International Journal of Differential Equations
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The latest articles from Hindawi Publishing Corporation
© 2016 , Hindawi Publishing Corporation . All rights reserved.

On Some Existence and Uniqueness Results for a Class of Equations of Order on Arbitrary Time Scales
Wed, 27 Jul 2016 13:43:33 +0000
http://www.hindawi.com/journals/ijde/2016/7327319/
This paper investigates the existence and uniqueness of solution for a class of nonlinear fractional differential equations of fractional order in arbitrary time scales. The results are established using extensions of KrasnoselskiiKrein, Rogers, and Kooi conditions.
Abdourazek Souahi, Assia GuezaneLakoud, and Rabah Khaldi
Copyright © 2016 Abdourazek Souahi et al. All rights reserved.

Boundary Layers and Shock Profiles for the Broadwell Model
Wed, 20 Jul 2016 05:50:57 +0000
http://www.hindawi.com/journals/ijde/2016/5801728/
We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.
Niclas Bernhoff
Copyright © 2016 Niclas Bernhoff. All rights reserved.

Improving Results on Solvability of a Class of thOrder Linear Boundary Value Problems
Mon, 18 Jul 2016 07:29:02 +0000
http://www.hindawi.com/journals/ijde/2016/3750530/
This paper presents a modification of a recursive method described in a previous paper of the authors, which yields necessary and sufficient conditions for the existence of solutions of a class of thorder linear boundary value problems, in the form of integral inequalities. Such a modification simplifies the assessment of the conditions on restricting the inequality to be verified to a single point instead of the full interval where the boundary value problem is defined. The paper also provides an error bound that needs to be considered in the integral inequalities of the previous paper when they are calculated numerically.
Pedro Almenar and Lucas Jódar
Copyright © 2016 Pedro Almenar and Lucas Jódar. All rights reserved.

Existence of Optimal Control for a NonlinearViscous Fluid Model
Mon, 27 Jun 2016 14:26:53 +0000
http://www.hindawi.com/journals/ijde/2016/9428128/
We consider the optimal control problem for a mathematical model describing steady flows of a nonlinearviscous incompressible fluid in a bounded threedimensional (or a twodimensional) domain with impermeable solid walls. The control parameter is the surface force at a given part of the flow domain boundary. For a given bounded set of admissible controls, we construct generalized (weak) solutions that minimize a given cost functional.
Evgenii S. Baranovskii and Mikhail A. Artemov
Copyright © 2016 Evgenii S. Baranovskii and Mikhail A. Artemov. All rights reserved.

Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions
Sun, 05 Jun 2016 14:00:00 +0000
http://www.hindawi.com/journals/ijde/2016/4726526/
This paper is motivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.
Khalid Hilal and Ahmed Kajouni
Copyright © 2016 Khalid Hilal and Ahmed Kajouni. All rights reserved.

Exact Solutions Superimposed with Nonlinear Plane Waves
Thu, 02 Jun 2016 16:07:51 +0000
http://www.hindawi.com/journals/ijde/2016/1846341/
The flow of fluid in atmosphere and ocean is governed by rotating stratified Boussinesq equations. Through the literature, we found that many researchers are trying to find the solutions of rotating stratified Boussinesq equations. In this paper, we have obtained special exact solutions and nonlinear plane waves. Finally, we provide exact solutions to rotating stratified Boussinesq equations with large scale motion superimposed with the nonlinear plane waves. In support of our investigations, we provided two examples: one described the special exact solution and in second example, we have determined the special exact solution superimposed with nonlinear plane wave. Also, we depicted some integral curves that represent the flow of an incompressible fluid particle on the plane as the particular case.
B. S. Desale and Vivek Sharma
Copyright © 2016 B. S. Desale and Vivek Sharma. All rights reserved.

Static Consensus in Passifiable Linear Networks
Wed, 18 May 2016 07:39:54 +0000
http://www.hindawi.com/journals/ijde/2016/9192127/
Sufficient conditions of consensus (synchronization) in networks described by digraphs and consisting of identical deterministic SIMO systems are derived. Identical and nonidentical control gains (positive arc weights) are considered. Connection between admissible digraphs and nonsmooth hypersurfaces (sufficient gain boundary) is established. Necessary and sufficient conditions for static consensus by output feedback in networks consisting of certain class of double integrators are rediscovered. Scalability for circle digraph in terms of gain magnitudes is studied. Examples and results of numerical simulations are presented.
Ibragim A. Junussov
Copyright © 2016 Ibragim A. Junussov. All rights reserved.

Numerical Solution of FirstOrder Linear Differential Equations in Fuzzy Environment by RungeKuttaFehlberg Method and Its Application
Thu, 12 May 2016 07:20:59 +0000
http://www.hindawi.com/journals/ijde/2016/8150497/
The numerical algorithm for solving “firstorder linear differential equation in fuzzy environment” is discussed. A scheme, namely, “RungeKuttaFehlberg method,” is described in detail for solving the said differential equation. The numerical solutions are compared with (i)gH and (ii)gH differential (exact solutions concepts) system. The method is also followed by complete error analysis. The method is illustrated by solving an example and an application.
Sankar Prasad Mondal, Susmita Roy, and Biswajit Das
Copyright © 2016 Sankar Prasad Mondal et al. All rights reserved.

Multiplicity of Positive Solutions for Fractional Differential Equation with Laplacian Boundary Value Problems
Thu, 12 May 2016 06:06:30 +0000
http://www.hindawi.com/journals/ijde/2016/6906049/
We investigate the existence of multiple positive solutions of fractional differential equations with Laplacian operator , , , , where , , , , , is a fixed integer, and , by applying Leggett–Williams fixed point theorems and fixed point index theory.
Sabbavarapu Nageswara Rao
Copyright © 2016 Sabbavarapu Nageswara Rao. All rights reserved.

Qualitative Behaviour of Solutions in Two Models of Thin Liquid Films
Thu, 05 May 2016 13:00:12 +0000
http://www.hindawi.com/journals/ijde/2016/4063740/
For the thinfilm model of a viscous flow which originates from lubrication approximation and has a full nonlinear curvature term, we prove existence of nonnegative weak solutions. Depending on initial data, we show algebraic or exponential dissipation of an energy functional which implies dissipation of the solution arc length that is a well known property for a HeleShaw flow. For the classical thinfilm model with linearized curvature term, under some restrictions on parameter and gradient values, we also prove analytically the arc length dissipation property for positive solutions. We compare the numerical solutions for both models, with nonlinear and with linearized curvature terms. In regimes when solutions develop finite time singularities, we explain the difference in qualitative behaviour of solutions.
Matthew Michal, Marina Chugunova, and Roman Taranets
Copyright © 2016 Matthew Michal et al. All rights reserved.

On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations
Tue, 16 Feb 2016 11:57:17 +0000
http://www.hindawi.com/journals/ijde/2016/7246027/
We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.
ElHassan ElJaoui and Said Melliani
Copyright © 2016 ElHassan ElJaoui and Said Melliani. All rights reserved.

Equivariant Hopf Bifurcation in a TimeDelayed Ring of Antigenic Variants
Sun, 14 Feb 2016 12:49:05 +0000
http://www.hindawi.com/journals/ijde/2016/5915768/
We consider an intrahost malaria model allowing for antigenic variation within a single species. The host’s immune response is compartmentalised into reactions to major and minor epitopes. We investigate the dynamics of the model, paying particular attention to bifurcation and stability of the uniform nonzero endemic equilibrium. We establish conditions for the existence of an equivariant Hopf bifurcation in a ring of antigenic variants, characterised by time delay.
Israel Ncube
Copyright © 2016 Israel Ncube. All rights reserved.

A New Result Concerning the Solvability of a Class of General Systems of Variational Equations with Nonmonotone Operators: Applications to Dirichlet and Neumann Nonlinear Problems
Tue, 19 Jan 2016 12:25:46 +0000
http://www.hindawi.com/journals/ijde/2016/1683759/
A new result of solvability for a wide class of systems of variational equations depending on parameters and governed by nonmonotone operators is found in a Banach real and reflexive space with applications to Dirichlet and Neumann problems related to nonlinear elliptic systems.
Luisa Toscano and Speranza Toscano
Copyright © 2016 Luisa Toscano and Speranza Toscano. All rights reserved.

Existence and Permanence in a Diffusive KiSS Model with Robust Numerical Simulations
Thu, 24 Dec 2015 09:24:59 +0000
http://www.hindawi.com/journals/ijde/2015/485860/
We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the longterm existence and permanence, the patch size denoted as must be greater than the critical patch size . It was also observed that the reactiondiffusion problem can be split into two parts: the linear and nonlinear terms. Hence, the use of two classical methods in space and time is permitted. We use spectral method in the area of mathematical community to remove the stiffness associated with the linear or diffusive terms. The resulting system is advanced with a modified exponential timedifferencing method whose formulation was based on the fourthorder RungeKutta scheme. With highorder method, this extends the onedimensional work and presents experiments for twodimensional problem. The complexity of the dynamical model is discussed theoretically and graphically simulated to demonstrate and compare the behavior of the timedependent density function.
Kolade M. Owolabi and Kailash C. Patidar
Copyright © 2015 Kolade M. Owolabi and Kailash C. Patidar. All rights reserved.

On Stability of Basis Property of Root Vectors System of the SturmLiouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular SamarskiiIonkin Type Problems
Mon, 14 Dec 2015 12:57:19 +0000
http://www.hindawi.com/journals/ijde/2015/641481/
We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of SamarskiiIonkin type boundary conditions.
N. S. Imanbaev
Copyright © 2015 N. S. Imanbaev. All rights reserved.

On Anisotropic Problems with Neumann Boundary Conditions
Sun, 13 Dec 2015 13:52:32 +0000
http://www.hindawi.com/journals/ijde/2015/238261/
This work is devoted to the study of a general class of anisotropic problems involving Laplace operator. Based on the variational method, we establish the existence of a nontrivial solution without AmbrosettiRabinowitz type conditions.
Anass Ourraoui
Copyright © 2015 Anass Ourraoui. All rights reserved.

On the SecondOrder Shape Derivative of the KohnVogelius Objective Functional Using the Velocity Method
Mon, 07 Dec 2015 14:23:48 +0000
http://www.hindawi.com/journals/ijde/2015/954836/
The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the socalled KohnVogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the secondorder shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.
Jerico B. Bacani and Julius Fergy T. Rabago
Copyright © 2015 Jerico B. Bacani and Julius Fergy T. Rabago. All rights reserved.

Existence and Iteration of Positive Solutions to ThirdOrder BVP for a Class of Laplacian Dynamic Equations on Time Scales
Thu, 03 Dec 2015 06:48:48 +0000
http://www.hindawi.com/journals/ijde/2015/567209/
We investigate the existence and iteration of positive solutions for the following thirdorder Laplacian dynamic equations on time scales: where is Laplacian operator; that is, , and By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions.
A. Kameswara Rao
Copyright © 2015 A. Kameswara Rao. All rights reserved.

Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm
Tue, 24 Nov 2015 09:28:08 +0000
http://www.hindawi.com/journals/ijde/2015/347864/
We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the stepsize for directly solving general secondorder initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous twostep hybrid trigonometrically fitted method with two offgrid points. The BHT is implemented in a blockbyblock fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictorcorrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.
F. F. Ngwane and S. N. Jator
Copyright © 2015 F. F. Ngwane and S. N. Jator. All rights reserved.

Entropy Solution for Doubly Nonlinear Elliptic Anisotropic Problems with Robin Boundary Conditions
Tue, 24 Nov 2015 07:27:45 +0000
http://www.hindawi.com/journals/ijde/2015/919608/
We study in this paper nonlinear anisotropic problems with Robin boundary conditions. We prove, by using the technic of monotone operators in Banach spaces, the existence of a sequence of weak solutions of approximation problems associated with the anisotropic Robin boundary value problem. For the existence and uniqueness of entropy
solutions, we prove that the sequence of weak solutions converges to a measurable function which is the entropy solution of the anisotropic Robin boundary value problem.
I. Ibrango and S. Ouaro
Copyright © 2015 I. Ibrango and S. Ouaro. All rights reserved.

Numerical Analysis of a Distributed Optimal Control Problem Governed by an Elliptic Variational Inequality
Mon, 23 Nov 2015 13:38:06 +0000
http://www.hindawi.com/journals/ijde/2015/407930/
The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energy . The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter goes to zero.
Mariela Olguín and Domingo A. Tarzia
Copyright © 2015 Mariela Olguín and Domingo A. Tarzia. All rights reserved.

Optimal Control of the IllPosed Cauchy Elliptic Problem
Mon, 23 Nov 2015 06:44:14 +0000
http://www.hindawi.com/journals/ijde/2015/468918/
We give a characterization of the control for illposed problems of oscillating solutions. More precisely, we study the control of Cauchy elliptic problems via a regularization approach which generates incomplete information. We obtain a singular optimality system characterizing the noregret control for the Cauchy problem.
A. Berhail and A. Omrane
Copyright © 2015 A. Berhail and A. Omrane. All rights reserved.

Bounds for Products of Zeros of Solutions to Nonhomogeneous ODE with Polynomial Coefficients
Wed, 18 Nov 2015 06:52:19 +0000
http://www.hindawi.com/journals/ijde/2015/690519/
We consider the equation , where is a polynomial and is an entire function. Let be the zeros of a solution to that equation. Lower estimates for the products are derived. In particular, they give us a bound for the zero free domain. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed.
Michael Gil’
Copyright © 2015 Michael Gil’. All rights reserved.

Redistribution of Nodes with Two Constraints in Meshless Method of Line to TimeDependent Partial Differential Equations
Thu, 05 Nov 2015 14:20:32 +0000
http://www.hindawi.com/journals/ijde/2015/762034/
Meshless method of line is a powerful device to solve timedependent partial differential equations. In integrating step, choosing a suitable set of points, such as adaptive nodes in spatial domain, can be useful, although in some cases this can cause illconditioning. In this paper, to produce smooth adaptive points in each step of the method, two constraints are enforced in Equidistribution algorithm. These constraints lead to two different meshes known as quasiuniform and locally bounded meshes. These avoid the illconditioning in applying radial basis functions. Moreover, to generate more smooth adaptive meshes another modification is investigated, such as using modified arclength monitor function in Equidistribution algorithm. Influence of them in growing the accuracy is investigated by some numerical examples. The results of consideration of two constraints are compared with each other and also with uniform meshes.
Jafar Biazar and Mohammad Hosami
Copyright © 2015 Jafar Biazar and Mohammad Hosami. All rights reserved.

Solvability of Nth Order Linear Boundary Value Problems
Thu, 29 Oct 2015 11:23:03 +0000
http://www.hindawi.com/journals/ijde/2015/230405/
This paper presents a method that provides necessary and sufficient conditions for the existence of solutions of nth order linear boundary value problems. The method is based on the recursive application of a linear integral operator to some functions and the comparison of the result with these same functions. The recursive comparison yields sequences of bounds of extremes that converge to the exact values of the extremes of the BVP for which a solution exists.
P. Almenar and L. Jódar
Copyright © 2015 P. Almenar and L. Jódar. All rights reserved.

Connections between Some Concepts of Polynomial Trichotomy for Noninvertible Evolution Operators in Banach Spaces
Thu, 29 Oct 2015 06:50:29 +0000
http://www.hindawi.com/journals/ijde/2015/241402/
The present paper treats three concepts of nonuniform polynomial trichotomies for noninvertible evolution operators acting on Banach spaces. The connections between these concepts are established through numerous examples and counterexamples for systems defined on the Banach space of squaresummable sequences.
MihaiGabriel Babuţia and Nicolae Marian Seimeanu
Copyright © 2015 MihaiGabriel Babuţia and Nicolae Marian Seimeanu. All rights reserved.

Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent
Wed, 28 Oct 2015 08:08:49 +0000
http://www.hindawi.com/journals/ijde/2015/494907/
We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland’s variational principle.
Mohammed El Mokhtar Ould El Mokhtar
Copyright © 2015 Mohammed El Mokhtar Ould El Mokhtar. All rights reserved.

On the InitialBoundaryValue Problem for the TimeFractional Diffusion Equation on the Real Positive Semiaxis
Wed, 07 Oct 2015 06:04:19 +0000
http://www.hindawi.com/journals/ijde/2015/439419/
We consider the timefractional derivative in the Caputo sense of order . Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in , two different initialboundaryvalue problems for the timefractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when of the respective solutions is analyzed, recovering the solutions of the classical boundaryvalue problems when α = 1, and the fractional diffusion equation becomes the heat equation.
D. Goos, G. Reyero, S. Roscani, and E. Santillan Marcus
Copyright © 2015 D. Goos et al. All rights reserved.

Propagation of Water Waves over Uneven Bottom under the Effect of Surface Tension
Wed, 30 Sep 2015 16:36:25 +0000
http://www.hindawi.com/journals/ijde/2015/805625/
We establish existence and uniqueness of solutions to the Cauchy problem associated with a new onedimensional weaklynonlinear, weaklydispersive system which arises as an asymptotical approximation of the full potential theory equations for modelling propagation of small amplitude water waves on the surface of a shallow channel with variable depth, taking into account the effect of surface tension. Furthermore, numerical schemes of spectral type are introduced for approximating the evolution in time of solutions of this system and its travelling wave solutions, in both the periodic and nonperiodic case.
Juan Carlos Muñoz Grajales
Copyright © 2015 Juan Carlos Muñoz Grajales. All rights reserved.

On the Convergence of a Nonlinear BoundaryValue Problem in a Perforated Domain
Wed, 30 Sep 2015 16:26:08 +0000
http://www.hindawi.com/journals/ijde/2015/392479/
We consider a family with respect to a small parameter of nonlinear boundaryvalue problems as well as the corresponding spectral problems in a domain perforated periodically along a
part of the boundary. We prove the convergence of solution of the original problems to the solution of the respective homogenized problem in this domain.
Yulia Koroleva
Copyright © 2015 Yulia Koroleva. All rights reserved.