Application of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate
|
|
- Diane Warren
- 6 years ago
- Views:
Transcription
1 Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite plate Mohame M. Mousa,, Aiarkhan Kaltayev Department of Basic Science, Benha High Institute of Technology, Benha University, 5, Egypt, r.eng.mmmm@gmail.com Department of Mechanics, al-farabi Kazakh National University, 9/47 Masanchi 5, Almaty, Kazakhstan Abstract The equations governing the flow of an electrically conucting, incompressible viscous flui over an infinite flat plate in the presence of a magnetic fiel are investigate using the homotopy perturbation metho (HPM) with Paé approximants (PA) an 4 th orer Runge Kutta metho (4RKM). Approximate analytical an numerical solutions for the velocity fiel an heat transfer are obtaine an compare with each other, showing excellent agreement. The effects of the magnetic parameter an Prantl number on velocity fiel, shear stress, temperature an heat transfer are iscusse as well. Keywors: Electrically conucting elastico-viscous flui; symmetry solution; Homotopy perturbation metho; Paé approximation; 4 th orer Runge Kutta; Maple.. Introuction The bounary layer flow of an electrically conucting, incompressible viscous flui over a continuously flat plate is often encountere in many engineering an inustrial processes such as polymer technology, aeroynamic extrusion of plastic sheets an so on. The problem of a fluctuating flow of a magneto-elastico-viscous flui along an infinite flat plate uner the conition of very small elastic parameter was stuie in [ ]. This type of problems may be approximate to a problem of fluctuating flow of a magneto-viscous flui in case of consieration a very small elastic parameter. Frater [] pointe out that the solution for the velocity shoul ten to the Newtonian value when the elastic parameter vanishes. In this paper the flow of an electrically conucting, incompressible elastico-viscous flui along a flat plate coinciing with the plane y= is consiere, such that the flow is confine to the region y>. The magnetic fiel is assume to be normal to the plate on which the bounary layer is forme. The main purpose of this work is to investigate the effects of the magnetic fiel parameter an Prantl number on the velocity an shear stress of the flui analytically using the classical homotopy perturbation metho (HPM) with the enhancement of Paé approximants (PA) an using the evelope HPM as well; an numerically using the well-known 4 th orer Runge Kutta metho (4RKM). The classical homotopy perturbation metho, base on series approximation, is one among the newly evelope analytical methos for strongly nonlinear problems an has been proven successful in solving a wie class of nonlinear ifferential equations [5 9]. The evelope HPM can be achieve by introucing aition linear operator(s) with unknown parameter(s) that can be chosen suitably to fulfill certain esirable criteria an ientifie optimally [ ]. In this paper, we are intereste in applying the classical HPM with PA technique, evelope HPM an 4RKM for obtaining analytical an numerical solutions of the bounary layer flow of an electrically conucting elastico-viscous flui along an infinite flat plat with heat transfer
2 4 M.M. Mousa & A. Kaltayev: Homotopy perturbation metho to magneto-elastico-viscous flui along semi-infinite plate in presence of a magnetic fiel normal to the plate. The comparison of the analytical solutions with the numerical solution has been mae an excellent agreement note.. Governing equations In terms of the stream function ψ the governing equations of a steay twoimensional incompressible flow of an electrically conucting elastico-viscous flui over a semi-infinite flat plate coinciing with the plane y=, such that the flow is confine to the region y> uner the influence of a constant transverse applie magnetic fiel normal to the plate on which the bounary layer is forme are given in [, ]. The magnetic Reynols number is assume to be small an negligible in comparison to the applie magnetic file. The governing equations escribe flui motion an temperature are given by ψ M ψ ψ ψ ψ ψ + x y x x k ψ ψ ψ ψ ψ ψ ψ ψ = + xy x x xy () 4 4, 4 T ψ T ψ T =, () Pr x x where M is the magnetic parameter, k is a small elastic parameter representing the non- Newtonian character of the flui an Pr is the Prantl number. The bounary conitions of the problem are: ψ ψ y = : =, =, T = T, () x ψ y : ψ, T, (4) where T an ψ are constants. Because the elastic parameter k is small an may be neglecte, the solution of the problem escribe by Eqs. () an () may be approximate to the solution of the Newtonian flui escribe by the following equations: ψ M ψ ψ ψ ψ ψ x x x + =, T ψ T ψ T + Pr, = x x (5) () uner the same bounary conitions. This approximation gives excellent results in case of small values of k as we will see in the next sections.. Invariant transformation Using one-parametric group transformation inclue in PDEtools package of Maple software, the two-inepenent variables PDEs (5) an () will be transforme into ODEs in only one-inepenent similarity variable... The complete set of invariants The invariants set obtaine by Maple are: gx ( ) ψ( xy, ) y η( xy, ) = x, F( η ) =, θη ( ) = Txy (, ), gx ( ) ( xy, ) = x, F( η) ψ ( xy, ), g( xt ) ( xy, ) θ ( η) = gx ( ) ( xy, ) = x, F( η) = ψ( xy, ), η = y, (7) (8) η y (9) θ ( η) = Txy (, ) exp, g( x) y gx ( ) ( xy, ) η( xy, ) = x, F( η) = ψ, x x x () θη= T xy,, ( ) ( ) where η is the similarity variable, F an θ are invariants of the epenent variables ψ an T respectively an g is an arbitrary function. From the invariants set (7) (), it is clear that the invariants in Eq. () are the only ones which make both of ψ an T a function in x an y.
3 ISSN: 55-9 International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 5 Therefore, we use Eq. () for oing the similarity transformation of PDEs (5) an ()... The orinary ifferential equations invariant transformation Substituting Eq. () into PDEs (5) an () yiels the following system of ODEs: F( η) + F( η) F( η) M F( η ) =, () θ( η) + Pr F ( η) θ( η) =. () By examining invariants in Eq. () an bounary conitions () an (4), function g (x) shoul be equal to zero in orer to make the left bounary point constant at y=. Therefore, the suitable similarity invariants of this problem are: η ( xy, ) = c, F( η) ( xy, ) y ψ + =, θη ( ) = T( x, y), x x () where c is an arbitrary constant (left bounary point of the similarity bounary problem). Hence, the appropriate corresponing conitions are: ( η ) F η = c: =, F ( η) =, θ( η) = T, (4) η ( ) F η η η :, θ( η). η ψ (5) It is obvious that Eq.() is the Blasius equation in the case of M = [4]. For convenience an comparison with results in [], let ψ = T =, c= an η =. 4. Analytical solution using the classical HPM with PA technique Following the stanar proceures of the HPM escribe in [5 9], the system () an () shoul be written in the classical homotopy form, F ( p) U( η ) + ( ) ( ) p U η U η U( η ) M U( ), η η η + = η () ( p) V( η ) θ + p V + Pr U V = where ( η ) ( η ) ( η ), ( ) ( ) ( ) θ( ) U F, V θ, F = U c = F c an θ = V c = c. (7) One can now try to obtain a solution of system () an () in the form of, ( η) ( η) ( η) ( η) U = U + pu + p U +..., (8) ( η) ( μ) ( η) ( η) V = V + pv + p V +..., (9) where U n an V n, n=,,, are functions yet to be etermine. Substituting Eqs. (8) an (9) into system () an (7), an arranging the coefficients of "p" powers yiels: p : U =, V =, p : U + U U MU =, V + PrU V =, p : U MU + UU + UU =, V + PrUV + PrUV =, () with corresponing initial conitions, ( ) ( ) ( ) V ( ) =, V ( ) = β, U =, U =, U = α, Un = U n = U n = Vn = V n =, at η =, () for n =,,,..., where unknown initial values α an β can be calculate using the bounary conitions in Eq. (5) after obtaining a close form expression to the solution.
4 M.M. Mousa & A. Kaltayev: Homotopy perturbation metho to magneto-elastico-viscous flui along semi-infinite plate We continue solving system () corresponing to initial conitions () for U n an V n, n=,,, until n= an hence obtaine a six-term approximation: F ( η ) = U an θ ( η) n= n = V. n= It is known that Paé approximations (PA) [] have the avantage of manipulating the polynomial approximation into a rational function of polynomials. This manipulation provies us with more information about the mathematical behavior of the solution. Besies that, a power series solution is not useful for large value of η. Therefore, the combination of the series solution through HBM or any other series solution metho with the Paé approximation provies an effective tool for hanling bounary value problems on semi-infinite omains. It is a known fact that Paé approximation converges on the entire real axis if the solution is free of singularities on the real axis. So, the more accurate analytical solutions will be obtaine after application of PA [M/N] to both F an θ such that M+N (highest power of η in the series solution). We have applie PA [/] to obtain the analytical solution for the problem, say F [/] an θ [/]. n 5. Analytical solution using evelope HPM Accoring to the evelope HPM [ ], a homotopy of the system () an () may be written as ( ) F η + a + () p F( η) F( η) M F( η) a =, θη ( ) b p ( ) ( ) P F η θη b + + r =, () where a an b are unknown constants to be further ientifie. Using p as an expaning parameter as that in the classic perturbation metho, we have : + FF MF =, ( ) = F ( ) = F ( ) =, ( ) ( ) ( ) ( ), θ ( ), p : F + a=, F = F =, F =, θ + b =, θ = = p F a F θ + Pr Fθ b =, θ( ) = θ( ) =. (4) Solving the system (4) an setting p =, we obtain a first-orer approximate solution which reas F F F M M Ma Ma + a + + a η + a + a η a η, θ ( η) = θ( η) + θ( η) = + bapr+ Pra + Prb + Pr η 8 4 Prbaη + Pr+ Pr a Prb bapr η ( η ) = ( η) + ( η) = + a a Ma η + η (5) + ba a + b η Pr Pr Pr. () There are many approaches for ientification of the unknown parameters in the obtaine solution. One of those methos is weighte resiuals, especially the least squares metho [ ]. For the present problem, we set RF RF =, an Rθ Rθ =, (7) a b to ientify the unknown constants a an b,where R F an R θ are the resiuals RF = F + FF MF, an R θ = θ + Pr F θ. (8)
5 ISSN: 55-9 International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 7. Results an iscussion With the analytical solution given by F [/] an θ [/] using the classical HPM with PA technique, approximate values of α= F () an β=θ () can be calculate using the conitions in Eq. (5). Some numerical results of α an β that are obtaine from F [/] (η )=ψ an θ [/] (η )= are presente in Table for ifferent values of M an Pr when η = an ψ =. M α Table Numerical values of α= F () an β=θ ()for ifferent values of M an Pr β Pr=.5 Pr=.7 Pr=. Pr=. Pr= (a) (b) (c) Fig. : Profiles of (a) stream function; (b) velocity an (c) shear stress using analytical results of F [/] ; F [/] an F [/] respectively an numerical results of 4RKM for various values of M at Pr=.7 (a) (b)
6 8 M.M. Mousa & A. Kaltayev: Homotopy perturbation metho to magneto-elastico-viscous flui along semi-infinite plate Fig. : Profiles of (a) temperature an (b) heat transfer using analytical results of θ [/] an θ [/] respectively an numerical results of 4RKM for various values of M at Pr=.7 (a) (b) Fig. : Profiles of (a) temperature an (b) heat transfer using analytical results of θ [/] an θ [/] respectively an numerical results of 4RKM for various values of Pr at M= (a) (b) (c) () Fig. 4: Profiles of (a) stream function; (b) velocity; (c) temperature an () heat transfer using evelope HPM analytical results an 4RKM numerical results for M= an M=.5 at Pr=.7
7 ISSN: 55-9 International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 9 With the first-orer approximate solution arising in Eqs. (5) an (), approximate values of unknown constants a an b are optimally ientifie using Eqs. (7) an (8) an presente in Table for M= an M=.5 at Pr=.7. Table. Numerical values of a an b for M= an M=.5 at Pr=.7 M a b In orer to obtain a numerical solution, we have solve the initial value problem of Eqs. () an () corresponing to conitions in Eq. (4) an the numerical values arise in Table using the well-known 4RKM. Figs. (a), (b) an (c) show the variations of the flui stream function, velocity an shear stress with η. As shown in Figs. (a) an (b), the stream function F an flui velocity F ecrease an come near to each other as the magnetic parameter M increases. In aition, Fig. (b) shows that the smaller the value of M, the faster it reaches the maximum value of F. From Fig. (c), it is clear that the behavior of the shear stress F epens on the magnetic parameter an the istance. In case of M=, the shear stress starts with the high value, an then ecreases with increasing istance. Oppositely for M>, the shear stress starts with a lower value, an then increases with the istance. Figs. (a) an (b) show the variations of the temperature an heat transfer with η. As shown in Fig. (a), the temperature θ increases with the increasing of M. For M=, the temperature almost linearly epens on η. From Fig. (b), it is clear that the heat transfer θ starts with a higher value for the lower values of M an then ecreases. In aition, for the higher values of M, the behavior of the heat transfer with η tens to be uniform an takes a horizontal shape. Figs. (a) an () illustrate the effect of Prantl number Pr on the temperature an heat transfer at M=. The results are obtaine for Pr =.5,, an. Form Fig., it clear that the temperature an heat transfer rapi ecrease as the Prantl number increases. Moreover, the rapi ecrease of θ an θ becomes more obvious for larger values of Pr. To emonstrate the acceptability an accuracy of evelope HPM results, even though we use only the first-orer approximate solution, the behaviors of the flui stream function, velocity, temperature an heat transfer using the close form solutions in Eqs. () an (), with the values in Table, are illustrate in Figs. 4(a), (b), (c) an () in a comparison with 4RKM results. It is obvious that the results of α an β obtaine by the classical HPM with PA technique are use for obtaining the numerical solution using 4RKM by converting the bounary value problem to an initial value one. Moreover, the analytical solutions using the classical HPM with PA technique an evelope HPM in great agree with the numerical solution using the 4 th orer Runge Kutta metho. The results obtaine in this investigation, in case of the elastic parameter k=, agree with that obtaine in [] in case of k=.. Hence, the problem of fluctuating flow of a magneto-elastico-viscous flui over a semi-infinite flat plate uner the conition of a very small elastic parameter k can be approximate to the problem of fluctuating flow of a magneto-viscous flui, i.e. k=. The present results of F for M= agree with that obtaine in [4] as well. 7. Conclusions The homotopy perturbation metho is applie to the system of nonlinear ifferential equations that escribe a magneto-viscous flui along a semi-infinite flat plate in presence of a magnetic fiel. The excellent agreement of the analytical solution with the 4RKM numerical one shows the reliability an efficiency of the HPM. The behaviors of flui stream function, velocity, shear stress, temperature an heat transfer illustrate by the graphs are consistent with the graphs obtaine in [, 4] an therefore further establish the reliability an effective-ness of the HPM. It has been emonstrate that the HPM can be applie avantageously even when
8 M.M. Mousa & A. Kaltayev: Homotopy perturbation metho to magneto-elastico-viscous flui along semi-infinite plate the flow is governe by a BVP consisting of more than one ifferential equation. References [] V.M. Sounalgekar, P. Puri, On fluctuating flow of an elastico-viscous flui past an infinite plate with variable suction, J. Flui Mech., 5 (99) [] K.R. Frater, On the solution of some bounary-value problems arising in elastic-viscous flui mechanics, Z. Angew. Math. Phys., (97) 4 7. [] M.M. Helal, M.B. Ab-el-Malek, Group metho analysis of magneto-elastico-viscous flow along a semi-infinite flat plate with heat transfer, J. Comput. Appl. Math., 7 (5) 99 [4] T. Cebeci, P. Brashaw, Momentum Transfer in Bounary Layers, Hemisphere Publishing Corporation, New York, 977. [5] J.H. He, Homotopy perturbation technique, Comput. Methos Appl. Mech. Eng., 78 (999) 57. [] J.H. He, Homotopy perturbation metho: a new nonlinear analytical technique, Appl. Math. Comput., 5 () [7] J.H. He, Application of homotopy perturbation metho to nonlinear wave equations, Chaos Solitons an Fractals (5) [8] J.H. He, Some asymptotic methos for strongly nonlinear equations, Int. J. Mo. Phys. B, () () [9] M.M. Mousa, S.F. Ragab, Application of the homotopy perturbation metho to linear an nonlinear schröinger equations, Z.Naturforsch., a (8) [] J.H. He, An Elementary introuction to recently evelope asymptotic methos an nanomechanics in textile engineering, Int. J. Mo. Phys. B, () (8) [] J.H. He, Recent evelopment of the homotopy perturbation metho, Topological Methos in Nonlinear Analysis, (8) 5 9. [] J.H. He, An elementary introuction to the homotopy perturbation metho, Computers & Mathematics with Applications, 57 (9) 4 4. [] G.A. Baker, Essentials of Paé Approximants, Acaemic press, New York, 975.
Analytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces
Cent. Eur. J. Eng. 4(4) 014 341-351 DOI: 10.478/s13531-013-0176-8 Central European Journal of Engineering Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive
More informationPhysics 505 Electricity and Magnetism Fall 2003 Prof. G. Raithel. Problem Set 3. 2 (x x ) 2 + (y y ) 2 + (z + z ) 2
Physics 505 Electricity an Magnetism Fall 003 Prof. G. Raithel Problem Set 3 Problem.7 5 Points a): Green s function: Using cartesian coorinates x = (x, y, z), it is G(x, x ) = 1 (x x ) + (y y ) + (z z
More informationHomotopy Perturbation Method for Solving Twelfth Order Boundary Value Problems
International Journal of Research an Reviews in Applie Sciences ISSN: 276-734X, EISSN: 276-7366 Volume 1, Issue 2(November 29) Homotopy Perturbation Metho for Solving Twelfth Orer Bounary Value Problems
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationDissipative numerical methods for the Hunter-Saxton equation
Dissipative numerical methos for the Hunter-Saton equation Yan Xu an Chi-Wang Shu Abstract In this paper, we present further evelopment of the local iscontinuous Galerkin (LDG) metho esigne in [] an a
More informationThermal Modulation of Rayleigh-Benard Convection
Thermal Moulation of Rayleigh-Benar Convection B. S. Bhaauria Department of Mathematics an Statistics, Jai Narain Vyas University, Johpur, Inia-3400 Reprint requests to Dr. B. S.; E-mail: bsbhaauria@reiffmail.com
More informationEXACT TRAVELING WAVE SOLUTIONS FOR A NEW NON-LINEAR HEAT TRANSFER EQUATION
THERMAL SCIENCE, Year 017, Vol. 1, No. 4, pp. 1833-1838 1833 EXACT TRAVELING WAVE SOLUTIONS FOR A NEW NON-LINEAR HEAT TRANSFER EQUATION by Feng GAO a,b, Xiao-Jun YANG a,b,* c, an Yu-Feng ZHANG a School
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationChapter 2 Governing Equations
Chapter 2 Governing Equations In the present an the subsequent chapters, we shall, either irectly or inirectly, be concerne with the bounary-layer flow of an incompressible viscous flui without any involvement
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationStability of Stratified Couple-Stress Dusty Fluid in the Presence of Magnetic Field through Porous Medium
vailable at http://pvamu.eu/aam ppl. ppl. Math. ISSN: 93-9466 Vol. 6, Issue (December ), pp. 5 5 pplications pplie Mathematics: n International Journal (M) Stability of Stratifie Couple-Stress Dusty Flui
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More information6. Friction and viscosity in gasses
IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner
More informationMATH , 06 Differential Equations Section 03: MWF 1:00pm-1:50pm McLaury 306 Section 06: MWF 3:00pm-3:50pm EEP 208
MATH 321-03, 06 Differential Equations Section 03: MWF 1:00pm-1:50pm McLaury 306 Section 06: MWF 3:00pm-3:50pm EEP 208 Instructor: Brent Deschamp Email: brent.eschamp@ssmt.eu Office: McLaury 316B Phone:
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationMomentum and Energy. Chapter Conservation Principles
Chapter 2 Momentum an Energy In this chapter we present some funamental results of continuum mechanics. The formulation is base on the principles of conservation of mass, momentum, angular momentum, an
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationStable and compact finite difference schemes
Center for Turbulence Research Annual Research Briefs 2006 2 Stable an compact finite ifference schemes By K. Mattsson, M. Svär AND M. Shoeybi. Motivation an objectives Compact secon erivatives have long
More informationANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS
ANALYSIS OF A GENERAL FAMILY OF REGULARIZED NAVIER-STOKES AND MHD MODELS MICHAEL HOLST, EVELYN LUNASIN, AND GANTUMUR TSOGTGEREL ABSTRACT. We consier a general family of regularize Navier-Stokes an Magnetohyroynamics
More informationChapter 9 Method of Weighted Residuals
Chapter 9 Metho of Weighte Resiuals 9- Introuction Metho of Weighte Resiuals (MWR) is an approimate technique for solving bounary value problems. It utilizes a trial functions satisfying the prescribe
More informationComputing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions
Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5
More informationLagrangian and Hamiltonian Dynamics
Lagrangian an Hamiltonian Dynamics Volker Perlick (Lancaster University) Lecture 1 The Passage from Newtonian to Lagrangian Dynamics (Cockcroft Institute, 22 February 2010) Subjects covere Lecture 2: Discussion
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationConservation laws a simple application to the telegraph equation
J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationThe effect of nonvertical shear on turbulence in a stably stratified medium
The effect of nonvertical shear on turbulence in a stably stratifie meium Frank G. Jacobitz an Sutanu Sarkar Citation: Physics of Fluis (1994-present) 10, 1158 (1998); oi: 10.1063/1.869640 View online:
More informationApplication of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate
Physics Letters A 37 007) 33 38 www.elsevier.com/locate/pla Application of He s homotopy perturbation method to boundary layer flow and convection heat transfer over a flat plate M. Esmaeilpour, D.D. Ganji
More informationDusty Plasma Void Dynamics in Unmoving and Moving Flows
7 TH EUROPEAN CONFERENCE FOR AERONAUTICS AND SPACE SCIENCES (EUCASS) Dusty Plasma Voi Dynamics in Unmoving an Moving Flows O.V. Kravchenko*, O.A. Azarova**, an T.A. Lapushkina*** *Scientific an Technological
More informationModeling time-varying storage components in PSpice
Moeling time-varying storage components in PSpice Dalibor Biolek, Zenek Kolka, Viera Biolkova Dept. of EE, FMT, University of Defence Brno, Czech Republic Dept. of Microelectronics/Raioelectronics, FEEC,
More informationA Short Note on Self-Similar Solution to Unconfined Flow in an Aquifer with Accretion
Open Journal o Flui Dynamics, 5, 5, 5-57 Publishe Online March 5 in SciRes. http://www.scirp.org/journal/oj http://x.oi.org/.46/oj.5.57 A Short Note on Sel-Similar Solution to Unconine Flow in an Aquier
More informationinflow outflow Part I. Regular tasks for MAE598/494 Task 1
MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the
More informationA Universal Model for Bingham Fluids with Two Characteristic Yield Stresses
A Universal Moel for Bingham Fluis with Two Characteristic Yiel Stresses NGDomostroeva 1 an NNTrunov DIMeneleyev Institute for Metrology Russia, StPetersburg 195 Moskovsky pr 19 February, 4, 9 Abstract:
More informationarxiv: v1 [math-ph] 5 May 2014
DIFFERENTIAL-ALGEBRAIC SOLUTIONS OF THE HEAT EQUATION VICTOR M. BUCHSTABER, ELENA YU. NETAY arxiv:1405.0926v1 [math-ph] 5 May 2014 Abstract. In this work we introuce the notion of ifferential-algebraic
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationCompletely passive natural convection
Early View publication on wileyonlinelibrary.com (issue an page numbers not yet assigne; citable using Digital Object Ientifier DOI) ZAMM Z. Angew. Math. Mech., 1 6 (2011) / DOI 10.1002/zamm.201000030
More informationA Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationEffect of Variable Viscosity on Hydro Magnetic Flow and Heat Transfer Over a Stretching Surface with Variable Temperature
37 Effect of Variable Viscosity on Hydro Magnetic Flow and Heat Transfer Over a Stretching Surface with Variable Temperature M. Y. Akl Department of Basic Science, Faculty of Engineering (Shopra Branch),
More informationPolynomial Inclusion Functions
Polynomial Inclusion Functions E. e Weert, E. van Kampen, Q. P. Chu, an J. A. Muler Delft University of Technology, Faculty of Aerospace Engineering, Control an Simulation Division E.eWeert@TUDelft.nl
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationHyperbolic Moment Equations Using Quadrature-Based Projection Methods
Hyperbolic Moment Equations Using Quarature-Base Projection Methos J. Koellermeier an M. Torrilhon Department of Mathematics, RWTH Aachen University, Aachen, Germany Abstract. Kinetic equations like the
More informationAnalytic Scaling Formulas for Crossed Laser Acceleration in Vacuum
October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945
More informationConvective heat transfer
CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case,
More informationTHE ACCURATE ELEMENT METHOD: A NEW PARADIGM FOR NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS
THE PUBISHING HOUSE PROCEEDINGS O THE ROMANIAN ACADEMY, Series A, O THE ROMANIAN ACADEMY Volume, Number /, pp. 6 THE ACCURATE EEMENT METHOD: A NEW PARADIGM OR NUMERICA SOUTION O ORDINARY DIERENTIA EQUATIONS
More informationPlanar sheath and presheath
5/11/1 Flui-Poisson System Planar sheath an presheath 1 Planar sheath an presheath A plasma between plane parallel walls evelops a positive potential which equalizes the rate of loss of electrons an ions.
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationAn Application of Homotopy Analysis Method for Estimation the Diaphragm Deflection in MEMS Capacitive Microphone
ISSN 1749-3889 (print, 1749-3897 (online International Journal of Nonlinear Science Vol.17(2014 No.1,pp.3-13 An Application of Homotopy Analysis Metho for Estimation the Diaphragm Deflection in MEMS Capacitive
More informationA Novel Decoupled Iterative Method for Deep-Submicron MOSFET RF Circuit Simulation
A Novel ecouple Iterative Metho for eep-submicron MOSFET RF Circuit Simulation CHUAN-SHENG WANG an YIMING LI epartment of Mathematics, National Tsing Hua University, National Nano evice Laboratories, an
More informationarxiv: v1 [physics.class-ph] 20 Dec 2017
arxiv:1712.07328v1 [physics.class-ph] 20 Dec 2017 Demystifying the constancy of the Ermakov-Lewis invariant for a time epenent oscillator T. Pamanabhan IUCAA, Post Bag 4, Ganeshkhin, Pune - 411 007, Inia.
More informationHe s Homotopy Perturbation Method for solving Linear and Non-Linear Fredholm Integro-Differential Equations
nternational Journal of Theoretical an Alie Mathematics 2017; 3(6): 174-181 htt://www.scienceublishinggrou.com/j/ijtam oi: 10.11648/j.ijtam.20170306.11 SSN: 2575-5072 (Print); SSN: 2575-5080 (Online) He
More informationChaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena
Chaos, Solitons an Fractals (7 64 73 Contents lists available at ScienceDirect Chaos, Solitons an Fractals onlinear Science, an onequilibrium an Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
More informationFluid Mechanics EBS 189a. Winter quarter, 4 units, CRN Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 Veihmeyer Hall.
Flui Mechanics EBS 189a. Winter quarter, 4 units, CRN 52984. Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 eihmeyer Hall. Course Description: xioms of flui mechanics, flui statics,
More informationEffect of Magnetic Field on Steady Boundary Layer Slip Flow Along With Heat and Mass Transfer over a Flat Porous Plate Embedded in a Porous Medium
Global Journal of Pure and Applied Mathematics. ISSN 973-768 Volume 3, Number 2 (27), pp. 647-66 Research India Publications http://www.ripublication.com Effect of Magnetic Field on Steady Boundary Layer
More information3-D FEM Modeling of fiber/matrix interface debonding in UD composites including surface effects
IOP Conference Series: Materials Science an Engineering 3-D FEM Moeling of fiber/matrix interface eboning in UD composites incluing surface effects To cite this article: A Pupurs an J Varna 2012 IOP Conf.
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationThe effect of dissipation on solutions of the complex KdV equation
Mathematics an Computers in Simulation 69 (25) 589 599 The effect of issipation on solutions of the complex KV equation Jiahong Wu a,, Juan-Ming Yuan a,b a Department of Mathematics, Oklahoma State University,
More informationarxiv: v1 [hep-ex] 4 Sep 2018 Simone Ragoni, for the ALICE Collaboration
Prouction of pions, kaons an protons in Xe Xe collisions at s =. ev arxiv:09.0v [hep-ex] Sep 0, for the ALICE Collaboration Università i Bologna an INFN (Bologna) E-mail: simone.ragoni@cern.ch In late
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationA STABILITY STUDY OF NON-NEWTONIAN FLUID FLOWS
U.P.B. Sci. Bull., Series A, Vol. 71, Iss. 4, 009 ISSN 13-707 A STABILITY STUDY OF NON-NEWTONIAN FLUID FLOWS Corina CIPU 1, Carmen PRICINĂ, Victor ŢIGOIU 3 Se stuiază problema e curgere a unui flui Olroy
More informationThermal conductivity of graded composites: Numerical simulations and an effective medium approximation
JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University
More information'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21
Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting
More informationTopological Sensitivity Analysis for Three-dimensional Linear Elasticity Problem
Topological Sensitivity Analysis for Three-imensional Linear Elasticity Problem A.A. Novotny, R.A. Feijóo, E. Taroco Laboratório Nacional e Computação Científica LNCC/MCT, Av. Getúlio Vargas 333, 25651-075
More informationProblems Governed by PDE. Shlomo Ta'asan. Carnegie Mellon University. and. Abstract
Pseuo-Time Methos for Constraine Optimization Problems Governe by PDE Shlomo Ta'asan Carnegie Mellon University an Institute for Computer Applications in Science an Engineering Abstract In this paper we
More informationFinal Exam Study Guide and Practice Problems Solutions
Final Exam Stuy Guie an Practice Problems Solutions Note: These problems are just some of the types of problems that might appear on the exam. However, to fully prepare for the exam, in aition to making
More informationINFLUENCE OF SURFACE ROUGHNESS THROUGH A SERIES OF FLOW FACTORS ON THE PERFORMANCE OF A LONGITUDINALLY ROUGH FINITE SLIDER BEARING
ANNALS of Faculty Engineering Huneoara International Journal of Engineering Tome XIV [2016] Fascicule 2 [May] ISSN: 1584-2665 [print; online] ISSN: 1584-2673 [CD-Rom; online] a free-accessmultiisciplinarypublication
More informationTHE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE
Journal of Soun an Vibration (1996) 191(3), 397 414 THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE E. M. WEINSTEIN Galaxy Scientific Corporation, 2500 English Creek
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationInfluence the Nozzle Shape on Local Heat Transfer in Impinging Jet
Issue 6, Volume 6, 12 Influence the Nozzle Shape on Local Heat Transfer in Impinging Jet M. Attalla an M. S. Ahme 1 Abstract The local Nusselt number istributions of circular nozzle on a heate flat plate
More informationEnergy behaviour of the Boris method for charged-particle dynamics
Version of 25 April 218 Energy behaviour of the Boris metho for charge-particle ynamics Ernst Hairer 1, Christian Lubich 2 Abstract The Boris algorithm is a wiely use numerical integrator for the motion
More informationSimilarity Flow Solution of MHD Boundary Layer Model for Non-Newtonian Power-Law Fluids over a Continuous Moving Surface
Gen. Math. Notes, Vol. 4, No., October 014, pp. 97-10 ISSN 19-7184; Copyright ICSRS Publication, 014 www.i-csrs.org Available free online at http://www.geman.in Similarity Flow Solution of MHD Boundary
More informationSolving the Schrödinger Equation for the 1 Electron Atom (Hydrogen-Like)
Stockton Univeristy Chemistry Program, School of Natural Sciences an Mathematics 101 Vera King Farris Dr, Galloway, NJ CHEM 340: Physical Chemistry II Solving the Schröinger Equation for the 1 Electron
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More informationSturm-Liouville Theory
LECTURE 5 Sturm-Liouville Theory In the three preceing lectures I emonstrate the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series are just the tip of the iceberg of the theory
More informationELECTRON DIFFRACTION
ELECTRON DIFFRACTION Electrons : wave or quanta? Measurement of wavelength an momentum of electrons. Introuction Electrons isplay both wave an particle properties. What is the relationship between the
More informationGeneralization of the persistent random walk to dimensions greater than 1
PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,
More informationarxiv:hep-th/ v1 3 Feb 1993
NBI-HE-9-89 PAR LPTHE 9-49 FTUAM 9-44 November 99 Matrix moel calculations beyon the spherical limit arxiv:hep-th/93004v 3 Feb 993 J. Ambjørn The Niels Bohr Institute Blegamsvej 7, DK-00 Copenhagen Ø,
More informationA simple model for the small-strain behaviour of soils
A simple moel for the small-strain behaviour of soils José Jorge Naer Department of Structural an Geotechnical ngineering, Polytechnic School, University of São Paulo 05508-900, São Paulo, Brazil, e-mail:
More informationarxiv:physics/ v2 [physics.ed-ph] 23 Sep 2003
Mass reistribution in variable mass systems Célia A. e Sousa an Vítor H. Rorigues Departamento e Física a Universiae e Coimbra, P-3004-516 Coimbra, Portugal arxiv:physics/0211075v2 [physics.e-ph] 23 Sep
More informationSwitching Time Optimization in Discretized Hybrid Dynamical Systems
Switching Time Optimization in Discretize Hybri Dynamical Systems Kathrin Flaßkamp, To Murphey, an Sina Ober-Blöbaum Abstract Switching time optimization (STO) arises in systems that have a finite set
More informationEvaporating droplets tracking by holographic high speed video in turbulent flow
Evaporating roplets tracking by holographic high spee vieo in turbulent flow Loïc Méès 1*, Thibaut Tronchin 1, Nathalie Grosjean 1, Jean-Louis Marié 1 an Corinne Fournier 1: Laboratoire e Mécanique es
More information6 Wave equation in spherical polar coordinates
6 Wave equation in spherical polar coorinates We now look at solving problems involving the Laplacian in spherical polar coorinates. The angular epenence of the solutions will be escribe by spherical harmonics.
More informationStudents need encouragement. So if a student gets an answer right, tell them it was a lucky guess. That way, they develop a good, lucky feeling.
Chapter 8 Analytic Functions Stuents nee encouragement. So if a stuent gets an answer right, tell them it was a lucky guess. That way, they evelop a goo, lucky feeling. 1 8.1 Complex Derivatives -Jack
More informationNUMERICAL STUDY OF THERMAL RADIATIONS AND THERMAL STRATIFICATION MECHANISMS IN MHD CASSON FLUID FLOW. and Sardar Muhammad BILAL c
NUMERICAL STUDY OF THERMAL RADIATIONS AND THERMAL STRATIFICATION MECHANISMS IN MHD CASSON FLUID FLOW Khalil Ur REHMAN b c * Noor Ul SABA b Iffat ZEHRA c Muhamma Yousaf MALIK ab an Sarar Muhamma BILAL c
More informationAn Approach for Design of Multi-element USBL Systems
An Approach for Design of Multi-element USBL Systems MIKHAIL ARKHIPOV Department of Postgrauate Stuies Technological University of the Mixteca Carretera a Acatlima Km. 2.5 Huajuapan e Leon Oaxaca 69000
More informationMagnetic field generated by current filaments
Journal of Phsics: Conference Series OPEN ACCESS Magnetic fiel generate b current filaments To cite this article: Y Kimura 2014 J. Phs.: Conf. Ser. 544 012004 View the article online for upates an enhancements.
More informationMHD Flow and Heat Transfer over an. Exponentially Stretching Sheet with Viscous. Dissipation and Radiation Effects
Applied Mathematical Sciences, Vol. 7, 3, no. 4, 67-8 MHD Flow and Heat Transfer over an Exponentially Stretching Sheet with Viscous Dissipation and Radiation Effects R. N. Jat and Gopi Chand Department
More informationOptimization of Geometries by Energy Minimization
Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997.
More informationTMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments
Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary
More informationZ. Elhadj. J. C. Sprott UDC
UDC 517. 9 ABOUT THE BOUNDEDNESS OF 3D CONTINUOUS-TIME QUADRATIC SYSTEMS ПРО ОБМЕЖЕНIСТЬ 3D КВАДРАТИЧНИХ СИСТЕМ З НЕПЕРЕРВНИМ ЧАСОМ Z. Elhaj Univ. f Tébessa 100), Algeria e-mail: zeraoulia@mail.univ-tebessa.z
More informationChapter-2. Steady Stokes flow around deformed sphere. class of oblate axi-symmetric bodies
hapter- Steay Stoes flow aroun eforme sphere. class of oblate axi-symmetric boies. General In physical an biological sciences, an in engineering, there is a wie range of problems of interest lie seimentation
More informationSemiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom
PHYSICAL REVIEW A 69, 063409 (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics,
More informationDelocalization of boundary states in disordered topological insulators
Journal of Physics A: Mathematical an Theoretical J. Phys. A: Math. Theor. 48 (05) FT0 (pp) oi:0.088/75-83/48//ft0 Fast Track Communication Delocalization of bounary states in isorere topological insulators
More information