International Journal of Partial Differential Equations The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Development of a Nonlinear Model Incorporating Strain and Rotation Parameters for Prediction of Complex Turbulent Flows Thu, 19 Feb 2015 09:47:11 +0000 The standard model has the deficiency of predicting swirling and vortical flows due to its isotropic assumption of eddy viscosity. In this study, a second-order nonlinear model is developed incorporating some new functions for the model coefficients to explore the models applicability to complex turbulent flows. Considering the realizability principle, the coefficient of eddy viscosity () is derived as a function of strain and rotation parameters. The coefficients of nonlinear quadratic term are estimated considering the anisotropy of turbulence in a simple shear layer. Analytical solutions for the fundamental properties of swirl jet are derived based on the nonlinear model, and the values of model constants are determined by tuning their values for the best-fitted comparison with the experiments. The model performance is examined for two test cases: (i) for an ideal vortex (Stuart vortex), the basic equations are solved numerically to predict the turbulent structures at the vortex center and the (ii) unsteady 3D simulation is carried out to calculate the flow field of a compound channel. It is observed that the proposed nonlinear model can successfully predict the turbulent structures at vortex center, while the standard model fails. The model is found to be capable of accounting the effect of transverse momentum transfer in the compound channel through generating the horizontal vortices at the interface. Md. Shahjahan Ali, Takashi Hosoda, and Ichiro Kimura Copyright © 2015 Md. Shahjahan Ali et al. All rights reserved. Weighted Pluricomplex Energy II Tue, 10 Feb 2015 09:42:02 +0000 We continue our study of the complex Monge-Ampère operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classes by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure is the Monge-Ampère of a unique function if and only if . Then we show that if for some then for some , where is given boundary data. If moreover the nonnegative Borel measure is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sublevel sets of the solutions. As a consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space. Slimane Benelkourchi Copyright © 2015 Slimane Benelkourchi. All rights reserved. Direct and Inverse Scattering Problems for Domains with Multiple Corners Mon, 26 Jan 2015 13:46:02 +0000 We proposed numerical methods for solving the direct and inverse scattering problems for domains with multiple corners. Both the near field and far field cases are considered. For the forward problem, the challenges of logarithmic singularity from Green’s functions and corner singularity are both taken care of. For the inverse problem, an efficient and robust direct imaging method is proposed. Multiple frequency data are combined to capture details while not losing robustness. Songming Hou, Yihong Jiang, and Yuan Cheng Copyright © 2015 Songming Hou et al. All rights reserved. Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities Sun, 21 Dec 2014 08:43:56 +0000 Working in a weighted Sobolev space, this paper is devoted to the study of the boundary value problem for the quasilinear parabolic equations with superlinear growth conditions in a domain of . Some conditions which guarantee the solvability of the problem are given. Zhong-Xiang Wang, Gao Jia, and Xiao-Juan Zhang Copyright © 2014 Zhong-Xiang Wang et al. All rights reserved. A New Method for Inextensible Flows of Timelike Curves in Minkowski Space-Time Tue, 16 Dec 2014 13:43:41 +0000 We construct a new method for inextensible flows of timelike curves in Minkowski space-time . Using the Frenet frame of the given curve, we present partial differential equations. We give some characterizations for curvatures of a timelike curve in Minkowski space-time . Talat Körpinar Copyright © 2014 Talat Körpinar. All rights reserved. Conservation Laws for a Degasperis Procesi Equation and a Coupled Variable-Coefficient Modified Korteweg-de Vries System in a Two-Layer Fluid Model via the Multiplier Approach Thu, 13 Nov 2014 07:49:11 +0000 We employ the multiplier approach (variational derivative method) to derive the conservation laws for the Degasperis Procesi equation and a coupled variable-coefficient modified Korteweg-de Vries system in a two-layer fluid model. Firstly, the multipliers are computed and then conserved vectors are obtained for each multiplier. E. Osman, M. Khalfallah, and H. Sapoor Copyright © 2014 E. Osman et al. All rights reserved. Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics Thu, 06 Nov 2014 00:00:00 +0000 The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs. Nemat Dalir Copyright © 2014 Nemat Dalir. All rights reserved. MHD Equations with Regularity in One Direction Mon, 27 Oct 2014 00:00:00 +0000 We consider the 3D MHD equations and prove that if one directional derivative of the fluid velocity, say, , with , , then the solution is in fact smooth.  This improves previous results greatly. Zujin Zhang Copyright © 2014 Zujin Zhang. All rights reserved. Spectral Bounds for Polydiagonal Jacobi Matrix Operators Sun, 19 Oct 2014 11:42:05 +0000 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 Schrödinger-type 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 Lieb-Thirring 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 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 Zakharov-Shabath 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 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 Sine-Gordon Equation by Modified Cubic B-Spline Collocation Method Sun, 10 Aug 2014 06:23:46 +0000 Modified cubic B-spline collocation method is discussed for the numerical solution of one-dimensional nonlinear sine-Gordon equation. The method is based on collocation of modified cubic B-splines 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 B-spline 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 SSP-RK54 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 Two-Way Propagation of Nonlinear Dispersive Waves Using Radial Basis Functions Sun, 03 Aug 2014 08:28:41 +0000 We obtain the numerical solution of a Boussinesq system for two-way 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 fourth-order Runge-Kutta 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 Time-Fractional Coupled Schrödinger System Tue, 15 Jul 2014 00:00:00 +0000 We present a new technique to obtain the solution of time-fractional 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 Bitsadze-Samarskii Nonlocal Boundary Value Problem for Poisson’s Two-Dimensional Equation Thu, 19 Jun 2014 13:33:37 +0000 The Bitsadze-Samarskii nonlocal boundary value problem is considered. Variational formulation is done. The domain decomposition and Schwarz-type 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 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 One-Dimensional Advection-Diffusion Equations in Heterogeneous Media Thu, 08 May 2014 11:18:52 +0000 We derive general bounds for the large time size of supnorm values of solutions to one-dimensional advection-diffusion 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 We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order 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 Dini-continuous 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 Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions Thu, 20 Feb 2014 07:02:08 +0000 We consider initial boundary value problem for a reaction-diffusion 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 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 The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free 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 disease-free 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 disease-free 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 Ventcel-Type Conditions on Fractal Boundaries Wed, 22 Jan 2014 08:09:38 +0000 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 The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure-velocity equation and the concentration equation. In this paper, we present a mixed finite volume element method (FVEM) for the approximation of the pressure-velocity 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 Well-Posedness of the Cauchy Problem for a Modified Two-Component Camassa-Holm System in Besov Spaces Tue, 31 Dec 2013 17:48:27 +0000 We consider the Cauchy problem for an integrable modified two-component Camassa-Holm system with cubic nonlinearity. By using the Littlewood-Paley decomposition, nonhomogeneous Besov spaces, and a priori estimates for linear transport equation, we prove that the Cauchy problem is locally well-posed 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 Initial-Boundary Value Problems with the Fractional Dynamic Boundary Condition Thu, 07 Nov 2013 09:00:04 +0000 In this paper, we analyze some initial-boundary value problems for the subdiffusion equation with a fractional dynamic boundary condition in a one-dimensional bounded domain. First, we establish the unique solvability in the Hölder space of the initial-boundary 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 We apply Rothe‚Äôs 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 In view of variational approach we discuss a nonlocal problem, that is, a Kirchhoff-type 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 Boussinesq-Like Equation Wed, 09 Oct 2013 12:00:32 +0000 The nonlinear dispersive Boussinesq-like 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 We consider the problem of determining an unknown source which depends only on one variable in two-dimensional 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. Xiao-Xiao Li and Dun-Gang Li Copyright © 2013 Xiao-Xiao Li and Dun-Gang Li. All rights reserved. Boundary Value Problems for the Classical and Mixed Integrodifferential Equations with Riemann-Liouville Operators Sun, 08 Sep 2013 08:54:18 +0000 By method of integral equations, unique solvability is proved for the solution of boundary value problems of loaded third-order integrodifferential equations with Riemann-Liouville operators. B. Islomov and U. I. Baltaeva Copyright © 2013 B. Islomov and U. I. Baltaeva. All rights reserved.