Exponential Decay for Nonlinear Damped Equation of Suspended String
|
|
- Kerry Carpenter
- 5 years ago
- Views:
Transcription
1 9 Inernionl Symoium on Comuing, Communicion, nd Conrol (ISCCC 9) Proc of CSIT vol () () IACSIT Pre, Singore Eonenil Decy for Nonliner Dmed Equion of Suended Sring Jiong Kemuwn Dermen of Mhemic, Fculy of Science, King Mongku Iniue of Technology dkrbng Abrc Thi er i concerned wih he energy decy of he globl oluion for IBVP o nonliner dmed equion of uended ring wih uniform deniy o which nonliner ouer force work For hi uroe, we emloyed he energy mehod [Wo-Y] nd derive he decy eime by he nonliner dming erm long he refined mehod of [M-Ik] Keyword: nonliner, decy eime, uended ring, dmed equion Inroducion In hi work, we will udy he energy decy of he globl oluion of hevy nd fleible ring wih uniform deniy uended from ceiling under grviy Suoe h nonliner ouer borbing force nd nonliner dming work o he ring in horizonl direcion in vericl lne For he derivion of he uended ring equion, ee [K-G-S], [Ko] nd [Y] e Ω be cylindricl domin (, ) (, T) nd be he finie lengh of he ring Then he bove nonliner roblem i formuled he following IBVP q u (,) u (,) u u = β u u, (,) Ω, (P) u (, ) =, (, T), u (,) = φ( ), u (,) = ψ( ), (, ), where i econd order differenil oeror of he form nd, β = (, ) = ( ), q > > re conn nd We noe h i he ecil ce of he differenil oeror μ (ee ) nd degenere he origin [Y] hve hown he eience of lmo eriodicc -oluion o IBVP for he liner equion of uended ring equion wih he qui-eriodic forcing erm, nd everl eriodic roblem of he nonliner uended ring equion hve been udied in [Y]-[Y4] nd [Y-N-M] [Wo-Y] udied he eience nd uniquene of ime-globl clicl oluion wih monoonou cubic nonliner erm u 3, when he iniil d re lrge The uroe of hi er i o how decy eime of globl oluion o he roblem (P) Noe h u q β u i he borbing erm when β < nd liner ce when β = In [Wo- Y], we roved he eience of ime-globl wek oluion o nonliner equion of uended ring wihou he dming erm by energy mehod bed on he oenil well Thi reerch, i o how he globl oluion of he roblem (P), Becue we hve he imilr energy ideniy o he roblem in [Wo-Y], Correonding uhor Tel: (83)8-94; f: (e 685) E-mil ddre: jiui@homilcom 39
2 herefore we cn conruc he oenil well for he globl oluion by he me wy [Wo-Y] To derive he decy eime of he globl oluion, we ly n roch of [M-Ik], which dicued he globl eience nd energy decy o he wve equion of Kirchhoff Tye wih Nonliner Dming Term Conrry o [M-Ik], we need o ly he roerie of he oeror nd ome Poincre-Sobolev ye inequliie on ome rorie funcion ce uible for our roblem (ee [Y]) Funcion Sce, Oeror, he Bic Inequliie Definiion of Funcion Sce In hi ecion, we briefly review noion nd reul, which will be emloyed ler e R nd Z be he e of nonnegive number nd he e of nonnegive ineger, reecively n e nd Z For ny oen e O in R ( O ) nd H ( O) re he uul ebegue nd Sobolev ce, reecively In he following we ume h ll funcion re rel-vlued e μ > We denoe (, ; μ ) by Bnch ce whoe elemen f ( ) re meurble in (, ) nd ify μ / f( ) (, ) I norm i / f ( ) (, ; ) μ f d μ = In riculr, (, ; μ ) i Hilber ce wih inner roduc ( f, g ) = μ f ( ) g ( ) d μ (, ; ) Denoe H (, ; μ ) by Hilber ce, whoe elemen f ( ) nd heir weighed derivive j/ ( j f ) ( ), j =,,, belong o, norm (, ; μ ), where ( j f ) ( ) men he j -h derivive of f ( ) I / μ j ( j) μ = ( ) H (, ; ) j= f f d Denoe W (, ; μ ) by Bnch ce, whoe elemen f ( ) nd heir weighed derivive j/ ( j f ) ( ), j =,,, belong o (, ; μ ) I norm i / μ j ( j), μ = ( ) W (, ; ) j= f f d e T > nd Ω= (, ) (, T) We denoe ( Ω ; μ ) by Bnch ce, whoe elemen f (, ) re meurble in Ω nd ify j/ j k μ / f(, ) ( Ω ) I norm i / f (, ) ( ; ) μ f dd μ = Ω Ω We denoe H ( Ω ; ) by Hilber ce, whoe elemen f (, ) nd heir weighed derivive μ f(, ), j k, belong o ( Ω ; μ ) I norm i / μ j j k (, ) Ω j k f μ = f dd H ( Ω; ) H (, μ ; ) i ubce of H (, ; μ ) whoe elemen f ( ) ify f( ) = Similrly, H ( Ω ; μ ) i ubce of H ( Ω ; μ ) whoe elemen f (, ) ify f(, ) = for lmo ll (, T) K (, ; μ ) i ubce of H (, ; ) whoe elemen f = f( ) ify j =,,[( ) / ] Noe h K μ μ (, ; ) = (, ; ), μ K(, ; ) = H μ (, ; ), μ μ μ K = H H (, ; ) (, ; ) (, ; ) j H μ μ (, ; ) for 3
3 The Bic Proerie of μ We recll he more generl uended ring oeror μ inroduced by [K-G-S], [Ko] for μ > : μ = ( ) μ Clerly, μ coincide wih our differenil oeror for μ = Prooiion ([Y]) e μ be in () for μ > Then we hve he following erion: For f K (, ; μ ) nd g K (, ; μ ), we hve Thi lemm i he Poincre ye inequliy in μ ( μ f, g) μ = ( ) ( ) (, ; ) f g d μ H (, μ ; ) emm [Y] e μ > Then, for u H (, ; μ ), we hve u u μ (, ; ) μ (, ; ) emm 3 [Y] e nd μ > Then, for (), u W (, ; μ ) we hve μ μ μ u ( ) d c( u ( ) d u( ) d), where he conn c > deend on μ,, emm 4 [N] e Φ ( ) be non-increing nd nonnegive funcion on [, T ], T < uch h r Φ() k( Φ() Φ ( )) on [, T ], where k i oiive conn, nd r i nonnegive conn Then, we hve: (i) if we ume r >, hen where = { } [ ] m, r / r Φ() ( Φ () k r[ ] ) on [, T ], k (ii) if r =, hen Φ() Φ() e [ ] on [, T ], where k = log( k/( k )) 3 Energy eime nd Poenil well of (P) The ol energy of (P) coni of he oenil energy nd kineic energy defined follow E( u ; ) = Eu ( (, ), u(, )) = Ku ( (, )) Ju ( (, )) e E ( uu, ) be denoed by E( uu, ) = u( ) d Ju ( ) β q The oenil energy i defined J ( w) = w w( ) d q (, ; ), where = ( ( )) for w H w w d which i equivlen o H nd he kineic energy i defined K( w) = ( w (, )) d for w H ( Ω ; ) The nonliner erm in (P) ifie he following condiion (A) [Wo-Y] (A) f ( u, ) i of C -cl in (, u) [, ] Ru nd monoone decreing in u R, nd ifie r (, ) r C u uf u C u for ny [, ] nd u R Here C, C > re conn nd r > From (A), here ei conn λ uch h J ( λ u) i monoone increing in λ (, λ ) for ny fied u By he me wy [Wo-Y], he oenil well W for (P) round he origin i defined by 3
4 W = { u H (, ; ); J( λu) < d, λ } e λ = λ ( u) > be he fir vlue of λ which J ( λ u ) r o decree ricly The deh d of he oenil well W i defined by d = inf J( λ ( u) u) u H (, ; )\{} We ee [Wo-Y] h < d < nd W re oen nd bounded in H (, ; ) We ume he following condiion on he iniil d (B) e φ W nd ψ (, ; ) φ nd ψ ify he following condiion ( ψ φ ) β q ( ) ( ) q w( ) d< d Min Theorem Aume (A) nd (B) Then roblem (P) h wek oluion u H ( Ω ; ) ifying u(, ) W for ll (, T) Furhermore, we hve he decy eime: if =, hen k E( u ( ), u ( )) Ce on [, ), nd if >, hen /( ) Eu ( ( ), u ( )) C( ) on [, ), where kc, nd C re ny oiive conn deending on iniil d Proof Min Theorem Mulilying eq (P) by u nd inegring over [, ] (, ), we hve (3) = β u u dd u udd u u dd u u dd e u conider he econd erm of eq(33) (3) u ud = u ( u) d = ( u) d = ( u) We ue Prooiion o obin (33) uudd = ( u) d u ( ) u ( ) = where u = ( ) u d From (3) - (33), we hve β q u dd = ( ( )) ( ) ( ) ( ) (34) u d u u u d q = E () E ( ) = D (), when we e E( ) = E( u, u ) Mulilying eq(p) by u nd inegring over [, ] (, ), we hve (35) ( ) = J () d u() dd (( u), u( ) u ( ), u( )) u() u(), u () d Noe h [ ] nd [ ] (36), /4 3/4, By lying Holder' inequliy o he econd erm of eq(35), we hve ( )/( ) /( ) ( )/( ) {( )}/ u() dd u() d d /( ) ( ) /( ) ( ) ( ) /( ) = me u d d = me D (, ) ( ) (, ) ( ) The hird erm of eq(35) i eimed by he men vlue heorem /{( )} (37) u( i) me(, ) D( ) From emm 3, he l erm of eq (35) i eimed by /( ) /( ) () () () () u u dd u d u d d () () = u u d q 3
5 , W C u u ( ) d, nd le u conider he oenil energy nd emm, E () Ju ( ()) = ( ) q ( ) ( ) u d β u d q hen we hve (38) (39) (3) u, W u u dd ce D From (35) - (38), we obin () () () () ( ) ( ) ( ) ( ) J () d me(, ) D() 4 me(, ) D() u u() c E() D() Hence, i follow from (36) nd (39) h E () d () (()) u d Ju d ( ( ) ( ) ( ) ( ) c D D E D E ( ) ) = 4 Acknowledgemen I would like o ere my dee griude o Profeor MYmguchi for hi vluble uggeion I would like o lo hnk Profeor TMuym for giving me he inigh of hi reerch work 5 Reference [] R A Adm, Sobolev Sce, Acdemic Pre, 975 [] B G Korenev, Beel Funcion nd heir Alicion, Tylor nd Frnci Inc, [3] N S Kohlykov, E V Gliner nd M M Smirnov, Differenil Equion of Mhemicl Phyic, Mocow, 96 (in Ruin) Englih Trnlion: Norh-Hollnd Publ Co, 964 [4] T Muym nd RIkeh, On Globl oluion nd energy decy for he wve equion of Kirchhoff Tye wih Nonliner Dming Term, J Mh Anl Al 4, (996), [5] M Nko, Aymoic biliy of he bounded or lmo eriodic oluion of he wve equion wih nonliner diiive erm, J Mh Anl Al 58 (977), [6] J Sher, The eience of globl clicl oluion of he iniil-boundry vlue roblem for u u 3 = f, Arch Rionl Mech Anl, (966), 9-37 [7] C J Trner, Beel Funcion wih Some Phyicl Alicion, Hr Publihing Co, New York, 969 [8] G N Won, Theory of Beel Funcion, Cmbridge Univeriy Pre, 96 [9] J Wongwdi nd M Ymguchi, Globl oluion of IBVP o nonliner equion of uended ring, Tokyo J Mh 3, No (7), [] J Wongwdi nd M Ymguchi, Globl clicl oluion of IBVP o nonliner equion of uended ring, Tokyo J Mh 3, No (8), [] M Ymguchi, Almo eriodic ocillion of uended ring under Quieriodic liner force, J Mh Anl Al 33, No (5), [] M Ymguchi, Free vibrion of nonliner equion of uended ring, rerin [3] M Ymguchi, Globl mooh oluion of IBVP o nonliner equion of uended ring, J Mh Anl Al 34, No (8), [4] M Ymguchi, Infiniely mny eriodic oluion of nonliner equion of uended ring, FUNKCIAAJ EKVACIOJ-SERIO INTERNACIA 5, No (8), [5] M Ymguchi, T Ngi nd K Mukne, Forced ocillion of nonliner dmed equion of uended ring, J Mh Anl Al 34, No (8),
Positive and negative solutions of a boundary value problem for a
Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM) ISSN: 2455-3689 www.ijrem.com Volume 2 Iue 9 ǁ Sepemer 28 ǁ PP 73-83 Poiive nd negive oluion of oundry vlue prolem for frcionl, -difference
More informationLAPLACE TRANSFORMS. 1. Basic transforms
LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming
More informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 6, 8 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More information5.1-The Initial-Value Problems For Ordinary Differential Equations
5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
More informationContraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of
More informationgraph of unit step function t
.5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion"
More informationA LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly
More informationApplications of Prüfer Transformations in the Theory of Ordinary Differential Equations
Irih Mh. Soc. Bullein 63 (2009), 11 31 11 Applicion of Prüfer Trnformion in he Theory of Ordinry Differenil Equion GEORGE CHAILOS Abrc. Thi ricle i review ricle on he ue of Prüfer Trnformion echnique in
More informationcan be viewed as a generalized product, and one for which the product of f and g. That is, does
Boyce/DiPrim 9 h e, Ch 6.6: The Convoluion Inegrl Elemenry Differenil Equion n Bounry Vlue Problem, 9 h eiion, by Willim E. Boyce n Richr C. DiPrim, 9 by John Wiley & Son, Inc. Someime i i poible o wrie
More informationResearch Article The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses
Hindwi Advnce in Mhemicl Phyic Volume 207, Aricle ID 309473, pge hp://doi.org/0.55/207/309473 Reerch Aricle The Generl Soluion of Differenil Equion wih Cpuo-Hdmrd Frcionl Derivive nd Noninnneou Impule
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More informationBipartite Matching. Matching. Bipartite Matching. Maxflow Formulation
Mching Inpu: undireced grph G = (V, E). Biprie Mching Inpu: undireced, biprie grph G = (, E).. Mching Ern Myr, Hrld äcke Biprie Mching Inpu: undireced, biprie grph G = (, E). Mflow Formulion Inpu: undireced,
More informationC 0 Approximation on the Spatially Homogeneous Boltzmann Equation for Maxwellian Molecules*
Alied Mhemics,,, 54-59 doi:.46/m..666 Published Online December (h://www.scip.org/journl/m) C Aroximion on he Silly Homogeneous Bolzmnn Equion or Mxwellin Molecules Absrc Minling Zheng School o Science,
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM
Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: 072-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE
More informationTransformations. Ordered set of numbers: (1,2,3,4) Example: (x,y,z) coordinates of pt in space. Vectors
Trnformion Ordered e of number:,,,4 Emple:,,z coordine of p in pce. Vecor If, n i i, K, n, i uni ecor Vecor ddiion +w, +, +, + V+w w Sclr roduc,, Inner do roduc α w. w +,.,. The inner produc i SCLR!. w,.,
More informationRefinements to Hadamard s Inequality for Log-Convex Functions
Alied Mhemics 899-93 doi:436/m7 Pulished Online Jul (h://wwwscirporg/journl/m) Refinemens o Hdmrd s Ineuli for Log-Convex Funcions Asrc Wdllh T Sulimn Dermen of Comuer Engineering College of Engineering
More informationApplication on Inner Product Space with. Fixed Point Theorem in Probabilistic
Journl of Applied Mhemics & Bioinformics, vol.2, no.2, 2012, 1-10 ISSN: 1792-6602 prin, 1792-6939 online Scienpress Ld, 2012 Applicion on Inner Produc Spce wih Fixed Poin Theorem in Probbilisic Rjesh Shrivsv
More informationFLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER
#A30 INTEGERS 10 (010), 357-363 FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER Nahan Kaplan Deparmen of Mahemaic, Harvard Univeriy, Cambridge, MA nkaplan@mah.harvard.edu Received: 7/15/09, Revied:
More informationSome New Dynamic Inequalities for First Order Linear Dynamic Equations on Time Scales
Applied Memicl Science, Vol. 1, 2007, no. 2, 69-76 Some New Dynmic Inequliie for Fir Order Liner Dynmic Equion on Time Scle B. İ. Yşr, A. Tun, M. T. Djerdi nd S. Küükçü Deprmen of Memic, Fculy of Science
More informationAnalysis of Boundedness for Unknown Functions by a Delay Integral Inequality on Time Scales
Inernaional Conference on Image, Viion and Comuing (ICIVC ) IPCSIT vol. 5 () () IACSIT Pre, Singaore DOI:.7763/IPCSIT..V5.45 Anali of Boundedne for Unknown Funcion b a Dela Inegral Ineuali on Time Scale
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationOn The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function
Turkish Journl o Anlysis nd Numer Theory, 4, Vol., No. 3, 85-89 Aville online h://us.scieu.com/jn//3/6 Science nd Educion Pulishing DOI:.69/jn--3-6 On The Hermie- Hdmrd-Fejér Tye Inegrl Ineuliy or Convex
More informationHermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals
Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, 355 366 Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for
More informationCitation Abstract and Applied Analysis, 2013, v. 2013, article no
Tile An Opil-Type Inequliy in Time Scle Auhor() Cheung, WS; Li, Q Ciion Arc nd Applied Anlyi, 13, v. 13, ricle no. 53483 Iued De 13 URL hp://hdl.hndle.ne/17/181673 Righ Thi work i licened under Creive
More informationConvergence of Singular Integral Operators in Weighted Lebesgue Spaces
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335-347 ISSN 1307-5543 www.ejpm.com Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue
More informationENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
More information, the. L and the L. x x. max. i n. It is easy to show that these two norms satisfy the following relation: x x n x = (17.3) max
ecure 8 7. Sabiliy Analyi For an n dimenional vecor R n, he and he vecor norm are defined a: = T = i n i (7.) I i eay o how ha hee wo norm aify he following relaion: n (7.) If a vecor i ime-dependen, hen
More informationHow to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
More information1. Introduction. 1 b b
Journl of Mhemicl Inequliies Volume, Number 3 (007), 45 436 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY N. ELEZOVIĆ, LJ. MARANGUNIĆ AND J. PEČARIĆ (communiced b A. Čižmešij) Absrc. In his pper some inequliies
More informationREAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
More informationJournal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle
J. Mh. Anl. Appl. 353 009) 43 48 Conens liss vilble ScienceDirec Journl of Mhemicl Anlysis nd Applicions www.elsevier.com/loce/jm Two normliy crieri nd he converse of he Bloch principle K.S. Chrk, J. Rieppo
More informationIntroduction to SLE Lecture Notes
Inroducion o SLE Lecure Noe May 13, 16 - The goal of hi ecion i o find a ufficien condiion of λ for he hull K o be generaed by a imple cure. I urn ou if λ 1 < 4 hen K i generaed by a imple curve. We will
More informationOptimality of Myopic Policy for a Class of Monotone Affine Restless Multi-Armed Bandit
Univeriy of Souhern Cliforni Opimliy of Myopic Policy for Cl of Monoone Affine Rele Muli-Armed Bndi Pri Mnourifrd USC Tr Jvidi UCSD Bhkr Krihnmchri USC Dec 0, 202 Univeriy of Souhern Cliforni Inroducion
More informationProcedia Computer Science
Procedi Compuer Science 00 (0) 000 000 Procedi Compuer Science www.elsevier.com/loce/procedi The Third Informion Sysems Inernionl Conference The Exisence of Polynomil Soluion of he Nonliner Dynmicl Sysems
More informationON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX
Journl of Applied Mhemics, Sisics nd Informics JAMSI), 9 ), No. ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX MEHMET ZEKI SARIKAYA, ERHAN. SET
More informationIntroduction to Congestion Games
Algorihmic Game Theory, Summer 2017 Inroducion o Congeion Game Lecure 1 (5 page) Inrucor: Thoma Keelheim In hi lecure, we ge o know congeion game, which will be our running example for many concep in game
More informationINTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).
INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely
More informationFlow networks. Flow Networks. A flow on a network. Flow networks. The maximum-flow problem. Introduction to Algorithms, Lecture 22 December 5, 2001
CS 545 Flow Nework lon Efra Slide courey of Charle Leieron wih mall change by Carola Wenk Flow nework Definiion. flow nework i a direced graph G = (V, E) wih wo diinguihed verice: a ource and a ink. Each
More informationPrice Discrimination
My 0 Price Dicriminion. Direc rice dicriminion. Direc Price Dicriminion uing wo r ricing 3. Indirec Price Dicriminion wih wo r ricing 4. Oiml indirec rice dicriminion 5. Key Inigh ge . Direc Price Dicriminion
More informationNetwork Flows: Introduction & Maximum Flow
CSC 373 - lgorihm Deign, nalyi, and Complexiy Summer 2016 Lalla Mouaadid Nework Flow: Inroducion & Maximum Flow We now urn our aenion o anoher powerful algorihmic echnique: Local Search. In a local earch
More informationAN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS
CHAPTER 5 AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS 51 APPLICATIONS OF DE MORGAN S LAWS A we have een in Secion 44 of Chaer 4, any Boolean Equaion of ye (1), (2) or (3) could be
More informationSLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO SOME PROBLEMS IN NUMBER THEORY
VOL. 8, NO. 7, JULY 03 ISSN 89-6608 ARPN Jourl of Egieerig d Applied Sciece 006-03 Ai Reerch Publihig Nework (ARPN). All righ reerved. www.rpjourl.com SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO
More informationGreen s Functions and Comparison Theorems for Differential Equations on Measure Chains
Green s Funcions nd Comprison Theorems for Differenil Equions on Mesure Chins Lynn Erbe nd Alln Peerson Deprmen of Mhemics nd Sisics, Universiy of Nebrsk-Lincoln Lincoln,NE 68588-0323 lerbe@@mh.unl.edu
More informationFlow Networks Alon Efrat Slides courtesy of Charles Leiserson with small changes by Carola Wenk. Flow networks. Flow networks CS 445
CS 445 Flow Nework lon Efr Slide corey of Chrle Leieron wih mll chnge by Crol Wenk Flow nework Definiion. flow nework i direced grph G = (V, E) wih wo diingihed erice: orce nd ink. Ech edge (, ) E h nonnegie
More informationThe Residual Graph. 12 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
More informationMotion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl
More informationTemperature Rise of the Earth
Avilble online www.sciencedirec.com ScienceDirec Procedi - Socil nd Behviorl Scien ce s 88 ( 2013 ) 220 224 Socil nd Behviorl Sciences Symposium, 4 h Inernionl Science, Socil Science, Engineering nd Energy
More informationMain Reference: Sections in CLRS.
Maximum Flow Reied 09/09/200 Main Reference: Secion 26.-26. in CLRS. Inroducion Definiion Muli-Source Muli-Sink The Ford-Fulkeron Mehod Reidual Nework Augmening Pah The Max-Flow Min-Cu Theorem The Edmond-Karp
More information..,..,.,
57.95. «..» 7, 9,,. 3 DOI:.459/mmph7..,..,., E-mil: yshr_ze@mil.ru -,,. -, -.. -. - - ( ). -., -. ( - ). - - -., - -., - -, -., -. -., - - -, -., -. : ; ; - ;., -,., - -, []., -, [].,, - [3, 4]. -. 3 (
More informationThe Residual Graph. 11 Augmenting Path Algorithms. Augmenting Path Algorithm. Augmenting Path Algorithm
Augmening Pah Algorihm Greedy-algorihm: ar wih f (e) = everywhere find an - pah wih f (e) < c(e) on every edge augmen flow along he pah repea a long a poible The Reidual Graph From he graph G = (V, E,
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More informationNew Inequalities in Fractional Integrals
ISSN 1749-3889 (prin), 1749-3897 (online) Inernionl Journl of Nonliner Science Vol.9(21) No.4,pp.493-497 New Inequliies in Frcionl Inegrls Zoubir Dhmni Zoubir DAHMANI Lborory of Pure nd Applied Mhemics,
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationMath Week 12 continue ; also cover parts of , EP 7.6 Mon Nov 14
Mh 225-4 Week 2 coninue.-.3; lo cover pr of.4-.5, EP 7.6 Mon Nov 4.-.3 Lplce rnform, nd pplicion o DE IVP, epecilly hoe in Chper 5. Tody we'll coninue (from l Wednedy) o fill in he Lplce rnform ble (on
More informationStability in Distribution for Backward Uncertain Differential Equation
Sabiliy in Diribuion for Backward Uncerain Differenial Equaion Yuhong Sheng 1, Dan A. Ralecu 2 1. College of Mahemaical and Syem Science, Xinjiang Univeriy, Urumqi 8346, China heng-yh12@mail.inghua.edu.cn
More informationFractional Ornstein-Uhlenbeck Bridge
WDS'1 Proceeding of Conribued Paper, Par I, 21 26, 21. ISBN 978-8-7378-139-2 MATFYZPRESS Fracional Ornein-Uhlenbeck Bridge J. Janák Charle Univeriy, Faculy of Mahemaic and Phyic, Prague, Czech Republic.
More informationFIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 136, Number 3, March 28, Page 99 918 S 2-9939(7)989-2 Aricle elecronically publihed on November 3, 27 FIXED POINTS AND STABILITY IN NEUTRAL DIFFERENTIAL
More informationFURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)
Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. 8 (997), 79{83 FUTHE GENEALIZATIONS OF INEQUALITIES FO AN INTEGAL QI Feng Using he Tylor's formul we prove wo inegrl inequliies, h generlize K. S. K. Iyengr's
More informationPhysics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationu(t) Figure 1. Open loop control system
Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference
More informationEndpoint Strichartz estimates
Endpoin Sricharz esimaes Markus Keel and Terence Tao (Amer. J. Mah. 10 (1998) 955 980) Presener : Nobu Kishimoo (Kyoo Universiy) 013 Paricipaing School in Analysis of PDE 013/8/6 30, Jeju 1 Absrac of he
More informationAdditional Methods for Solving DSGE Models
Addiional Mehod for Solving DSGE Model Karel Meren, Cornell Univeriy Reference King, R. G., Ploer, C. I. & Rebelo, S. T. (1988), Producion, growh and buine cycle: I. he baic neoclaical model, Journal of
More informationHermite-Hadamard and Simpson Type Inequalities for Differentiable Quasi-Geometrically Convex Functions
Trkish Jornl o Anlysis nd Nmer Theory, 4, Vol, No, 4-46 Aville online h://ssciecom/jn/// Science nd Edcion Plishing DOI:69/jn--- Hermie-Hdmrd nd Simson Tye Ineliies or Dierenile Qsi-Geomericlly Convex
More informationAnalytical Solution of Time-Fractional Advection Dispersion Equation
Aville h://vu.edu/ Al. Al. Mh. ISSN: 93-9466 Vol. 4 Iue (June 9). 76 88 (Previoul Vol. 4 No. ) Alicion nd Alied Mheic: An Inernionl Journl (AAM) Anlicl Soluion of Tie-Frcionl Advecion Dierion Equion Triq
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddhrh Ngr, Nrynvnm Rod 5758 QUESTION BANK (DESCRIPTIVE) Subjec wih Code :Engineering Mhemic-I (6HS6) Coure & Brnch: B.Tech Com o ll Yer & Sem:
More informationLaplace Transform. Inverse Laplace Transform. e st f(t)dt. (2)
Laplace Tranform Maoud Malek The Laplace ranform i an inegral ranform named in honor of mahemaician and aronomer Pierre-Simon Laplace, who ued he ranform in hi work on probabiliy heory. I i a powerful
More informationMODELING OF DYNAMIC FORCES OF A TRACTOR AND THREE-POINT HITCHED IMPLEMENT IN THE MATLAB-SIMULINK PROGRAM ENVIRONMENT 1
TEKA Kom. Mo. Energ. Roln., 006, 6, 9 33 MODELING OF DYNAMIC FORCES OF A TRACTOR AND THREE-POINT HITCHED IMPLEMENT IN THE MATLAB-SIMULINK PROGRAM ENVIRONMENT Konny Juiucuk, Zbigniew Kmińki Dermen of Auomoive
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informationExplicit form of global solution to stochastic logistic differential equation and related topics
SAISICS, OPIMIZAION AND INFOMAION COMPUING Sa., Opim. Inf. Compu., Vol. 5, March 17, pp 58 64. Publihed online in Inernaional Academic Pre (www.iapre.org) Explici form of global oluion o ochaic logiic
More informationON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX.
ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX. MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Asrc. In his noe, we oin new some ineuliies
More information2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )
Scholrs Journl of Engineering d Technology SJET) Sch. J. Eng. Tech., ; A):8-6 Scholrs Acdemic d Scienific Publisher An Inernionl Publisher for Acdemic d Scienific Resources) www.sspublisher.com ISSN -X
More informationThe order of reaction is defined as the number of atoms or molecules whose concentration change during the chemical reaction.
www.hechemisryguru.com Re Lw Expression Order of Recion The order of recion is defined s he number of oms or molecules whose concenrion chnge during he chemicl recion. Or The ol number of molecules or
More informationIntegral Transform. Definitions. Function Space. Linear Mapping. Integral Transform
Inegrl Trnsform Definiions Funcion Spce funcion spce A funcion spce is liner spce of funcions defined on he sme domins & rnges. Liner Mpping liner mpping Le VF, WF e liner spces over he field F. A mpping
More informationGENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS
- TAMKANG JOURNAL OF MATHEMATICS Volume 5, Number, 7-5, June doi:5556/jkjm555 Avilble online hp://journlsmhkueduw/ - - - GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS MARCELA
More informationFUZZY n-inner PRODUCT SPACE
Bull. Korean Mah. Soc. 43 (2007), No. 3, pp. 447 459 FUZZY n-inner PRODUCT SPACE Srinivaan Vijayabalaji and Naean Thillaigovindan Reprined from he Bullein of he Korean Mahemaical Sociey Vol. 43, No. 3,
More informationSystems Variables and Structural Controllability: An Inverted Pendulum Case
Reserch Journl of Applied Sciences, Engineering nd echnology 6(: 46-4, 3 ISSN: 4-7459; e-issn: 4-7467 Mxwell Scienific Orgniion, 3 Submied: Jnury 5, 3 Acceped: Mrch 7, 3 Published: November, 3 Sysems Vribles
More informationLet. x y. denote a bivariate time series with zero mean.
Linear Filer Le x y : T denoe a bivariae ime erie wih zero mean. Suppoe ha he ime erie {y : T} i conruced a follow: y a x The ime erie {y : T} i aid o be conruced from {x : T} by mean of a Linear Filer.
More informationON DIFFERENTIATION OF A LEBESGUE INTEGRAL WITH RESPECT TO A PARAMETER
Mh. Appl. 1 (2012, 91 116 ON DIFFERENTIATION OF A LEBESGUE INTEGRAL WITH RESPECT TO A PARAMETER JIŘÍ ŠREMR Abr. The im of hi pper i o diu he bolue oninuiy of erin ompoie funion nd differeniion of Lebegue
More informationOn New Inequalities of Hermite-Hadamard-Fejér Type for Harmonically s-convex Functions via Fractional Integrals
Krelm en ve Müh. Derg. 6(:879 6 Krelm en ve Mühendili Dergii Jornl home ge: h://fd.en.ed.r eerch Aricle n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl Keirli İnegrller
More informationCHAPTER 7: SECOND-ORDER CIRCUITS
EEE5: CI RCUI T THEORY CHAPTER 7: SECOND-ORDER CIRCUITS 7. Inroducion Thi chaper conider circui wih wo orage elemen. Known a econd-order circui becaue heir repone are decribed by differenial equaion ha
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More information3. Renewal Limit Theorems
Virul Lborories > 14. Renewl Processes > 1 2 3 3. Renewl Limi Theorems In he inroducion o renewl processes, we noed h he rrivl ime process nd he couning process re inverses, in sens The rrivl ime process
More informationGEOMETRIC EFFECTS CONTRIBUTING TO ANTICIPATION OF THE BEVEL EDGE IN SPREADING RESISTANCE PROFILING
GEOMETRIC EFFECTS CONTRIBUTING TO ANTICIPATION OF THE BEVEL EDGE IN SPREADING RESISTANCE PROFILING D H Dickey nd R M Brennn Solecon Lbororie, Inc Reno, Nevd 89521 When preding reince probing re mde prior
More informationMath 2142 Homework 2 Solutions. Problem 1. Prove the following formulas for Laplace transforms for s > 0. a s 2 + a 2 L{cos at} = e st.
Mth 2142 Homework 2 Solution Problem 1. Prove the following formul for Lplce trnform for >. L{1} = 1 L{t} = 1 2 L{in t} = 2 + 2 L{co t} = 2 + 2 Solution. For the firt Lplce trnform, we need to clculte:
More informationNEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory and Application to Heat Transfer Model
Angn, A., e l.: New Frcionl Derivives wih Non-Locl nd THERMAL SCIENCE, Yer 216, Vol. 2, No. 2, pp. 763-769 763 NEW FRACTIONAL DERIVATIVES WITH NON-LOCAL AND NON-SINGULAR KERNEL Theory nd Applicion o He
More informationON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS
Hceepe Journl of Mhemics nd Sisics Volume 45) 0), 65 655 ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS M Emin Özdemir, Ahme Ock Akdemir nd Erhn Se Received 6:06:0 : Acceped
More informationSOLUTIONS TO ASSIGNMENT 2 - MATH 355. with c > 3. m(n c ) < δ. f(t) t. g(x)dx =
SOLUTIONS TO ASSIGNMENT 2 - MATH 355 Problem. ecall ha, B n {ω [, ] : S n (ω) > nɛ n }, and S n (ω) N {ω [, ] : lim }, n n m(b n ) 3 n 2 ɛ 4. We wan o show ha m(n c ). Le δ >. We can pick ɛ 4 n c n wih
More information1 Motivation and Basic Definitions
CSCE : Deign and Analyi of Algorihm Noe on Max Flow Fall 20 (Baed on he preenaion in Chaper 26 of Inroducion o Algorihm, 3rd Ed. by Cormen, Leieron, Rive and Sein.) Moivaion and Baic Definiion Conider
More informationBernoulli numbers. Francesco Chiatti, Matteo Pintonello. December 5, 2016
UNIVERSITÁ DEGLI STUDI DI PADOVA, DIPARTIMENTO DI MATEMATICA TULLIO LEVI-CIVITA Bernoulli numbers Francesco Chiai, Maeo Pinonello December 5, 206 During las lessons we have proved he Las Ferma Theorem
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More informationSeptember 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
More informationINTRODUCTION TO INERTIAL CONFINEMENT FUSION
INTODUCTION TO INETIAL CONFINEMENT FUSION. Bei Lecure 7 Soluion of he imple dynamic igniion model ecap from previou lecure: imple dynamic model ecap: 1D model dynamic model ρ P() T enhalpy flux ino ho
More informationCSC 373: Algorithm Design and Analysis Lecture 9
CSC 373: Algorihm Deign n Anlyi Leure 9 Alln Boroin Jnury 28, 2013 1 / 16 Leure 9: Announemen n Ouline Announemen Prolem e 1 ue hi Friy. Term Te 1 will e hel nex Mony, Fe in he uoril. Two nnounemen o follow
More informationANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2
ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise -: Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen
More informationSolutions of half-linear differential equations in the classes Gamma and Pi
Soluions of hlf-liner differenil equions in he clsses Gmm nd Pi Pvel Řehák Insiue of Mhemics, Acdemy of Sciences CR CZ-6662 Brno, Czech Reublic; Fculy of Educion, Msryk Universiy CZ-60300 Brno, Czech Reublic
More informationAlgorithmic Discrete Mathematics 6. Exercise Sheet
Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap
More informationChapter Direct Method of Interpolation
Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o
More informationNetwork Flow. Data Structures and Algorithms Andrei Bulatov
Nework Flow Daa Srucure and Algorihm Andrei Bulao Algorihm Nework Flow 24-2 Flow Nework Think of a graph a yem of pipe We ue hi yem o pump waer from he ource o ink Eery pipe/edge ha limied capaciy Flow
More information