OPEN NEWTON  COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS


 Eugene Todd
 2 years ago
 Views:
Transcription
1 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 96 pissn: 78 OPEN NEWTON  COTES QUADRATURE WITH MIDPOINT DERIVATIVE FOR INTEGRATION OF ALGEBRAIC FUNCTIONS T. Rmchndrn R.Priml Assistnt Professor Deprtment of Mthemtics M.V.Muthih Government Arts College for Women Tmil Ndu Indi Reserch Scholr PG &Reserch Deprtment of Mthemtics Government Arts College (Autonomous) Tmil Ndu Indi Astrct Mny methods re ville for pproximting the integrl to the desired precision in Numericl integrtion. A new set of numericl integrtion formul of Open NewtonCotes Qudrture with Midpoint Derivtive type is suggested which is the modified form of Open NewtonCotes Qudrture. This new midpoint derivtive sed formul increse the two order of precision thn the clssicl Open NewtonCotes formul nd lso gives more ccurcy thn the existing formul. Further the error terms re otined nd compred with the existing methods. Finlly the effectiveness of the proposed lgorithm is illustrted y mens of numericl exmple. Key Words: Numericl Integrtion Open NewtonCotes formul Midpoint Derivtive Numericl Exmples *** INTRODUCTION Numericl integrtion is the process of computing the vlue of definite integrl from set of numericl vlues of the integrnd. The process of evlution of integrtion of function of single vrile is sometimes clled Mechnicl Qudrture. The computtion of doule integrl of function of two independent vriles is clled Mechnicl Cuture. There re mny methods re ville for numericl integrtion []. Consider the definite integrl I f where the function f(x) is continuous in the closed intervl [] so tht the integrl I(f) exists. An efficient formul is developed for computing pproximte vlue of the integrl using only vlues of the integrnd f(x) t points x []. To pproximte the integrl I(f)we integrte exctly piecewise polynomil pproximtions of f(x) on the intervl []. Generlly qudrture rule hs the form n w i f x i () i Where there (n) distinct points x < x < < x n nd (n) weights w w... w n within the intervl []. The error of pproximtion is given s E n f n w i f x i () i. Definition An integrtion method of the form ( ) is sid to e of order P if it produces exct results (En[f] ) for ll polynomils of degree less thn or equl to P []. In Open Newtoncotes rule the end points of the intervl is excluded in the function evlutioni.e w i f x i () i x Volume: Issue: Oct n x n for given n distinct points x < x <... < xn nd n weights w w... wn over the intervl ( ) with xi ( i ) h i...n nd h n [ 7 ]. the Open Newton cotes rules re given s follows If n ; where ξ. If n ; f f ( ) (ξ) () f ( ) (ξ) (6) 6
2 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 96 pissn: 78 where ξ. If n ; where ξ. If n ; where ξ. f f f f ξ (7) f f 9 f f f ξ (8) In the closed Newtoncotes qudrture the endpoints re included; wheres the open Newton cotes qudrture only the interior points re included. The corrected open Newtoncotes qudrture hs higher precision thn the clssicl qudrture rule. There re so mny works hs een done on the numericl improvement of Newtoncotes formuls. Dehghn et l. presented n improvement of open semiopen closed first nd second kind [ 6 8 9] Cheyshev Newton cotes qudrture rules. In the recent yers Clrence O.E Burg nd his compnions introduced new fmily of derivtive sed rules [ 7]. In Weijing Zho nd Hongxing Li [] introduced new fmily of closed Newtoncotes qudrture with Midpoint derivtive rules. In this pper new fmily of open Newtoncotes qudrture rule is descried which uses the derivtive v l u e t the Midpoint with their error terms. Also some numericl exmples re given with their results nd comprison. The result shows tht the new formuls give etter solution thn the clssicl ones..open NEWTON  COTES QUADRATURE WITH MIDPOINT DERIVATIVE A new Open Newton  Cotes Qudrture rules with Midpoint Derivtive is explined elow which g i v e s higher precision thn the clssicl NewtonCotes Qudrture rules.. Theorem Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is Volume: Issue: Oct f ( ) The precision of this method is. (9) Since the rule (9) hs the degree of precision. Now we verify tht the rule (9) is exct for f (x) x x. When f x x x dx ; n When f x x x dx ; n 6.. Qudrture with Midpoint  Derivtive is.. Theorem Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is f ( ) 6 f The precision of this method is. () Since the rule () hs the degree of precision.now we verify tht the rule () is exct for f (x) x x. When f x x x dx ; n ( ) 6.
3 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 96 pissn: 78 When f x x x dx ; n 6( ) 6. Qudrture with Midpoint  Derivtive is.. Theorem Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is f f f f The precision of this method is. () Since the rule () hs the degree of precision.now we verify tht the rule () is exct for f (x) x x. When f x x x dx ; n 6. When f x x x dx ; n Qudrture with Midpoint  Derivtive is. f f 9 The precision of this method is. f f f () Since the rule () hs the degree of precision.now we verify tht the rule () is exct for f (x) x x. When f x x x dx ; n 8. When f x x x dx ; n Qudrture with Midpoint  Derivtive is.. THE ERROR TERMS OF OPEN NEWTONCOTES WITH MIDPOINT DERIVATIVE QUADRATURE The Error terms of Open NewtonCotes Qudrture with Midpoint derivtive re given elow. The Error terms re the difference etween the exct vlue nd the qudrture rule.. Theorem The Error term of Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is. Theorem Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is Volume: Issue: Oct f ( ) ( ) 9 f() (ξ) ()
4 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 96 pissn: 78 where ξ ( ).This is fifth order ccurte with the error term is E f 9 f ξ. Let f x x! f! x dx ;!.6!.6 ( ) 9. Therefore the Error term is. Theorem E f 9 f ξ. The Error term of Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is ( ) 6 f f 9( ) 888 f () (ξ) ( ) where ξ ( ).This is fifth order ccurte with the error term is 9 E f f ξ. 888 Let f x x! f! f x dx ; Therefore the Error term is. Theorem E f ( ) f ξ. 888 The Error term of Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is f f f f. ( )7 888 f(6) (ξ) () where ξ ( ).This is seventh order ccurte with the error term is E f Let f x x6 f 7. ( 7 7 ) 7. ( )7 888 f(6) (ξ). f x 6 dx f f 7 7 ; Therefore the Error term is E f ( )7 888 ( )7 888 f(6) (ξ). Volume: Issue: Oct
5 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 96 pissn: 78. Theorem The Error term of Open Newton  Cotes Qudrture with Midpoint  Derivtive for (n) is f f 9 8( ) f f f f (6) (ξ) (6) where ξ ( ).This is seventh order ccurte with the error term is 8( )7 E f 68. f (6) (ξ).. Tle : Exct vlue of e x dx n vlue App. vlue Error App. vlue Error n n n n Tle : Exct vlue of n vlue dx x App. vlue Error App. vlue Error Let f x x6 x 6 f f f 7. ( 7 7 ) 7. dx f 7 7 ; 9 f Therefore the Error term is E f 8( ) ( ) NUMERICAL RESULTS f (6) (ξ). An pproximte vlue of the following exmples using the open Newton cotes qudrture with Midpoint Derivtive rules re determined nd presented. To demonstrte the ccurcy of the results we evlute the exmples nd the Comprison of results is shown in Tles.. n n n n Tle : Exct vlue of n vlue x dx. App. vlue Error App. vlue Error n n n n Tle : Exct vlue of e x dx n vlue App. vlue Error App. vlue Error n n n n From the results presented in Tles  it is o served tht the Open NewtonCotes q u d r t u r e with midpoint derivtive g i v e s more ccurcy thn the clssicl ones. Volume: Issue: Oct
6 IJRET: Interntionl Journl of Reserch in Engineering nd Technology eissn: 96 pissn: 78. CLUSION In this work we pplied the Open Ne wtoncotes Q u d r t u r e with Midpoint derivtive over finite intervl []. The Error terms gives two orders of precision thn the S t n d r d m e t h o d s.a numericl exmple is given to clrify the proposed Algorithm. 6. REFERENCES [] K.E.Atkinson An Introduction to Numericl Anlysis John wiley nd Sons New York NY USA Second Edition 989. [] Clrence O.E.Burg Derivtivesed closed Newtoncotes numericl qudrture Applied Mthemtics nd Computtions vol.8 pp [] Clrence O.E.Burg nd Ezechiel Degny Derivtivesed midpoint qudrture rule Applied Mthemtics nd Computtions vol. pp. 8. [] M.Dehghn M.MsjedJmei nd M.R.Eslhchi On numericl improvementof closed NewtonCotes qudrture rules Applied Mthemtics nd Computtions vol.6 pp. 6. [] M.Dehghn M.MsjedJmei nd M.R.Eslhchi The semiopen NewtonCotes qudrture rule nd its numericl improvement Applied Mthemtics nd Computtions vol.7 pp. 9. [6] M.Dehghn M.MsjedJmei nd M.R.Eslhchi On numericl improvement of open NewtonCotes qudrture rules Applied Mthemtics nd Computtions vol.7 pp [7] Fiz ZfrSir Sleem nd Clrence O.E.Burg New Derivive sed open Newton cotes qudrture rules Astrct nd Applied AnlysisVolume Article ID 98 6 pges. [8] S.M.Hshemiprst M.R.EslhchiM.Dehghn M.MsjedJmei On numericl improvement of the first kind CheyshevNewtonCotes qudrture rules Applied Mthemtics nd Computtions vol.7 pp [9]S.M.HshemiprstM.MsjedJmei M.R.EslhchiM.Dehghn On numericl improvement of the second kind CheyshevNewtonCotes qudrture rules(open type) Applied Mthemtics nd Computtionsvol.8 pp []M.K.JinS.R.K.Iyengr nd R.K.Jin Numericl methods for Scientific nd Computtion New Age Interntionl (P) limited Fifth Edition 7. [] Weijing Zho nd Hongxing Midpoint DerivtiveBsed Closed Newton Cotes Qudrture Astrct nd Applied Anlysis vol. Article ID 97 pges Volume: Issue: Oct
Harmonic Mean Derivative  Based Closed Newton Cotes Quadrature
IOSR Journl of Mthemtics (IOSRJM) eissn:  pissn: 9X. Volume Issue Ver. IV (My.  Jun. 0) PP  www.iosrjournls.org Hrmonic Men Derivtive  Bsed Closed Newton Cotes Qudrture T. Rmchndrn D.Udykumr nd
More informationArithmetic Mean Derivative Based Midpoint Rule
Applied Mthemticl Sciences, Vol. 1, 018, no. 13, 65633 HIKARI Ltd www.mhikri.com https://doi.org/10.1988/ms.018.858 Arithmetic Men Derivtive Bsed Midpoint Rule Rike Mrjulis 1, M. Imrn, Symsudhuh Numericl
More informationCOSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III)  Gauss Quadrature and Adaptive Quadrature
COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III  Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting
More informationDOING PHYSICS WITH MATLAB MATHEMATICAL ROUTINES
DOIG PHYSICS WITH MATLAB MATHEMATICAL ROUTIES COMPUTATIO OF OEDIMESIOAL ITEGRALS In Cooper School of Physics, University of Sydney in.cooper@sydney.edu.u DOWLOAD DIRECTORY FOR MATLAB SCRIPTS mth_integrtion_1d.m
More informationQUADRATURE is an oldfashioned word that refers to
World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = 2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationLab 11 Approximate Integration
Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties
More information5.1 How do we Measure Distance Traveled given Velocity? Student Notes
. How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the xxis & yxis
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationZ b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...
Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics ENotes, 5(005), 5360 c ISSN 1607510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
More informationNumerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden
Numericl Anlysis 10th ed R L Burden, J D Fires, nd A M Burden Bemer Presenttion Slides Prepred by Dr. Annette M. Burden Youngstown Stte University July 9, 2015 Chpter 4.1: Numericl Differentition 1 ThreePoint
More informationAN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir
RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An
More informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
More informationUndergraduate Research
Undergrdute Reserch A Trigonometric Simpson s Rule By Ctherine Cusimno Kirby nd Sony Stnley Biogrphicl Sketch Ctherine Cusimno Kirby is the dughter of Donn nd Sm Cusimno. Originlly from Vestvi Hills, Albm,
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationConstruction of Gauss Quadrature Rules
Jim Lmbers MAT 772 Fll Semester 201011 Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht NewtonCotes qudrture
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationCOT4501 Spring Homework VII
COT451 Spring 1 Homework VII The ssignment is due in clss on Thursdy, April 19, 1. There re five regulr problems nd one computer problem (using MATLAB). For written problems, you need to show your work
More informationLecture 12: Numerical Quadrature
Lecture 12: Numericl Qudrture J.K. Ryn@tudelft.nl WI3097TU Delft Institute of Applied Mthemtics Delft University of Technology 5 December 2012 () Numericl Qudrture 5 December 2012 1 / 46 Outline 1 Review
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationNumerical Integration
Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. LmiAthens Lmi 3500 Greece Abstrct Using
More informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationp(x) = 3x 3 + x n 3 k=0 so the right hand side of the equality we have to show is obtained for r = b 0, s = b 1 and 2n 3 b k x k, q 2n 3 (x) =
Norwegin University of Science nd Technology Deprtment of Mthemticl Sciences Pge 1 of 5 Contct during the exm: Elen Celledoni, tlf. 73593541, cell phone 48238584 PLESE NOTE: this solution is for the students
More informationFredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method
Int. Journl of Mth. Anlysis, Vol. 5, 211, no. 19, 93594 Fredholm Integrl Equtions of the First Kind Solved by Using the Homotopy Perturbtion Method Seyyed Mhmood Mirzei Deprtment of Mthemtics, Fculty
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus MATH 151 Clculus for Mngement J. Robert Buchnn Deprtment of Mthemtics Fll 2018 Objectives Define nd evlute definite integrls using the concept of re. Evlute definite integrls
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationLECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and
LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only
More informationSection 5.4 Fundamental Theorem of Calculus 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus 1
Section 5.4 Fundmentl Theorem of Clculus 2 Lectures College of Science MATHS : Clculus (University of Bhrin) Integrls / 24 Definite Integrl Recll: The integrl is used to find re under the curve over n
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationA Generalized Inequality of Ostrowski Type for Twice Differentiable Bounded Mappings and Applications
Applied Mthemticl Sciences, Vol. 8, 04, no. 38, 88990 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.988/ms.04.4 A Generlized Inequlity of Ostrowski Type for Twice Differentile Bounded Mppings nd Applictions
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationApplication Chebyshev Polynomials for Determining the Eigenvalues of SturmLiouville Problem
Applied nd Computtionl Mthemtics 5; 4(5): 369373 Pulished online Septemer, 5 (http://www.sciencepulishinggroup.com//cm) doi:.648/.cm.545.6 ISSN: 38565 (Print); ISSN: 38563 (Online) Appliction Cheyshev
More informationAdomian Decomposition Method with Green s. Function for Solving TwelfthOrder Boundary. Value Problems
Applied Mthemticl Sciences, Vol. 9, 25, no. 8, 353368 HIKARI Ltd, www.mhikri.com http://dx.doi.org/.2988/ms.25.486 Adomin Decomposition Method with Green s Function for Solving TwelfthOrder Boundry
More informationInterpolation. Gaussian Quadrature. September 25, 2011
Gussin Qudrture September 25, 2011 Approximtion of integrls Approximtion of integrls by qudrture Mny definite integrls cnnot be computed in closed form, nd must be pproximted numericlly. Bsic building
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More information7. Numerical evaluation of definite integrals
7. Numericl evlution of definite integrls Tento učení text yl podpořen z Operčního progrmu Prh  Adptilit Hn Hldíková Numericl pproximtion of definite integrl is clled numericl qudrture, the formuls re
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of
More informationComposite Mendeleev s Quadratures for Solving a Linear Fredholm Integral Equation of The Second Kind
Globl Journl of Pure nd Applied Mthemtics. ISSN 09731768 Volume 12, Number (2016), pp. 393 398 Reserch Indi Publictions http://www.ripubliction.com/gjpm.htm Composite Mendeleev s Qudrtures for Solving
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationB.Sc. in Mathematics (Ordinary)
R48/0 DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 B.Sc. in Mthemtics (Ordinry) SUPPLEMENTAL EXAMINATIONS 01 Numericl Methods Dr. D. Mckey Dr. C. Hills Dr. E.A. Cox Full mrks for complete nswers
More informationTHE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS
THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2Dimensionl Gussin Qudrture 20 4
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationA Companion of Ostrowski Type Integral Inequality Using a 5Step Kernel with Some Applications
Filomt 30:3 06, 360 36 DOI 0.9/FIL6360Q Pulished y Fculty of Sciences nd Mthemtics, University of Niš, Seri Aville t: http://www.pmf.ni.c.rs/filomt A Compnion of Ostrowski Type Integrl Inequlity Using
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NONCALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NONCALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More information1. GaussJacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. GussJcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More informationResearch Article Composite GaussLegendre Formulas for Solving Fuzzy Integration
Hindwi Pulishing Corportion Mthemticl Prolems in Engineering, Article ID 873498, 7 pges http://dx.doi.org/0.55/04/873498 Reserch Article Composite GussLegendre Formuls for Solving Fuzzy Integrtion Xioin
More informationCLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTONCOTES QUADRATURES
Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED
More informationNumerical Integration. 1 Introduction. 2 Midpoint Rule, Trapezoid Rule, Simpson Rule. AMSC/CMSC 460/466 T. von Petersdorff 1
AMSC/CMSC 46/466 T. von Petersdorff 1 umericl Integrtion 1 Introduction We wnt to pproximte the integrl I := f xdx where we re given, b nd the function f s subroutine. We evlute f t points x 1,...,x n
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at UrbanaChampaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t UrbnChmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationLecture 23: Interpolatory Quadrature
Lecture 3: Interpoltory Qudrture. Qudrture. The computtion of continuous lest squres pproximtions to f C[, b] required evlutions of the inner product f, φ j = fxφ jx dx, where φ j is polynomil bsis function
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 3 Polynomial Interpolation: Numerical Differentiation and Integration.
Advnced Computtionl Fluid Dynmics AA215A Lecture 3 Polynomil Interpoltion: Numericl Differentition nd Integrtion Antony Jmeson Winter Qurter, 2016, Stnford, CA Lst revised on Jnury 7, 2016 Contents 3 Polynomil
More information10. AREAS BETWEEN CURVES
. AREAS BETWEEN CURVES.. Ares etween curves So res ove the xxis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in
More informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
More informationLecture Note 4: Numerical differentiation and integration. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 4: Numericl differentition nd integrtion Xioqun Zng Sngi Jio Tong University Lst updted: November, 0 Numericl Anlysis. Numericl differentition.. Introduction Find n pproximtion of f (x 0 ),
More informationNumerical integration. Quentin Louveaux (ULiège  Institut Montefiore) Numerical analysis / 10
Numericl integrtion Quentin Louveux (ULiège Institut Montefiore) Numericl nlysis 2018 1 / 10 Numericl integrtion We consider definite integrls Z b f (x)dx better to it use if known! A We do not ssume tht
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationTangent Line and Tangent Plane Approximations of Definite Integral
RoseHulmn Undergrdute Mthemtics Journl Volume 16 Issue 2 Article 8 Tngent Line nd Tngent Plne Approximtions of Definite Integrl Meghn Peer Sginw Vlley Stte University Follow this nd dditionl works t:
More informationMETHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS
Journl of Young Scientist Volume III 5 ISSN 448; ISSN CDROM 449; ISSN Online 445; ISSNL 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationThe Shortest Confidence Interval for the Mean of a Normal Distribution
Interntionl Journl of Sttistics nd Proility; Vol. 7, No. 2; Mrch 208 ISSN 9277032 EISSN 9277040 Pulished y Cndin Center of Science nd Eduction The Shortest Confidence Intervl for the Men of Norml Distriution
More informationNumerical quadrature based on interpolating functions: A MATLAB implementation
SEMINAR REPORT Numericl qudrture bsed on interpolting functions: A MATLAB implementtion by Venkt Ayylsomyjul A seminr report submitted in prtil fulfillment for the degree of Mster of Science (M.Sc) in
More informationSome estimates on the HermiteHadamard inequality through quasiconvex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13693 Some estimtes on the HermiteHdmrd inequlity through qusiconvex functions Dniel Alexndru Ion Abstrct. In this pper
More informationSolutions of Klein  Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSRJM) eissn: 22785728, pissn: 2319765X. Volume 13, Issue 6 Ver. IV (Nov.  Dec. 2017), PP 1924 www.iosrjournls.org Solutions of Klein  Gordn equtions, using Finite Fourier
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationQuadrature Rules for Evaluation of Hyper Singular Integrals
Applied Mthemticl Sciences, Vol., 01, no. 117, 53955 HIKARI Ltd, www.mhikri.com http://d.doi.org/10.19/ms.01.75 Qudrture Rules or Evlution o Hyper Singulr Integrls Prsnt Kumr Mohnty Deprtment o Mthemtics
More informationNumerical Integration. Newton Cotes Formulas. Quadrature. Newton Cotes Formulas. To approximate the integral b
Numericl Integrtion Newton Cotes Formuls Given function f : R R nd two rel numbers, b R, < b, we clculte (pproximtely) the integrl I(f,, b) = f (x) dx K. Frischmuth (IfM UR) Numerics for CSE 08/09 8 /
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationKey words. Numerical quadrature, piecewise polynomial, convergence rate, trapezoidal rule, midpoint rule, Simpson s rule, spectral accuracy.
O SPECTRA ACCURACY OF QUADRATURE FORMUAE BASED O PIECEWISE POYOMIA ITERPOATIO A KURGAOV AD S TSYKOV Abstrct It is wellknown tt te trpezoidl rule, wile being only secondorder ccurte in generl, improves
More informationTrapezoidal Rule, n = 1, x 0 = a, x 1 = b, h = b a. f (x)dx = h 2 (f (x 0) + f (x 1 )) h3
Trpezoidl Rule, n = 1, x 0 =, x 1 = b, h = b f (x)dx = h 2 (f (x 0) + f (x 1 )) h3 12 f (ξ). Simpson s Rule: n = 3, x 0 =, x 1 = +b 2, x 2 = b, h = b 2. Qudrture Rule, double node t x 1 P 3 (x)dx = f (x
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More information