Mehmet Pakdemirli* Precession of a Planet with the Multiple Scales Lindstedt Poincare Technique (2)
|
|
- Christine McDowell
- 6 years ago
- Views:
Transcription
1 Z. Natrforsch. 05; aop Mehmet Pakemirli* Precession of a Planet with the Mltiple Scales Linstet Poincare Techniqe DOI 0.55/zna Receive May, 05; accepte Jly 5, 05 Abstract: The recently evelope mltiple scales Linstet Poincare (MSLP) techniqe is sccessflly applie to a mathematical moel of planet motion. The eqation is originally evelope to precisely nerstan the orbital motion of the planet Mercry aron the Sn an the precession of the orbit e to the relativistic effects. The qaratic nonlinear eqation is solve by the classical Linstet Poincare metho (LP) an then by the newly evelope mltiple scales Linstet Poincare metho (MSLP). Both approximate soltions are contraste with the nmerical simlations. When the relativistic effects are small, all three soltions coincie with each other. When the pertrbation effects are increase, the MSLP soltions agree better with the nmerical soltions than the LP soltions. The precession of the perihelion of the planet is calclate an compare for the approximate soltions. Keywors: Linstet Poincare Metho; Mltiple Scales Linstet Poincare Metho; Pertrbation Analysis; Planet Motion. Introction Using the Einstein gravitation theory, the precession of the planet Mercry was moelle with the ifferential system [, ] + = + ε a () (0) = b, (0) = 0 () *Corresponing athor: Mehmet Pakemirli, Applie Mathematics an Comptation Center, Celal Bayar University, 4540, Mraiye, Manisa, mpak@cb.e.tr where is the angle coorinate an is the reciprocal of the imensionless isplacement. = (3) r / r r is the istance between Mercry an Sn an r is the average istance ( r = m). The remaining imensionless qantities are as follows: GMr 3GM a =, ε = h c r where G is the niversal gravitational constant, M is the mass of the Sn, h is the anglar momentm per nit mass for Mercry, c is the spee of light in the vacm, b is the reciprocal of the isplacement when the planet is closest to Sn. Initially, the planet is assme to be in the closest istance to Sn (perihelion). For the planet Mercry, a = 0.98, b =.0. Accoring to the theory, the qaratic nonlinear term moels the small pertrbations from the orbit of the Mercry an has the orer of magnite ε O(0 7 ) or more precisely ε = []. Other than the implicit integral form that is evalate nmerically or expressions in Jacobi elliptic fnctions [], the moel oes not possess a simple analytical close form soltion. The ifferential system is solve first by the Linstet Poincare techniqe (LP) that is evelope by the astronomers to nerstan the orbital motion of the planets an small pertrbations in the trajectories. Next, the recently evelope mltiple scales Linstet Poincare techniqe (MSLP) [3 ] is employe in search of approximate analytical soltions. The metho combines the elimination mechanisms of seclar terms of both the classical Linstet Poincare techniqe an the classical mltiple scales techniqe an increases the possibility of sccess for obtaining niform soltions for the nonlinear ynamical systems. Another featre of the metho is the incorporation of an nsal freqency expansion sggeste by H [7] an H an Xiong [8], which leas to the niformly vali soltions for the strongly nonlinear systems. In a pioneering work, for a classical anharmonic oscillator with rescale pertrbation series, soltions vali for strongly nonlinear systems were also obtaine [9] by employing a similar reasoning. Finally, the (4) Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM
2 M. Pakemirli: Precession of a Planet system is also solve nmerically by sing a variable step size Rnge Ktta algorithm. As in the case of the planet Mercry, when the pertrbation parameter is qite small, all three soltions coincie with each other an the precession angles are the same. However, when the pertrbation parameter is increase, the MSLP soltions are in close agreement with the nmerical soltions, whereas the LP soltions iverge from the others. The precession angles also start ifferentiating from each other. Pertrbation Analysis In this section, the system () an () is solve by the Linstet Poincare metho first an then by the mltiple scales Linstet Poincare metho.. The Linstet Poincare Metho In this metho, first the coorinate transformation is inserte into the eqation = (5) + = a+ ε () where is an introce transformation freqency. The epenent variable an the freqency are expane as the following pertrbation series: ( ; ε) = ( ) + ε ( ) + ε ( ) + (7) 0 = + ε + ε + (8) By applying into () an separate with respect to similar orers, the following eqations are obtaine. 0 + = = 0 a, (0) b, (0) = 0 (9) = 0 + =, (0) 0, (0) = 0 (0) = +. () 0 Since at the last level of approximation, only the seclar terms will be eliminate, an will not be calclate openly, one oes not nee the conitions at this level [0]. At the first-orer, the soltion is = ( b a)cos + a 0 () which pon sbstittion into the next level an elimination of the seclar term yiels = a (3) The soltion at this level is = a + ( b a) ( cos ) + ( b a) ( cos ) 3 (4) Finally, at the last level, 0 an are sbstitte an only the seclarities are eliminate. 5 = a ( b a ) (5) Combining the terms an retrning back to the original variables yiels = ( b a)cos + a+ ε a + ( b a) ( cos ) 3 () + ( b a ) ( cos ) + O ( ε ) = ε ε + 5 a a ( b a ), (7) which is the esire LP soltion for the problem. Smith [] gave a pertrbation soltion of the problem sing both the Krylov Bogolibov techniqe an the mltiple scales techniqe which is = ( b a)cos + a+ O ( ε) (8) = εa (9) It is to be note that the above soltion is a first approximation of the fnction an the freqency of the LP techniqe.. The Mltiple Scales Linstet Poincare Metho In this metho, similar to the LP metho, initially, a coorinate transformation is introce. = (0) an sbstitte into the original eqation Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM
3 M. Pakemirli: Precession of a Planet 3 ε + = a + () As in the classical mltiple scales metho, the ifferent coorinate scales T =, T = ε, T = ε () 0 are efine an the associate erivatives are written in terms of the new variables = D + D + D + = D + DD + ε ( D + D D ) + ε ε, ε (3) where D n = / T n. The epenent variable is expane in a series in terms of the new inepenent variables ( ; ε) = ( T, T, T ) + ε ( T, T, T ) + ε ( T, T, T ) 0 (4) Instea of the transformation freqency, the normalise freqency is expane in this metho [3 ]. To be consistent, the parameter a shol also be expane in a similar manner. = ε ε + (5) a= a ( ε ε ) + () Sbstitting all into (), separating with respect to orers yiels ( D + ) = a, (0) = b, D (0) = 0 (7) The initial conitions ictate b (0) = b a, β (0) = 0 (33) Sbstitting (30) into the next level an eliminating the seclarities reqire i D B+ B+ ab = 0 (34) The metho gives priority to the elimination mechanism of LP first. Hence, pon selecting D B = 0, if is real, the choice is amissible which is exactly the case in (34) an hence B = BT ( ), = a (35) The soltion at this level is = E( T, T )exp( it ) + cc ( B exp( it ) + cc) ( BB a ) where The real form of is (3) E = e exp( i γ) (37) = + γ b e cos( T ) b cos(t + β) a The initial conitions reqire (38) ( D + ) = D D + + a, (0) = 0, ( D + D )(0) = (8) a e (0) = ( b a ), γ(0) = 0 (39) 3 ( D + ) = D D ( D + D D ) a The soltion at the leaing orer is (9) = B( T, T )exp( it ) + cc + a (30) 0 0 where cc stans for the complex conjgates an B is given by the following polar form. B = b exp( i β) (3) At the final level, only seclarities are eliminate. Sbstitting (3) an (30) into (9) an eliminating seclarities reqires 0 i D E i DB+ E+ B+ BB 3 a + B + ae = 0 (40) Withot loss of generality, D E = 0 can be selecte. For D B = 0, trns ot to be real an LP mechanism of eliminating seclarities is sfficient. Hence The soltion in real form is = b cos( T + β) + a (3) = = = E const, B const, b a, β = 0, γ = 0 (4) Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM
4 4 M. Pakemirli: Precession of a Planet The freqency expansion now reas ε 5 = εa + a ( b a ) (4) It is to be note that now appears on both sies an has to be solve. Combining all the information an solving for the real an positive vale, the final approximate soltion in terms of the original variables an parameters is ε = ( b a)cos + a+ a + ( b a) ( cos ) 3 + ( b a ) ( cos ) + O ( ε ) (43) 5 = εa+ ( εa) 4ε a + ( b a) (44) Now, the freqency is slightly ifferent than the one obtaine by the LP metho. Frthermore, while the coefficient of the correction term is ε in the LP, here, the coefficient is ε/. It will be shown in the next section by comparing with the nmerical simlations that these slight ifferences in the freqencies an coefficients lea to more precise soltions Figre : Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = ) Nmerical LP MSLP 3 Comparisons The orinary ifferential system () an () is solve nmerically by employing an aaptive step size Rnge Ktta algorithm an compare with the analytical soltions. In Figre, the analytical soltions of both methos are contraste with the nmerical ones for the planet Mercry for ε = All soltions coincie with each other. When the pertrbation parameter is of orer, oscillation type soltions cannot be retrieve nmerically. If one assmes that the relativistic effects on the motion became larger somehow, the pertrbation parameter shol be increase from its originally calclate vale. In Figre, the pertrbation parameter is again small, bt relatively larger than its original vale, that is ε = While the nmerical an MSLP soltions agree with each other, LP soltions can be istingishe from both of them. In Figre 3, the LP, MSLP, an the one term KB soltions (i.e. 8 an 9) are contraste with the nmerical Figre : Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = 0.05). soltion for a fairly large vale of the pertrbation parameter. The MSLP an the nmerical soltions agree well with each other. The qantitative iscrepancy is observe for the LP soltion. For the one term KB soltion given in [], there is both qalitative an qantitative iscrepancy compare to the other soltions. Finally, the parameter is increase p to the limit of oscillatory soltions where ε = 0. in Figre 4. While all three soltions are separate from each other, the MSLP better preicts the maximm amplites an freqencies compare to the LP. It is note that the motion of the planet is a qasiperioic motion becase the orbit itself oes not remain Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM
5 M. Pakemirli: Precession of a Planet LP MSLP KB Nmerical Table : Precession angles for varios pertrbation parameters. Precession angle/centry ε KB LP MSLP Figre 3: Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = 0.) Nmerical LP MSLP Figre 4: Reciprocal of the imensionless isplacement verss angle (a = 0.98, b =.0, ε = 0.). fixe in the space rather the major axis of the ellipse rotates within time. The perihelion ths precesses by an increibly small amont for which the accmlate anglar vale can be measre in a time interval of a centry. For one perio of motion, the precession angle in raians is [] PA = π( ) rev (45) where the freqency is sbstitte from (9) for KB, from (7) for LP an from (44) for MSLP. For the planet Mercry, this is a rather small qantity an the accmlate angle for a centry is calclate as follows: PA ra rev 355 ays = π( ) centry rev 88 ays centry 80 eg 300 s π ra eg (4) where the planet Mercry completes one revoltion in approximately 88 earth ays. All three methos give the same precession angle of abot 40 s of angle for the pertrbation parameter of ε = which is a rather small qantity (Tab. ). This vale is compatible with the astronomical observations []. When the pertrbation parameter is increase, first, the KB soltion ifferentiates from the others, an then, all soltions ifferentiate slightly from each other for a sfficiently large parameter. Base on the figres, the highest precision is obtaine by the MSLP. 4 Conclsion A ifferential system erive sing Einstein gravitation theory is consiere. The moel incorporate the relativistic effects on the orbital motion of the planet Mercry aron the Sn. Approximate soltions are fon sing the classical LP techniqe commonly se in astronomy an the recently evelope mltiple scales LP techniqe. The approximate analytical soltions are contraste with the nmerical soltions. For the stanar pertrbation parameter of ε = for the planet Mercry, all three soltions agree well with each other. While the moel is originally evelope for Mercry, it can be applie to other planets also. It may happen that the pertrbation parameter, which represents the relativistic effects, may become mch larger than the reporte vale, an hence, the parameter is increase p to the limit of oscillatory soltions. It is fon that the MSLP metho preicts the isplacements better compare to the LP metho. In the previos sties [3 ], the MSLP is sccessflly applie to the strongly nonlinear systems procing compatible soltions with the nmerical simlations. In this sty, the strong nonlinear system oes not proce oscillatory soltions, an hence, the pertrbation parameter is increase near to the oscillatory limiting vale. Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM
6 M. Pakemirli: Precession of a Planet References [] D. R. Smith, Singlar-Pertrbation Theory, Cambrige University Press, New York 985. [] H. Golstein, Classical Mechanics, Aison-Wesley, New York 980. [3] M. Pakemirli, M. M. F. Karahan, an H. Boyacı, Math. Compt. Appl. 4, 3 (009). [4] M. Pakemirli an M. M. F. Karahan, Math. Methos Appl. Sci. 33, 704 (00). [5] M. Pakemirli, M. M. F. Karahan, an H. Boyacı, Math. Compt. Appl., 879 (0). [] M. Pakemirli an G. Sarı, Math. Compt. Appl. 0, 37 (05). [7] H. H, J. Son Vib. 9, 409 (004). [8] H. H an Z. G. Xiong, J. Son Vib. 78, 437 (004). [9] K. Banerjee, J. K. Bhattacharjee, an H. S. Mani, Phys. Rev. A. 30, 8 (984). [0] A. H. Nayfeh, Introction to Pertrbation Techniqes, John Wiley an Sons, New York 985. Athenticate mpak@cb.e.tr athor's copy Downloa Date 8/7/5 9: AM
Theorem (Change of Variables Theorem):
Avance Higher Notes (Unit ) Prereqisites: Integrating (a + b) n, sin (a + b) an cos (a + b); erivatives of tan, sec, cosec, cot, e an ln ; sm/ifference rles; areas ner an between crves. Maths Applications:
More informationNumerical simulation on wind pressure transients in high speed train tunnels
Compters in ailways XI 905 Nmerical simlation on win pressre transients in high spee train tnnels S.-W. Nam Department of High Spee Train, Korea ailroa esearch Institte, Korea Abstract When a train passes
More informationModeling of a Self-Oscillating Cantilever
Moeling of a Self-Oscillating Cantilever James Blanchar, Hi Li, Amit Lal, an Doglass Henerson University of Wisconsin-Maison 15 Engineering Drive Maison, Wisconsin 576 Abstract A raioisotope-powere, self-oscillating
More informationn s n Z 0 on complex-valued functions on the circle. Then sgn n + 1 ) n + 1 2) s
. What is the eta invariant? The eta invariant was introce in the famos paper of Atiyah, Patoi, an Singer see [], in orer to proce an inex theorem for manifols with bonary. The eta invariant of a linear
More informationRainer Friedrich
Rainer Frierich et al Rainer Frierich rfrierich@lrztme Holger Foysi Joern Sesterhenn FG Stroemngsmechanik Technical University Menchen Boltzmannstr 5 D-85748 Garching, Germany Trblent Momentm an Passive
More informationOn the Total Duration of Negative Surplus of a Risk Process with Two-step Premium Function
Aailable at http://pame/pages/398asp ISSN: 93-9466 Vol, Isse (December 7), pp 7 (Preiosly, Vol, No ) Applications an Applie Mathematics (AAM): An International Jornal Abstract On the Total Dration of Negatie
More informationOptimization of pile design for offshore wind turbine jacket foundations
Downloae from orbit.t.k on: May 11, 2018 Optimization of pile esign for offshore win trbine jacket fonations Sanal, Kasper; Zania, Varvara Pblication ate: 2016 Docment Version Peer reviewe version Link
More information2.13 Variation and Linearisation of Kinematic Tensors
Section.3.3 Variation an Linearisation of Kinematic ensors.3. he Variation of Kinematic ensors he Variation In this section is reviewe the concept of the variation, introce in Part I, 8.5. he variation
More informationStagnation Analysis in Particle Swarm Optimisation or What Happens When Nothing Happens
Stagnation Analysis in Particle Swarm Optimisation or What Happens When Nothing Happens Marice Clerc To cite this version: Marice Clerc. Stagnation Analysis in Particle Swarm Optimisation or What Happens
More informationMath 273b: Calculus of Variations
Math 273b: Calcls of Variations Yacob Kreh Homework #3 [1] Consier the 1D length fnctional minimization problem min F 1 1 L, or min 1 + 2, for twice ifferentiable fnctions : [, 1] R with bonary conitions,
More informationOptimal Operation by Controlling the Gradient to Zero
Optimal Operation by Controlling the Graient to Zero Johannes Jäschke Sigr Skogesta Department of Chemical Engineering, Norwegian University of Science an Technology, NTNU, Tronheim, Norway (e-mail: {jaschke}{skoge}@chemeng.ntn.no)
More informationThroughput Maximization for Tandem Lines with Two Stations and Flexible Servers
Throghpt Maximization for Tanem Lines with Two Stations an Flexible Servers Sigrún Anraóttir an Hayriye Ayhan School of Instrial an Systems Engineering Georgia Institte of Technology Atlanta GA 30332-0205
More informationAdaptive partial state feedback control of the DC-to-DC Ćuk converter
5 American Control Conference Jne 8-, 5. Portlan, OR, USA FrC7.4 Aaptive partial state feeback control of the DC-to-DC Ćk converter Hgo Rorígez, Romeo Ortega an Alessanro Astolfi Abstract The problem of
More informationDesign Method for RC Building Structure Controlled by Elasto-Plastic Dampers Using Performance Curve
Design Metho or RC Biling Strctre Controlle by Elasto-Plastic Dampers Using Perormance Crve W. P Whan University o Technology, China K. Kasai Tokyo Institte o Technology, Japan SUMMARY: This paper proposes
More informationSetting The K Value And Polarization Mode Of The Delta Undulator
LCLS-TN-4- Setting The Vale And Polarization Mode Of The Delta Undlator Zachary Wolf, Heinz-Dieter Nhn SLAC September 4, 04 Abstract This note provides the details for setting the longitdinal positions
More informationLogarithmic, Exponential and Other Transcendental Functions
Logarithmic, Eponential an Other Transcenental Fnctions 5: The Natral Logarithmic Fnction: Differentiation The Definition First, yo mst know the real efinition of the natral logarithm: ln= t (where > 0)
More informationModel Predictive Control Lecture VIa: Impulse Response Models
Moel Preictive Control Lectre VIa: Implse Response Moels Niet S. Kaisare Department of Chemical Engineering Inian Institte of Technolog Maras Ingreients of Moel Preictive Control Dnamic Moel Ftre preictions
More informationRamp Metering Control on the Junction of Freeway and Separated Connecting Collector-Distributor Roadway
Proceeings of the 8th WSEAS International Conference on APPLIED MATHEMATICS, Tenerife, Spain, December 16-18, 5 (pp45-5) Ramp Metering Control on the Jnction of Freeway an Separate Connecting Collector-Distribtor
More informationChapter 17. Weak Interactions
Chapter 17 Weak Interactions The weak interactions are meiate by W ± or (netral) Z exchange. In the case of W ±, this means that the flavors of the qarks interacting with the gage boson can change. W ±
More informationThe spreading residue harmonic balance method for nonlinear vibration of an electrostatically actuated microbeam
J.L. Pan W.Y. Zh Nonlinear Sci. Lett. Vol.8 No. pp.- September The spreading reside harmonic balance method for nonlinear vibration of an electrostatically actated microbeam J. L. Pan W. Y. Zh * College
More informationHOMEWORK 2 SOLUTIONS
HOMEWORK 2 SOLUTIONS PHIL SAAD 1. Carroll 1.4 1.1. A qasar, a istance D from an observer on Earth, emits a jet of gas at a spee v an an angle θ from the line of sight of the observer. The apparent spee
More informationNEURAL CONTROL OF NONLINEAR SYSTEMS: A REFERENCE GOVERNOR APPROACH
NEURAL CONTROL OF NONLINEAR SYSTEMS: A REFERENCE GOVERNOR APPROACH L. Schnitman Institto Tecnológico e Aeronática.8-900 - S.J. os Campos, SP Brazil leizer@ele.ita.cta.br J.A.M. Felippe e Soza Universiae
More informationComplementing the Lagrangian Density of the E. M. Field and the Surface Integral of the p-v Vector Product
Applie Mathematics,,, 5-9 oi:.436/am..4 Pblishe Online Febrary (http://www.scirp.org/jornal/am) Complementing the Lagrangian Density of the E. M. Fiel an the Srface Integral of the p- Vector Proct Abstract
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationLight flavor asymmetry of polarized quark distributions in thermodynamical bag model
Inian Jornal of Pre & Applie Physics Vol. 5, April 014, pp. 19-3 Light flavor asymmetry of polarize qark istribtions in thermoynamical bag moel K Ganesamrthy a & S Mrganantham b* a Department of Physics,
More informationSEG Houston 2009 International Exposition and Annual Meeting
Fonations of the metho of M fiel separation into pgoing an ongoing parts an its application to MCSM ata Michael S. Zhanov an Shming Wang*, Universit of Utah Smmar The renee interest in the methos of electromagnetic
More informationFULL-SCALE DYNAMIC TESTS AND ANALYTICAL VERIFICATION OF A FORCE-RESTRICTED TUNED VISCOUS MASS DAMPER
FULL-SCALE DYNAMIC TESTS AND ANALYTICAL VERIFICATION OF A FORCE-RESTRICTED TUNED VISCOUS MASS DAMPER Y. Watanabe THK Co., Lt., Tokyo, Japan K. Ikago & N. Inoe Tohok University, Senai, Japan H. Kia, S.
More informationA Note on Irreducible Polynomials and Identity Testing
A Note on Irrecible Polynomials an Ientity Testing Chanan Saha Department of Compter Science an Engineering Inian Institte of Technology Kanpr Abstract We show that, given a finite fiel F q an an integer
More informationDesert Mountain H. S. Math Department Summer Work Packet
Corse #50-51 Desert Montain H. S. Math Department Smmer Work Packet Honors/AP/IB level math corses at Desert Montain are for stents who are enthsiastic learners of mathematics an whose work ethic is of
More informationA NEW ENTROPY FORMULA AND GRADIENT ESTIMATES FOR THE LINEAR HEAT EQUATION ON STATIC MANIFOLD
International Jornal of Analysis an Applications ISSN 91-8639 Volme 6, Nmber 1 014, 1-17 http://www.etamaths.com A NEW ENTROPY FORULA AND GRADIENT ESTIATES FOR THE LINEAR HEAT EQUATION ON STATIC ANIFOLD
More informationSolution to Tutorial 8
Soltion to Ttorial 8 01/013 Semester I MA464 Game Theory Ttor: Xiang Sn October, 01 1 eview A perfect Bayesian eqilibrim consists of strategies an beliefs satisfying eqirements 1 throgh 4. eqirement 1:
More informationHongliang Yang and Michael Pollitt. September CWPE 0741 and EPRG 0717
Distingishing Weak an Strong Disposability among Unesirable Otpts in DEA: The Example of the Environmental Efficiency of Chinese Coal-Fire Power Plants Hongliang Yang an Michael Pollitt September 2007
More informationUNIT IV BOUNDARY LAYER AND FLOW THROUGH PIPES Definition of bonary layer Thickness an classification Displacement an momentm Thickness Development of laminar an trblent flows in circlar pipes Major an
More informationOptimal Contract for Machine Repair and Maintenance
Optimal Contract for Machine Repair an Maintenance Feng Tian University of Michigan, ftor@mich.e Peng Sn Dke University, psn@ke.e Izak Denyas University of Michigan, enyas@mich.e A principal hires an agent
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 informationApproximate Solution of Convection- Diffusion Equation by the Homotopy Perturbation Method
Gen. Math. Notes, Vol. 1, No., December 1, pp. 18-114 ISSN 19-7184; Copyright ICSRS Pblication, 1 www.i-csrs.org Available free online at http://www.geman.in Approximate Soltion of Convection- Diffsion
More information08.06 Shooting Method for Ordinary Differential Equations
8.6 Shooting Method for Ordinary Differential Eqations After reading this chapter, yo shold be able to 1. learn the shooting method algorithm to solve bondary vale problems, and. apply shooting method
More informationGravitational Instability of a Nonrotating Galaxy *
SLAC-PUB-536 October 25 Gravitational Instability of a Nonrotating Galaxy * Alexander W. Chao ;) Stanford Linear Accelerator Center Abstract Gravitational instability of the distribtion of stars in a galaxy
More informationTHE DISPLACEMENT GRADIENT AND THE LAGRANGIAN STRAIN TENSOR Revision B
HE DISPLACEMEN GRADIEN AND HE LAGRANGIAN SRAIN ENSOR Revision B By om Irvine Email: tom@irvinemail.org Febrary, 05 Displacement Graient Sppose a boy having a particlar configration at some reference time
More informationFormal Methods for Deriving Element Equations
Formal Methods for Deriving Element Eqations And the importance of Shape Fnctions Formal Methods In previos lectres we obtained a bar element s stiffness eqations sing the Direct Method to obtain eact
More informationStudy on the impulsive pressure of tank oscillating by force towards multiple degrees of freedom
EPJ Web of Conferences 80, 0034 (08) EFM 07 Stdy on the implsive pressre of tank oscillating by force towards mltiple degrees of freedom Shigeyki Hibi,* The ational Defense Academy, Department of Mechanical
More informationSolving Ordinary differential equations with variable coefficients
Jornal of Progreive Reearch in Mathematic(JPRM) ISSN: 2395-218 SCITECH Volme 1, Ie 1 RESEARCH ORGANISATION Pblihe online: November 3, 216 Jornal of Progreive Reearch in Mathematic www.citecreearch.com/jornal
More informationKaons and long-distance meson mixing from lattice QCD
Kaons an long-istance meson mixing from lattice QCD Stephen R. Sharpe University of Washington S. Sharpe, Kaons & long-istance meson mixing 3/7/14 @ Lattice meets Experiment 2014, FNAL 1 /34 Base partly
More informationA NOTE ON PERELMAN S LYH TYPE INEQUALITY. Lei Ni. Abstract
A NOTE ON PERELAN S LYH TYPE INEQUALITY Lei Ni Abstract We give a proof to the Li-Ya-Hamilton type ineqality claime by Perelman on the fnamental soltion to the conjgate heat eqation. The rest of the paper
More informationElectromagnet 1 Electromagnet 2. Rotor. i 2 + e 2 - V 2. - m R 2. x -x. zero bias. Force. low bias. Control flux
Low-Bias Control of AMB's Sbject to Satration Constraints anagiotis Tsiotras an Efstathios Velenis School of Aerospace Engineering Georgia Institte of Technology, Atlanta, GA 333-5, USA p.tsiotras@ae.gatech.e,
More informationSolving a System of Equations
Solving a System of Eqations Objectives Understand how to solve a system of eqations with: - Gass Elimination Method - LU Decomposition Method - Gass-Seidel Method - Jacobi Method A system of linear algebraic
More informationSection 7.4: Integration of Rational Functions by Partial Fractions
Section 7.4: Integration of Rational Fnctions by Partial Fractions This is abot as complicated as it gets. The Method of Partial Fractions Ecept for a few very special cases, crrently we have no way to
More informationHomotopy Perturbation Method for Solving Linear Boundary Value Problems
International Jornal of Crrent Engineering and Technolog E-ISSN 2277 4106, P-ISSN 2347 5161 2016 INPRESSCO, All Rights Reserved Available at http://inpressco.com/categor/ijcet Research Article Homotop
More informationResearch Article Vertical Velocity Distribution in Open-Channel Flow with Rigid Vegetation
e Scientific Worl Jornal, Article ID 89, pages http://x.oi.org/.//89 Research Article Vertical Velocity Distribtion in Open-Channel Flow with Rigi Vegetation Changjn Zh, Wenlong Hao, an Xiangping Chang
More informationu t v t v t c a u t b a v t u t v t b a
Nonlinear Dynamical Systems In orer to iscuss nonlinear ynamical systems, we must first consier linear ynamical systems. Linear ynamical systems are just systems of linear equations like we have been stuying
More informationMathematical Model for Vibrations Analysis of The Tram Wheelset
TH ARCHIVS OF TRANSPORT VOL. XXV-XXVI NO - 03 Mathematical Moel for Vibrations Analysis of The Tram Wheelset Anrzej Grzyb Karol Bryk ** Marek Dzik *** Receive March 03 Abstract This paper presents a mathematical
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 informationChapter 2 Difficulties associated with corners
Chapter Difficlties associated with corners This chapter is aimed at resolving the problems revealed in Chapter, which are cased b corners and/or discontinos bondar conditions. The first section introdces
More informationCurves - Foundation of Free-form Surfaces
Crves - Fondation of Free-form Srfaces Why Not Simply Use a Point Matrix to Represent a Crve? Storage isse and limited resoltion Comptation and transformation Difficlties in calclating the intersections
More informationLecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.
b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference
More informationRate-Compatible Puncturing of Low-Density Parity-Check Codes
84 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 11, NOVEMBER 004 Rate-Compatible Pnctring of Low-Density Parity-Check Coes Jeongseok Ha, Jaehong Kim, an Steven W. McLaghlin, Senior Member, IEEE
More informationEnergy-Efficient Resource Allocation for Multi-User Mobile Edge Computing
Energy-Efficient Resorce Allocation for Mlti-User Mobile Ege Compting Jnfeng Go, Ying Ci, Zhi Li Abstract Designing mobile ege compting MEC systems by jointly optimizing commnication an comptation resorces
More information4 Exact laminar boundary layer solutions
4 Eact laminar bondary layer soltions 4.1 Bondary layer on a flat plate (Blasis 1908 In Sec. 3, we derived the bondary layer eqations for 2D incompressible flow of constant viscosity past a weakly crved
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More informationPIPELINE MECHANICAL DAMAGE CHARACTERIZATION BY MULTIPLE MAGNETIZATION LEVEL DECOUPLING
PIPELINE MECHANICAL DAMAGE CHARACTERIZATION BY MULTIPLE MAGNETIZATION LEVEL DECOUPLING INTRODUCTION Richard 1. Davis & 1. Brce Nestleroth Battelle 505 King Ave Colmbs, OH 40201 Mechanical damage, cased
More informationPhysicsAndMathsTutor.com
C Integration - By sbstittion PhysicsAndMathsTtor.com. Using the sbstittion cos +, or otherwise, show that e cos + sin d e(e ) (Total marks). (a) Using the sbstittion cos, or otherwise, find the eact vale
More information7. Differentiation of Trigonometric Function
7. Differentiation of Trigonoetric Fnction RADIAN MEASURE. Let s enote the length of arc AB intercepte y the central angle AOB on a circle of rais r an let S enote the area of the sector AOB. (If s is
More informationVibrational modes of a rotating string
Vibrational modes of a rotating string Theodore J. Allen Department of Physics, University of Wisconsin-Madison, 1150 University Avene, Madison, WI 53706 USA and Department of Physics, Eaton Hall Hobart
More informationA Survey of the Implementation of Numerical Schemes for Linear Advection Equation
Advances in Pre Mathematics, 4, 4, 467-479 Pblished Online Agst 4 in SciRes. http://www.scirp.org/jornal/apm http://dx.doi.org/.436/apm.4.485 A Srvey of the Implementation of Nmerical Schemes for Linear
More informationCalculations involving a single random variable (SRV)
Calclations involving a single random variable (SRV) Example of Bearing Capacity q φ = 0 µ σ c c = 100kN/m = 50kN/m ndrained shear strength parameters What is the relationship between the Factor of Safety
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 informationNonparametric Identification and Robust H Controller Synthesis for a Rotational/Translational Actuator
Proceedings of the 6 IEEE International Conference on Control Applications Mnich, Germany, October 4-6, 6 WeB16 Nonparametric Identification and Robst H Controller Synthesis for a Rotational/Translational
More informationTRANSIENT FREE CONVECTION MHD FLOW BETWEEN TWO LONG VERTICAL PARALLEL PLATES WITH VARIABLE TEMPERATURE AND UNIFORM MASS DIFFUSION IN A POROUS MEDIUM
VOL. 6, O. 8, AUGUST ISS 89-668 ARP Jornal of Engineering an Applie Sciences 6- Asian Research Pblishing etork (ARP). All rights reserve. TRASIET FREE COVECTIO MD FLOW BETWEE TWO LOG VERTICAL PARALLEL
More informationarxiv: v1 [physics.flu-dyn] 7 Oct 2014
9 th Symposim on Naval Hyroynamics Gothenbrg, Sween, 6-3 Agst arxiv:4.896v physics.fl-yn] 7 Oct 4 Nmerical Simlation of Internal Tie Generation at a Continental Shelf Brea Lara K. Brant, James W. Rottman,
More informationA Decomposition Method for Volume Flux. and Average Velocity of Thin Film Flow. of a Third Grade Fluid Down an Inclined Plane
Adv. Theor. Appl. Mech., Vol. 1, 8, no. 1, 9 A Decomposition Method for Volme Flx and Average Velocit of Thin Film Flow of a Third Grade Flid Down an Inclined Plane A. Sadighi, D.D. Ganji,. Sabzehmeidani
More informationMultivariable functions approximation using a single quantum neuron
Mltivariable fnctions approximation sing a single qantm neron A.A. Ezhov,, A.G. Khromov, an G.P. Berman Theoretical Division an CNS, os Alamos National aboratory, os Alamos, New Mexico 87545 Troitsk Institte
More informationLIPSCHITZ SEMIGROUP FOR AN INTEGRO DIFFERENTIAL EQUATION FOR SLOW EROSION
QUARTERLY OF APPLIED MATHEMATICS VOLUME, NUMBER 0 XXXX XXXX, PAGES 000 000 S 0033-569X(XX)0000-0 LIPSCHITZ SEMIGROUP FOR AN INTEGRO DIFFERENTIAL EQUATION FOR SLOW EROSION By RINALDO M. COLOMBO (Dept. of
More informationReducing Conservatism in Flutterometer Predictions Using Volterra Modeling with Modal Parameter Estimation
JOURNAL OF AIRCRAFT Vol. 42, No. 4, Jly Agst 2005 Redcing Conservatism in Fltterometer Predictions Using Volterra Modeling with Modal Parameter Estimation Rick Lind and Joao Pedro Mortaga University of
More informationA Single Species in One Spatial Dimension
Lectre 6 A Single Species in One Spatial Dimension Reading: Material similar to that in this section of the corse appears in Sections 1. and 13.5 of James D. Mrray (), Mathematical Biology I: An introction,
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 informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
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 informationElectric Dipole Moments on the lattice with higher-dimensional operators
Introction Electric Dipole Moments on the lattice with higher-imensional operators Los Alamos National Laboratory Santa Fe Institte Oct 1, 2015 EDMs from O im>4 Introction Effective Fiel Theory Introction
More informationG y 0nx (n. G y d = I nu N (5) (6) J uu 2 = and. J ud H1 H
29 American Control Conference Hatt Regenc Riverfront, St Lois, MO, USA Jne 1-12, 29 WeC112 Conve initialization of the H 2 -optimal static otpt feeback problem Henrik Manm, Sigr Skogesta, an Johannes
More informationIJSER. =η (3) = 1 INTRODUCTION DESCRIPTION OF THE DRIVE
International Jornal of Scientific & Engineering Research, Volme 5, Isse 4, April-014 8 Low Cost Speed Sensor less PWM Inverter Fed Intion Motor Drive C.Saravanan 1, Dr.M.A.Panneerselvam Sr.Assistant Professor
More informationAscertainment of The Certain Fundamental Units in a Specific Type of Real Quadratic Fields
J. Ana. Nm. Theor. 5, No., 09-3 (07) 09 Jorna of Anaysis & Nmber Theory An Internationa Jorna http://x.oi.org/0.8576/jant/05004 Ascertainment of The Certain Fnamenta Units in a Specific Type of Rea Qaratic
More informationQUARK WORKBENCH TEACHER NOTES
QUARK WORKBENCH TEACHER NOTES DESCRIPTION Stents se cleverly constrcte pzzle pieces an look for patterns in how those pieces can fit together. The pzzles pieces obey, as mch as possible, the Stanar Moel
More informationWhen are Two Numerical Polynomials Relatively Prime?
J Symbolic Comptation (1998) 26, 677 689 Article No sy980234 When are Two Nmerical Polynomials Relatively Prime? BERNHARD BECKERMANN AND GEORGE LABAHN Laboratoire d Analyse Nmériqe et d Optimisation, Université
More informationAMS 212B Perturbation Methods Lecture 05 Copyright by Hongyun Wang, UCSC
AMS B Pertrbation Methods Lectre 5 Copright b Hongn Wang, UCSC Recap: we discssed bondar laer of ODE Oter epansion Inner epansion Matching: ) Prandtl s matching ) Matching b an intermediate variable (Skip
More informationThe dynamics of the simple pendulum
.,, 9 G. Voyatzis, ept. of Physics, University of hessaloniki he ynamics of the simple penulum Analytic methos of Mechanics + Computations with Mathematica Outline. he mathematical escription of the moel.
More informationChapter 9 Flow over Immersed Bodies
57:00 Mechanics o Flids and Transport Processes Chapter 9 Proessor Fred Stern Fall 01 1 Chapter 9 Flow over Immersed Bodies Flid lows are broadly categorized: 1. Internal lows sch as dcts/pipes, trbomachinery,
More informationNonlinear parametric optimization using cylindrical algebraic decomposition
Proceedings of the 44th IEEE Conference on Decision and Control, and the Eropean Control Conference 2005 Seville, Spain, December 12-15, 2005 TC08.5 Nonlinear parametric optimization sing cylindrical algebraic
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 informationSingle Particle Closed Orbits in Yukawa Potential
Single Particle Closed Orbits in Ykawa Potential Rpak Mkherjee 3, Sobhan Sonda 3 arxiv:75.444v [physics.plasm-ph] 6 May 7. Institte for Plasma Research, HBNI, Gandhinagar, Gjarat, India.. Ramakrishna Mission
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 informationarxiv: v3 [gr-qc] 29 Jun 2015
QUANTITATIVE DECAY RATES FOR DISPERSIVE SOLUTIONS TO THE EINSTEIN-SCALAR FIELD SYSTEM IN SPHERICAL SYMMETRY JONATHAN LUK AND SUNG-JIN OH arxiv:402.2984v3 [gr-qc] 29 Jn 205 Abstract. In this paper, we stdy
More informationSimplified Identification Scheme for Structures on a Flexible Base
Simplified Identification Scheme for Strctres on a Flexible Base L.M. Star California State University, Long Beach G. Mylonais University of Patras, Greece J.P. Stewart University of California, Los Angeles
More informationSTEP Support Programme. STEP III Hyperbolic Functions: Solutions
STEP Spport Programme STEP III Hyperbolic Fnctions: Soltions Start by sing the sbstittion t cosh x. This gives: sinh x cosh a cosh x cosh a sinh x t sinh x dt t dt t + ln t ln t + ln cosh a ln ln cosh
More informationLecture Notes: Finite Element Analysis, J.E. Akin, Rice University
9. TRUSS ANALYSIS... 1 9.1 PLANAR TRUSS... 1 9. SPACE TRUSS... 11 9.3 SUMMARY... 1 9.4 EXERCISES... 15 9. Trss analysis 9.1 Planar trss: The differential eqation for the eqilibrim of an elastic bar (above)
More informationOverview of particle physics
Overview of particle physics The big qestions of particle physics are 1. What is the niverse mae of? 2. How is it hel together? We can start at orinary istances an work or way own. Macroscopic stff is
More informationRestricted Three-Body Problem in Different Coordinate Systems
Applied Mathematics 3 949-953 http://dx.doi.org/.436/am..394 Pblished Online September (http://www.scirp.org/jornal/am) Restricted Three-Body Problem in Different Coordinate Systems II-In Sidereal Spherical
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 informationGLOBAL PHASE PORTRAITS OF SOME REVERSIBLE CUBIC CENTERS WITH NONCOLLINEAR SINGULARITIES
This is a preprint of: Global phase portraits of some reversible cbic centers with noncollinear singlarities, Magalena Cabergh, Joan Torregrosa, Internat. J. Bifr. Chaos Appl. Sci. Engrg., vol. 23(9),
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationApplication of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate
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
More information