International Journal of Partial Differential Equations
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© 2014 , Hindawi Publishing Corporation . All rights reserved.

Spectral Bounds for Polydiagonal Jacobi Matrix Operators
Sun, 19 Oct 2014 11:42:05 +0000
http://www.hindawi.com/journals/ijpde/2014/920695/
The research on spectral inequalities for discrete Schrödinger operators has proved fruitful in the last decade. Indeed, several authors analysed the operator’s canonical relation to a tridiagonal Jacobi matrix operator. In this paper, we consider a generalisation of this relation with regard to connecting higher order Schrödingertype operators with symmetric matrix operators with arbitrarily many nonzero diagonals above and below the main diagonal. We thus obtain spectral bounds for such matrices, similar in nature to the LiebThirring inequalities.
Arman Sahovic
Copyright © 2014 Arman Sahovic. All rights reserved.

On Construction of Solutions of Evolutionary Nonlinear Schrödinger Equation
Tue, 07 Oct 2014 00:00:00 +0000
http://www.hindawi.com/journals/ijpde/2014/830413/
In this work we present an application of a theory of vessels to a solution of the evolutionary nonlinear Schrödinger (NLS) equation. The classes of functions for which the initial value problem is solvable rely on the existence of an analogue of the inverse scattering theory for the usual NLS equation. This approach is similar to the classical approach of ZakharovShabath for solving evolutionary NLS equation but has an advantage of simpler formulas and new techniques and notions to understand the solutions.
Andrey Melnikov
Copyright © 2014 Andrey Melnikov. All rights reserved.

Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay
Thu, 02 Oct 2014 13:22:58 +0000
http://www.hindawi.com/journals/ijpde/2014/787092/
This paper is divided in two parts. In the first part we study a second order neutral partial differential equation with state dependent delay and noninstantaneous impulses. The conditions for existence and uniqueness of the mild solution are investigated via Hausdorff measure of noncompactness and Darbo Sadovskii fixed point theorem. Thus we remove the need to assume the compactness assumption on the associated family of operators. The conditions for approximate controllability are investigated for the neutral second order system with respect to the approximate controllability of the corresponding linear system in a Hilbert space. A simple range condition is used to prove approximate controllability. Thereby, we remove the need to assume the invertibility of a controllability operator used by authors in (Balachandran and Park, 2003), which fails to exist in infinite dimensional spaces if the associated semigroup is compact. Our approach also removes the need to check the invertibility of the controllability Gramian operator and associated limit condition used by the authors in (Dauer and Mahmudov, 2002), which are practically difficult to verify and apply. Examples are provided to illustrate the presented theory.
Sanjukta Das, Dwijendra N. Pandey, and N. Sukavanam
Copyright © 2014 Sanjukta Das et al. All rights reserved.

Numerical Solution of Nonlinear SineGordon Equation by Modified Cubic BSpline Collocation Method
Sun, 10 Aug 2014 06:23:46 +0000
http://www.hindawi.com/journals/ijpde/2014/343497/
Modified cubic Bspline collocation method is discussed for the numerical solution of onedimensional nonlinear sineGordon equation. The method is based on collocation of modified cubic Bsplines over finite elements, so we have continuity of the dependent variable and its first two derivatives throughout the solution range. The given equation is decomposed into a system of equations and modified cubic Bspline basis functions have been used for spatial variable and its derivatives, which gives results in amenable system of ordinary differential equations. The resulting system of equation has subsequently been solved by SSPRK54 scheme. The efficacy of the proposed approach has been confirmed with numerical experiments, which shows that the results obtained are acceptable and are in good agreement with earlier studies.
R. C. Mittal and Rachna Bhatia
Copyright © 2014 R. C. Mittal and Rachna Bhatia. All rights reserved.

Numerical Solutions of TwoWay Propagation of Nonlinear Dispersive Waves Using Radial Basis Functions
Sun, 03 Aug 2014 08:28:41 +0000
http://www.hindawi.com/journals/ijpde/2014/407387/
We obtain the numerical solution of a Boussinesq system for
twoway propagation of nonlinear dispersive waves by using the meshless
method, based on collocation with radial basis functions. The system of
nonlinear partial differential equation is discretized in space by approximating
the solution using radial basis functions. The discretization leads to a
system of coupled nonlinear ordinary differential equations. The equations
are then solved by using the fourthorder RungeKutta method. A stability
analysis is provided and then the accuracy of method is tested by comparing
it with the exact solitary solutions of the Boussinesq system. In addition, the
conserved quantities are calculated numerically and compared to an exact
solution. The numerical results show excellent agreement with the analytical
solution and the calculated conserved quantities.
Pablo U. Suárez and J. Héctor Morales
Copyright © 2014 Pablo U. Suárez and J. Héctor Morales. All rights reserved.

An Efficient Method for TimeFractional Coupled Schrödinger System
Tue, 15 Jul 2014 00:00:00 +0000
http://www.hindawi.com/journals/ijpde/2014/137470/
We present a new technique to obtain the solution of timefractional coupled Schrödinger system. The fractional derivatives are considered in Caputo sense. The proposed scheme is based on Laplace transform and new homotopy perturbation method. To illustrate the power and reliability of the method some examples are provided. The results obtained by the proposed method show that the approach is very efficient and simple and can be applied to other partial differential equations.
Hossein Aminikhah, A. Refahi Sheikhani, and Hadi Rezazadeh
Copyright © 2014 Hossein Aminikhah et al. All rights reserved.

Variational Statement and Domain Decomposition Algorithms for BitsadzeSamarskii Nonlocal Boundary Value Problem for Poisson’s TwoDimensional Equation
Thu, 19 Jun 2014 13:33:37 +0000
http://www.hindawi.com/journals/ijpde/2014/680760/
The BitsadzeSamarskii nonlocal boundary value problem is considered. Variational formulation is done. The domain decomposition and Schwarztype iterative methods are used. The parallel algorithm as well as sequential ones is investigated.
Temur Jangveladze, Zurab Kiguradze, and George Lobjanidze
Copyright © 2014 Temur Jangveladze et al. All rights reserved.

Existence of the Mild Solution for Impulsive Semilinear Differential Equation
Sun, 18 May 2014 11:11:20 +0000
http://www.hindawi.com/journals/ijpde/2014/640931/
We study the existence of solutions of impulsive semilinear differential equation in a Banach space in which impulsive condition is not instantaneous. We establish the existence of a mild solution by using the Hausdorff measure of noncompactness and a fixed point theorem for the convex power condensing operator.
Alka Chadha and Dwijendra N. Pandey
Copyright © 2014 Alka Chadha and Dwijendra N. Pandey. All rights reserved.

General Asymptotic Supnorm Estimates for Solutions of OneDimensional AdvectionDiffusion Equations in Heterogeneous Media
Thu, 08 May 2014 11:18:52 +0000
http://www.hindawi.com/journals/ijpde/2014/450417/
We derive general bounds for the large time size of supnorm values of solutions to onedimensional advectiondiffusion equations with initial data for some and arbitrary bounded advection speeds , introducing new techniques based on suitable energy arguments. Some open problems and related results are also given.
José A. Barrionuevo, Lucas S. Oliveira, and Paulo R. Zingano
Copyright © 2014 José A. Barrionuevo et al. All rights reserved.

Explicit Estimates for Solutions of Mixed Elliptic Problems
Mon, 31 Mar 2014 11:15:41 +0000
http://www.hindawi.com/journals/ijpde/2014/845760/
We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic secondorder equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in () of class . The existence of and estimates is assured for and any (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive estimates for different ranges of the exponent depending on the fact that the coefficient is either Dinicontinuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.
Luisa Consiglieri
Copyright © 2014 Luisa Consiglieri. All rights reserved.

A ReactionDiffusion System with Nonlinear Nonlocal Boundary Conditions
Thu, 20 Feb 2014 07:02:08 +0000
http://www.hindawi.com/journals/ijpde/2014/523656/
We consider initial boundary value problem for a reactiondiffusion system with nonlinear and nonlocal boundary conditions and nonnegative initial data. We prove local existence, uniqueness, and nonuniqueness of solutions.
Alexander Gladkov and Alexandr Nikitin
Copyright © 2014 Alexander Gladkov and Alexandr Nikitin. All rights reserved.

A Note on the Painlevé Property of Coupled KdV Equations
Wed, 19 Feb 2014 11:23:52 +0000
http://www.hindawi.com/journals/ijpde/2014/125821/
We prove that one system of coupled KdV equations, claimed by Hirota et al. to pass the Painlevé test for integrability, actually fails the test at the highest resonance of the generic branch and therefore must be nonintegrable.
Sergei Sakovich
Copyright © 2014 Sergei Sakovich. All rights reserved.

Partial Differential Equations of an Epidemic Model with Spatial Diffusion
Mon, 10 Feb 2014 09:59:55 +0000
http://www.hindawi.com/journals/ijpde/2014/186437/
The aim of this paper is to study the dynamics of a reactiondiffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reactiondiffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the diseasefree equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the diseasefree equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then diseasefree equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.
El Mehdi Lotfi, Mehdi Maziane, Khalid Hattaf, and Noura Yousfi
Copyright © 2014 El Mehdi Lotfi et al. All rights reserved.

Semilinear Evolution Problems with VentcelType Conditions on Fractal Boundaries
Wed, 22 Jan 2014 08:09:38 +0000
http://www.hindawi.com/journals/ijpde/2014/461046/
A semilinear parabolic transmission problem with Ventcel's boundary conditions on a fractal interface or the corresponding prefractal interface is studied. Regularity results for the solution in both cases are proved. The asymptotic behaviour of the solutions of the approximating problems to the solution of limit fractal problem is analyzed.
Maria Rosaria Lancia and Paola Vernole
Copyright © 2014 Maria Rosaria Lancia and Paola Vernole. All rights reserved.

Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media
Sun, 19 Jan 2014 00:00:00 +0000
http://www.hindawi.com/journals/ijpde/2014/245086/
The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear
partial differential equations, the pressurevelocity equation and the concentration equation. In this paper, we present a mixed finite volume element
method (FVEM) for the approximation of the
pressurevelocity equation. Since modified method of characteristics (MMOC) minimizes the grid orientation effect, for the approximation of the concentration
equation, we apply a standard FVEM combined with MMOC. A priori error estimates in norm are derived for velocity, pressure and concentration.
Numerical results are presented to substantiate the validity of the theoretical results.
Sarvesh Kumar and Sangita Yadav
Copyright © 2014 Sarvesh Kumar and Sangita Yadav. All rights reserved.

On the Local WellPosedness of the Cauchy Problem for a Modified TwoComponent CamassaHolm System in Besov Spaces
Tue, 31 Dec 2013 17:48:27 +0000
http://www.hindawi.com/journals/ijpde/2013/834912/
We consider the Cauchy problem for an integrable modified twocomponent CamassaHolm system with cubic nonlinearity. By using the LittlewoodPaley decomposition, nonhomogeneous Besov spaces, and a priori estimates for linear transport equation, we prove that the Cauchy problem is locally wellposed in Besov spaces with , and .
Jiangbo Zhou, Lu Yao, Lixin Tian, and Wenbin Zhang
Copyright © 2013 Jiangbo Zhou et al. All rights reserved.

Existence and Uniqueness of the Solutions for Some InitialBoundary Value Problems with the Fractional Dynamic Boundary Condition
Thu, 07 Nov 2013 09:00:04 +0000
http://www.hindawi.com/journals/ijpde/2013/796430/
In this paper, we analyze some initialboundary value problems for
the subdiffusion equation with a fractional dynamic boundary condition in a
onedimensional bounded domain. First, we establish the unique solvability
in the Hölder space of the initialboundary value problems for the equation , , where L is a uniformly elliptic
operator with smooth coefficients with the fractional dynamic boundary condition. Second, we apply the contraction theorem to prove the existence and
uniqueness locally in time in the Hölder classes of the solution to the corresponding nonlinear problems.
Mykola Krasnoschok and Nataliya Vasylyeva
Copyright © 2013 Mykola Krasnoschok and Nataliya Vasylyeva. All rights reserved.

Approximate Controllability of a Semilinear Heat Equation
Sun, 03 Nov 2013 14:56:58 +0000
http://www.hindawi.com/journals/ijpde/2013/424309/
We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: in on , where is a bounded domain in , , is an open nonempty subset of , denotes the characteristic function of the set , the distributed control belongs to , and the nonlinear function is smooth enough, and there are , and such that for all Under this condition, we prove the following statement: for all open nonempty subset of , the system is approximately controllable on . Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state to an neighborhood of the final state at time .
Hugo Leiva, N. Merentes, and J. Sanchez
Copyright © 2013 Hugo Leiva et al. All rights reserved.

Solutions of Nonlocal Laplacian Equations
Thu, 10 Oct 2013 15:55:46 +0000
http://www.hindawi.com/journals/ijpde/2013/364251/
In view of variational approach we discuss a nonlocal problem, that is, a Kirchhofftype equation involving Laplace operator. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.
Mustafa Avci and Rabil Ayazoglu (Mashiyev)
Copyright © 2013 Mustafa Avci and Rabil Ayazoglu (Mashiyev). All rights reserved.

Single Peak Solitons for the BoussinesqLike Equation
Wed, 09 Oct 2013 12:00:32 +0000
http://www.hindawi.com/journals/ijpde/2013/732809/
The nonlinear dispersive Boussinesqlike equation , which exhibits single peak solitons, is investigated. Peakons, cuspons and smooth soliton solutions are obtained by setting the equation under inhomogeneous boundary condition. Asymptotic behavior and numerical simulations are provided for these three types of single peak soliton solutions of the equation.
Lina Zhang, Shumin Li, and Aiyong Chen
Copyright © 2013 Lina Zhang et al. All rights reserved.

A Posteriori Regularization Parameter Choice Rule for Truncation Method for Identifying the Unknown Source of the Poisson Equation
Thu, 19 Sep 2013 10:53:25 +0000
http://www.hindawi.com/journals/ijpde/2013/590737/
We consider the problem of determining an unknown source which depends only on one variable in twodimensional Poisson equation. We prove a conditional stability for this problem. Moreover, we propose a truncation regularization method combined with an a posteriori regularization parameter choice rule to deal with this problem and give the corresponding convergence estimate. Numerical results are presented to illustrate the accuracy and efficiency of this method.
XiaoXiao Li and DunGang Li
Copyright © 2013 XiaoXiao Li and DunGang Li. All rights reserved.

Boundary Value Problems for the Classical and Mixed Integrodifferential Equations with RiemannLiouville Operators
Sun, 08 Sep 2013 08:54:18 +0000
http://www.hindawi.com/journals/ijpde/2013/157947/
By method of integral equations, unique solvability is proved for the solution of boundary value problems of loaded thirdorder integrodifferential equations with RiemannLiouville operators.
B. Islomov and U. I. Baltaeva
Copyright © 2013 B. Islomov and U. I. Baltaeva. All rights reserved.

Analysis of a Singular Convection Diffusion System Arising in Turbulence Modelling
Mon, 26 Aug 2013 13:28:18 +0000
http://www.hindawi.com/journals/ijpde/2013/940924/
We shall study some singular stationary convection diffusion system governing the steady state of a turbulence model closely related to the one. We shall establish existence, positivity, and regularity results in a very general framework.
P. Dreyfuss
Copyright © 2013 P. Dreyfuss. All rights reserved.

An Initial Boundary Value Problem for the Zakharov Equation
Thu, 25 Jul 2013 09:49:17 +0000
http://www.hindawi.com/journals/ijpde/2013/748761/
This paper studies an inhomogeneous initial boundary value problem for the onedimensional Zakharov equation. Existence and uniqueness of the global strong solution are proved by Galerkin’s method and integral estimates.
Quankang Yang and Charles Bu
Copyright © 2013 Quankang Yang and Charles Bu. All rights reserved.

A Numerical Method for Solving 3D Elasticity Equations with SharpEdged Interfaces
Sun, 21 Jul 2013 10:56:18 +0000
http://www.hindawi.com/journals/ijpde/2013/476873/
Interface problems occur frequently when two or more materials meet. Solving elasticity equations with sharpedged interfaces in three dimensions is a very complicated and challenging problem for most existing methods. There are several difficulties: the coupled elliptic system, the matrix coefficients, the sharpedged interface, and three dimensions. An accurate and efficient method is desired. In this paper, an efficient nontraditional finite element method with nonbodyfitting grids is proposed to solve elasticity equations with sharpedged interfaces in three dimensions. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up).
Liqun Wang, Songming Hou, and Liwei Shi
Copyright © 2013 Liqun Wang et al. All rights reserved.

Probabilistic Representations for the Solution of Higher Order Differential Equations
Wed, 17 Jul 2013 08:25:25 +0000
http://www.hindawi.com/journals/ijpde/2013/297857/
A probabilistic representation for the solution of the partial differential equation , is constructed in terms of the expectation with respect to the measure associated to a complexvalued stochastic process.
S. Mazzucchi
Copyright © 2013 S. Mazzucchi. All rights reserved.

Properties of Some Partial Dynamic Equations on Time Scales
Mon, 01 Jul 2013 13:53:57 +0000
http://www.hindawi.com/journals/ijpde/2013/345697/
The main objective of the paper is to study the properties of the solution of a certain partial dynamic equation on time scales. The tools employed are based on the application of the Banach fixedpoint theorem and a certain integral inequality with explicit estimates on time scales.
Deepak B. Pachpatte
Copyright © 2013 Deepak B. Pachpatte. All rights reserved.

Numerical Approximation for Nonlinear Gas Dynamic Equation
Sun, 16 Jun 2013 16:04:37 +0000
http://www.hindawi.com/journals/ijpde/2013/846749/
Laplace transform and new homotopy perturbation methods are adopted to study gas dynamic equation analytically. The solutions introduced in this study can be used to obtain the closed form of the solutions if they are required. The combined method needs less work in comparison with the other homotopy perturbation methods and decreases volume of calculations considerably. Results show that the new method is more effective and convenient to use, and is high accuracy evident.
Hossein Aminikhah and Ali Jamalian
Copyright © 2013 Hossein Aminikhah and Ali Jamalian. All rights reserved.

Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations
Sun, 28 Apr 2013 14:09:34 +0000
http://www.hindawi.com/journals/ijpde/2013/207581/
Let be a generator of an exponentially stable
operator semigroup in a Banach space,
and let be a linear bounded variable operator.
Assuming that is sufficiently small in a certain sense for the equation
, we derive exponential stability conditions.
Besides, we do not require that for each , the “frozen”
autonomous equation is stable. In particular, we consider
evolution equations with periodic operator coefficients.
These results are applied to partial differential equations.
Michael Gil'
Copyright © 2013 Michael Gil'. All rights reserved.