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

A Hybrid Natural Transform Homotopy Perturbation Method for Solving Fractional Partial Differential Equations
Tue, 27 Sep 2016 15:42:46 +0000
http://www.hindawi.com/journals/ijde/2016/9207869/
A hybrid analytical method for solving linear and nonlinear fractional partial differential equations is presented. The proposed analytical approach is an elegant combination of the Natural Transform Method (NTM) and a wellknown method, Homotopy Perturbation Method (HPM). In this analytical method, the fractional derivative is computed in Caputo sense and the nonlinear term is calculated using He’s polynomial. The proposed analytical method reduces the computational size and avoids roundoff errors. Exact solution of linear and nonlinear fractional partial differential equations is successfully obtained using the analytical method.
Shehu Maitama
Copyright © 2016 Shehu Maitama. All rights reserved.

Symmetry Classification and Exact Solutions of a Variable Coefficient SpaceTime Fractional Potential Burgers’ Equation
Sun, 25 Sep 2016 12:24:40 +0000
http://www.hindawi.com/journals/ijde/2016/4270724/
We investigate the symmetry properties of a variable coefficient spacetime fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.
Manoj Gaur and K. Singh
Copyright © 2016 Manoj Gaur and K. Singh. All rights reserved.

HomogeneousLike Generalized Cubic Systems
Mon, 05 Sep 2016 11:26:55 +0000
http://www.hindawi.com/journals/ijde/2016/7640340/
We consider properties and center conditions for plane polynomial systems of the forms , where , and , are polynomials of degrees and , respectively, for integers . We restrict our attention to those systems for which . In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems . From this we show that any center condition of a homogeneous system for a given can be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation.
G. R. Nicklason
Copyright © 2016 G. R. Nicklason. All rights reserved.

On Oscillatory and Asymptotic Behavior of a SecondOrder Nonlinear Damped Neutral Differential Equation
Mon, 29 Aug 2016 16:25:02 +0000
http://www.hindawi.com/journals/ijde/2016/3746368/
This paper discusses oscillatory and asymptotic properties of solutions of a class of secondorder nonlinear damped neutral differential equations. Some new sufficient conditions for any solution of the equation to be oscillatory or to converge to zero are given. The results obtained extend and improve some of the related results reported in the literature. The results are illustrated with examples.
Ercan Tunç and Said R. Grace
Copyright © 2016 Ercan Tunç and Said R. Grace. All rights reserved.

Estimates on the Lower Bound of the Eigenvalue of the Smallest Modulus Associated with a General Weighted SturmLiouville Problem
Mon, 29 Aug 2016 09:28:25 +0000
http://www.hindawi.com/journals/ijde/2016/7396951/
We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular SturmLiouville problem. The main motivation for this study is the result obtained by Mingarelli (1988).
Mervis Kikonko and Angelo Bernado Mingarelli
Copyright © 2016 Mervis Kikonko and Angelo Bernado Mingarelli. All rights reserved.

Interval Oscillation Criteria for Forced SecondOrder Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales
Wed, 24 Aug 2016 17:53:57 +0000
http://www.hindawi.com/journals/ijde/2016/3298289/
We are concerned with the interval oscillation of general type of forced secondorder nonlinear dynamic equation with oscillatory potential of the form , on a time scale . We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced secondorder nonlinear delay differential equation and the forced secondorder nonlinear delay difference equation. Finally, we give some examples to illustrate our results.
Hassan A. Agwa, Ahmed M. M. Khodier, and Heba A. Hassan
Copyright © 2016 Hassan A. Agwa et al. All rights reserved.

Semianalytic Solution of SpaceTime Fractional Diffusion Equation
Mon, 08 Aug 2016 09:59:49 +0000
http://www.hindawi.com/journals/ijde/2016/2371837/
We study the spacetime fractional diffusion equation with spatial RieszFeller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior.
A. Elsaid, S. Shamseldeen, and S. Madkour
Copyright © 2016 A. Elsaid et al. All rights reserved.

A Comparison Principle for the Mean Curvature Flow Equation with Discontinuous Coefficients
Thu, 04 Aug 2016 14:17:44 +0000
http://www.hindawi.com/journals/ijde/2016/3627896/
We study the level set equation in a bounded domain when the velocity of the interface is given by the mean curvature plus a discontinuous velocity. We prove a comparison principle for the initialboundary value problem whose consequence is uniqueness of continuous solutions and well posedness of the level set method.
Cecilia De Zan and Pierpaolo Soravia
Copyright © 2016 Cecilia De Zan and Pierpaolo Soravia. All rights reserved.

Positive Stabilization of Linear Differential Algebraic Equation System
Wed, 03 Aug 2016 07:06:10 +0000
http://www.hindawi.com/journals/ijde/2016/6346780/
We study in this paper the existence of a feedback for linear differential algebraic equation system such that the closedloop system is positive and stable. A necessary and sufficient condition for such existence has been established. This result can be used to detect the existence of a state feedback law that makes the linear differential algebraic equation system in closed loop positive and stable.
Muhafzan
Copyright © 2016 Muhafzan. All rights reserved.

Periodicity, Stability, and Boundedness of Solutions to Certain Second Order Delay Differential Equations
Mon, 01 Aug 2016 12:56:35 +0000
http://www.hindawi.com/journals/ijde/2016/2843709/
The behaviour of solutions to certain second order nonlinear delay differential equations with variable deviating arguments is discussed. The main procedure lies in the properties of a complete Lyapunov functional which is used to obtain suitable criteria to guarantee existence of unique solutions that are periodic, uniformly asymptotically stable, and uniformly ultimately bounded. Obtained results are new and also complement related ones that have appeared in the literature. Moreover, examples are given to illustrate the feasibility and correctness of the main results.
A. T. Ademola, B. S. Ogundare, M. O. Ogundiran, and O. A. Adesina
Copyright © 2016 A. T. Ademola et al. 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 CarathĂ©odory 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 Leggettâ€“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.