Power Series Solutions to Generalized Abel Integral Equations
|
|
- Chester Stanley
- 5 years ago
- Views:
Transcription
1 Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp Power Series Solutios to Geerlized Abel Itegrl Equtios Rufi Abdulli * Deprtmet of Physics d Mthemtics, Sterlitm Brch of the Bshir Stte Uiversity, Sterlitm, Russi Abstrct Eve though they hve rther specilized structure, Abel equtios form importt clss of itegrl equtios i pplictios. This hppes becuse completely idepedet problems led to the solutio of such equtios. I this pper we cosider the geerlized Abel itegrl equtio of the first d secod id. Authors hve bee proposed ew method for costructig solutios of Abel by power series. Keywords Geerlized Abel Itegrl Equtios, Itegrl Equtio, Power Series Received: My, 5 / Accepted: My 7, 5 / Published olie: Jue 3, 5 The Authors. Published by Americ Istitute of Sciece. This Ope Access rticle is uder the CC BY-NC licese. Itroductio The rel world problems i scietific fields such s solid stte physics, plsm physics, fluid mechics, chemicl ietics d mthemticl biology re olier i geerl whe formulted s prtil differetil equtios or itegrl equtios. Abel s itegrl equtio occurs i my brches of scietific fields [], [] such s microscopy, seismology, rdio stroomy, electro emissio, tomic sctterig, rdr rgig, plsm digostics, X-ry rdiogrphy, d opticl fiber evlutio. Abel s itegrl equtio is the erliest emple of itegrl equtio [5], [9]. I this pper, we use the method of geerlized power series, to solve lier Volterr itegrl equtios of the first d secod id. This power series re udetermied coefficiets method, or method bsed o the pplictio of the Tylor series. The result obtied i the form of geerlized power series solutio further coverted to the iversio formul of the itegrl equtio. Oe such method is the represettio of the solutio of the equtio i the form of power series [8], []. Moreover, the bsic theorems of this sectio re give without proof.. Mi Results Cosider the geerlized Abel itegrl equtio of the first id: φ() t f ( ), () ( t) where < < rbitrry rel costt, f( ):[, b] R is give fuctio. Let s ssume tht the fuctio f( ) c be represeted s follows: f c c c () ( ) ( ) ( )... c( )... We shll see for solutio equtio i the form of the followig geerlized power series: φ() t ( t ) ( t ) ( t )... ( t )..., (3) where the uow coefficiets tht must be determied. Substitutig the power series (), (3) ito equtio (), we * Correspodig uthor E-mil ddress: rufi.bdulli94@mil.ru
2 Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp obti: ( ( ) ( )... ( )...) c c( )... c ( )... m m t t t ( t) ( t ) ( t) m t ( ) z ( ) dz t ( )( z) z z dz t ( ) z t z t z m m m( ) B(, m ) cm ( ). m m m m m( ) ( ) m (4) Β Euler bet fuctio, ( ) Here (, b) fuctio. Γ Euler gmm Let, the, equtig the terms with the sme power of i (4) yields: cm mb(, m ) cm m. B(, m ) If we substitute obtied coefficiets i (3), the by simple clcultios we obti: c( ) Γ ( ) c B(, ) Γ( ) Γ ( ) φ( ) ( )! si! c( ) ( ) ( )( ) ( ) c( ) Γ Γ si! c( ) si Γ( ) Γ ( ) c( ) ( ) Γ ( ) t z, dz, si m t с( ) ( z) z dz z, z t, t z t. ( ( )... ( )... ) si c c t c t si f( t) ( t) ( t) We re thus led to the solutio: si () φ( ) t f t. (5) ( ) This solutio is ideticl to the solutio tht is obtied i [4]. Theorem. If f ( t ) C [, b], the there eists uique solutio of equtio Abel of first id, which is epressed i the form (5). The theorem is proved by direct verifictio tht the formul (5) is solutio of equtio (). Let s ow cosider geerlized Abel itegrl equtio of the secod id: ( ) y( ) y t f( ). (6) t The solutio of this equtio will be sought i the form of sum of two geerlized power series: b (7) y( )
3 5 Rufi Abdulli: Power Series Solutios to Geerlized Abel Itegrl Equtios Similrly to the first cse, substitutig the power series (), (7) ito equtio (6), performig the clcultios, obti b t c b t z b dz b t z t t t z t z ( ) ( ) b z z dz b z z dz 3 b Β, b Β, The by equtig the terms with ideticl powers of we get: m bmβ, m, b c, 3 3 b Β, m c b Β, m Β, m c m m m m m p! Γ ( ) Γ( p ) p! ( p )! Γ ( p ) p p p p p b c c p p, p p c Β (, p ) p ( p )! p p!, c c Β (, ) ( )!! p p p p c! p p 3 p Γ ( ) ( p Γ Γ ) Γ p p c p 3 ( p )! 3 Γ Γ p p p 3 p c Β, Β, p p p ( p )! c Β, c Β, Β,! d this implies tht our solutio c be writte s y( ) c Β, c Β, Β,!
4 Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp ! c c Β (, ) F( ) c c Β, f( ) ct ( t) ct ( ) f( ) f t f( t) t 3 I c 3 3 (, ),,! c! Β Β Β 3 c t t c Β t t 3 ( ), ( )!! t ( t) ( t) 3 c c t Β,!! c t e c t Β ( t) ( t) 3,! ( ) ( t) t [ ] e f( t) F( t) f( t) e F( t) Ad s fil result: f( t) ( t) ( ) ( ) ( ) y f e F t. (8) t This solutio is ideticl to the solutio tht is obtied i [4]. Theorem. If f ( t ) C [, b], the there eist uique solutio of equtio Abel of secod id, which is epressed i the form (7). As i first cse, the theorem is proved by direct verifictio tht the formul (8) is solutio of equtio (6). 3. Coclusios I this pper, solutio is obtied by power series method. This my be used i more combitoril wy to obti solutio of higher degree o-lier itegrl equtios. Acowledgmets My sicere ths go to Prof. Adrey Aimov from the Sterlitm Brch of the Bshir Stte Uiversity for his help d dvice. Refereces [] R. Goreflo d S. Vessell, Abel itegrl Equtios: Alysis d Applictio, Spriger-Verlg, Berli-New Yor, 99. [] Jerri, A Itroductio to Itegrl Equtios with Applictios. Wiley, New Yor, 999. [3] Dvis, H. T. Itroductio to Nolier Differetil d Itegrl Equtios, st ed., Dover Publictios, Ic., New Yor, 96. [4] A. V. Mzhirov d A. D. Polyi, Hdboo of Itegrl Equtios: Solutio Methods [i Russi], Fctoril Press, Moscow,. [5] Deutsch, M, Beimiy, I Derivtive-free Iversio of AbeVs Itegrl Equtio. Applied Physics Letters 4: pp. 7-8, 98. [6] Abel, NH (86) Auflosug eier Mechische Aufgbe. Jourl für Reie Agewe Mthemti : pp , 86. [7] Adersse, RS Stble Procedures for the Iversio of AbeVs Equtio. Jourl of the Istitute of Mthemtics d its Applictios 7: pp , 976. [8] Mierbo, GN, Levy, ME Iversio of AbeVs Itegrl Equtio by Mes of Orthogol Polyomils. SIAM Jourl o Numericl Alysis 6: pp , 969. [9] J. D. Tmri, O itegrble solutios of Abel s itegrl equtio, Als of Mthemtics, vol. 3, o., pp. 9 9, 93.
5 54 Rufi Abdulli: Power Series Solutios to Geerlized Abel Itegrl Equtios [] A.C. Pipi, A Course o Itegrl Equtios, Spriger Verlg, New Yor, 99.
Double Sums of Binomial Coefficients
Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u
More informationVariational Iteration Method for Solving Volterra and Fredholm Integral Equations of the Second Kind
Ge. Mth. Notes, Vol. 2, No. 1, Jury 211, pp. 143-148 ISSN 2219-7184; Copyright ICSRS Publictio, 211 www.i-csrs.org Avilble free olie t http://www.gem.i Vritiol Itertio Method for Solvig Volterr d Fredholm
More informationNumerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials
Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,
More informationSOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM
Jourl of Applied Mthemtics d Computtiol Mechics () 57-6 SOUION O DIERENIA EQUAION OR HE EUER-ERNOUI EAM Izbel Zmorsk Istitute of Mthemtics Czestochow Uiversit of echolog Częstochow Pold izbel.zmorsk@im.pcz.pl
More informationAPPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES
Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics,
More informationSOME SHARP OSTROWSKI-GRÜSS TYPE INEQUALITIES
Uiv. Beogrd. Publ. Elektroteh. Fk. Ser. Mt. 8 006 4. Avilble electroiclly t http: //pefmth.etf.bg.c.yu SOME SHARP OSTROWSKI-GRÜSS TYPE INEQUALITIES Zheg Liu Usig vrit of Grüss iequlity to give ew proof
More informationConvergence rates of approximate sums of Riemann integrals
Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki
More informationSome New Iterative Methods Based on Composite Trapezoidal Rule for Solving Nonlinear Equations
Itertiol Jourl of Mthemtics d Sttistics Ivetio (IJMSI) E-ISSN: 31 767 P-ISSN: 31-759 Volume Issue 8 August. 01 PP-01-06 Some New Itertive Methods Bsed o Composite Trpezoidl Rule for Solvig Nolier Equtios
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationAbel Resummation, Regularization, Renormalization & Infinite Series
Prespcetime Jourl August 3 Volume 4 Issue 7 pp. 68-689 Moret, J. J. G., Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series Abel Resummtio, Regulriztio, Reormliztio & Ifiite Series 68 Article Jose
More informationIntegral Operator Defined by k th Hadamard Product
ITB Sci Vol 4 A No 35-5 35 Itegrl Opertor Deied by th Hdmrd Product Msli Drus & Rbh W Ibrhim School o Mthemticl Scieces Fculty o sciece d Techology Uiversiti Kebgs Mlysi Bgi 436 Selgor Drul Ehs Mlysi Emil:
More informationTHE SOLUTION OF THE FRACTIONAL DIFFERENTIAL EQUATION WITH THE GENERALIZED TAYLOR COLLOCATIN METHOD
IJRRAS August THE SOLUTIO OF THE FRACTIOAL DIFFERETIAL EQUATIO WITH THE GEERALIZED TAYLOR COLLOCATI METHOD Slih Ylçıbş Ali Kourlp D. Dömez Demir 3* H. Hilmi Soru 4 34 Cell Byr Uiversity Fculty of Art &
More informationCertain sufficient conditions on N, p n, q n k summability of orthogonal series
Avilble olie t www.tjs.com J. Nolier Sci. Appl. 7 014, 7 77 Reserch Article Certi sufficiet coditios o N, p, k summbility of orthogol series Xhevt Z. Krsiqi Deprtmet of Mthemtics d Iformtics, Fculty of
More informationCALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS
CALCULATION OF CONVOLUTION PRODUCTS OF PIECEWISE DEFINED FUNCTIONS AND SOME APPLICATIONS Abstrct Mirce I Cîru We preset elemetry proofs to some formuls ive without proofs by K A West d J McClell i 99 B
More information2. Infinite Series 3. Power Series 4. Taylor Series 5. Review Questions and Exercises 6. Sequences and Series with Maple
Chpter II CALCULUS II.4 Sequeces d Series II.4 SEQUENCES AND SERIES Objectives: After the completio of this sectio the studet - should recll the defiitios of the covergece of sequece, d some limits; -
More informationNumerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials
Jourl of Al-Nhri Uiversity Vol.5 (), Mrch,, pp.-9 Sciece Numericl Solutio of Fuzzy Fredholm Itegrl Equtios of the Secod Kid usig Berstei Polyomils Srmd A. Altie Deprtmet of Computer Egieerig d Iformtio
More informationReduction of Higher Order Linear Ordinary Differential Equations into the Second Order and Integral Evaluation of Exact Solutions
Mthemtic Aeter, Vol. 4, 04, o., 75-89 Reductio o Higher Order Lier Ordiry Dieretil Equtios ito the Secod Order d Itegrl Evlutio o Ect Solutios Guw Nugroho* Deprtmet o Egieerig Physics, Istitut Tekologi
More informationEigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):
Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )
More informationA general theory of minimal increments for Hirsch-type indices and applications to the mathematical characterization of Kosmulski-indices
Mlysi Jourl of Librry & Iformtio Sciece, Vol. 9, o. 3, 04: 4-49 A geerl theory of miiml icremets for Hirsch-type idices d pplictios to the mthemticl chrcteriztio of Kosmulski-idices L. Egghe Uiversiteit
More informationGreen s Function Approach to Solve a Nonlinear Second Order Four Point Directional Boundary Value Problem
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 3 Ver. IV (My - Jue 7), PP -8 www.iosrjourls.org Gree s Fuctio Approch to Solve Nolier Secod Order Four Poit Directiol
More information=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property
Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h(
More informationSome Properties of Brzozowski Derivatives of Regular Expressions
Itertiol Jourl of Computer Treds d Techology (IJCTT) volume 13 umber 1 Jul 014 Some Properties of Brzozoski erivtives of Regulr Expressios NMuruges #1, OVShmug Sudrm * #1 Assistt Professor, ept of Mthemtics,
More informationConvergence rates of approximate sums of Riemann integrals
Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationKing Fahd University of Petroleum & Minerals
Kig Fhd Uiversity of Petroleum & Mierls DEPARTMENT OF MATHEMATICAL CIENCE Techicl Report eries TR 434 April 04 A Direct Proof of the Joit Momet Geertig Fuctio of mple Me d Vrice Awr H. Jorder d A. Lrdji
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationOn The Homogeneous Quintic Equation with Five Unknowns
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-78,p-ISSN: 319-76X, Volume 7, Issue 3 (Jul. - Aug. 013), PP 7-76 www.iosrjourls.org O The Homogeeous Quitic Equtio with Five Ukows y y 3 3 ( y ) 3(( y)( z w
More informationINTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION
Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: 9-006 Volume: 04 Issue: Mr -017 www.irjet.et p-issn: 9-007 INTEGRL SOLUTIONS OF THE TERNRY CUBIC EQUTION y ) 4y y ) 97z G.Jki 1, C.Sry,* ssistt
More informationf(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
More informationReversing the Arithmetic mean Geometric mean inequality
Reversig the Arithmetic me Geometric me iequlity Tie Lm Nguye Abstrct I this pper we discuss some iequlities which re obtied by ddig o-egtive expressio to oe of the sides of the AM-GM iequlity I this wy
More informationRemarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
More informationDiscrete Mathematics I Tutorial 12
Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationMultiplication and Translation Operators on the Fock Spaces for the q-modified Bessel Function *
Advces i Pure Mthemtics 0-7 doi:0436/pm04039 Pulished Olie July 0 (http://wwwscirporg/jourl/pm) Multiplictio d Trsltio Opertors o the Fock Spces or the -Modiied Bessel Fuctio * Astrct Fethi Solti Higher
More informationAge-Structured Population Projection of Bangladesh by Using a Partial Differential Model with Quadratic Polynomial Curve Fitting
Ope Jourl of Applied Scieces, 5, 5, 54-55 Published Olie September 5 i SciRes. http://www.scirp.org/jourl/ojpps http://dx.doi.org/.436/ojpps.5.595 Age-Structured Popultio Projectio of Bgldesh by Usig Prtil
More informationA GENERALIZATION OF GAUSS THEOREM ON QUADRATIC FORMS
A GENERALIZATION OF GAU THEOREM ON QUADRATIC FORM Nicole I Brtu d Adi N Cret Deprtmet of Mth - Criov Uiversity, Romi ABTRACT A origil result cocerig the extesio of Guss s theorem from the theory of biry
More informationLinear Programming. Preliminaries
Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationUpper Bound of Partial Sums Determined by Matrix Theory
Turish Jourl of Alysis d Nuber Theory, 5, Vol, No 6, 49-5 Avilble olie t http://pubssciepubco/tjt//6/ Sciece d Eductio Publishig DOI:69/tjt--6- Upper Boud of Prtil Sus Deteried by Mtrix Theory Rbh W Ibrhi
More informationInternational Journal of Mathematical Archive-5(1), 2014, Available online through ISSN
Itertiol Jourl of Mthemticl Archive-5( 4 93-99 Avilble olie through www.ijm.ifo ISSN 9 546 GENERALIZED FOURIER TRANSFORM FOR THE GENERATION OF COMPLE FRACTIONAL MOMENTS M. Gji F. Ghrri* Deprtmet of Sttistics
More informationPrior distributions. July 29, 2002
Prior distributios Aledre Tchourbov PKI 357, UNOmh, South 67th St. Omh, NE 688-694, USA Phoe: 4554-64 E-mil: tchourb@cse.ul.edu July 9, Abstrct This documet itroduces prior distributios for the purposes
More informationChapter 25 Sturm-Liouville problem (II)
Chpter 5 Sturm-Liouville problem (II Speer: Lug-Sheg Chie Reerece: [] Veerle Ledou, Study o Specil Algorithms or solvig Sturm-Liouville d Schrodiger Equtios. [] 王信華教授, chpter 8, lecture ote o Ordiry Dieretil
More informationResearch Article The Applications of Cardinal Trigonometric Splines in Solving Nonlinear Integral Equations
ISRN Applied Mthemtics Volume 214, Article ID 21399, 7 pges http://d.doi.org/1.1155/214/21399 Reserch Article The Applictios of Crdil Trigoometric Splies i Solvig Nolier Itegrl Equtios Ji Xie, 1 Xioy Liu,
More informationParticle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise
Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The
More informationImportant Facts You Need To Know/Review:
Importt Fcts You Need To Kow/Review: Clculus: If fuctio is cotiuous o itervl I, the its grph is coected o I If f is cotiuous, d lim g Emple: lim eists, the lim lim f g f g d lim cos cos lim 3 si lim, t
More informationLecture 38 (Trapped Particles) Physics Spring 2018 Douglas Fields
Lecture 38 (Trpped Prticles) Physics 6-01 Sprig 018 Dougls Fields Free Prticle Solutio Schrödiger s Wve Equtio i 1D If motio is restricted to oe-dimesio, the del opertor just becomes the prtil derivtive
More informationAN INEQUALITY OF GRÜSS TYPE FOR RIEMANN-STIELTJES INTEGRAL AND APPLICATIONS FOR SPECIAL MEANS
RGMIA Reserch Report Collectio, Vol., No., 998 http://sci.vut.edu.u/ rgmi/reports.html AN INEQUALITY OF GRÜSS TYPE FOR RIEMANN-STIELTJES INTEGRAL AND APPLICATIONS FOR SPECIAL MEANS S.S. DRAGOMIR AND I.
More informationABEL RESUMMATION, REGULARIZATION, RENORMALIZATION AND INFINITE SERIES
ABEL RESUMMATION, REGULARIZATION, RENORMALIZATION AND INFINITE SERIES Jose Jvier Grci Moret Grdute studet of Physics t the UPV/EHU (Uiversity of Bsque coutry) I Solid Stte Physics Addres: Prctictes Ad
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationSolutions to Problem Set 7
8.0 Clculus Jso Strr Due by :00pm shrp Fll 005 Fridy, Dec., 005 Solutios to Problem Set 7 Lte homework policy. Lte work will be ccepted oly with medicl ote or for other Istitute pproved reso. Coopertio
More informationSOLUTION OF SYSTEM OF LINEAR EQUATIONS. Lecture 4: (a) Jacobi's method. method (general). (b) Gauss Seidel method.
SOLUTION OF SYSTEM OF LINEAR EQUATIONS Lecture 4: () Jcobi's method. method (geerl). (b) Guss Seidel method. Jcobi s Method: Crl Gustv Jcob Jcobi (804-85) gve idirect method for fidig the solutio of system
More informationCalculus II Homework: The Integral Test and Estimation of Sums Page 1
Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More informationSupplemental Handout #1. Orthogonal Functions & Expansions
UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede Supplemetl Hdout #1 Orthogol Fuctios & Epsios Cosider fuctio f ( ) which is defied o the itervl. The fuctio f ( )
More informationChapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Dr. Hamid R. Rabiee Fall 2013
Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Dr. Hmid R. Rbiee Fll 03 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Outlie Itroductio to the -Trsform Properties
More informationResearch Article Refinements of Aczél-Type Inequality and Their Applications
Hidwi Pulishig Corportio Jourl of Applied Mthetics Volue 04, Article ID 58354, 7 pges http://dxdoiorg/055/04/58354 Reserch Article Refieets of Aczél-Type Iequlity d Their Applictios Jigfeg Ti d We-Li Wg
More informationThe Elementary Arithmetic Operators of Continued Fraction
Americ-Eursi Jourl of Scietific Reserch 0 (5: 5-63, 05 ISSN 88-6785 IDOSI Pulictios, 05 DOI: 0.589/idosi.ejsr.05.0.5.697 The Elemetry Arithmetic Opertors of Cotiued Frctio S. Mugssi d F. Mistiri Deprtmet
More informationCanonical Form and Separability of PPT States on Multiple Quantum Spaces
Coicl Form d Seprbility of PPT Sttes o Multiple Qutum Spces Xio-Hog Wg d Sho-Mig Fei, 2 rxiv:qut-ph/050445v 20 Apr 2005 Deprtmet of Mthemtics, Cpitl Norml Uiversity, Beijig, Chi 2 Istitute of Applied Mthemtics,
More informationAbel type inequalities, complex numbers and Gauss Pólya type integral inequalities
Mthemticl Commuictios 31998, 95-101 95 Abel tye iequlities, comlex umbers d Guss Póly tye itegrl iequlities S. S. Drgomir, C. E. M. Perce d J. Šude Abstrct. We obti iequlities of Abel tye but for odecresig
More informationIsolating the Polynomial Roots with all Zeros Real
th WSEAS Itertiol Coferece o COMPUTERS Herlio Greece July 3-5 008 Isoltig the Polyomil Roots with ll Zeros Rel MURESAN ALEXE CALIN Deprtmet of Mthemtics Uiversity Petroleum d Gs Ploiesti Romi Str Democrtiei
More informationA VERSION OF THE KRONECKER LEMMA
UPB Sci Bull, Series A, Vol 70, No 2, 2008 ISSN 223-7027 A VERSION OF THE KRONECKER LEMMA Gheorghe BUDIANU I lucrre se prezit o vrit lemei lui Kroecer reltiv l siruri si serii de umere rele Rezulttele
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationFrequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationQuadrature Methods for Numerical Integration
Qudrture Methods for Numericl Itegrtio Toy Sd Istitute for Cle d Secure Eergy Uiversity of Uth April 11, 2011 1 The Need for Numericl Itegrtio Nuemricl itegrtio ims t pproximtig defiite itegrls usig umericl
More information[Ismibayli*, 5(3): March, 2016] ISSN: (I2OR), Publication Impact Factor: 3.785
IJERT INTERNATIONAL JOURNAL OF ENGINEERING CIENCE & REEARCH TECHNOLOGY IMULATION OF ELECTROMAGNETIC FIELD FROM MICROWAVE RECTANGULAR WAVEGUIDE TO CIRCULAR IN TRANITION DEVICE E.G.Ismibyli, I.J.Islmov,
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationDefinite Integral. The Left and Right Sums
Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give
More informationNumerical method for solving system of Fredhlom integral equations using Chebyshev cardinal functions
2014 2014) 1-13 Avilble olie t www.ispcs.com/cte Volume 2014, Yer 2014 Article ID cte-00165, 13 Pges doi:10.5899/2014/cte-00165 Reserch Article umericl method for solvig system of Fredhlom itegrl equtios
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationToeplitz and Translation Operators on the q-fock Spaces *
Advces i Pure Mthemtics 35-333 doi:436/pm659 Published Olie November (http://wwwscirporg/jourl/pm) Toeplit d Trsltio Opertors o the -Foc Spces * Abstrct Fethi Solti Higher College o Techology d Iormtics
More informationComputer Algebra Algorithms for Orthogonal Polynomials and Special Functions
Computer Alger Algorithms for Orthogol Polyomils d Specil Fuctios Prof. Dr. Wolfrm Koepf Deprtmet of Mthemtics Uiversity of Kssel oepf@mthemti.ui-ssel.de http://www.mthemti.ui-ssel.de/~oepf Olie Demostrtios
More informationDETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1
NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit
More informationCOMPOSITE TRAPEZOID RULE FOR THE RIEMANN-STIELTJES INTEGRAL AND ITS RICHARDSON EXTRAPOLATION FORMULA
itli jourl of pure d pplied mthemtics. 5 015 (11 18) 11 COMPOSITE TRAPEZOID RULE FOR THE RIEMANN-STIELTJES INTEGRAL AND ITS RICHARDSON EXTRAPOLATION FORMULA Weijig Zho 1 College of Air Trffic Mgemet Civil
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationFourier Series and Applications
9/7/9 Fourier Series d Applictios Fuctios epsio is doe to uderstd the better i powers o etc. My iportt probles ivolvig prtil dieretil equtios c be solved provided give uctio c be epressed s iiite su o
More informationOrthogonal functions - Function Approximation
Orthogol uctios - Fuctio Approimtio - he Problem - Fourier Series - Chebyshev Polyomils he Problem we re tryig to pproimte uctio by other uctio g which cosists o sum over orthogol uctios Φ weighted by
More information1 Tangent Line Problem
October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,
More informationUNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)
UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte
More informationStatistics for Financial Engineering Session 1: Linear Algebra Review March 18 th, 2006
Sttistics for Ficil Egieerig Sessio : Lier Algebr Review rch 8 th, 6 Topics Itroductio to trices trix opertios Determits d Crmer s rule Eigevlues d Eigevectors Quiz The cotet of Sessio my be fmilir to
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationReview of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
More informationsin m a d F m d F m h F dy a dy a D h m h m, D a D a c1cosh c3cos 0
Q1. The free vibrtio of the plte is give by By ssumig h w w D t, si cos w W x y t B t Substitutig the deflectio ito the goverig equtio yields For the plte give, the mode shpe W hs the form h D W W W si
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationClosed Newton-Cotes Integration
Closed Newto-Cotes Itegrtio Jmes Keeslig This documet will discuss Newto-Cotes Itegrtio. Other methods of umericl itegrtio will be discussed i other posts. The other methods will iclude the Trpezoidl Rule,
More information(1 q an+b ). n=0. n=0
AN ELEMENTARY DERIVATION OF THE ASYMPTOTICS OF PARTITION FUNCTIONS Diel M Ke Abstrct Let S,b { + b : 0} where is iteger Let P,b deote the umber of prtitios of ito elemets of S,b I prticulr, we hve the
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric
More information9.5. Alternating series. Absolute convergence and conditional convergence
Chpter 9: Ifiite Series I this Chpter we will be studyig ifiite series, which is just other me for ifiite sums. You hve studied some of these i the pst whe you looked t ifiite geometric sums of the form:
More information