we can take $\stackrel{^}{K}(\xi )=({\xi }^{\mathrm{2}}-{\xi }_{\mathrm{0}}^{\mathrm{2}})/({\xi }^{\mathrm{2}}+\mathrm{1})$ . We
prove the global in time existence and the large time asymptotic
behavior of solutions.
</p>projecteuclid.org/euclid.ijde/1487905254_20170718220210Tue, 18 Jul 2017 22:02 EDTExistence and Uniqueness of Solutions for BVP of Nonlinear Fractional
Differential Equation
http://projecteuclid.org/euclid.ijde/1487905255
<strong>Cheng-Min Su</strong>, <strong>Jian-Ping Sun</strong>, <strong>Ya-Hong Zhao</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 7 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the existence and uniqueness of solutions for the
following boundary value problem of nonlinear fractional differential
equation: $({}^{C}{D}_{\mathrm{0}+}^{q}u)(t)=f(t,u(t))$ , $t\in (\mathrm{0,1})$ , $u(\mathrm{0})={u}^{\mathrm{\prime }\mathrm{\prime }}(\mathrm{0})=\mathrm{0}, ({}^{C}{D}_{\mathrm{0}+}^{{\sigma }_{\mathrm{1}}}u)(\mathrm{1})=\lambda ({I}_{\mathrm{0}+}^{{\sigma }_{\mathrm{2}}}u)(\mathrm{1})$ , where $\mathrm{2}<q<\mathrm{3}$ , $\mathrm{0}<{\sigma }_{\mathrm{1}}\le \mathrm{1}$ , ${\sigma }_{\mathrm{2}}>\mathrm{0}$ , and $\lambda \ne \mathrm{\Gamma }(\mathrm{2}+{\sigma }_{\mathrm{2}})/\mathrm{\Gamma }(\mathrm{2}-{\sigma }_{\mathrm{1}})$ .
The main tools used are nonlinear alternative of Leray-Schauder type
and Banach contraction principle.
</p>projecteuclid.org/euclid.ijde/1487905255_20170718220210Tue, 18 Jul 2017 22:02 EDTAn Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed
Linear Delay Differential Equations
http://projecteuclid.org/euclid.ijde/1491962518
<strong>Süleyman Cengizci</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 8 pages.</p><p><strong>Abstract:</strong><br/>
In this work, approximations to the solutions of singularly perturbed
second-order linear delay differential equations are studied. We
firstly use two-term Taylor series expansion for the delayed
convection term and obtain a singularly perturbed ordinary
differential equation (ODE). Later, an efficient and simple asymptotic
method so called Successive Complementary Expansion Method (SCEM) is
employed to obtain a uniformly valid approximation to this
corresponding singularly perturbed ODE. As the final step, we employ a
numerical procedure to solve the resulting equations that come from
SCEM procedure. In order to show efficiency of this
numerical-asymptotic hybrid method, we compare the results with exact
solutions if possible; if not we compare with the results that are
obtained by other reported methods.
</p>projecteuclid.org/euclid.ijde/1491962518_20170718220210Tue, 18 Jul 2017 22:02 EDTApproximate Controllability of Semilinear Control System Using Tikhonov
Regularization
http://projecteuclid.org/euclid.ijde/1491962519
<strong>Ravinder Katta</strong>, <strong>N. Sukavanam</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 6 pages.</p><p><strong>Abstract:</strong><br/>
For an approximately controllable semilinear system, the problem of
computing control for a given target state is converted into an
equivalent problem of solving operator equation which is ill-posed. We
exhibit a sequence of regularized controls which steers the semilinear
control system from an arbitrary initial state ${x}^{\mathrm{0}}$ to an $\mathrm{ϵ}$ neighbourhood of the
target state ${x}_{\tau }$
at time $\tau >\mathrm{0}$ under the assumption that the nonlinear
function $f$ is Lipschitz
continuous. The convergence of the sequences of regularized controls
and the corresponding mild solutions are shown under some assumptions
on the system operators. It is also proved that the target state
corresponding to the regularized control is close to the actual state
to be attained.
</p>projecteuclid.org/euclid.ijde/1491962519_20170718220210Tue, 18 Jul 2017 22:02 EDTCritical Oscillation Constant for Euler Type Half-Linear Differential
Equation Having Multi-Different Periodic Coefficients
http://projecteuclid.org/euclid.ijde/1491962520
<strong>Adil Misir</strong>, <strong>Banu Mermerkaya</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 8 pages.</p><p><strong>Abstract:</strong><br/>
We compute explicitly the oscillation constant for Euler type half-linear
second-order differential equation having multi-different periodic
coefficients.
</p>projecteuclid.org/euclid.ijde/1491962520_20170718220210Tue, 18 Jul 2017 22:02 EDTAn Analysis of the Replicator Dynamics for an Asymmetric Hawk-Dove
Game
http://projecteuclid.org/euclid.ijde/1494468060
<strong>Ikjyot Singh Kohli</strong>, <strong>Michael C. Haslam</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 7 pages.</p><p><strong>Abstract:</strong><br/>
We analyze, using a dynamical systems approach, the replicator dynamics
for the asymmetric Hawk-Dove game in which there is a set of four pure
strategies with arbitrary payoffs. We give a full account of the
equilibrium points and their stability and derive the Nash equilibria.
We also give a detailed account of the local bifurcations that the
system exhibits based on choices of the typical Hawk-Dove parameters
$v$ and $c$ . We also give
details on the connections between the results found in this work and
those of the standard two-strategy Hawk-Dove game. We conclude the
paper with some examples of numerical simulations that further
illustrate some global behaviours of the system.
</p>projecteuclid.org/euclid.ijde/1494468060_20170718220210Tue, 18 Jul 2017 22:02 EDTExistence and Uniqueness of Solution of Stochastic Dynamic Systems with
Markov Switching and Concentration Points
http://projecteuclid.org/euclid.ijde/1494468061
<strong>Taras Lukashiv</strong>, <strong>Igor Malyk</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 5 pages.</p><p><strong>Abstract:</strong><br/>
In this article the problem of existence and uniqueness of solutions of
stochastic differential equations with jumps and concentration points
are solved. The theoretical results are illustrated by one
example.
</p>projecteuclid.org/euclid.ijde/1494468061_20170718220210Tue, 18 Jul 2017 22:02 EDTOn a Singular Second-Order Multipoint Boundary Value Problem at
Resonance
http://projecteuclid.org/euclid.ijde/1500429716
<strong>S. A. Iyase</strong>, <strong>O. F. Imaga</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 6 pages.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to derive existence results for a second-order
singular multipoint boundary value problem at resonance using
coincidence degree arguments.
</p>projecteuclid.org/euclid.ijde/1500429716_20170718220210Tue, 18 Jul 2017 22:02 EDTCollocation Method Based on Genocchi Operational Matrix for Solving
Generalized Fractional Pantograph Equations
http://projecteuclid.org/euclid.ijde/1500429717
<strong>Abdulnasir Isah</strong>, <strong>Chang Phang</strong>, <strong>Piau Phang</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 10 pages.</p><p><strong>Abstract:</strong><br/>
An effective collocation method based on Genocchi operational matrix for
solving generalized fractional pantograph equations with initial and
boundary conditions is presented. Using the properties of Genocchi
polynomials, we derive a new Genocchi delay operational matrix which
we used together with the Genocchi operational matrix of fractional
derivative to approach the problems. The error upper bound for the
Genocchi operational matrix of fractional derivative is also shown.
Collocation method based on these operational matrices is applied to
reduce the generalized fractional pantograph equations to a system of
algebraic equations. The comparison of the numerical results with some
existing methods shows that the present method is an excellent
mathematical tool for finding the numerical solutions of generalized
fractional pantograph equations.
</p>projecteuclid.org/euclid.ijde/1500429717_20170718220210Tue, 18 Jul 2017 22:02 EDTIdentifying Initial Condition in Degenerate Parabolic Equation with
Singular Potential
http://projecteuclid.org/euclid.ijde/1500429718
<strong>K. Atifi</strong>, <strong>Y. Balouki</strong>, <strong>El-H. Essoufi</strong>, <strong>B. Khouiti</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 17 pages.</p><p><strong>Abstract:</strong><br/>
A hybrid algorithm and regularization method are proposed, for the first
time, to solve the one-dimensional degenerate inverse heat conduction
problem to estimate the initial temperature distribution from point
measurements. The evolution of the heat is given by a degenerate
parabolic equation with singular potential. This problem can be
formulated in a least-squares framework, an iterative procedure which
minimizes the difference between the given measurements and the value
at sensor locations of a reconstructed field. The mathematical model
leads to a nonconvex minimization problem. To solve it, we prove the
existence of at least one solution of problem and we propose two
approaches: the first is based on a Tikhonov regularization, while the
second approach is based on a hybrid genetic algorithm (married
genetic with descent method type gradient). Some numerical experiments
are given.
</p>projecteuclid.org/euclid.ijde/1500429718_20170718220210Tue, 18 Jul 2017 22:02 EDTCyclic Growth and Global Stability of Economic Dynamics of Kaldor Type in Two Dimensionshttp://projecteuclid.org/euclid.ijde/1502762491<strong>Aka Fulgence Nindjin</strong>, <strong>Albin Tetchi N’Guessan</strong>, <strong>Hypolithe Okou</strong>, <strong>Kessé Thiban Tia</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 12 pages.</p><p><strong>Abstract:</strong><br/>
This article proposes nonlinear economic dynamics continuous in two dimensions of Kaldor type, the saving rate and the investment rate, which are functions of ecological origin verifying the nonwasting properties of the resources and economic assumption of Kaldor. The important results of this study contain the notions of bounded solutions, the existence of an attractive set, local and global stability of equilibrium, the system permanence, and the existence of a limit cycle.
</p>projecteuclid.org/euclid.ijde/1502762491_20170814220141Mon, 14 Aug 2017 22:01 EDTFractional Variational Iteration Method for Solving Fractional Partial
Differential Equations with Proportional Delayhttps://projecteuclid.org/euclid.ijde/1491962508<strong>Brajesh Kumar Singh</strong>, <strong>Pramod Kumar</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 11 pages.</p><p><strong>Abstract:</strong><br/>
This paper deals with an alternative approximate analytic solution to time
fractional partial differential equations (TFPDEs) with proportional delay,
obtained by using fractional variational iteration method, where the fractional
derivative is taken in Caputo sense. The proposed series solutions are found to
converge to exact solution rapidly. To confirm the efficiency and validity of
FRDTM, the computation of three test problems of TFPDEs with proportional delay
was presented. The scheme seems to be very reliable, effective, and efficient
powerful technique for solving various types of physical models arising in
science and engineering.
</p>projecteuclid.org/euclid.ijde/1491962508_20170918220157Mon, 18 Sep 2017 22:01 EDTAnalysis of a Predator-Prey Model with Switching and Stage-Structure for
Predatorhttps://projecteuclid.org/euclid.ijde/1507687407<strong>T. Suebcharoen</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 11 pages.</p><p><strong>Abstract:</strong><br/>
This paper studies the behavior of a predator-prey model with switching
and stage-structure for predator. Bounded positive solution,
equilibria, and stabilities are determined for the system of delay
differential equation. By choosing the delay as a bifurcation
parameter, it is shown that the positive equilibrium can be
destabilized through a Hopf bifurcation. Some numerical simulations
are also given to illustrate our results.
</p>projecteuclid.org/euclid.ijde/1507687407_20171010220341Tue, 10 Oct 2017 22:03 EDTSolving Nonlinear Fourth-Order Boundary Value Problems Using a Numerical Approach: $(m+1)$ th-Step Block Methodhttps://projecteuclid.org/euclid.ijde/1510801548<strong>Oluwaseun Adeyeye</strong>, <strong>Zurni Omar</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 9 pages.</p><p><strong>Abstract:</strong><br/>
Nonlinear boundary value problems (BVPs) are more tedious to solve than their linear counterparts. This is observed in the extra computation required when determining the missing conditions in transforming BVPs to initial value problems. Although a number of numerical approaches are already existent in literature to solve nonlinear BVPs, this article presents a new block method with improved accuracy to solve nonlinear BVPs. A $(m+\mathrm{1})\mathrm{t}\mathrm{h}$ -step block method is developed using a modified Taylor series approach to directly solve fourth-order nonlinear boundary value problems (BVPs) where $m$ is the order of the differential equation under consideration. The schemes obtained were combined to simultaneously produce solution to the fourth-order nonlinear BVPs at $m+\mathrm{1}$ points iteratively. The derived block method showed improved accuracy in comparison to previously existing authors when solving the same problems. In addition, the suitability of the $(m+\mathrm{1})\mathrm{t}\mathrm{h}$ -step block method was displayed in the solution for magnetohydrodynamic squeezing flow in porous medium.
</p>projecteuclid.org/euclid.ijde/1510801548_20171115220602Wed, 15 Nov 2017 22:06 ESTThe Morbidity of Multivariable Grey Model MGM$(\mathrm 1,m)$https://projecteuclid.org/euclid.ijde/1513220604<strong>Haixia Wang</strong>, <strong>Lingdi Zhao</strong>, <strong>Mingzhao Hu</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 5 pages.</p><p><strong>Abstract:</strong><br/>
This paper proposes the morbidity of the multivariable grey prediction MGM$(\mathrm 1,m)$ model. Based on the morbidity of the differential equations, properties of matrix, and Gerschgorin Panel Theorem, we analyze the factors that affect the morbidity of the multivariable grey model and give a criterion to justify the morbidity of MGM$(\mathrm 1,m)$. Finally, an example is presented to illustrate the practicality of our results.
</p>projecteuclid.org/euclid.ijde/1513220604_20171213220340Wed, 13 Dec 2017 22:03 ESTFinite Time Synchronization of Extended Nonlinear Dynamical Systems Using Local Couplinghttps://projecteuclid.org/euclid.ijde/1515467020<strong>A. Acosta</strong>, <strong>P. García</strong>, <strong>H. Leiva</strong>, <strong>A. Merlitti</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 7 pages.</p><p><strong>Abstract:</strong><br/>
We consider two reaction-diffusion equations connected by one-directional coupling function and study the synchronization problem in the case where the coupling function affects the driven system in some specific regions. We derive conditions that ensure that the evolution of the driven system closely tracks the evolution of the driver system at least for a finite time. The framework built to achieve our results is based on the study of an abstract ordinary differential equation in a suitable Hilbert space. As a specific application we consider the Gray-Scott equations and perform numerical simulations that are consistent with our main theoretical results.
</p>projecteuclid.org/euclid.ijde/1515467020_20180108220359Mon, 08 Jan 2018 22:03 ESTOptimization of the Two Fishermen’s Profits Exploiting Three Competing Species Where Prices Depend on Harvesthttps://projecteuclid.org/euclid.ijde/1515467021<strong>Imane Agmour</strong>, <strong>Meriem Bentounsi</strong>, <strong>Naceur Achtaich</strong>, <strong>Youssef El Foutayeni</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2017, 17 pages.</p><p><strong>Abstract:</strong><br/>
Bioeconomic modeling of the exploitation of biological resources such as fisheries has gained importance in recent years. In this work we propose to define and study a bioeconomic equilibrium model for two fishermen who catch three species taking into consideration the fact that the prices of fish populations vary according to the quantity harvested; these species compete with each other for space or food; the natural growth of each species is modeled using a logistic law. The main purpose of this work is to define the fishing effort that maximizes the profit of each fisherman, but all of them have to respect two constraints: the first one is the sustainable management of the resources and the second one is the preservation of the biodiversity. The existence of the steady states and their stability are studied using eigenvalue analysis. The problem of determining the equilibrium point that maximizes the profit of each fisherman leads to Nash equilibrium problem; to solve this problem we transform it into a linear complementarity problem (LCP); then we prove that the obtained problem (LCP) admits a unique solution that represents the Nash equilibrium point of our problem. We close our paper with some numerical simulations.
</p>projecteuclid.org/euclid.ijde/1515467021_20180108220359Mon, 08 Jan 2018 22:03 ESTThe Impact of Price on the Profits of Fishermen Exploiting Tritrophic Prey-Predator Fish Populationshttps://projecteuclid.org/euclid.ijde/1518577243<strong>Meriem Bentounsi</strong>, <strong>Imane Agmour</strong>, <strong>Naceur Achtaich</strong>, <strong>Youssef El Foutayeni</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 13 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1518577243_20180213220045Tue, 13 Feb 2018 22:00 ESTLinearization of Fifth-Order Ordinary Differential Equations by Generalized Sundman Transformationshttps://projecteuclid.org/euclid.ijde/1518577244<strong>Supaporn Suksern</strong>, <strong>Kwanpaka Naboonmee</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 17 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1518577244_20180213220045Tue, 13 Feb 2018 22:00 ESTExistence of Global Solutions for Nonlinear Magnetohydrodynamics with Finite Larmor Radius Correctionshttps://projecteuclid.org/euclid.ijde/1518577245<strong>Fariha Elsrrawi</strong>, <strong>Harumi Hattori</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 11 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1518577245_20180213220045Tue, 13 Feb 2018 22:00 ESTConvergent Power Series of $\mathrm{sech}(x)$ and Solutions to Nonlinear Differential Equationshttps://projecteuclid.org/euclid.ijde/1521252034<strong>U. Al Khawaja</strong>, <strong>Qasem M. Al-Mdallal</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1521252034_20180316220037Fri, 16 Mar 2018 22:00 EDTUniqueness Results for Higher Order Elliptic Equations in Weighted Sobolev Spaceshttps://projecteuclid.org/euclid.ijde/1523498445<strong>Loredana Caso</strong>, <strong>Patrizia Di Gironimo</strong>, <strong>Sara Monsurrò</strong>, <strong>Maria Transirico</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1523498445_20180411220054Wed, 11 Apr 2018 22:00 EDTSpatiotemporal Dynamics of an HIV Infection Model with Delay in Immune Response Activationhttps://projecteuclid.org/euclid.ijde/1523498446<strong>Mehdi Maziane</strong>, <strong>Khalid Hattaf</strong>, <strong>Noura Yousfi</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 9 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1523498446_20180411220054Wed, 11 Apr 2018 22:00 EDTAn Analytical and Approximate Solution for Nonlinear Volterra Partial Integro-Differential Equations with a Weakly Singular Kernel Using the Fractional Differential Transform Methodhttps://projecteuclid.org/euclid.ijde/1525744831<strong>Rezvan Ghoochani-Shirvan</strong>, <strong>Jafar Saberi-Nadjafi</strong>, <strong>Morteza Gachpazan</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1525744831_20180507220036Mon, 07 May 2018 22:00 EDTApplications of Parameterized Nonlinear Ordinary Differential Equations and Dynamic Systems: An Example of the Taiwan Stock Indexhttps://projecteuclid.org/euclid.ijde/1525744832<strong>Meng-Rong Li</strong>, <strong>Tsung-Jui Chiang-Lin</strong>, <strong>Yong-Shiuan Lee</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 11 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1525744832_20180507220036Mon, 07 May 2018 22:00 EDTHigh-Speed Transmission in Long-Haul Electrical Systemshttps://projecteuclid.org/euclid.ijde/1525744833<strong>Beatriz Juárez-Campos</strong>, <strong>Elena I. Kaikina</strong>, <strong>Pavel I. Naumkin</strong>, <strong>Héctor Francisco Ruiz-Paredes</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 13 pages.</p><p><strong>Abstract:</strong><br/>
We study the equations governing the high-speed transmission in long-haul electrical systems $i{\partial }_{t}u-(\mathrm{1}/\mathrm{3}){|{\partial }_{x}|}^{\mathrm{3}}u=i\lambda {\partial }_{x}({|u|}^{\mathrm{2}}u)$ , $(t,x)\in {\mathbb{R}}^{+}\times\mathbb{R}$ , $u(\mathrm{0},x)={u}_{\mathrm{0}}(x)$ , $x\in \mathbb{R},$ where $\lambda \in \mathbb{R},\text{\hspace\{0.17em\}\hspace\{0.17em\}}{|{\partial }_{x}|}^{\alpha }={\mathcal{F}}^{-\mathrm{1}}{|\xi |}^{\alpha }\mathcal{F}$ , 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 $\int {u}_{\mathrm{0}}(x)dx\ne \mathrm{0}.$
</p>projecteuclid.org/euclid.ijde/1525744833_20180507220036Mon, 07 May 2018 22:00 EDTLinear Analysis of an Integro-Differential Delay Equation Modelhttps://projecteuclid.org/euclid.ijde/1528855325<strong>Anael Verdugo</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 6 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1528855325_20180612220213Tue, 12 Jun 2018 22:02 EDTNumerical Simulations of Water Quality Measurement Model in an Opened-Closed Reservoir with Contaminant Removal Mechanismhttps://projecteuclid.org/euclid.ijde/1528855358<strong>Kaboon Thongtha</strong>, <strong>Jaipong Kasemsuwan</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 12 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1528855358_20180918220050Tue, 18 Sep 2018 22:00 EDTNonlinear Evolution of Benjamin-Bona-Mahony Wave Packet due to an Instability of a Pair of Modulationshttps://projecteuclid.org/euclid.ijde/1531360941<strong>Vera Halfiani</strong>, <strong>Dwi Fadhiliani</strong>, <strong>Harish Abdillah Mardi</strong>, <strong>Marwan Ramli</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1531360941_20180918220050Tue, 18 Sep 2018 22:00 EDTAffine Discontinuous Galerkin Method Approximation of Second-Order Linear Elliptic Equations in Divergence Form with Right-Hand Side in ${L}^{1}$https://projecteuclid.org/euclid.ijde/1537322435<strong>Abdeluaab Lidouh</strong>, <strong>Rachid Messaoudi</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 15 pages.</p><p><strong>Abstract:</strong><br/>
We consider the standard affine discontinuous Galerkin method approximation of the second-order linear elliptic equation in divergence form with coefficients in ${L}^{\mathrm{\infty }}(\mathrm{\Omega })$ and the right-hand side belongs to ${L}^{\mathrm{1}}(\mathrm{\Omega })$ ; 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 ${W}_{\mathrm{0}}^{\mathrm{1},q}(\mathrm{\Omega })$ for every $q$ with $\mathrm{1}\le q<d/(d-\mathrm{1})$ ( $d=\mathrm{2}$ 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 ${W}_{\mathrm{0}}^{\mathrm{1},q}(\mathrm{\Omega })$ when the right-hand side $f$ belongs to ${L}^{r}(\mathrm{\Omega })$ verifying ${T}_{k}(f)\in {H}^{\mathrm{1}}(\mathrm{\Omega })$ for every $k>\mathrm{0}$ , for some $r>\mathrm{1}.$
</p>projecteuclid.org/euclid.ijde/1537322435_20180918220050Tue, 18 Sep 2018 22:00 EDTApplication of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flowhttps://projecteuclid.org/euclid.ijde/1537322436<strong>Anas Arafa</strong>, <strong>Ghada Elmahdy</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 10 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1537322436_20180918220050Tue, 18 Sep 2018 22:00 EDTNumerical Simulation of Dispersed Particle-Blood Flow in the Stenosed Coronary Arterieshttps://projecteuclid.org/euclid.ijde/1537322437<strong>Mongkol Kaewbumrung</strong>, <strong>Somsak Orankitjaroen</strong>, <strong>Pichit Boonkrong</strong>, <strong>Buraskorn Nuntadilok</strong>, <strong>Benchawan Wiwatanapataphee</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 16 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1537322437_20180918220050Tue, 18 Sep 2018 22:00 EDTExistence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equationshttps://projecteuclid.org/euclid.ijde/1537322438<strong>Junfei Cao</strong>, <strong>Zaitang Huang</strong>, <strong>Gaston M. N’Guérékata</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 23 pages.</p><p><strong>Abstract:</strong><br/>
This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations ${\mathrm{D}}_{t}^{\alpha }x(t)=Ax(t)+{\mathrm{D}}_{t}^{\alpha -\mathrm{1}}F(t,x(t),Bx(t)), t\in \mathbb{R},$ where $\mathrm{1}<\alpha <\mathrm{2}$ , $A$ 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.
</p>projecteuclid.org/euclid.ijde/1537322438_20180918220050Tue, 18 Sep 2018 22:00 EDTExistence of Weak Solutions for Fractional Integrodifferential Equations with Multipoint Boundary Conditionshttps://projecteuclid.org/euclid.ijde/1539136934<strong>Haide Gou</strong>, <strong>Baolin Li</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 8 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1539136934_20181009220230Tue, 09 Oct 2018 22:02 EDTGlobal and Local Structures of Bifurcation Curves of ODE with Nonlinear Diffusionhttps://projecteuclid.org/euclid.ijde/1539136935<strong>Tetsutaro Shibata</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 7 pages.</p><p><strong>Abstract:</strong><br/>
We consider the nonlinear eigenvalue problem ${[D(u){u}^{\mathrm{\prime }}]}^{\mathrm{\prime }}+\lambda f(u)=\mathrm{0}$ , $u(t)>\mathrm{0}$ , $t\in I≔(\mathrm{0,1})$ , $u(\mathrm{0})=u(\mathrm{1})=\mathrm{0}$ , where $D(u)={u}^{k}$ , $f(u)={u}^{\mathrm{2}n-k-\mathrm{1}}+\mathrm{sin}u$ , and $\lambda >\mathrm{0}$ is a bifurcation parameter. Here, $n\in \mathbb{N}$ and $k$ ( $\mathrm{0}\le k<\mathrm{2}n-\mathrm{1}$ ) 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 $\mathrm{sin}u$ , 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.
</p>projecteuclid.org/euclid.ijde/1539136935_20181009220230Tue, 09 Oct 2018 22:02 EDTThe Existence of Strong Solutions for a Class of Stochastic Differential Equationshttps://projecteuclid.org/euclid.ijde/1542337348<strong>Junfei Zhang</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 5 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1542337348_20181115220246Thu, 15 Nov 2018 22:02 ESTExistence of Solutions for Unbounded Elliptic Equations with Critical Natural Growthhttps://projecteuclid.org/euclid.ijde/1544756544<strong>Aziz Bouhlal</strong>, <strong>Abderrahmane El Hachimi</strong>, <strong>Jaouad Igbida</strong>, <strong>El Mostafa Sadek</strong>, <strong>Hamad Talibi Alaoui</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 7 pages.</p><p><strong>Abstract:</strong><br/>
We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is $\{-\mathrm{d}\mathrm{i}\mathrm{v}[(\mathrm{1}+{|u|}^{q})\nabla u]+u=\gamma ({|\nabla u|}^{\mathrm{2}}/{(\mathrm{1}+|u|)}^{\mathrm{1}-q})+f \mathrm{i}\mathrm{n} \mathrm{\Omega }, u=\mathrm{0} \mathrm{o}\mathrm{n} \partial \mathrm{\Omega },\}$ , where $\mathrm{0}<q<\mathrm{1}$ , $\gamma >\mathrm{0}$ 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.
</p>projecteuclid.org/euclid.ijde/1544756544_20181213220241Thu, 13 Dec 2018 22:02 ESTOn Solving the Poisson Equation with Discontinuities on Irregular Interfaces: GFM and VIMhttps://projecteuclid.org/euclid.ijde/1542337333<strong>Ásdís Helgadóttir</strong>, <strong>Arthur Guittet</strong>, <strong>Frédéric Gibou</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 8 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1542337333_20190109220127Wed, 09 Jan 2019 22:01 ESTOn FTCS Approach for Box Model of Three-Dimension Advection-Diffusion Equationhttps://projecteuclid.org/euclid.ijde/1544756593<strong>Jeffry Kusuma</strong>, <strong>Agustinus Ribal</strong>, <strong>Andi Galsan Mahie</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 9 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1544756593_20190109220127Wed, 09 Jan 2019 22:01 ESTNumerical Method for Solving Nonhomogeneous Backward Heat Conduction Problemhttps://projecteuclid.org/euclid.ijde/1547089277<strong>LingDe Su</strong>, <strong>TongSong Jiang</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2018, 11 pages.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.ijde/1547089277_20190109220127Wed, 09 Jan 2019 22:01 ESTBlow-Up Solution of Modified-Logistic-Diffusion Equationhttps://projecteuclid.org/euclid.ijde/1551150359<strong>P. Sitompul</strong>, <strong>Y. Soeharyadi</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2019, 6 pages.</p><p><strong>Abstract:</strong><br/>
Modified-Logistic-Diffusion Equation ${u}_{t}=D{u}_{xx}+u|\mathrm{1}-u|$ with Neumann boundary condition has a global solution, if the given initial condition $\psi $ satisfies $\psi (x)\le \mathrm{1}$ , for all $x\in [\mathrm{0,1}]$ . Other initial conditions can lead to another type of solutions; i.e., an initial condition that satifies ${\int }_{\mathrm{0}}^{\mathrm{1}}\psi (x)dx>\mathrm{1}$ will cause the solution to blow up in a finite time. Another initial condition will result in another kind of solution, which depends on the diffusion coefficient $D$ . In this paper, we obtained the lower bound of $D$ , so that the solution of Modified-Logistic-Diffusion Equation with a given initial condition will have a global solution.
</p>projecteuclid.org/euclid.ijde/1551150359_20190225220603Mon, 25 Feb 2019 22:06 ESTFréchet Differentiability for a Damped Kirchhoff-Type Equation and Its Application to Bilinear Minimax Optimal Control Problemshttps://projecteuclid.org/euclid.ijde/1552615260<strong>Jin-soo Hwang</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2019, 16 pages.</p><p><strong>Abstract:</strong><br/>
We consider a damped Kirchhoff-type equation with Dirichlet boundary conditions. The objective is to show the Fréchet differentiability of a nonlinear solution map from a bilinear control input to the solution of a Kirchhoff-type equation. We use this result to formulate the minimax optimal control problem. We show the existence of optimal pairs and find their necessary optimality conditions.
</p>projecteuclid.org/euclid.ijde/1552615260_20190314220110Thu, 14 Mar 2019 22:01 EDTPositive Solutions for a Coupled System of Nonlinear Semipositone Fractional Boundary Value Problemshttps://projecteuclid.org/euclid.ijde/1552615261<strong>S. Nageswara Rao</strong>, <strong>M. Zico Meetei</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2019, 9 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we consider a four-point coupled boundary value problem for system of the nonlinear semipositone fractional differential equation ${D}_{{\mathrm{0}}^{+}}^{\alpha }u(t)+\lambda f(t,u(t),v(t))=\mathrm{0}, \mathrm{0}<t<\mathrm{1}$ , ${D}_{{\mathrm{0}}^{+}}^{\alpha }v(t)+\mu g(t,u(t),v(t))=\mathrm{0}, \mathrm{0}<t<\mathrm{1}$ , $u(\mathrm{0})=v(\mathrm{0})=\mathrm{0}, {a}_{\mathrm{1}}{D}_{{\mathrm{0}}^{+}}^{\beta }u(\mathrm{1})={b}_{\mathrm{1}}{D}_{{\mathrm{0}}^{+}}^{\beta }v(\xi )$ , ${a}_{\mathrm{2}}{D}_{{\mathrm{0}}^{+}}^{\beta }v(\mathrm{1})={b}_{\mathrm{2}}{D}_{{\mathrm{0}}^{+}}^{\beta }u(\eta ), \eta ,\xi \in (\mathrm{0,1}),$ where the coefficients ${a}_{i},{b}_{i},i=\mathrm{1,2}$ are real positive constants, $\alpha \in (\mathrm{1,2}],\beta \in (\mathrm{0,1}],$ ${D}_{{\mathrm{0}}^{+}}^{\alpha }$ , ${D}_{{\mathrm{0}}^{+}}^{\beta }$ are the standard Riemann-Liouville derivatives. Values of the parameters $\lambda $ and $\mu $ are determined for which boundary value problem has positive solution by utilizing a fixed point theorem on cone.
</p>projecteuclid.org/euclid.ijde/1552615261_20190314220110Thu, 14 Mar 2019 22:01 EDTNumerical Simulation of an Air Pollution Model on Industrial Areas by Considering the Influence of Multiple Point Sourceshttps://projecteuclid.org/euclid.ijde/1552615262<strong>Pravitra Oyjinda</strong>, <strong>Nopparat Pochai</strong>. <p><strong>Source: </strong>International Journal of Differential Equations, Volume 2019, 10 pages.</p><p><strong>Abstract:</strong><br/>
A numerical simulation on a two-dimensional atmospheric diffusion equation of an air pollution measurement model is proposed. The considered area is separated into two parts that are an industrial zone and an urban zone. In this research, the air pollution measurement by releasing the pollutant from multiple point sources above an industrial zone to the other area is simulated. The governing partial differential equation of air pollutant concentration is approximated by using a finite difference technique. The approximate solutions of the air pollutant concentration on both areas are compared. The air pollutant concentration levels influenced by multiple point sources are also analyzed.
</p>projecteuclid.org/euclid.ijde/1552615262_20190314220110Thu, 14 Mar 2019 22:01 EDT