Search results for: parabolic equations

Authors: Abdulrahman Abdulrahman

Abstract:

A side weir is a hydraulic structure set into the side of a channel. This structure is used for water level control in channels, to divert flow from a main channel into a side channel when the water level in the main channel exceeds a specific limit and as storm overflows from urban sewerage system. Computation of water surface over the side weirs is essential to determine the flow rate of the side weir. Analytical solutions for water surface profile along rectangular side weir are available only for the special cases of rectangular and trapezoidal channels considering constant specific energy. In this paper, a rectangular side weir located in a combined (trapezoidal with exponential) channel was considered. Expanding binominal series of integer and fraction powers and the using of reduction formula of cosine function integrals, a general analytical formula was obtained for water surface profile along a side weir in a combined (trapezoidal with exponential) channel. Since triangular, rectangular, trapezoidal and parabolic cross-sections are special cases of the combined cross section, the derived formula, is applicable to triangular, rectangular, trapezoidal cross-sections as analytical solution and semi-analytical solution to parabolic cross-section with maximum relative error smaller than 0.76%. The proposed solution should be a useful engineering tool for the evaluation and design of side weirs in open channel.

Keywords: Analytical solution, combined channel, exponential channel, side weirs, trapezoidal channel, water surface profile.

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Authors: N. M. Kamoh, D. G. Gyemang, M. C. Soomiyol

Abstract:

This paper compared the efficiency of Simpson’s 1/3 and 3/8 rules for the numerical solution of first order Volterra integro-differential equations. In developing the solution, collocation approximation method was adopted using the shifted Legendre polynomial as basis function. A block method approach is preferred to the predictor corrector method for being self-starting. Experimental results confirmed that the Simpson’s 3/8 rule is more efficient than the Simpson’s 1/3 rule.

Keywords: Collocation shifted Legendre polynomials, Simpson’s rule and Volterra integro-differential equations.

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Authors: Rana Khalid Naeem, Asif Mansoor, Waseem Ahmed Khan, Aurangzaib

Abstract:

The exact solutions of the equations describing the steady plane motion of an incompressible fluid of variable viscosity for an arbitrary state equation are determined in the (ξ,ψ) − or (η,ψ )- coordinates where ψ(x,y) is the stream function, ξ and η are the parts of the analytic function, ϖ =ξ( x,y )+iη( x,y ). Most of the solutions involve arbitrary function/ functions indicating  that the flow equations possess an infinite set of solutions. 

Keywords: Exact solutions, Fluid of variable viscosity, Navier-Stokes equations, Steady plane flows

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Authors: Amit Goyal, Alka, Rama Gupta, C. Nagaraja Kumar

Abstract:

We have solved the Burgers-Fisher (BF) type equations, with time-dependent coefficients of convection and reaction terms, by using the auxiliary equation method. A class of solitary wave solutions are obtained, and some of which are derived for the first time. We have studied the effect of variable coefficients on physical parameters (amplitude and velocity) of solitary wave solutions. In some cases, the BF equations could be solved for arbitrary timedependent coefficient of convection term.

Keywords: Solitary wave solution, Variable coefficient Burgers- Fisher equation, Auxiliary equation method.

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Authors: Serifenur Cebesoy, Yelda Aygar, Elgiz Bairamov

Abstract:

In this study, we examine some spectral properties of non-selfadjoint matrix-valued difference equations consisting of a polynomial-type Jost solution. The aim of this study is to investigate the eigenvalues and spectral singularities of the difference operator L which is expressed by the above-mentioned difference equation. Firstly, thanks to the representation of polynomial type Jost solution of this equation, we obtain asymptotics and some analytical properties. Then, using the uniqueness theorems of analytic functions, we guarantee that the operator L has a finite number of eigenvalues and spectral singularities.

Keywords: Difference Equations, Jost Functions, Asymptotics, Eigenvalues, Continuous Spectrum, Spectral Singularities.

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Authors: Ameleh Aghajanloo, Moharam Dolatshahi Pirouz, Masoud Montazeri Namin

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In this paper, a two-dimensional (2D) numerical model for the tidal currents simulation in Persian Gulf is presented. The model is based on the depth averaged equations of shallow water which consider hydrostatic pressure distribution. The continuity equation and two momentum equations including the effects of bed friction, the Coriolis effects and wind stress have been solved. To integrate the 2D equations, the Alternative Direction Implicit (ADI) technique has been used. The base of equations discritization was finite volume method applied on rectangular mesh. To evaluate the model validation, a dam break case study including analytical solution is selected and the comparison is done. After that, the capability of the model in simulation of tidal current in a real field is represented by modeling the current behavior in Persian Gulf. The tidal fluctuations in Hormuz Strait have caused the tidal currents in the area of study. Therefore, the water surface oscillations data at Hengam Island on Hormoz Strait are used as the model input data. The check point of the model is measured water surface elevations at Assaluye port. The comparison between the results and the acceptable agreement of them showed the model ability for modeling marine hydrodynamic.

Keywords: Persian Gulf, Tidal Currents, Shallow Water Equations, Finite Volumes

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Authors: J. Geiser, R. Röhle

Abstract:

In this paper we present discretization and decomposition methods for a multi-component transport model of a chemical vapor deposition (CVD) process. CVD processes are used to manufacture deposition layers or bulk materials. In our transport model we simulate the deposition of thin layers. The microscopic model is based on the heavy particles, which are derived by approximately solving a linearized multicomponent Boltzmann equation. For the drift-process of the particles we propose diffusionreaction equations as well as for the effects of heat conduction. We concentrate on solving the diffusion-reaction equation with analytical and numerical methods. For the chemical processes, modelled with reaction equations, we propose decomposition methods and decouple the multi-component models to simpler systems of differential equations. In the numerical experiments we present the computational results of our proposed models.

Keywords: Chemical reactions, chemical vapor deposition, convection-diffusion-reaction equations, decomposition methods, multi-component transport.

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Authors: Vineet K. Srivastava, Mukesh K. Awasthi, Mohammad Tamsir

Abstract:

A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.

Keywords: Burgers’ equation, Implicit Finite-difference method, Newton’s method, Gauss elimination with partial pivoting.

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Authors: Z. N. Haji, S. O. Oyadiji, H. Samami, O. Farrell

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The prediction of the vibration response of rubber products by analytical or numerical method depends mainly on the predefined intrinsic material properties such as Young’s modulus, damping factor and Poisson’s ratio. Such intrinsic properties are determined experimentally by subjecting a bonded rubber sample to compression tests. The compression tests on such a sample yield an apparent Young’s modulus which is greater in magnitude than the intrinsic Young’s modulus of the rubber. As a result, many analytical equations have been developed to determine Young’s modulus from an apparent Young’s modulus of bonded rubber materials. In this work, the applicability of some of these analytical equations is assessed via experimental testing. The assessment is based on testing of vulcanized nitrile butadiene rubber (NBR70) samples using tensile test and compression test methods. The analytical equations are used to determine the intrinsic Young’s modulus from the apparent modulus that is derived from the compression test data of the bonded rubber samples. Then, these Young’s moduli are compared with the actual Young’s modulus that is derived from the tensile test data. The results show significant discrepancy between the Young’s modulus derived using the analytical equations and the actual Young’s modulus.

Keywords: Bonded rubber, quasi-static test, shape factor, apparent Young’s modulus.

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Authors: J. Sulaiman, M. Othman, M. K. Hasan

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Recently, the findings on the MEG iterative scheme has demonstrated to accelerate the convergence rate in solving any system of linear equations generated by using approximation equations of boundary value problems. Based on the same scheme, the aim of this paper is to investigate the capability of a family of four-point block iterative methods with a weighted parameter, ω such as the 4 Point-EGSOR, 4 Point-EDGSOR, and 4 Point-MEGSOR in solving two-dimensional elliptic partial differential equations by using the second-order finite difference approximation. In fact, the formulation and implementation of three four-point block iterative methods are also presented. Finally, the experimental results show that the Four Point MEGSOR iterative scheme is superior as compared with the existing four point block schemes.

Keywords: MEG iteration, second-order finite difference, weighted parameter.

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Authors: Sara Sallam, Mohamed Taqi, Naoual Belouaggadia

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Solar thermal power plants using parabolic trough collectors (PTC) are currently a powerful technology for generating electricity. Most of these solar power plants use thermal oils as heat transfer fluid. The latter is heated in the solar field and transfers the heat absorbed in an oil-water heat exchanger for the production of steam driving the turbines of the power plant. Currently, we are seeking to develop PTCs with direct steam generation (DSG). This process consists of circulating water under pressure in the receiver tube to generate steam directly into the solar loop. This makes it possible to reduce the investment and maintenance costs of the PTCs (the oil-water exchangers are removed) and to avoid the environmental risks associated with the use of thermal oils. The pressure drops in these systems are an important parameter to ensure their proper operation. The determination of these losses is complex because of the presence of the two phases, and most often we limit ourselves to describing them by models using empirical correlations. A comparison of these models with experimental data was performed. Our calculations focused on the evolution of the pressure of the liquid-vapor mixture along the receiver tube of a PTC-DSG for pressure values and inlet flow rates ranging respectively from 3 to 10 MPa, and from 0.4 to 0.6 kg/s. The comparison of the numerical results with experience allows us to demonstrate the validity of some models according to the pressures and the flow rates of entry in the PTC-DSG receiver tube. The analysis of these two parameters’ effects on the evolution of the pressure along the receiving tub, shows that the increase of the inlet pressure and the decrease of the flow rate lead to minimal pressure losses.

Keywords: Direct steam generation, parabolic trough collectors, pressure drop.

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Authors: R. A. Petre, S. E. Nichifor, A. Craifaleanu, I. Stroe

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The relative motion of a robotic arm formed by homogeneous bars of different lengths and masses, hinged to each other is investigated. The first bar of the mechanism is articulated on a platform, considered initially fixed on the surface of the Earth, while for the second case the platform is considered to be in rotation with respect to the Earth. For both analyzed cases the motion equations are determined using the Lagrangian formalism, applied in its traditional form, valid with respect to an inertial reference system, conventionally considered as fixed. However, in the second case, a generalized form of the formalism valid with respect to a non-inertial reference frame will also be applied. The numerical calculations were performed using a MATLAB program.

Keywords: Lagrange equations, relative motion, inertial or non-inertial reference frame.

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Authors: Rudolf Csikja, Janos Toth

Abstract:

Methods to detect and localize time singularities of polynomial and quasi-polynomial ordinary differential equations are systematically presented and developed. They are applied to examples taken form different fields of applications and they are also compared to better known methods such as those based on the existence of linear first integrals or Lyapunov functions.

Keywords: blow up, finite escape time, polynomial ODE, singularity, Lotka–Volterra equation, Painleve analysis, Ψ-series, global existence

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Authors: O. Miraliyari

Abstract:

This article attempts to analyze functionally graded beam thermal buckling along with piezoelectric layers applying based on the third order shearing deformation theory considering various boundary conditions. The beam properties are assumed to vary continuously from the lower surface to the upper surface of the beam. The equilibrium equations are derived using the total potential energy equations, Euler equations, piezoelectric material constitutive equations and third order shear deformation theory assumptions. In order to fulfill such an aim, at first functionally graded beam with piezoelectric layers applying the third order shearing deformation theory along with clamped -clamped boundary conditions are thoroughly analyzed, and then following making sure of the correctness of all the equations, the very same beam is analyzed with piezoelectric layers through simply-simply and simply-clamped boundary conditions. In this article buckling critical temperature for functionally graded beam is derived in two different ways, without piezoelectric layer and with piezoelectric layer and the results are compared together. Finally, all the conclusions obtained will be compared and contrasted with the same samples in the same and distinguished conditions through tables and charts. It would be noteworthy that in this article, the software MAPLE has been applied in order to do the numeral calculations.

Keywords: Thermal buckling, functionally graded beam, piezoelectric layer, various boundary conditions.

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Authors: M. Ashaque Meah, M. Shah Noor, M Asif Arefin, Md. Fazlul Karim

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A numerical technique in a boundary-fitted curvilinear grid model is developed to simulate the extent of inland inundation along the coastal belts of Peninsular Malaysia and Southern Thailand due to 2004 Indian ocean tsunami. Tsunami propagation and run-up are also studied in this paper. The vertically integrated shallow water equations are solved by using the method of lines (MOL). For this purpose the boundary-fitted grids are generated along the coastal and island boundaries and the other open boundaries of the model domain. A transformation is used to the governing equations so that the transformed physical domain is converted into a rectangular one. The MOL technique is applied to the transformed shallow water equations and the boundary conditions so that the equations are converted into ordinary differential equations initial value problem. Finally the 4^th order Runge-Kutta method is used to solve these ordinary differential equations. The moving boundary technique is applied instead of fixed sea side wall or fixed coastal boundary to ensure the movement of the coastal boundary. The extent of intrusion of water and associated tsunami propagation are simulated for the 2004 Indian Ocean tsunami along the west coast of Peninsular Malaysia and southern Thailand. The simulated results are compared with the results obtained from a finite difference model and the data available in the USGS website. All simulations show better approximation than earlier research and also show excellent agreement with the observed data.

Keywords: Open boundary condition, moving boundary condition, boundary-fitted curvilinear grids, far field tsunami, Shallow Water Equations, tsunami source, Indonesian tsunami of 2004.

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Authors: Simon Brown

Abstract:

Glomerular filtration rate (GFR) is a measure of kidney function. It is usually estimated from serum concentrations of cystatin C or creatinine although there has been considerable debate in the literature about (i) the best equation to use and (ii) the variability in the correlation between the concentrations of creatinine and cystatin C. The equations for GFR can be written in a general form and from these I calculate the error of the GFR estimates associated with analyte measurement error. These show that the error of the GFR estimates is such that it is not possible to distinguish between the equations over much of the concentration range of either analyte. The general forms of the equations are also used to derive an expression for the concentration of cystatin C as a function of the concentration of creatinine. This equation shows that these analyte concentrations are not linearly related. Clinical reports of cystatin C and creatinine concentration are consistent with the expression derived.

Keywords: creatinine, cystatin C, error analysis, glomerularfiltration rate, measurement error.

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Authors: Fuziyah Ishak, Mohamed B. Suleiman, Zanariah A. Majid, Khairil I. Othman

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This paper considers the development of a two-point predictor-corrector block method for solving delay differential equations. The formulae are represented in divided difference form and the algorithm is implemented in variable stepsize variable order technique. The block method produces two new values at a single integration step. Numerical results are compared with existing methods and it is evident that the block method performs very well. Stability regions of the block method are also investigated.

Keywords: block method, delay differential equations, predictor-corrector, stability region, variable stepsize variable order.

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Authors: Nurhakimah Ab. Rahman, Lazim Abdullah

Abstract:

According to fuzzy arithmetic, dual fuzzy polynomials cannot be replaced by fuzzy polynomials. Hence, the concept of ranking method is used to find real roots of dual fuzzy polynomial equations. Therefore, in this study we want to propose an interval type-2 dual fuzzy polynomial equation (IT2 DFPE). Then, the concept of ranking method also is used to find real roots of IT2 DFPE (if exists). We transform IT2 DFPE to system of crisp IT2 DFPE. This transformation performed with ranking method of fuzzy numbers based on three parameters namely value, ambiguity and fuzziness. At the end, we illustrate our approach by two numerical examples.

Keywords: Dual fuzzy polynomial equations, Interval type-2, Ranking method, Value.

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Authors: Xin Luo, Jin Huang, Chuan-Long Wang

Abstract:

The mechanical quadrature methods for solving the boundary integral equations of the anisotropic Darcy-s equations with Dirichlet conditions in smooth domains are presented. By applying the collectively compact theory, we prove the convergence and stability of approximate solutions. The asymptotic expansions for the error show that the methods converge with the order O (h3), where h is the mesh size. Based on these analysis, extrapolation methods can be introduced to achieve a higher convergence rate O (h5). An a posterior asymptotic error representation is derived in order to construct self-adaptive algorithms. Finally, the numerical experiments show the efficiency of our methods.

Keywords: Darcy's equation, anisotropic, mechanical quadrature methods, extrapolation methods, a posteriori error estimate.

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Authors: Naveed Ahmed, Gunar Matthies

Abstract:

We deal with the numerical solution of time-dependent convection-diffusion-reaction equations. We combine the local projection stabilization method for the space discretization with two different time discretization schemes: the continuous Galerkin-Petrov (cGP) method and the discontinuous Galerkin (dG) method of polynomial of degree k. We establish the optimal error estimates and present numerical results which shows that the cGP(k) and dG(k)- methods are accurate of order k +1, respectively, in the whole time interval. Moreover, the cGP(k)-method is superconvergent of order 2k and dG(k)-method is of order 2k +1 at the discrete time points. Furthermore, the dependence of the results on the choice of the stabilization parameter are discussed and compared.

Keywords: Convection-diffusion-reaction equations, stabilized finite elements, discontinuous Galerkin, continuous Galerkin-Petrov.

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Authors: M. Behbahani-Nejad, Y. Shekari

Abstract:

A reduced order modeling approach for natural gas transient flow in pipelines is presented. The Euler equations are considered as the governing equations and solved numerically using the implicit Steger-Warming flux vector splitting method. Next, the linearized form of the equations is derived and the corresponding eigensystem is obtained. Then, a few dominant flow eigenmodes are used to construct an efficient reduced-order model. A well-known test case is presented to demonstrate the accuracy and the computational efficiency of the proposed method. The results obtained are in good agreement with those of the direct numerical method and field data. Moreover, it is shown that the present reduced-order model is more efficient than the conventional numerical techniques for transient flow analysis of natural gas in pipelines.

Keywords: Eigenmode, Natural Gas, Reduced Order Modeling, Transient Flow.

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Authors: Reza Abazari, Rasool Abazari

Abstract:

In this paper, the two-dimension differential transformation method (DTM) is employed to obtain the closed form solutions of the three famous coupled partial differential equation with physical interest namely, the coupled Korteweg-de Vries(KdV) equations, the coupled Burgers equations and coupled nonlinear Schrödinger equation. We begin by showing that how the differential transformation method applies to a linear and non-linear part of any PDEs and apply on these coupled PDEs to illustrate the sufficiency of the method for this kind of nonlinear differential equations. The results obtained are in good agreement with the exact solution. These results show that the technique introduced here is accurate and easy to apply.

Keywords: Coupled Korteweg-de Vries(KdV) equation, Coupled Burgers equation, Coupled Schrödinger equation, differential transformation method.

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Authors: N. A. Samat, D. F. Percy

Abstract:

The main aim of this study is to describe and introduce a method of numerical analysis in obtaining approximate solutions for the SIR-SI differential equations (susceptible-infectiverecovered for human populations; susceptible-infective for vector populations) that represent a model for dengue disease transmission. Firstly, we describe the ordinary differential equations for the SIR-SI disease transmission models. Then, we introduce the numerical analysis of solutions of this continuous time, discrete space SIR-SI model by simplifying the continuous time scale to a densely populated, discrete time scale. This is followed by the application of this numerical analysis of solutions of the SIR-SI differential equations to the estimation of relative risk using continuous time, discrete space dengue data of Kuala Lumpur, Malaysia. Finally, we present the results of the analysis, comparing and displaying the results in graphs, table and maps. Results of the numerical analysis of solutions that we implemented offers a useful and potentially superior model for estimating relative risks based on continuous time, discrete space data for vector borne infectious diseases specifically for dengue disease. 

Keywords: Dengue disease, disease mapping, numerical analysis, SIR-SI differential equations.

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Authors: Jessada Tariboon

Abstract:

In this article, using distribution kernel, we study the nonlinear equations with n-dimensional telegraph operator iterated k-times.

Keywords: Telegraph operator, Elementary solution, Distribution kernel.

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Authors: Dylan M. Copeland

Abstract:

We present new finite element methods for Helmholtz and Maxwell equations on general three-dimensional polyhedral meshes, based on domain decomposition with boundary elements on the surfaces of the polyhedral volume elements. The methods use the lowest-order polynomial spaces and produce sparse, symmetric linear systems despite the use of boundary elements. Moreover, piecewise constant coefficients are admissible. The resulting approximation on the element surfaces can be extended throughout the domain via representation formulas. Numerical experiments confirm that the convergence behavior on tetrahedral meshes is comparable to that of standard finite element methods, and equally good performance is attained on more general meshes.

Keywords: Boundary elements, finite elements, Helmholtz equation, Maxwell equations.

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Authors: Li Xiguang

Abstract:

In this paper, by constructing a special set and utilizing fixed point index theory, we study the existence of solution for the boundary value problem of second-order singular differential equations in Banach space, which improved and generalize the result of related paper.

Keywords: Banach space, cone, fixed point index, singular equation.

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Authors: Kholod M. Abu-Alnaja

Abstract:

In this work, we derive two numerical schemes for solving a class of nonlinear partial differential equations. The first method is of second order accuracy in space and time directions, the scheme is unconditionally stable using Von Neumann stability analysis, the scheme produced a nonlinear block system where Newton-s method is used to solve it. The second method is of fourth order accuracy in space and second order in time. The method is unconditionally stable and Newton's method is used to solve the nonlinear block system obtained. The exact single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitary waves for different parameters are also discussed.

Keywords: Crank-Nicolson Scheme, Douglas Scheme, Partial Differential Equations

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Authors: Faranak Rabiei, Fudziah Ismail, S. Norazak, Saeid Emadi

Abstract:

In this paper we developed the Improved Runge-Kutta Nystrom (IRKN) method for solving second order ordinary differential equations. The methods are two step in nature and require lower number of function evaluations per step compared with the existing Runge-Kutta Nystrom (RKN) methods. Therefore, the methods are computationally more efficient at achieving the higher order of local accuracy. Algebraic order conditions of the method are obtained and the third and fourth order method are derived with two and three stages respectively. The numerical results are given to illustrate the efficiency of the proposed method compared to the existing RKN methods.

Keywords: Improved Runge-Kutta Nystrom method, Two step method, Second-order ordinary differential equations, Order conditions

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Authors: Li Xiguang

Abstract:

In this paper, by constructing a special set and utilizing fixed point theory, we study the existence and multiplicity of the positive solutions for systems of nonlinear third-order differential equations with p-laplacian, which improve and generalize the result of related paper.

Keywords: p-Laplacian, cone, fixed point theorem, positive solution.

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Authors: Felix Che Shu

Abstract:

We give an explicit formula for the general solution of a one dimensional linear delay differential equation with multiple delays, which are integer multiples of the smallest delay. For an equation of this class with two delays, we derive two equations with single delays, whose stability is sufficient for the stability of the equation with two delays. This presents a new approach to the study of the stability of such systems. This approach avoids requirement of the knowledge of the location of the characteristic roots of the equation with multiple delays which are generally more difficult to determine, compared to the location of the characteristic roots of equations with a single delay.

Keywords: Delay Differential Equation, Explicit Solution, Exponential Stability, Lyapunov Exponents, Multiple Delays.

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