Two-step Extrapolated Newton s Method with High Efficiency Index

Size: px
Start display at page:

Download "Two-step Extrapolated Newton s Method with High Efficiency Index"

Transcription

1 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 Two-step Etrapolated Newto s Method with High Efficiecy Ide V.B. Kumar Vatti Dept. of Egieerig Mathematics Adhra Uiversity Visakhapatam Idia. drvattivbk@yahoo.co.i Ramadevi Sri Dept. of Egieerig Mathematics Adhra Uiversity Visakhapatam Idia. ramadevisri0@yahoo.com M.S. Kumar Mylapalli Dept. of Mathematics Gitam Uiversity Visakhapatam Idia. mairaj9@yahoo.co.i Subimal Deb Dept. of Physics Gitam Uiversity Visakhapatam Idia. subimal.deb@gmail.com Abstract--- I this paper two steps of the three step eighteeth order coverget method developed by Vattiet.al is cosidered ad it is show that this method has high efficiecy ide. Few umerical eamples are cosidered for compariso. AMS Subject Classificatio: 1A 6K0 6H0. Keywords--- Iterative Method Noliear Equatio Newto s Method Covergece Aalysis Higher Order Covergece. I. Itroductio For fidig the zero s of a oliear equatio f ( ) 0 (1.1) Where f : D R R may be algebraic trascedetal or combied of both the Eighteeth Order Coverget Method (EOCM) developed by Vattiet.al [11] is give by: For a give compute 0 is a scalar fuctio o a ope iterval D ad f by the iterative schemes + 1 ( ) ( ) f ( y ) f w f f w 1 w f w 1 ˆ ρ (1.) 1 y 1 + f y 1 1 ρ + ( ) f ( w ) Where ˆ f w ρ (1.) f ( w ) f ( ) f ( y ) ad ρ f ( y ) (1.) The eighteeth order three step Predictor-corrector Newto s Halley method (PCNH) developed by Mohamed see 1 to 10 is give by: ad Hafiz ( [ ]) For a give 0 we compute by usig + 1 ISSN 19-0X 8

2 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 ( ) ( ) ( ) f ( w ) f ( w ) f ( w ) f w f f w f w w (1.) f ( ) f ( ) f ( ) y 1 + f ( ) f ( ) ( ) The ith order two step Predictor-corrector Halley method (PCH) developed by Mohamed ad Hafiz [8] is give by: For a give 0 we compute by usig + 1 f ( ) f ( ) f f f f ( ) f ( ) f ( ) y 1 + f ( ) f ( ) ( ) (1.) I sectio of this paper we cosider the two step Etrapolated Newto s method ad discuss the covergece criteria of the method. Few umerical eamples are cosidered i the cocludig sectio for compariso. II. Two-Step Etrapolated Newto s Method (TENM) Igorig the first step ad cosiderig the last two steps of the method (1.) we have f ( w ) 1 w f ( w ) 1 ˆ ρ f ( y ) 1 y 1 + f ( y ) 1+ 1 ρ ( ) where ˆ ρ ad ρ are as give i (1.) ad (1.). Replacig w by i the above we thus have the followig algorithm: (.1) Algorithm.1: For a give compute 0 ( ) f ( y ) by the iterative schemes + 1 f 1 f 1 ˆ ρ 1 y 1 + f y 1+ 1 ρ ( ) (.) ISSN 19-0X 9

3 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 Where ˆ ρ ad ρ f f f ( ) ( ) ( ) f f y f y y This algorithm ca be called as Two-step Etrapolated Newto s Method (TENM) ad it requires si fuctioal evaluatios. III. Covergece Criteria Theorem.1.Let (.) (.) Dbe a sigle zero of a sufficietly differetiable fuctio f : D R Rfor a ope iterval D. If is i the viciity of the the algorithm (.1) has ith order covergece. 0 Proof: Let be a sigle zero of (1.1) ad If be the we have Ad Now + e (.1) The f ( ) 0 (.) th approimate to the root of (1.1) the epadig f ( ) about usig Taylor s epasio ( ) ( ) e ( e ) ( ) f f + e f + f f...! +! + 1 f ( ) 1 f ( ) 1 f v ( ) f e + e e e...! +! +! + f ( ) f ( ) f ( ) f ( ) e ce ce ce (.) wherec j ( ) j 1 f... j! f ( j ) f 1 f ce ce ce (.) ( ) f f c + 6ce 1 + ce 0ce ce ( ) ( c 7cc + c ) e + oe ( ) f e ce c c e f ˆ ρ ad 1 f f f ( ) (.) (.6) ISSN 19-0X 10

4 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No c e + ( 6c + c ) e + ( 1c + 8c c) e + ( 0c 1 6 )... + c c + c e + + ( 1 6 ) + ( 6 8 ) ( 80c 1c c 8c c 7c 10c ) e... 1 c e c c e c c c c e S e S e S e S e (.7) where S c S c + c S 8c 1cc + 6c 1 S 0c c 0c cc 1c + 10c Now ( ) 1 1 ˆ ρ 1 1 S e S e S e S e where Q c Qe Qe Qe Qe (.8) Q c + c Q 8c 1cc + 6c Q 0c c 0c cc 1c + 10c Multiplyig (.6) with (.8) we obtai 1 ρ 1 + ( ) + ( ) ( Q cq Q( c c ) c 7cc c ) e... 1 f e Q c e Q cq c c e f Thus usig (.1) ad (.9) we have e + c c e oe... + (.9) y + T (.10) where T ( c c ) e. (.11) Now epadig f ( ) f ( ) f ( ) about by usig (.10) we obtai f y f T ct ct ct f ( y ) f ( ) 1+ ct +... ct + ct + ct + c + 6cT+ 1cT f ( ) f ( ) 0cT ct ( ) ad ρ ( ) ( ) f f y f y y c T + ( 6c + c ) T + ( 1c + 8c c ) T + ( 0c 1 6 )... + c c + c T + c + ( 1c 6c ) + ( 6c c c 8c ) ( 80c 1c c 8c c 7c 10c ) T... 1 T T T (.1) (.1) (.1) ISSN 19-0X 11

5 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 PT+ PT + PT + PT +... (.1) 1 Where P 1 P 6c 6c P 16c 8cc + 1c P 100c c 0c 0cc 0c + 0c c + Now P P ρ PT P T P PP 1 T P + P PP P P P T Ad 1 M T M T M T M T Where M 1 M c c c P P P P PP P T + T T 1+ 1 ρ P P PP 1 P P P T (.16) (.17) M c + 8cc 6 c M c c + c + 1cc c 10c Now agai ( ) + ( ) 1 1 ρ + ( M + MM M M M + M ) T +... MT M M T MM M M T N T + N T + N T + N T +... (.18) 1 where N c N c + c N c cc + 6 c 1 N 1c c + c cc + 1c + 10c From (.1) ad (.1) we obtai f ( y ) T ct c c T 7cc c c T ot + (.19) + + f y From (.18) ad (.19) we obtai f ( y ) ˆ ρ f ( y ) + T + [ 0] T + c T + T N c T N Nc c c T N Nc N 1 1 ( c c) 7 cc c c + + T 1 + ot N Nc N 1( c c ) 7 cc c + + c T ot + (.0) Hece usig (.1) ad (.0) we have + T T+ ct ISSN 19-0X 1

6 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 i.e. e c( c c ) e + oe ( ) ct +... c ( c c ) e Hece this method has ith order covergece ad its efficiecy ide is Case.1: By epadig 1 ρ (.1) appearig i the deomiator of the secod step of algorithm.1 we obtai ( y ) ρ ρ f 1 y 1 f y + ρ f ( y ) ρ 1 ρ ρ y f ( y ) 8 where ρ is as give i (.). (.) Cosiderig the first degree ad secod degree terms of the epressio lyig withi the square brackets of the formula (.) ad from the algorithm.1 we have the followig two more algorithms. Algorithm.1: For a give 0 compute by the iterative schemes + 1 f ( ) 1 f ( ) 1 ˆ ρ Where f ( ) ˆ f ρ f ( ) f ( ) f ( ) f ( ) y 1 ( ) + f ( ) f ( ) Which is same as the ith order coverget method proposed by Mohammed ad Hafiz [8]. Algorithm.: For a give 0 compute by the iterative schemes + 1 f ( ) 1 f ( ) 1 ˆ ρ Where f ( ) ˆ f ρ f ( ) ( ) f y f f y 1 + f f f ( ) f ( ) f ( y ) 9 As doe i covergece criteria above i sectio oe ca easily obtai e α e + 1. Therefore the algorithm (.) has ith order covergece ad hece its efficiecy ide is which is same as that of the method (1.6) ad better tha those of the methods (1.) ad (1.) both have efficiecy ide as ISSN 19-0X 1

7 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 IV. Numerical Eamples We cosider the same eamples cosidered by Mohammed ad Hafiz [10] ad compared TENM with PCH PCNH ad EOCM methods. The computatios are carried out by usig mpmath-python software programmig 01 ad compariso of umber of iteratios for these methods are obtaied such that 1 < + 10 ad 01 f ( + 1) < 10. Table 1: Compariso of Differet Methods Method The equatio Iitial Guess f ( ) f ( ) f + 10 PCH E-01 E-199 PCNH E-01 E-199 EOCM E-01 E-198 TENM E E f si + 1 PCH E-01 E-00 PCNH E-01 E-00 EOCM E-00 E-00 TENM E-00 E-199 f e + PCH E E-01 PCNH E E-01 EOCM E E-01 TENM E E-01 - f e + PCH E E-01 PCNH E E-01 EOCM E E-01 TENM E E f cos PCH E-01. E-01 PCNH E -01. E-01 EOCM E-01. E-01 TENM E-01.E-01. f ( 1) 1 PCH PCNH EOCM TENM f 10 PCH E -00 PCNH E -00 EOCM E E E E-199 TENM E E-198 It is evidet from the tabulated values that TENM is superior to the method (1.6) cosiderig the umber of iteratios ad the rate at which TENM coverged is almost same as that of the methods (1.) ad (1.)although the ISSN 19-0X 1

8 Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 method (.1) is a ith order method. This is because the efficiecy ide of algorithm (.1) is more tha those of the methods (1.) ad (1.). Refereces [1] Argyros I.K. ad Khattri S.K. A improved semilocal covergece aalysis for the Chebyshev method. Joural of Applied Mathematics ad Computig (1-) (01) [] Hafiz M.A. Solvig oliear equatios usig Steffese-type methods with optimal order of covergece. Palestie Joural of Mathematics (1) (01) [] Hafiz M.A. A ew combied bracketig method for solvig oliear equatios. Joural of Mathematical ad Computatioal Sciece (1) (01) [] Hafiz M.A. ad Al-Goria S.M. Solvig oliear equatios usig a ew teth-ad seveth-order methods free from secod derivative. Iteratioal Joural of Differetial Equatios ad Applicatios 1 () (01). [] Khattri S.K. ad Log T. Costructig third-order derivative-free iterative methods. Iteratioal Joural of Computer Mathematics 88 (7) (011) [6] Khattri S.K. Quadrature based optimal iterative methods with applicatios i high-precisio computig. Numerical Mathematics: Theory Methods ad Applicatios (0) (01) [7] Khattri S.K. ad Steihaug T. Algorithm for formig derivative-free optimal methods. Numerical Algorithms 6 () (01) [8] Bahgat M.S. New Two-step predictor-corrector method with ith-order covergece for solvig oliear equatios. Joural: Joural of Advaces i Mathematics () (01). [9] Noor K.I. Noor M.A. ad Momai S. Modified Householder iterative method for oliear equatios. Applied mathematics ad computatio 190 () (007) [10] Bahgat M.S. ad Hafiz M.A. Three-Step Iterative Method with Eighteeth Order Covergece for Solvig Noliear Equatios. Iteratioal Joural of pure ad applied Mathematics 9 1 (01) 8-9. [11] Vatti V.K. Sri R. ad Mylapalli M.K. Eighteeth Order Coverget Method for Solvig No-Liear Equatios. Orietal Joural of Computer Sciece ad Techology 10 (1) (017) ISSN 19-0X 1

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem

More information

Some New Iterative Methods for Solving Nonlinear Equations

Some New Iterative Methods for Solving Nonlinear Equations World Applied Scieces Joural 0 (6): 870-874, 01 ISSN 1818-495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida

More information

Solving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods

Solving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods Applied ad Computatioal Mathematics 07; 6(6): 38-4 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 38-5605 (Prit); ISSN: 38-563 (Olie) Solvig a Noliear Equatio Usig a New Two-Step

More information

Numerical Solution of Non-linear Equations

Numerical Solution of Non-linear Equations Numerical Solutio of Noliear Equatios. INTRODUCTION The most commo reallife problems are oliear ad are ot ameable to be hadled by aalytical methods to obtai solutios of a variety of mathematical problems.

More information

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations Copyright, Darbose Iteratioal Joural o Applied Mathematics ad Computatio Volume (), pp -6, 9 http//: ijamc.darbose.com Some Variats o Newto's Method with Fith-Order ad Fourth-Order Covergece or Solvig

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series

More information

Solution of Differential Equation from the Transform Technique

Solution of Differential Equation from the Transform Technique Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from

More information

-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION

-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective

More information

Research Article A New Second-Order Iteration Method for Solving Nonlinear Equations

Research Article A New Second-Order Iteration Method for Solving Nonlinear Equations Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif

More information

International Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through ISSN

International Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through   ISSN Iteratioal Joural o Matheatical Archive-4(9), 03, -5 Available olie through www.ija.io ISSN 9 5046 THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED NEWTON RAPHSON METHOD FOR SOLVING NONLINEAR

More information

Newton Homotopy Solution for Nonlinear Equations Using Maple14. Abstract

Newton Homotopy Solution for Nonlinear Equations Using Maple14. Abstract Joural of Sciece ad Techology ISSN 9-860 Vol. No. December 0 Newto Homotopy Solutio for Noliear Equatios Usig Maple Nor Haim Abd. Rahma, Arsmah Ibrahim, Mohd Idris Jayes Faculty of Computer ad Mathematical

More information

Modification of Weerakoon-Fernando s Method with Fourth-Order of Convergence for Solving Nonlinear Equation

Modification of Weerakoon-Fernando s Method with Fourth-Order of Convergence for Solving Nonlinear Equation ISSN: 50-08 Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 Modiicatio o Weerakoo-Ferado s Method with Fourth-Order o Covergece or Solvig Noliear Equatio

More information

CS537. Numerical Analysis and Computing

CS537. Numerical Analysis and Computing CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be

More information

CS321. Numerical Analysis and Computing

CS321. Numerical Analysis and Computing CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca

More information

wavelet collocation method for solving integro-differential equation.

wavelet collocation method for solving integro-differential equation. IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 5-3, ISSN (p): 78-879 Vol. 5, Issue 3 (arch. 5), V3 PP -7 www.iosrje.org wavelet collocatio method for solvig itegro-differetial equatio. Asmaa Abdalelah Abdalrehma

More information

DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan

DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed

More information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

IN many scientific and engineering applications, one often

IN many scientific and engineering applications, one often INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 5 Secod Degree Refiemet Jacobi Iteratio Method for Solvig System of Liear Equatio Tesfaye Kebede Abstract Several

More information

Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials

Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 15), PP 1-11 www.iosrjourals.org Numerical Solutios of Secod Order Boudary Value Problems

More information

A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation

A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios

More information

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations

Chapter 2 The Solution of Numerical Algebraic and Transcendental Equations Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said

More information

NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 595 NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS by Yula WANG *, Da TIAN, ad Zhiyua LI Departmet of Mathematics,

More information

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 3

2.29 Numerical Fluid Mechanics Fall 2011 Lecture 3 Numerical Fluid Mechaics Fall 2011 Lecture 3 REVIEW Lectures 1-2 Approximatio ad roud-off errors ˆx a xˆ Absolute ad relative errors: E a xˆ a ˆx, a ˆx a xˆ Iterative schemes ad stop criterio: ˆx 1 a ˆx

More information

Finite Difference Approximation for Transport Equation with Shifts Arising in Neuronal Variability

Finite Difference Approximation for Transport Equation with Shifts Arising in Neuronal Variability Iteratioal Joural of Sciece ad Research (IJSR) ISSN (Olie): 39-764 Ide Copericus Value (3): 64 Impact Factor (3): 4438 Fiite Differece Approimatio for Trasport Equatio with Shifts Arisig i Neuroal Variability

More information

Root Finding COS 323

Root Finding COS 323 Root Fidig COS 323 Remider Sig up for Piazza Assigmet 0 is posted, due Tue 9/25 Last time.. Floatig poit umbers ad precisio Machie epsilo Sources of error Sesitivity ad coditioig Stability ad accuracy

More information

Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations

Uniform Strict Practical Stability Criteria for Impulsive Functional Differential Equations Global Joural of Sciece Frotier Research Mathematics ad Decisio Scieces Volume 3 Issue Versio 0 Year 03 Type : Double Blid Peer Reviewed Iteratioal Research Joural Publisher: Global Jourals Ic (USA Olie

More information

Finite Difference Approximation for First- Order Hyperbolic Partial Differential Equation Arising in Neuronal Variability with Shifts

Finite Difference Approximation for First- Order Hyperbolic Partial Differential Equation Arising in Neuronal Variability with Shifts Iteratioal Joural of Scietific Egieerig ad Research (IJSER) wwwiseri ISSN (Olie): 347-3878, Impact Factor (4): 35 Fiite Differece Approimatio for First- Order Hyperbolic Partial Differetial Equatio Arisig

More information

New Variants of Newton s Method for Nonlinear Unconstrained Optimization Problems

New Variants of Newton s Method for Nonlinear Unconstrained Optimization Problems Itelliget Iformatio Maagemet, 00,, 40-45 doi:0.436/iim.00.005 Pulished Olie Jauary 00 (http://www.scirp.org/joural/iim) ew Variats of ewto s Method for oliear Ucostraied Optimizatio Prolems Astract V.

More information

The Sumudu transform and its application to fractional differential equations

The Sumudu transform and its application to fractional differential equations ISSN : 30-97 (Olie) Iteratioal e-joural for Educatio ad Mathematics www.iejem.org vol. 0, No. 05, (Oct. 03), 9-40 The Sumudu trasform ad its alicatio to fractioal differetial equatios I.A. Salehbhai, M.G.

More information

The Phi Power Series

The Phi Power Series The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.

More information

An Account of Congruences Mod p k Using Halley s Method

An Account of Congruences Mod p k Using Halley s Method World Applied Scieces Joural 16 (11): 166-160, 01 ISSN 1818-49 IDOSI Pulicatios, 01 A Accout of Cogrueces Mod p Usig Halley s Method M. Khalid Mahmood ad M. Aslam Mali Departmet of Mathematics, Uiversity

More information

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat

More information

μ are complex parameters. Other

μ are complex parameters. Other A New Numerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. J. Suday * ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria.

More information

M.Jayalakshmi and P. Pandian Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India.

M.Jayalakshmi and P. Pandian Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India. M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: 48-96 www.iera.com Vol., Issue 4, July-August 0, pp.47-54 A New Method for Fidig a Optimal Fuzzy Solutio For

More information

Applied Mathematics Letters

Applied Mathematics Letters Applied Mathematics Letters 5 (01) 03 030 Cotets lists available at SciVerse ScieceDirect Applied Mathematics Letters joural homepage: www.elsevier.com/locate/aml O ew computatioal local orders of covergece

More information

Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan

Arkansas Tech University MATH 2924: Calculus II Dr. Marcel B. Finan Arkasas Tech Uiversity MATH 94: Calculus II Dr Marcel B Fia 85 Power Series Let {a } =0 be a sequece of umbers The a power series about x = a is a series of the form a (x a) = a 0 + a (x a) + a (x a) +

More information

Application of Homotopy Analysis Method for Solving various types of Problems of Ordinary Differential Equations

Application of Homotopy Analysis Method for Solving various types of Problems of Ordinary Differential Equations Iteratioal Joural o Recet ad Iovatio Treds i Coputig ad Couicatio IN: 31-8169 Volue: 5 Issue: 5 16 Applicatio of Hootopy Aalysis Meod for olvig various types of Probles of Ordiary Differetial Equatios

More information

Notes on iteration and Newton s method. Iteration

Notes on iteration and Newton s method. Iteration Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f

More information

Research Article Two Expanding Integrable Models of the Geng-Cao Hierarchy

Research Article Two Expanding Integrable Models of the Geng-Cao Hierarchy Abstract ad Applied Aalysis Volume 214, Article ID 86935, 7 pages http://d.doi.org/1.1155/214/86935 Research Article Two Epadig Itegrable Models of the Geg-Cao Hierarchy Xiurog Guo, 1 Yufeg Zhag, 2 ad

More information

TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction

TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES. 1. Introduction Math Appl 6 2017, 143 150 DOI: 1013164/ma201709 TRACES OF HADAMARD AND KRONECKER PRODUCTS OF MATRICES PANKAJ KUMAR DAS ad LALIT K VASHISHT Abstract We preset some iequality/equality for traces of Hadamard

More information

Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics

Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.

More information

Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations

Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-

More information

THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES

THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES Joural of Mathematical Aalysis ISSN: 17-341, URL: http://iliriascom/ma Volume 7 Issue 4(16, Pages 13-19 THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES ABEDALLAH RABABAH, AYMAN AL

More information

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece

More information

BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics

BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics Mohd Yusuf Yasi / BIBECHANA 8 (2012) 31-36 : BMHSS, p. 31 BIBECHANA A Multidiscipliary Joural of Sciece, Techology ad Mathematics ISSN 2091-0762 (olie) Joural homepage: http://epjol.ifo/ide.php/bibechana

More information

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =! .8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

More information

Numerical solution of Bagley-Torvik equation using Chebyshev wavelet operational matrix of fractional derivative

Numerical solution of Bagley-Torvik equation using Chebyshev wavelet operational matrix of fractional derivative It. J. Adv. Appl. Math. ad Mech. () (04) 83-9 ISSN: 347-59 Available olie at www.iaamm.com Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics Numerical solutio of Bagley-Torvik equatio usig

More information

NTMSCI 5, No. 1, (2017) 26

NTMSCI 5, No. 1, (2017) 26 NTMSCI 5, No. 1, - (17) New Treds i Mathematical Scieces http://dx.doi.org/1.85/tmsci.17.1 The geeralized successive approximatio ad Padé approximats method for solvig a elasticity problem of based o the

More information

Calculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.)

Calculus 2 - D. Yuen Final Exam Review (Version 11/22/2017. Please report any possible typos.) Calculus - D Yue Fial Eam Review (Versio //7 Please report ay possible typos) NOTE: The review otes are oly o topics ot covered o previous eams See previous review sheets for summary of previous topics

More information

2 The LCP problem and preliminary discussion

2 The LCP problem and preliminary discussion Iteratioal Mathematical Forum, Vol. 6, 011, o. 6, 191-196 Polyomial Covergece of a Predictor-Corrector Iterior-Poit Algorithm for LCP Feixiag Che College of Mathematics ad Computer Sciece, Chogqig Three

More information

Control chart for number of customers in the system of M [X] / M / 1 Queueing system

Control chart for number of customers in the system of M [X] / M / 1 Queueing system Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 3297: 07 Certified Orgaiatio) Cotrol chart for umber of customers i the system of M [X] / M / Queueig system T.Poogodi, Dr.

More information

Complex Analysis Spring 2001 Homework I Solution

Complex Analysis Spring 2001 Homework I Solution Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle

More information

Benaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco

Benaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha

More information

Stability of fractional positive nonlinear systems

Stability of fractional positive nonlinear systems Archives of Cotrol Scieces Volume 5(LXI), 15 No. 4, pages 491 496 Stability of fractioal positive oliear systems TADEUSZ KACZOREK The coditios for positivity ad stability of a class of fractioal oliear

More information

f t dt. Write the third-degree Taylor polynomial for G

f t dt. Write the third-degree Taylor polynomial for G AP Calculus BC Homework - Chapter 8B Taylor, Maclauri, ad Power Series # Taylor & Maclauri Polyomials Critical Thikig Joural: (CTJ: 5 pts.) Discuss the followig questios i a paragraph: What does it mea

More information

Computation for Jacobi-Gauss Lobatto Quadrature Based on Derivative Relation

Computation for Jacobi-Gauss Lobatto Quadrature Based on Derivative Relation Computatio for acobi-auss obatto Quadrature Based o Derivative Relatio Z.S. Zheg uaghui Huag Abstract. The three-term recurrece relatio for derivatives of acobi-type polyomial is derived ad the auss-obatto

More information

Numerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION

Numerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty

More information

Numerical Solution of the First-Order Hyperbolic Partial Differential Equation with Point-Wise Advance

Numerical Solution of the First-Order Hyperbolic Partial Differential Equation with Point-Wise Advance Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): 39-74 Ide Copericus Value (3): 4 Impact Factor (3): 4438 Numerical Solutio of the First-Order Hyperbolic Partial Differetial Equatio with Poit-Wise

More information

Root Finding COS 323

Root Finding COS 323 Root Fidig COS 33 Why Root Fidig? Solve or i ay equatio: b where? id root o g b 0 Might ot be able to solve or directly e.g., e -0. si3-0.5 Evaluatig might itsel require solvig a dieretial equatio, ruig

More information

Third-order Composite Runge Kutta Method for Solving Fuzzy Differential Equations

Third-order Composite Runge Kutta Method for Solving Fuzzy Differential Equations Global Joural of Pure ad Applied Mathematics. ISSN 097-768 Volume Number (06) pp. 7-76 Research Idia Publicatios http://www.ripublicatio.com/gjpam.htm Third-order Composite Ruge Kutta Method for Solvig

More information

Five Steps Block Predictor-Block Corrector Method for the Solution of ( )

Five Steps Block Predictor-Block Corrector Method for the Solution of ( ) Applied Mathematics, 4,, -66 Published Olie May 4 i SciRes. http://www.scirp.org/oural/am http://dx.doi.org/.46/am.4.87 Five Steps Block Predictor-Block Corrector y = f x, y, y Method for the Solutio of

More information

SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS

SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS 96 Egieerig Mathematics through Applicatios 5 SOLUTIONS OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS aaaaa 5. INTRODUCTION The equatios of the form f() = 0 where f() is purely a polyomial i. e.g. 6 4 3 =

More information

In this section, we show how to use the integral test to decide whether a series

In this section, we show how to use the integral test to decide whether a series Itegral Test Itegral Test Example Itegral Test Example p-series Compariso Test Example Example 2 Example 3 Example 4 Example 5 Exa Itegral Test I this sectio, we show how to use the itegral test to decide

More information

A NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS

A NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet

More information

Quiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.

Quiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not. Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where

More information

A Pseudo Spline Methods for Solving an Initial Value Problem of Ordinary Differential Equation

A Pseudo Spline Methods for Solving an Initial Value Problem of Ordinary Differential Equation Joural of Matematics ad Statistics 4 (: 7-, 008 ISSN 549-3644 008 Sciece Publicatios A Pseudo Splie Metods for Solvig a Iitial Value Problem of Ordiary Differetial Equatio B.S. Ogudare ad G.E. Okeca Departmet

More information

POWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS

POWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity

More information

Pellian sequence relationships among π, e, 2

Pellian sequence relationships among π, e, 2 otes o umber Theory ad Discrete Mathematics Vol. 8, 0, o., 58 6 Pellia sequece relatioships amog π, e, J. V. Leyedekkers ad A. G. Shao Faculty of Sciece, The Uiversity of Sydey Sydey, SW 006, Australia

More information

Teaching Mathematics Concepts via Computer Algebra Systems

Teaching Mathematics Concepts via Computer Algebra Systems Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) E-ISSN: 4767 P-ISSN: - 4759 Volume 4 Issue 7 September. 6 PP-- Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw,

More information

Some Oscillation Properties of Third Order Linear Neutral Delay Difference Equations

Some Oscillation Properties of Third Order Linear Neutral Delay Difference Equations ISSN (e): 50 3005 Volume, 05 Issue, 07 July 05 Iteratioal Joural of Computatioal Egieerig Research (IJCER) Some Oscillatio Properties of Third Order Liear Neutral Delay Differece Equatios AGeorge Maria

More information

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314

More information

Oscillation and Property B for Third Order Difference Equations with Advanced Arguments

Oscillation and Property B for Third Order Difference Equations with Advanced Arguments Iter atioal Joural of Pure ad Applied Mathematics Volume 3 No. 0 207, 352 360 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu ijpam.eu Oscillatio ad Property B for Third

More information

A 2nTH ORDER LINEAR DIFFERENCE EQUATION

A 2nTH ORDER LINEAR DIFFERENCE EQUATION A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy

More information

Modified Logistic Maps for Cryptographic Application

Modified Logistic Maps for Cryptographic Application Applied Mathematics, 25, 6, 773-782 Published Olie May 25 i SciRes. http://www.scirp.org/joural/am http://dx.doi.org/.4236/am.25.6573 Modified Logistic Maps for Cryptographic Applicatio Shahram Etemadi

More information

Solving third order boundary value problem with fifth order block method

Solving third order boundary value problem with fifth order block method Matematical Metods i Egieerig ad Ecoomics Solvig tird order boudary value problem wit it order bloc metod A. S. Abdulla, Z. A. Majid, ad N. Seu Abstract We develop a it order two poit bloc metod or te

More information

Section 5.5. Infinite Series: The Ratio Test

Section 5.5. Infinite Series: The Ratio Test Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches

More information

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist.

Topic 5 [434 marks] (i) Find the range of values of n for which. (ii) Write down the value of x dx in terms of n, when it does exist. Topic 5 [44 marks] 1a (i) Fid the rage of values of for which eists 1 Write dow the value of i terms of 1, whe it does eist Fid the solutio to the differetial equatio 1b give that y = 1 whe = π (cos si

More information

In exercises 1 and 2, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers _

In exercises 1 and 2, (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers _ Chapter 9 Curve I eercises ad, (a) write the repeatig decimal as a geometric series ad (b) write its sum as the ratio of two itegers _.9.976 Distace A ball is dropped from a height of 8 meters. Each time

More information

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor

IJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor IJITE Vol Issue-, (November 4) ISSN: 3-776 ATTRACTIVITY OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Guagfeg Liu School of Zhagjiagag Jiagsu Uiversit of Sciece ad Techolog, Zhagjiagag, Jiagsu 56,PR

More information

Using An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically

Using An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad Mid-Poit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab

More information

NEW CLOSE FORM APPROXIMATIONS OF ln(1 + x) Sanjay Kumar Khattri. 1. Introduction

NEW CLOSE FORM APPROXIMATIONS OF ln(1 + x) Sanjay Kumar Khattri. 1. Introduction THE TEACHING OF MATHEMATICS 009 Vol XII pp 7 4 NEW CLOSE FORM APPROXIMATIONS OF l + x) Sajay Kumar Khattri Abstract Based o Newto-Cotes ad Gaussia quadrature rules we develop several closed form approximatios

More information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 Calculus II Basic Convergence Tests for Series MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real

More information

Common Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces

Common Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex

More information

Stability Analysis of the Euler Discretization for SIR Epidemic Model

Stability Analysis of the Euler Discretization for SIR Epidemic Model Stability Aalysis of the Euler Discretizatio for SIR Epidemic Model Agus Suryato Departmet of Mathematics, Faculty of Scieces, Brawijaya Uiversity, Jl Vetera Malag 6545 Idoesia Abstract I this paper we

More information

College of Art and Sciences Universiti Utara Sintok Kedah, 0060, MALAYSIA

College of Art and Sciences Universiti Utara Sintok Kedah, 0060, MALAYSIA Iteratioal Joural of Pure ad Applied Mathematics Volume 2 No. 3 207, 497-57 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: 0.2732/ijpam.v2i3.5 PAijpam.eu GENERALIZED

More information

THE ZETA FUNCTION AND THE RIEMANN HYPOTHESIS. Contents 1. History 1

THE ZETA FUNCTION AND THE RIEMANN HYPOTHESIS. Contents 1. History 1 THE ZETA FUNCTION AND THE RIEMANN HYPOTHESIS VIKTOR MOROS Abstract. The zeta fuctio has bee studied for ceturies but mathematicias are still learig about it. I this paper, I will discuss some of the zeta

More information

SCORE. Exam 2. MA 114 Exam 2 Fall 2016

SCORE. Exam 2. MA 114 Exam 2 Fall 2016 MA 4 Exam Fall 06 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

A New Hybrid in the Nonlinear Part of Adomian Decomposition Method for Initial Value Problem of Ordinary Differential Equation

A New Hybrid in the Nonlinear Part of Adomian Decomposition Method for Initial Value Problem of Ordinary Differential Equation Joural of Matematics Researc; Vol No ; ISSN - E-ISSN - Publised b Caadia Ceter of Sciece ad Educatio A New Hbrid i te Noliear Part of Adomia Decompositio Metod for Iitial Value Problem of Ordiar Differetial

More information

TEACHING THE IDEAS BEHIND POWER SERIES. Advanced Placement Specialty Conference. LIN McMULLIN. Presented by

TEACHING THE IDEAS BEHIND POWER SERIES. Advanced Placement Specialty Conference. LIN McMULLIN. Presented by Advaced Placemet Specialty Coferece TEACHING THE IDEAS BEHIND POWER SERIES Preseted by LIN McMULLIN Sequeces ad Series i Precalculus Power Series Itervals of Covergece & Covergece Tests Error Bouds Geometric

More information

Bounds for the Extreme Eigenvalues Using the Trace and Determinant

Bounds for the Extreme Eigenvalues Using the Trace and Determinant ISSN 746-7659, Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp 49-55 Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics,

More information

Section 1 of Unit 03 (Pure Mathematics 3) Algebra

Section 1 of Unit 03 (Pure Mathematics 3) Algebra Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course

More information

Generating Functions for Laguerre Type Polynomials. Group Theoretic method

Generating Functions for Laguerre Type Polynomials. Group Theoretic method It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, 257-266 Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said

More information

Lecture 8. Nonlinear Device Stamping

Lecture 8. Nonlinear Device Stamping PRINCIPLES OF CIRCUIT SIMULATION Lecture 8. Noliear Device Stampig Guoyog Shi, PhD shiguoyog@ic.sjtu.edu.c School of Microelectroics Shaghai Jiao Tog Uiversity Fall -- Slide Outlie Solvig a oliear circuit

More information

On a class of convergent sequences defined by integrals 1

On a class of convergent sequences defined by integrals 1 Geeral Mathematics Vol. 4, No. 2 (26, 43 54 O a class of coverget sequeces defied by itegrals Dori Adrica ad Mihai Piticari Abstract The mai result shows that if g : [, ] R is a cotiuous fuctio such that

More information

Numerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets

Numerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie

More information

HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia

HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS Viko Petričević Uiversity of Zagre, Croatia Astract There are umerous methods for ratioal approximatio of real umers

More information

Harmonic Number Identities Via Euler s Transform

Harmonic Number Identities Via Euler s Transform 1 2 3 47 6 23 11 Joural of Iteger Sequeces, Vol. 12 2009), Article 09.6.1 Harmoic Number Idetities Via Euler s Trasform Khristo N. Boyadzhiev Departmet of Mathematics Ohio Norther Uiversity Ada, Ohio 45810

More information