Search results for: First Order Ordinary Differential Equations

Authors: Mostafa Raki, Mahdi Hamzehei

Abstract:

Equilibrium and stability equations of a thin rectangular plate with length a, width b, and thickness h(x)=C1x+C2, made of functionally graded materials under thermal loads are derived based on the first order shear deformation theory. It is assumed that the material properties vary as a power form of thickness coordinate variable z. The derived equilibrium and buckling equations are then solved analytically for a plate with simply supported boundary conditions. One type of thermal loading, uniform temperature rise and gradient through the thickness are considered, and the buckling temperatures are derived. The influences of the plate aspect ratio, the relative thickness, the gradient index and the transverse shear on buckling temperature difference are all discussed.

Keywords: Stability of plate, thermal buckling, rectangularplate, functionally graded material, first order shear deformationtheory.

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Authors: Mohd Ariff Admon, Abdul Rahman Mohd Kasim, Sharidan Shafie

Abstract:

This paper considers the effect of heat generation proportional l to (T - T∞ )p , where T is the local temperature and T∞ is the ambient temperature, in unsteady free convection flow near the stagnation point region of a three-dimensional body. The fluid is considered in an ambient fluid under the assumption of a step change in the surface temperature of the body. The non-linear coupled partial differential equations governing the free convection flow are solved numerically using an implicit finite-difference method for different values of the governing parameters entering these equations. The results for the flow and heat characteristics when p ≤ 2 show that the transition from the initial unsteady-state flow to the final steadystate flow takes place smoothly. The behavior of the flow is seen strongly depend on the exponent p.

Keywords: Free convection, Boundary layer flow, Stagnationpoint, Heat generation

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Authors: Mehrdad Mahmudizad, Roya Ahmadi Ahangar

Abstract:

This paper presents the method of designing the type 2 fuzzy PID controllers in order to solve the problem of Load Frequency Control (LFC). The Harmony Search (HS) algorithm is used to regulate the measurement factors and the effect of uncertainty of membership functions of Interval Type 2 Fuzzy Proportional Integral Differential (IT2FPID) controllers in order to reduce the frequency deviation resulted from the load oscillations. The simulation results implicitly show that the performance of the proposed IT2FPID LFC in terms of error, settling time and resistance against different load oscillations is more appropriate and preferred than PID and Type 1 Fuzzy Proportional Integral Differential (T1FPID) controllers.

Keywords: Load Frequency Control, Fuzzy-PID controller, Type 2 fuzzy system, Harmony Search algorithm.

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Authors: Wararit Panichkitkosolkul

Abstract:

The unit root tests based on the robust estimator for the first-order autoregressive process are proposed and compared with the unit root tests based on the ordinary least squares (OLS) estimator. The percentiles of the null distributions of the unit root test are also reported. The empirical probabilities of Type I error and powers of the unit root tests are estimated via Monte Carlo simulation. Simulation results show that all unit root tests can control the probability of Type I error for all situations. The empirical power of the unit root tests based on the robust estimator are higher than the unit root tests based on the OLS estimator.

Keywords: Autoregressive, Ordinary least squares, Type I error, Power of the test, Monte Carlo simulation.

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Authors: V.V. Lyahov, V.M. Neshchadim

Abstract:

The structural stability of the model of a nonelectroneutral current sheath is investigated. The stationary model of a current sheath represents the system of four connected nonlinear differential first-order equations and thus they should manifest structural instability property, i.e. sensitivity to the infinitesimal changes of parameters and starting conditions. Domains of existence of the solutions of current sheath type are found. Those solutions of the current sheath type are realized only in some regions of sevendimensional space of parameters of the problem. The phase volume of those regions is small in comparison with the whole phase volume of the definition range of those parameters. It is shown that the offered model of a nonelectroneutral current sheath is applicable for theoretical interpretation of the bifurcational current sheaths observed in the magnetosphere.

Keywords: Distribution function, electromagnetic field, magnetoactive plasma, nonelectroneutral current sheath, structural instability, bifurcational current sheath.

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Authors: Hassan Saberi-Nik, Mahin Golchaman

Abstract:

This paper applies the homotopy analysis method method to a nonlinear differential-difference equation arising in nanotechnology. Continuum hypothesis on nanoscales is invalid, and a differential-difference model is considered as an alternative approach to describing discontinued problems. Comparison of the approximate solution with the exact one reveals that the method is very effective.

Keywords: Homotopy analysis method, differential-difference, nanotechnology.

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Authors: S.H. Mirtalaie, M.A. Hajabasi

Abstract:

In this paper, an analytical approach is used to study the coupled lateral-torsional vibrations of laminated composite beam. It is known that in such structures due to the fibers orientation in various layers, any lateral displacement will produce a twisting moment. This phenomenon is modeled by the bending-twisting material coupling rigidity and its main feature is the coupling of lateral and torsional vibrations. In addition to the material coupling, the effects of shear deformation and rotary inertia are taken into account in the definition of the potential and kinetic energies. Then, the governing differential equations are derived using the Hamilton-s principle and the mathematical model matches the Timoshenko beam model when neglecting the effect of bending-twisting rigidity. The equations of motion which form a system of three coupled PDEs are solved analytically to study the free vibrations of the beam in lateral and rotational modes due to the bending, as well as the torsional mode caused by twisting. The analytic solution is carried out in three steps: 1) assuming synchronous motion for the kinematic variables which are the lateral, rotational and torsional displacements, 2) solving the ensuing eigenvalue problem which contains three coupled second order ODEs and 3) imposing different boundary conditions related to combinations of simply, clamped and free end conditions. The resulting natural frequencies and mode shapes are compared with similar results in the literature and good agreement is achieved.

Keywords: Free vibration, laminated composite beam, material coupling, state space.

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Authors: Hansoo Kim, Seunghak Shin

Abstract:

The differential column shortening in tall buildings can be reduced by improving material and structural characteristics of the structural systems. This paper proposes structural methods to reduce differential column shortening in reinforced concrete tall buildings; connecting columns with rigidly jointed horizontal members, using outriggers, and placing additional reinforcement at the columns. The rigidly connected horizontal members including outriggers reduce the differential shortening between adjacent vertical members. The axial stiffness of columns with greater shortening can be effectively increased by placing additional reinforcement at the columns, thus the differential column shortening can be reduced in the design stage. The optimum distribution of additional reinforcement can be determined by applying a gradient based optimization technique.

Keywords: Column shortening, long-term behavior, optimization, tall building.

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Authors: Li Jie, Zhang Wei

Abstract:

In this paper, the problem of posture stabilization for a kinematic model of differential drive robots is studied. A more complex model of the kinematics of differential drive robots is used for the design of stabilizing control. This model is formulated in terms of the physical parameters of the system such as the radius of the wheels, and velocity of the wheels are the control inputs of it. In this paper, the framework of Lyapunov-based control design has been used to solve posture stabilization problem for the comprehensive model of differential drive robots. The results of the simulations show that the devised controller successfully solves the posture regulation problem. Finally, robustness and performance of the controller have been studied under system parameter uncertainty.

Keywords: Differential drive robots, nonlinear control, Lyapunov-based control design, posture regulation.

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Authors: Marcin Sowa

Abstract:

The paper discusses the subinterval-based numerical method for fractional derivative computations. It is now referred to by its acronym – SubIval. The basis of the method is briefly recalled. The ability of the method to be applied in time stepping solvers is discussed. The possibility of implementing a time step size adaptive solver is also mentioned. The solver is tested on a transient circuit example. In order to display the accuracy of the solver – the results have been compared with those obtained by means of a semi-analytical method called gcdAlpha. The time step size adaptive solver applying SubIval has been proven to be very accurate as the results are very close to the referential solution. The solver is currently able to solve FDE (fractional differential equations) with various derivative orders for each equation and any type of source time functions.

Keywords: Numerical method, SubIval, fractional calculus, numerical solver, circuit analysis.

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Authors: Krzysztof Walczak

Abstract:

Fractional delay FIR filters design method based on the differential evolution algorithm is presented. Differential evolution is an evolutionary algorithm for solving a global optimization problems in the continuous search space. In the proposed approach, an evolutionary algorithm is used to determine the coefficients of a fractional delay FIR filter based on the Farrow structure. Basic differential evolution is enhanced with a restricted mating technique, which improves the algorithm performance in terms of convergence speed and obtained solution. Evolutionary optimization is carried out by minimizing an objective function which is based on the amplitude response and phase delay errors. Experimental results show that the proposed algorithm leads to a reduction in the amplitude response and phase delay errors relative to those achieved with the Least-Squares method.

Keywords: Fractional Delay Filters, Farrow Structure, Evolutionary Computation, Differential Evolution

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Authors: Li Xiguang

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In this paper, by constructing a special operator and using fixed point index theorem of cone, we get the sufficient conditions for symmetric positive solution of a class of nonlinear singular boundary value problems with p-Laplace operator, which improved and generalized the result of related paper.

Keywords: Banach space, cone, fixed point index, singular differential equation, p-Laplace operator, symmetric solutions.

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Authors: I. Algul, G. Akgun, H. Kurtaran

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Dynamic analysis of composite doubly curved panels with variable thickness subjected to different pulse types using Generalized Differential Quadrature method (GDQ) is presented in this study. Panels with variable thickness are used in the construction of aerospace and marine industry. Giving variable thickness to panels can allow the designer to get optimum structural efficiency. For this reason, estimating the response of variable thickness panels is very important to design more reliable structures under dynamic loads. Dynamic equations for composite panels with variable thickness are obtained using virtual work principle. Partial derivatives in the equation of motion are expressed with GDQ and Newmark average acceleration scheme is used for temporal discretization. Several examples are used to highlight the effectiveness of the proposed method. Results are compared with finite element method. Effects of taper ratios, boundary conditions and loading type on the response of composite panel are investigated.

Keywords: Generalized differential quadrature method, doubly curved panels, laminated composite materials, small displacement.

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Authors: Sara Bilal, Abdi Omar Shuriye, Raihan Othman

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Numerical Methods is a course that can be conducted using workshops and group discussion. This study has been implemented on undergraduate students of level two at the Faculty of Engineering, International Islamic University Malaysia. The Numerical Method course has been delivered to two Sections 1 and 2 with 44 and 22 students in each section, respectively. Systematic steps have been followed to apply the student centered learning approach in teaching Numerical Method course. Initially, the instructor has chosen the topic which was Euler’s Method to solve Ordinary Differential Equations (ODE) to be learned. The students were then divided into groups with five members in each group. Initial instructions have been given to the group members to prepare their subtopics before meeting members from other groups to discuss the subtopics in an expert group inside the classroom. For the time assigned for the classroom discussion, the setting of the classroom was rearranged to accommodate the student centered learning approach. Teacher strength was by monitoring the process of learning inside and outside the class. The students have been assessed during the migrating to the expert groups, recording of a video explanation outside the classroom and during the final examination. Euler’s Method to solve the ODE was set as part of Question 3(b) in the final exam. It is observed that none of the students from both sections obtained a zero grade in Q3(b), compared to Q3(a) and Q3(c). Also, for Section 1(44 students), 29 students obtained the full mark of 7/7, while only 10 obtained 7/7 for Q3(a) and no students obtained 6/6 for Q3(c). Finally, we can recommend that the Numerical Method course be moved toward more student-centered Learning classrooms where the students will be engaged in group discussion rather than having a teacher one man show.

Keywords: Teacher centered learning, student centered learning, mathematic, numerical methods.

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Authors: A. Selmi

Abstract:

Free vibration analysis of homogenous and axially functionally graded simply supported beams within the context of Euler-Bernoulli beam theory is presented in this paper. The material properties of the beams are assumed to obey the linear law distribution. The effective elastic modulus of the composite was predicted by using the rule of mixture. Here, the complexities which appear in solving differential equation of transverse vibration of composite beams which limit the analytical solution to some special cases are overcome using a relatively new approach called the Differential Transformation Method. This technique is applied for solving differential equation of transverse vibration of axially functionally graded beams. Natural frequencies and corresponding normalized mode shapes are calculated for different Young’s modulus ratios. MATLAB code is designed to solve the transformed differential equation of the beam. Comparison of the present results with the exact solutions proves the effectiveness, the accuracy, the simplicity, and computational stability of the differential transformation method. The effect of the Young’s modulus ratio on the normalized natural frequencies and mode shapes is found to be very important.

Keywords: Differential transformation method, functionally graded material, mode shape, natural frequency.

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Authors: Tayeb Bensattalah, Mohamed Zidour, Mohamed Ait Amar Meziane, Tahar Hassaine Daouadji, Abdelouahed Tounsi

Abstract:

In this paper, the Differential Transform Method (DTM) is employed to predict and to analysis the non-local critical buckling loads of carbon nanotubes with various end conditions and the non-local Timoshenko beam described by single differential equation. The equation differential of buckling of the nanobeams is derived via a non-local theory and the solution for non-local critical buckling loads is finding by the DTM. The DTM is introduced briefly. It can easily be applied to linear or nonlinear problems and it reduces the size of computational work. Influence of boundary conditions, the chirality of carbon nanotube and aspect ratio on non-local critical buckling loads are studied and discussed. Effects of nonlocal parameter, ratios L/d, the chirality of single-walled carbon nanotube, as well as the boundary conditions on buckling of CNT are investigated.

Keywords: Boundary conditions, buckling, non-local, the differential transform method.

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Authors: M. A. Sohaly, M. A. Elfouly

Abstract:

Parkinson's disease (PD) is a heterogeneous movement disorder that often appears in the elderly. PD is induced by a loss of dopamine secretion. Some drugs increase the secretion of dopamine. In this paper, we will simply study the stability of PD models as a nonlinear delay differential equation. After a period of taking drugs, these act as positive feedback and increase the tremors of patients, and then, the differential equation has positive coefficients and the system is unstable under these conditions. We will present a set of suggested modifications to make the system more compatible with the biodynamic system. When giving a set of numerical examples, this research paper is concerned with the mathematical analysis, and no clinical data have been used.

Keywords: Parkinson's disease, stability, simulation, two delay differential equation.

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Authors: Mohammad Taghi Darvishi, Maliheh Najafi, Mohammad Najafi

Abstract:

In this work, we obtain some analytic solutions for the (3+1)-dimensional breaking soliton after obtaining its Hirota-s bilinear form. Our calculations show that, three-wave method is very easy and straightforward to solve nonlinear partial differential equations.

Keywords: (3+1)-dimensional breaking soliton equation, Hirota'sbilinear form.

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Authors: jianhua Hou, Changqing Yang, and Beibo Qin

Abstract:

A numerical method for solving nonlinear Fredholm integral equations of second kind is proposed. The Fredholm type equations which have many applications in mathematical physics are then considered. The method is based on hybrid function  approximations. The properties of hybrid of block-pulse functions and Chebyshev polynomials are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of nonlinear. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method.

Keywords: Hybrid functions, Fredholm integral equation, Blockpulse, Chebyshev polynomials, product operational matrix.

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Authors: Kayvan Ahmadi, Hossein Shamsi

Abstract:

This paper proposes a low-voltage and low-power fully integrated digitally tuned continuous-time channel selection filter for WiMAX applications. A 5th-order elliptic low-pass filter is realized in a Gm-C topology. The bandwidth of the fully differential filter is reconfigurable from 2.5MHz to 20MHz (8x) for different requirements in WiMAX applications. The filter is simulated in a standard 90nm CMOS process. Simulation results show the THD (@Vout =100mVpp) is less than -66dB. The in-band ripple of the filter is about 0.15dB. The filter consumes 1.5mW from a supply voltage of 0.9V.

Keywords: Common-mode feedback, continuous-time, fully differential transconductor, Gm-C topology, low-voltage

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Authors: A.R.M. Kasim, N.F. Mohammad, Aurangzaib, S. Sharidan

Abstract:

The present paper considers the steady free convection boundary layer flow of a viscoelastic fluid on solid sphere with Newtonian heating. The boundary layer equations are an order higher than those for the Newtonian (viscous) fluid and the adherence boundary conditions are insufficient to determine the solution of these equations completely. Thus, the augmentation an extra boundary condition is needed to perform the numerical computational. The governing boundary layer equations are first transformed into non-dimensional form by using special dimensionless group and then solved by using an implicit finite difference scheme. The results are displayed graphically to illustrate the influence of viscoelastic K and Prandtl Number Pr parameters on skin friction, heat transfer, velocity profiles and temperature profiles. Present results are compared with the published papers and are found to concur very well.

Keywords: boundary layer flow, Newtonian heating, sphere, viscoelastic fluid.

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Authors: Tudor Barbu

Abstract:

We present a robust nonlinear parabolic partial differential equation (PDE)-based denoising scheme in this article. Our approach is based on a second-order anisotropic diffusion model that is described first. Then, a consistent and explicit numerical approximation algorithm is constructed for this continuous model by using the finite-difference method. Finally, our restoration experiments and method comparison, which prove the effectiveness of this proposed technique, are discussed in this paper.

Keywords: Image denoising and restoration, nonlinear PDE model, anisotropic diffusion, numerical approximation scheme, finite differences.

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Authors: Vikas Kumar

Abstract:

The present study is carried out to investigate the magneto-viscous effects on incompressible ferrofluid flow over a porous rotating disc with suction or injection on the surface of the disc subjected to a magnetic field. The flow under consideration is axi-symmetric steady ferrofluid flow of electrically non-conducting fluid. Karman’s transformation is used to convert the governing boundary layer equations involved in the problem to a system of non linear coupled differential equations. The solution of this system is obtained by using power series approximation. The flow characteristics i.e. radial, tangential, axial velocities and boundary layer displacement thickness are calculated for various values of MFD (magnetic field dependent) viscosity and for different values of suction injection parameter. Besides this, skin friction coefficients are also calculated on the surface of the disk. The results thus obtained are presented numerically and graphically in the paper.

Keywords: Axi-symmetric, ferrofluid, magnetic field, porous rotating disk.

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Authors: Sameerah Jamal, Abdul Hamid Kara, Ashfaque H. Bokhari

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Equations on curved manifolds display interesting properties in a number of ways. In particular, the symmetries and, therefore, the conservation laws reduce depending on how curved the manifold is. Of particular interest are the wave and Gordon-type equations; we study the symmetry properties and conservation laws of these equations on the Milne and Bianchi type III metrics. Properties of reduction procedures via symmetries, variational structures and conservation laws are more involved than on the well known flat (Minkowski) manifold.

Keywords: Bianchi metric, conservation laws, Milne metric, symmetries.

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Authors: Sachin Kumar, Y. K. Gupta, J. R. Sharma

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In the present article, a new class of solutions of Einstein field equations is investigated for a spherically symmetric space-time when the source of gravitation is a perfect fluid. All the solutions have been derived by making some suitable arrangements in the field equations. The solutions so obtained have been seen to describe Schwarzschild interior solutions. Most of the solutions are subjected to the reality conditions. As far as the authors are aware the solutions are new.

Keywords: Einstein's equations, General Relativity, PerfectFluid, Spherical symmetric.

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Authors: Shoeib Mahjoub, Ali Mahtabroshan

Abstract:

The steady incompressible flow has been solved in cylindrical coordinates in both vapour region and wick structure. The governing equations in vapour region are continuity, Navier-Stokes and energy equations. These equations have been solved using SIMPLE algorithm. For study of parameters variation on heat pipe operation, a benchmark has been chosen and the effect of changing one parameter has been analyzed when the others have been fixed.

Keywords: Vapour region, conventional heat pipe, numerical simulation.

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Authors: M. A. Talha, M. Osman Gani, M. Ferdows

Abstract:

This paper focuses on the study of two dimensional magnetohydrodynamic (MHD) steady incompressible viscous Williamson nanofluid with exponential internal heat generation containing gyrotactic microorganism over a stretching sheet. The governing equations and auxiliary conditions are reduced to a set of non-linear coupled differential equations with the appropriate boundary conditions using similarity transformation. The transformed equations are solved numerically through spectral relaxation method. The influences of various parameters such as Williamson parameter γ, power constant λ, Prandtl number P_r, magnetic field parameter M, Peclet number P_e, Lewis number Le, Bioconvection Lewis number Lb, Brownian motion parameter Nb, thermophoresis parameter Nt, and bioconvection constant σ are studied to obtain the momentum, heat, mass and microorganism distributions. Moment, heat, mass and gyrotactic microorganism profiles are explored through graphs and tables. We computed the heat transfer rate, mass flux rate and the density number of the motile microorganism near the surface. Our numerical results are in better agreement in comparison with existing calculations. The Residual error of our obtained solutions is determined in order to see the convergence rate against iteration. Faster convergence is achieved when internal heat generation is absent. The effect of magnetic parameter M decreases the momentum boundary layer thickness but increases the thermal boundary layer thickness. It is apparent that bioconvection Lewis number and bioconvection parameter has a pronounced effect on microorganism boundary. Increasing brownian motion parameter and Lewis number decreases the thermal boundary layer. Furthermore, magnetic field parameter and thermophoresis parameter has an induced effect on concentration profiles.

Keywords: Convection flow, internal heat generation, similarity, spectral method, numerical analysis, Williamson nanofluid.

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Authors: N. Kumaresan, J. Kavikumar, M. Kumudthaa, Kuru Ratnavelu

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In this paper, solution of fuzzy differential equation under general differentiability is obtained by genetic programming (GP). The obtained solution in this method is equivalent or very close to the exact solution of the problem. Accuracy of the solution to this problem is qualitatively better. An illustrative numerical example is presented for the proposed method.

Keywords: Fuzzy differential equation, Generalized differentiability, Genetic programming and H-difference.

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Authors: Changhong Guo, Shaomei Fang, Yong He

Abstract:

In this paper, fractional Black-Scholes models for the European option pricing were established based on the fractional G-Brownian motion (fGBm), which generalizes the concepts of the classical Brownian motion, fractional Brownian motion and the G-Brownian motion, and that can be used to be a tool for considering the long range dependence and uncertain volatility for the financial markets simultaneously. A generalized fractional Black-Scholes equation (FBSE) was derived by using the Taylor’s series of fractional order and the theory of absence of arbitrage. Finally, some explicit option pricing formulas for the European call option and put option under the FBSE were also solved, which extended the classical option pricing formulas given by F. Black and M. Scholes.

Keywords: European option pricing, fractional Black-Scholes equations, fractional G-Brownian motion, Taylor’s series of fractional order, uncertain volatility.

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Authors: Hamid A. Jalab, Rabha W. Ibrahim

Abstract:

This paper introduces an image denoising algorithm based on generalized Srivastava-Owa fractional differential operator for removing Gaussian noise in digital images. The structures of nxn fractional masks are constructed by this algorithm. Experiments show that, the capability of the denoising algorithm by fractional differential-based approach appears efficient to smooth the Gaussian noisy images for different noisy levels. The denoising performance is measured by using peak signal to noise ratio (PSNR) for the denoising images. The results showed an improved performance (higher PSNR values) when compared with standard Gaussian smoothing filter.

Keywords: Fractional calculus, fractional differential operator, fractional mask, fractional filter.

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