**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**1257

# Search results for: Advection equations

##### 1257 Semi-Lagrangian Method for Advection Equation on GPU in Unstructured R3 Mesh for Fluid Dynamics Application

**Authors:**
Irakli V. Gugushvili,
Nickolay M. Evstigneev

**Abstract:**

Numerical integration of initial boundary problem for advection equation in 3 ℜ is considered. The method used is conditionally stable semi-Lagrangian advection scheme with high order interpolation on unstructured mesh. In order to increase time step integration the BFECC method with limiter TVD correction is used. The method is adopted on parallel graphic processor unit environment using NVIDIA CUDA and applied in Navier-Stokes solver. It is shown that the calculation on NVIDIA GeForce 8800 GPU is 184 times faster than on one processor AMDX2 4800+ CPU. The method is extended to the incompressible fluid dynamics solver. Flow over a Cylinder for 3D case is compared to the experimental data.

**Keywords:**
Advection equations,
CUDA technology,
Flow overthe 3D Cylinder,
Incompressible Pressure Projection Solver,
Parallel computation.

##### 1256 An Efficient Backward Semi-Lagrangian Scheme for Nonlinear Advection-Diffusion Equation

**Authors:**
Soyoon Bak,
Sunyoung Bu,
Philsu Kim

**Abstract:**

In this paper, a backward semi-Lagrangian scheme combined with the second-order backward difference formula is designed to calculate the numerical solutions of nonlinear advection-diffusion equations. The primary aims of this paper are to remove any iteration process and to get an efficient algorithm with the convergence order of accuracy 2 in time. In order to achieve these objects, we use the second-order central finite difference and the B-spline approximations of degree 2 and 3 in order to approximate the diffusion term and the spatial discretization, respectively. For the temporal discretization, the second order backward difference formula is applied. To calculate the numerical solution of the starting point of the characteristic curves, we use the error correction methodology developed by the authors recently. The proposed algorithm turns out to be completely iteration free, which resolves the main weakness of the conventional backward semi-Lagrangian method. Also, the adaptability of the proposed method is indicated by numerical simulations for Burgers’ equations. Throughout these numerical simulations, it is shown that the numerical results is in good agreement with the analytic solution and the present scheme offer better accuracy in comparison with other existing numerical schemes.

**Keywords:**
Semi-Lagrangian method,
Iteration free method,
Nonlinear advection-diffusion equation.

##### 1255 Finite Volume Model to Study the Effect of Buffer on Cytosolic Ca2+ Advection Diffusion

**Authors:**
Brajesh Kumar Jha,
Neeru Adlakha,
M. N. Mehta

**Abstract:**

**Keywords:**
Ca2+ profile,
buffer,
Astrocytes,
Advection diffusion,
FVM

##### 1254 Finite Volume Model to Study The Effect of Voltage Gated Ca2+ Channel on Cytosolic Calcium Advection Diffusion

**Authors:**
Brajesh Kumar Jha,
Neeru Adlakha,
M. N. Mehta

**Abstract:**

**Keywords:**
Ca2+ Profile,
Advection Diffusion,
VOC,
FVM.

##### 1253 A Novel System of Two Coupled Equations for the Longitudinal Components of the Electromagnetic Field in a Waveguide

**Authors:**
Arti Vaish,
Harish Parthasarathy

**Abstract:**

**Keywords:**
Electromagnetism,
Maxwell's Equations,
Anisotropic permittivity,
Wave equation,
Matrix Equation,
Permittivity tensor.

##### 1252 Application of the Hybrid Methods to Solving Volterra Integro-Differential Equations

**Authors:**
G.Mehdiyeva,
M.Imanova,
V.Ibrahimov

**Abstract:**

**Keywords:**
Integro-differential equations,
initial value
problem,
hybrid methods,
predictor-corrector method

##### 1251 New Insight into Fluid Mechanics of Lorenz Equations

**Authors:**
Yu-Kai Ting,
Jia-Ying Tu,
Chung-Chun Hsiao

**Abstract:**

New physical insights into the nonlinear Lorenz equations related to flow resistance is discussed in this work. The chaotic dynamics related to Lorenz equations has been studied in many papers, which is due to the sensitivity of Lorenz equations to initial conditions and parameter uncertainties. However, the physical implication arising from Lorenz equations about convectional motion attracts little attention in the relevant literature. Therefore, as a first step to understand the related fluid mechanics of convectional motion, this paper derives the Lorenz equations again with different forced conditions in the model. Simulation work of the modified Lorenz equations without the viscosity or buoyancy force is discussed. The time-domain simulation results may imply that the states of the Lorenz equations are related to certain flow speed and flow resistance. The flow speed of the underlying fluid system increases as the flow resistance reduces. This observation would be helpful to analyze the coupling effects of different fluid parameters in a convectional model in future work.

**Keywords:**
Galerkin method,
Lorenz equations,
Navier-Stokes
equations.

##### 1250 Analysis of Three-Dimensional Longitudinal Rolls Induced by Double Diffusive Poiseuille-Rayleigh-Benard Flows in Rectangular Channels

**Authors:**
O. Rahli,
N. Mimouni,
R. Bennacer,
K. Bouhadef

**Abstract:**

**Keywords:**
Heat and mass transfer,
mixed convection,
Poiseuille-Rayleigh-Benard flow,
rectangular duct.

##### 1249 The Approximate Solution of Linear Fuzzy Fredholm Integral Equations of the Second Kind by Using Iterative Interpolation

**Authors:**
N. Parandin,
M. A. Fariborzi Araghi

**Abstract:**

**Keywords:**
Fuzzy function integral equations,
Iterative method,
Linear systems,
Parametric form of fuzzy number.

##### 1248 A First Course in Numerical Methods with “Mathematica“

**Authors:**
Andrei A. Kolyshkin

**Abstract:**

**Keywords:**
Numerical methods,
"Mathematica",
e-learning.

##### 1247 An Efficient Computational Algorithm for Solving the Nonlinear Lane-Emden Type Equations

**Authors:**
Gholamreza Hojjati,
Kourosh Parand

**Abstract:**

In this paper we propose a class of second derivative multistep methods for solving some well-known classes of Lane- Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. These methods, which have good stability and accuracy properties, are useful in deal with stiff ODEs. We show superiority of these methods by applying them on the some famous Lane-Emden type equations.

**Keywords:**
Lane-Emden type equations,
nonlinear ODE,
stiff problems,
multistep methods,
astrophysics.

##### 1246 Equations of Pulse Propagation in Three-Layer Structure of As2S3 Chalcogenide Plasmonic Nano-Waveguides

**Authors:**
Leila Motamed-Jahromi,
Mohsen Hatami,
Alireza Keshavarz

**Abstract:**

This research aims at obtaining the equations of pulse propagation in nonlinear plasmonic waveguides created with As_{2}S_{3} chalcogenide materials. Via utilizing Helmholtz equation and first-order perturbation theory, two components of electric field are determined within frequency domain. Afterwards, the equations are formulated in time domain. The obtained equations include two coupled differential equations that considers nonlinear dispersion.

**Keywords:**
Nonlinear optics,
propagation equation,
plasmonic waveguide.

##### 1245 Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations

**Authors:**
Fuziyah Ishak,
Siti Norazura Ahmad

**Abstract:**

Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.

**Keywords:**
Accuracy,
extended trapezoidal method,
numerical solution,
Volterra integro-differential equations.

##### 1244 The Pell Equation x2 − (k2 − k)y2 = 2t

**Authors:**
Ahmet Tekcan

**Abstract:**

**Keywords:**
Pell equation,
solutions of Pell equation.

##### 1243 Analysis of One Dimensional Advection Diffusion Model Using Finite Difference Method

**Authors:**
Vijay Kumar Kukreja,
Ravneet Kaur

**Abstract:**

**Keywords:**
Consistency,
Crank-Nicolson scheme,
Gerschgorin
circle,
Lax-Richtmyer theorem,
Peclet number,
stability.

##### 1242 The Euler Equations of Steady Flow in Terms of New Dependent and Independent Variables

**Authors:**
Peiangpob Monnuanprang

**Abstract:**

In this paper we study the transformation of Euler equations 1 , u u u Pf t (ρ ∂) + ⋅∇ = − ∇ + ∂ G G G G ∇⋅ = u 0, G where (ux, t) G G is the velocity of a fluid, P(x, t) G is the pressure of a fluid andρ (x, t) G is density. First of all, we rewrite the Euler equations in terms of new unknown functions. Then, we introduce new independent variables and transform it to a new curvilinear coordinate system. We obtain the Euler equations in the new dependent and independent variables. The governing equations into two subsystems, one is hyperbolic and another is elliptic.

**Keywords:**
Euler equations,
transformation,
hyperbolic,
elliptic

##### 1241 Automatic Iterative Methods for the Multivariate Solution of Nonlinear Algebraic Equations

**Authors:**
Rafat Alshorman,
Safwan Al-Shara',
I. Obeidat

**Abstract:**

**Keywords:**
Nonlinear Algebraic Equations,
Iterative Methods,
Homotopy
Analysis Method.

##### 1240 A New Approach to the Approximate Solutions of Hamilton-Jacobi Equations

**Authors:**
Joe Imae,
Kenjiro Shinagawa,
Tomoaki Kobayashi,
Guisheng Zhai

**Abstract:**

We propose a new approach on how to obtain the approximate solutions of Hamilton-Jacobi (HJ) equations. The process of the approximation consists of two steps. The first step is to transform the HJ equations into the virtual time based HJ equations (VT-HJ) by introducing a new idea of ‘virtual-time’. The second step is to construct the approximate solutions of the HJ equations through a computationally iterative procedure based on the VT-HJ equations. It should be noted that the approximate feedback solutions evolve by themselves as the virtual-time goes by. Finally, we demonstrate the effectiveness of our approximation approach by means of simulations with linear and nonlinear control problems.

**Keywords:**
Nonlinear Control,
Optimal Control,
Hamilton-Jacobi Equation,
Virtual-Time

##### 1239 Integrable Heisenberg Ferromagnet Equations with Self-Consistent Potentials

**Authors:**
Gulgassyl Nugmanova,
Zhanat Zhunussova,
Kuralay Yesmakhanova,
Galya Mamyrbekova,
Ratbay Myrzakulov

**Abstract:**

**Keywords:**
Spin systems,
equivalent counterparts,
integrable
reductions,
self-consistent potentials.

##### 1238 Exp-Function Method for Finding Some Exact Solutions of Rosenau Kawahara and Rosenau Korteweg-de Vries Equations

**Authors:**
Ehsan Mahdavi

**Abstract:**

In this paper, we apply the Exp-function method to Rosenau-Kawahara and Rosenau-KdV equations. Rosenau-Kawahara equation is the combination of the Rosenau and standard Kawahara equations and Rosenau-KdV equation is the combination of the Rosenau and standard KdV equations. These equations are nonlinear partial differential equations (NPDE) which play an important role in mathematical physics. Exp-function method is easy, succinct and powerful to implement to nonlinear partial differential equations arising in mathematical physics. We mainly try to present an application of Exp-function method and offer solutions for common errors wich occur during some of the recent works.

**Keywords:**
Exp-function method,
Rosenau Kawahara equation,
Rosenau Korteweg-de Vries equation,
nonlinear partial differential
equation.

##### 1237 Membrane Distillation Process Modeling: Dynamical Approach

**Authors:**
Fadi Eleiwi,
Taous Meriem Laleg-Kirati

**Abstract:**

This paper presents a complete dynamic modeling of a membrane distillation process. The model contains two consistent dynamic models. A 2D advection-diffusion equation for modeling the whole process and a modified heat equation for modeling the membrane itself. The complete model describes the temperature diffusion phenomenon across the feed, membrane, permeate containers and boundary layers of the membrane. It gives an online and complete temperature profile for each point in the domain. It explains heat conduction and convection mechanisms that take place inside the process in terms of mathematical parameters, and justify process behavior during transient and steady state phases. The process is monitored for any sudden change in the performance at any instance of time. In addition, it assists maintaining production rates as desired, and gives recommendations during membrane fabrication stages. System performance and parameters can be optimized and controlled using this complete dynamic model. Evolution of membrane boundary temperature with time, vapor mass transfer along the process, and temperature difference between membrane boundary layers are depicted and included. Simulations were performed over the complete model with real membrane specifications. The plots show consistency between 2D advection-diffusion model and the expected behavior of the systems as well as literature. Evolution of heat inside the membrane starting from transient response till reaching steady state response for fixed and varying times is illustrated.

**Keywords:**
Membrane distillation,
Dynamical modeling,
Advection-diffusion equation,
Thermal equilibrium,
Heat
equation.

##### 1236 Numerical Solution for Integro-Differential Equations by Using Quartic B-Spline Wavelet and Operational Matrices

**Authors:**
Khosrow Maleknejad,
Yaser Rostami

**Abstract:**

In this paper, Semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions

**Keywords:**
Integro-differential equations,
Quartic B-spline
wavelet,
Operational matrices.

##### 1235 Burning Rate Response of Solid Fuels in Laminar Boundary Layer

**Authors:**
A. M. Tahsini

**Abstract:**

**Keywords:**
Extinction,
Oscillation,
Regression rate,
Response,
Transient burning.

##### 1234 Strict Stability of Fuzzy Differential Equations with Impulse Effect

**Authors:**
Sanjay K.Srivastava,
Bhanu Gupta

**Abstract:**

In this paper some results on strict stability heve beeb extended for fuzzy differential equations with impulse effect using Lyapunov functions and Razumikhin technique.

**Keywords:**
Fuzzy differential equations,
Impulsive differential equations,
Strict stability,
Lyapunov function,
Razumikhin technique.

##### 1233 Toward a New Simple Analytical Formulation of Navier-Stokes Equations

**Authors:**
Gunawan Nugroho,
Ahmed M. S. Ali,
Zainal A. Abdul Karim

**Abstract:**

**Keywords:**
Navier-Stokes Equations,
potential function,
turbulent flows.

##### 1232 Solving Linear Matrix Equations by Matrix Decompositions

**Authors:**
Yongxin Yuan,
Kezheng Zuo

**Abstract:**

In this paper, a system of linear matrix equations is considered. A new necessary and sufficient condition for the consistency of the equations is derived by means of the generalized singular-value decomposition, and the explicit representation of the general solution is provided.

**Keywords:**
Matrix equation,
Generalized inverse,
Generalized
singular-value decomposition.

##### 1231 An Overview of Some High Order and Multi-Level Finite Difference Schemes in Computational Aeroacoustics

**Authors:**
Appanah Rao Appadu,
Muhammad Zaid Dauhoo

**Abstract:**

In this paper, we have combined some spatial derivatives with the optimised time derivative proposed by Tam and Webb in order to approximate the linear advection equation which is given by = 0. Ôêé Ôêé + Ôêé Ôêé x f t u These spatial derivatives are as follows: a standard 7-point 6 th -order central difference scheme (ST7), a standard 9-point 8 th -order central difference scheme (ST9) and optimised schemes designed by Tam and Webb, Lockard et al., Zingg et al., Zhuang and Chen, Bogey and Bailly. Thus, these seven different spatial derivatives have been coupled with the optimised time derivative to obtain seven different finite-difference schemes to approximate the linear advection equation. We have analysed the variation of the modified wavenumber and group velocity, both with respect to the exact wavenumber for each spatial derivative. The problems considered are the 1-D propagation of a Boxcar function, propagation of an initial disturbance consisting of a sine and Gaussian function and the propagation of a Gaussian profile. It is known that the choice of the cfl number affects the quality of results in terms of dissipation and dispersion characteristics. Based on the numerical experiments solved and numerical methods used to approximate the linear advection equation, it is observed in this work, that the quality of results is dependent on the choice of the cfl number, even for optimised numerical methods. The errors from the numerical results have been quantified into dispersion and dissipation using a technique devised by Takacs. Also, the quantity, Exponential Error for Low Dispersion and Low Dissipation, eeldld has been computed from the numerical results. Moreover, based on this work, it has been found that when the quantity, eeldld can be used as a measure of the total error. In particular, the total error is a minimum when the eeldld is a minimum.

**Keywords:**
Optimised time derivative,
dissipation,
dispersion,
cfl number,
Nomenclature: k : time step,
h : spatial step,
β :advection velocity,
r: cfl/Courant number,
hkrβ= ,
w =θ,
h : exact wave number,
n :time level,
RPE : Relative phase error per unit time step,
AFM :modulus of amplification factor

##### 1230 Laplace Technique to Find General Solution of Differential Equations without Initial Conditions

**Authors:**
Adil Al-Rammahi

**Abstract:**

Laplace transformations have wide applications in engineering and sciences. All previous studies of modified Laplace transformations depend on differential equation with initial conditions. The purpose of our paper is to solve the linear differential equations (not initial value problem) and then find the general solution (not particular) via the Laplace transformations without needed any initial condition. The study involves both types of differential equations, ordinary and partial.

**Keywords:**
Differential Equations,
Laplace Transformations.

##### 1229 New Application of EHTA for the Generalized(2+1)-Dimensional Nonlinear Evolution Equations

**Authors:**
Mohammad Taghi Darvishi,
Maliheh Najafi,
Mohammad Najafi

**Abstract:**

In this paper, the generalized (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (shortly CBS) equations are investigated. We employ the Hirota-s bilinear method to obtain the bilinear form of CBS equations. Then by the idea of extended homoclinic test approach (shortly EHTA), some exact soliton solutions including breather type solutions are presented.

**Keywords:**
EHTA,
(2+1)-dimensional CBS equations,
(2+1)-dimensional breaking solution equation,
Hirota's bilinear form.

##### 1228 Bifurcation Method for Solving Positive Solutions to a Class of Semilinear Elliptic Equations and Stability Analysis of Solutions

**Authors:**
Hailong Zhu,
Zhaoxiang Li

**Abstract:**

Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In this paper, bifurcation method is applied to solving semilinear elliptic equations which are with homogeneous Dirichlet boundary conditions in 2D. Using this method, nontrivial numerical solutions will be computed and visualized in many different domains (such as square, disk, annulus, dumbbell, etc).

**Keywords:**
Semilinear elliptic equations,
positive solutions,
bifurcation method,
isotropy subgroups.