A Computational Method for Solving Two Point Boundary Value Problems of Order Four
|
|
- Sophia Parsons
- 6 years ago
- Views:
Transcription
1 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 A Computatonal Metod for Solvng Two Pont Boundary Value Problem of Order Four Yoge Gupta Department of Matematc Unted College of Engg and Management Allaabad-00(U.P.) Inda E-mal: yogegupta@unted.ac.n Pankaj Kumar Srvatava Department of Matematc Jaypee Inttute of Informaton Tecnology Noda-00(U.P.) Inda E-mal: pankaj.rvatava@jt.ac.n Abtract Preent paper portray a computatonal metod ung cubc B-plne to olve fourt order boundary value problem. Te propoed ceme frt appled for oluton of pecal cae fourt order boundary value problem. Te metod, ten, extended to oluton of non-lnear and ngular problem. Selected Example from te lterature are olved numercally ung computer program n MATLAB. Key word: Fourt order boundary value problem, Sngular boundary value problem, Cubc B-plne, Nodal pont, Maxmum abolute error.. Introducton Engneer are reearcng oluton to reolve many of today tecncal callenge. Numercal tecnque are ued to olve te matematcal model n engneerng problem. Many of te matematcal model of engneerng problem are expreed n term of Boundary Value Problem, wc are ordnary dfferental equaton wt boundary condton. Fourt-order Boundary Value Problem are n te matematcal modelng of two-dmenonal cannel wt porou wall, vcoelatc and nelatc flow, deformaton of beam, plate deflecton teory, beam element teory and a number of oter engneerng and appled matematc applcaton. Solvng uc type of boundary value problem analytcally poble only n very rare cae. Many reearcer worked for te numercal oluton of fourt order boundary value problem. Some numercal metod uc a fnte dfference metod, dfferental tranformaton metod, Adoman' decompoton metod, omotopy perturbaton metod, varatonal teraton metod, plne metod ave been developed for olvng uc boundary value problem. Two-pont and mult-pont boundary value problem for fourt order ordnary dfferental equaton ave attracted a lot of attenton recently. Many autor ave tuded te beam equaton under varou boundary condton and by dfferent approace. Conder moot approxmaton to te problem of bendng a rectangular clamped beam of lengt l retng on elatc foundaton. Te vertcal deflecton w of te beam atfe te ytem d w ( / ) ( ), k Dw D + = qx dx () w(0) = wl ( ) = w'(0) = w'( l) = 0. were D te flexural rgdty of te beam, and k te prng contant of te elatc foundaton, and te load qx ( ) act vertcally downward per unt lengt of te beam. Te detal of te mecancal nterpretaton are gven n []. Matematcally, te ytem () belong to a general cla of boundary problem of te form d y f( x) yx ( ) gx ( ), a x b dx + = < < () ya ( ) = A, yb ( ) = A, y'( a) = B, y'( b) = B () IJCTA SEPT-OCT 0 Avalable onlne@
2 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 were f( x ) and gx ( ) are contnuou on [ ab, ] and A, B are fnte real arbtrary contant. Te analytcal oluton of () for arbtrary coce of f( x ) and gx ( ) cannot be determned. So, numercal metod are developed to overcome t lmtaton. Uman [] a formulated a mple condton tat guarantee te unquene of te oluton of te problem () and (). Among many numercal metod, a enumerated above, Splne metod ave been wdely appled for te approxmaton oluton of boundary value problem ncludng fourt order boundary value problem. (See [, ] and reference teren). Alo, Cubc B-plne a been ued to olve boundary value problem and ytem of boundary value problem [5,, 7], ngular boundary value problem [8] and alo, econd order perturbaton problem by ome autor. In te preent paper, we wll ue cubc B-plne to olve fourt order boundary value problem. T paper organzed a follow. In Secton, prelmnary reult and dervaton of cubc B- plne are preented; Secton contan cubc B-plne oluton of te pecal lnear fourt order boundary value problem baed on te reult n ecton ; general cae of te boundary value problem dcued n ecton ; ecton 5 deal wt non-lnear problem; wle ecton contan treatment of ngular problem. Secton 7 conclude te paper.. Dervaton for Cubc B-plne Te gven range of ndependent varable [ ab., ] For t range we cooe equdtant pont gven by π = { a = x0, x, x,... x n = b}.e. b a x = a + ( = 0,,... n) were =. Let u n defne S { ( π ) = pt () C [ ab, ] } uc tat pt () reduce to cubc polynomal on eac ub-nterval ( x, x + ).Te ba functon defned a ( x x ), f x [ x, x ] + ( x x ) + ( x x ) ( x x ), f x [ x, x] ( ) = ( + + ) + ( + ) ( x+ x), f x [ x, x+ ] x+ x f x x+ x+ B x x x x x ( ), [, ] 0, oterwe. Let u ntroduce four addtonal knot x < x < x0 and x > x > xn. From te above expreon, t obvou tat eac B ( x) C ( ). Alo, te value of B( x), B '( x) and B "( x ) at nodal pont are gven by Table I. Table I: Value of B ( x), B '( x) and B "( x ) at node B ( x ) B '( x ) B "( x ) x x / / / x / 0 / x + / / / x Snce eac B ( x) alo a pecewe cubc polynomal wt knot at π, eac B ( x) S( π ). Let Ω= { B, B0, B,..., B n + } and let B ( π ) = panω. Te functon n Ω are lnearly ndependent on[ ab,, ] tu B ( π ) ( n + ) - dmenonal. Alo B( π) = S( π) [9]. Let xbe ( ) te B-plne nterpolatng functon at te nodal pont and x ( ) B( π ).Ten xcan ( ) be wrtten a.terefore, for a gven functon yx ( ), = tere ext a unque cubc plne atfyng te nterpolatng condton: ( x) = cb ( x) = x ( ) = yx ( ) and '( a) = y'( a), '( b) = y'( b) () IJCTA SEPT-OCT 0 Avalable onlne@ 7
3 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 Let m = '( x ) and M = "( x ), we ave [0] ( v) m = '( x) y'( x) y ( x) (5) 80 ( v) ( v) M = "( x) y"( x) y ( x) + y ( x) 0 () M can be appled to contruct numercal dfference formulae for ( ) ( v) y ( x ), y ( x ) ( =,,..., n ) and ( v y ) ( x ) ( =,..., n ) a follow; ( ) ( ) M M ( x ) ( x ) ( ) ( v) = y ( x) + y ( x) (7) ( ) ( ) M M M ( x ) ( x ) = ( v) ( v) y ( x) y ( x) 70 M+ M+ + M M = M M M M M M y ( v) ( x ) (8) (9) Now, nce, ung Table I and above = equaton, we get approxmate value of ( yx ( ), y'( x ), y"( x ), ) ( v y ( x ) and y ) ( x ) a c + c + c+ yx ( ) = x ( ) (0) c+ c y'( x) = '( x) () c c + c+ y"( x) = "( x) () ( ) ( ) ( ) c ( ) + c + + c c y x x = () ( v) ( v) c+ c+ + c c + c y ( x) = ( x) (). Soluton of pecal cae fourt order boundary value problem Let yx ( ) = oluton of BVP = be te approxmate ( v y ) ( x) + f( x) yx ( ) = gx ( ) (5) Dcretzng BVP at te knot, we get ( v y ) ( x) + f( x) yx ( ) = gx ( ) ( =,... n ) () Puttng value n term of we get c ung equaton (0, ), c + c + + c c + c c c c f = g (7) Were f = f( x) an gd = gx ( ) are te value of f( x) and gxat ( ) te knot x. Smplfyng (7) become = ( c c c c c ) f( c c c ) g (8) T gve a ytem of ( n ) lnear equaton for ( =,... n ) n ( n + ) unknown vz. c ( =,0,... ). Remanng four equaton wll be obtaned ung te boundary condton a follow; ya ( ) = A c + c0 + c = A (9) yb ( ) = A cn + cn + c = A (0) y '( a) = B c + c = B () y '( b) = B c + c = B () n Te approxmate oluton yx ( ) = = obtaned by olvng te above ytem of ( n + ) lnear equaton n ( n + ) unknown ung equaton (8) and (9) to (). IJCTA SEPT-OCT 0 Avalable onlne@ 8
4 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 Numercal Example In t ecton we llutrate te numercal tecnque dcued n te prevou ecton by te followng two boundary-value problem: Problem. ( v) y + xy = (8 + 7 x + x ) e wt y(0) = y() = 0, y'(0) =, y'() = x () Te analytcal oluton yx ( ) = x( xe ) x.table II compare te numercal reult for problem of preent metod and numercal metod n []. Problem. ( v) y + y = wt y( ) = y() = 0, n n y'( ) = y'() = (co + co ) () Gven fourt order boundary value problem a analytcal oluton a nnn xn x+ cococo xco x yx ( ) = 0.5 co + co Comparon of numercal reult by preent metod and metod of [] demontrated n Table III. Table II: Max abolute error e for problem Preent metod Metod n[] /8.7E 8.5E 5 / 5.75E 9.9 E /.7E 9.5 E 8 Table III: Max abolute error e for problem Preent metod Metod n[] / 7.5E 9.0E. General cae lnear t order boundary value problem Conder te boundary value problem ( v) ( ) y ( x) + pxy ( ) ( x) + qxy ( ) " + rxy ( ) '( x) + txyx ( ) ( ) = ux ( ) (5) Subject to boundary condton gven by (). Let yx ( ) = be te approxmate = oluton of BVP. Dcretzg at knot ( v) ( ) y ( x) + py ( x) + qy "( x) + ry '( x) + tyx ( ) = u () Were p = px ( ), q = qx ( ), r = rx ( ), t = tx ( ), u = ux ( ). Puttng te value of dervatve ung (0-), we get c+ c+ + c c + c c+ c+ + c c + p c c + c+ c+ c c + c + c+ + q + r + t = u (7) On mplfcaton, t become c (c + c c + c ) + ( c p c + c c ) q( c c c+ ) r( c+ c ) t = ( c c c ) u (8) Now, te approxmate oluton obtaned by olvng te ytem gven by (8) and (9-). 5. Non-lnear t order boundary value problem Conder non-lnear fourt order BVP of te form ( v) ( ) y ( x) = f( x, y( x), y'( x), y"( x), y ( x)) (9) Subject to boundary condton gven n (). Let yx ( ) = be te approxmate = oluton of BVP. It mut atfy te BVP at knot. So, we ave ( v) ( ) y ( x) = f( x, y( x), y'( x), y"( x), y ( x)) (0) Ung (0-),we get /8.9E 7.8E 5 /.08E 8.7 E IJCTA SEPT-OCT 0 Avalable onlne@ 9
5 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN:9-09 c+ c+ + c c + c = c + c + c+ c+ c x,,, () f c c + c+ c+ c+ + c c, T eqn () togeter wt eqn (9-) gve a nonlnear ytem of equaton, wc olved to get te requred oluton of BVP.. Sngular t order boundary value problem Conder ngular fourt order BVP of te form ( v) γ ( ) y ( x) + y ( x) = f( xyx, ( )); x 0 x () Under te boundary condton y(0) = A, y () = A, y () = B, y (0) = 0. () Snce x = 0 ngular pont of eqn (), we frt modfy t at x = 0 to get tranformed problem a, ( v) ( ) y ( x) + pxy ( ) ( x) = rxy (, ) () were 0 x = 0 px ( ) = γ x 0 x (5) And f(0, y) x = 0 rxy (, ) = γ + f( xy, ) x 0 () Now, a n prevou ecton, let yx ( ) = be te approxmate oluton = of BVP. Dcretzg at knot, we get ( v) ( ) y ( x) + p( x) y ( x) = r( x, y( x)) (7) Puttng te value of dervatve ung (0-), c+ c+ + c c + c c+ c+ + c c + p c + c + c+ = rx (, ) (8) And boundary condton provde, y(0) = A c + c0 + c = A (9) y () = A c + c = A (0) n n n y () = B c c + c = B () T eqn (8) togeter wt eqn (9-) gve a nonlnear ytem of equaton, wc olved to get te requred oluton of BVP (). Numercal example Problem. d y y = e ( + x), 0 < x<, dx y(0) = 0, y() = ln, y'(0) =, y'() = /. () Te maxmum abolute error by our metod and by fnte dfference metod (Twzell []) for problem are preented n followng Table IV. Problem. d y d y 5 + = 5 y ( x y ) ( 7 x y ), x (0,), dx x dx () y(0) =, y () =, y () =, y (0) = Comparon of numercal reult by preent metod and tat of [] demontrated n Table V. Table IV: Max abolute error e for problem Preent metod Metod n[] /8.E-5 0.E- / 7.97E-7 0.E-5 / 5.8E-8 0.7E- Table IV: Max abolute error e for problem Preent metod Metod n[] /8.7E-.0E-0 /.E-7.75E-05 /.9E-8.8E-0 y (0) = 0 c c + c c = 0 () IJCTA SEPT-OCT 0 Avalable onlne@ 0
6 Yoge Gupta et al, Int. J. Comp. Tec. Appl., Vol (5), - ISSN: Concluon A numercal algortm for oluton of fourt order boundary value problem a been envaged. Te propoed metod a been extended to olve nonlnear and ngular problem a well. Te numercal reult demontrate tat te preent metod approxmate oluton better tan prevouly appled metod wt ame number of nterval. [] R. K. Sarma, C.P. Gupta, Iteratve oluton to nonlnear fourt-order dfferental equaton troug mult ntegral metod, Internatonal Journal of Computer Matematc, 8(989) 9. Reference [] E.L. Re, A.J. Callegar, D.S. Aluwala, Ordnary Dfferental Equaton wt Applcaton, Holt, Rneart and Wnton, New Cork, 97. [] R. A. Uman, Dcrete metod for boundary-value problem wt Engneerng applcaton, Matematc of Computaton, (978) [] M. Kumar, P. K. Srvatava, Computatonal tecnque for olvng dfferental equaton by cubc, quntc and extc plne, Internatonal Journal for Computatonal Metod n Engneerng Scence & Mecanc, 0( ) (009) [] M. Kumar, P. K. Srvatava, Computatonal tecnque for olvng dfferental equaton by quadratc, quartc and octc Splne, Advance n Engneerng Software 9 (008) -5. [5] N. Caglar, H. Caglar, B-plne metod for olvng lnear ytem of econd order boundary value problem, Computer and Matematc wt Applcaton 57 (009) [] H. Caglar, N. Caglar, K. Elfatur, B-plne nterpolaton compared wt fnte dfference, fnte element and fnte volume metod wc appled to two pont boundary value problem, Appled Matematc and Computaton 75 (00) [7] M. Degan, M. Laketan, Numercal oluton of nonlnear ytem of econd-order boundary value problem ung cubc B-plne calng functon, Internatonal Journal of Computer Matematc, 85(9) [8] M. Kumar, Y. Gupta, Metod for olvng ngular boundary value problem ung plne: a revew, Journal of Appled Matematc and Computng (00) [9] P. M. Prenter, Splne and varaton metod, Jon Wley & on, New York, 989 [0] F. Lang, Xao-png Xu, A new cubc B-plne metod for lnear fft order boundary value problem, Journal of Appled Matematc and Computng (0) 0-. [] Sraj-ul-Ilam, Ikram A. Trmz, Saadat Araf, A cla of metod baed on non-polynomal plne functon for te oluton of a pecal fourt-order boundary value problem wt engneerng applcaton, Appled Matematc and Computaton 7 (00) 9-80 [] E. H. Twzell, A two-grd, fourt order metod for nonlnear fourt order boundary value problem, Brunel Unverty department of matematc and Stattc Tecncal report TR//85 (985). IJCTA SEPT-OCT 0 Avalable onlne@
Application of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems
Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &
More informationSolution for singularly perturbed problems via cubic spline in tension
ISSN 76-769 England UK Journal of Informaton and Computng Scence Vol. No. 06 pp.6-69 Soluton for sngularly perturbed problems va cubc splne n tenson K. Aruna A. S. V. Rav Kant Flud Dynamcs Dvson Scool
More informationNumerical Simulation of One-Dimensional Wave Equation by Non-Polynomial Quintic Spline
IOSR Journal of Matematcs (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 14, Issue 6 Ver. I (Nov - Dec 018), PP 6-30 www.osrournals.org Numercal Smulaton of One-Dmensonal Wave Equaton by Non-Polynomal
More informationSolving Singularly Perturbed Differential Difference Equations via Fitted Method
Avalable at ttp://pvamu.edu/aam Appl. Appl. Mat. ISSN: 193-9466 Vol. 8, Issue 1 (June 013), pp. 318-33 Applcatons and Appled Matematcs: An Internatonal Journal (AAM) Solvng Sngularly Perturbed Dfferental
More informationTR/95 February Splines G. H. BEHFOROOZ* & N. PAPAMICHAEL
TR/9 February 980 End Condtons for Interpolatory Quntc Splnes by G. H. BEHFOROOZ* & N. PAPAMICHAEL *Present address: Dept of Matematcs Unversty of Tabrz Tabrz Iran. W9609 A B S T R A C T Accurate end condtons
More informationMethod Of Fundamental Solutions For Modeling Electromagnetic Wave Scattering Problems
Internatonal Workhop on MehFree Method 003 1 Method Of Fundamental Soluton For Modelng lectromagnetc Wave Scatterng Problem Der-Lang Young (1) and Jhh-We Ruan (1) Abtract: In th paper we attempt to contruct
More informationThe Finite Element Method: A Short Introduction
Te Fnte Element Metod: A Sort ntroducton Wat s FEM? Te Fnte Element Metod (FEM) ntroduced by engneers n late 50 s and 60 s s a numercal tecnque for solvng problems wc are descrbed by Ordnary Dfferental
More informationInitial-value Technique For Singularly Perturbed Two Point Boundary Value Problems Via Cubic Spline
Unversty of Central Florda Electronc Teses and Dssertatons Masters Tess (Open Access) Intal-value Tecnque For Sngularly Perturbed Two Pont Boundary Value Problems Va Cubc Splne 010 Lus G. Negron Unversty
More informationA Spline based computational simulations for solving selfadjoint singularly perturbed two-point boundary value problems
ISSN 746-769 England UK Journal of Informaton and Computng Scence Vol. 7 No. 4 pp. 33-34 A Splne based computatonal smulatons for solvng selfadjont sngularly perturbed two-pont boundary value problems
More informationImprovements on Waring s Problem
Improvement on Warng Problem L An-Png Bejng, PR Chna apl@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th paper, we wll gve ome mprovement for Warng problem Keyword: Warng Problem,
More informationMaejo International Journal of Science and Technology
Maejo Int. J. Sc. Technol. () - Full Paper Maejo Internatonal Journal of Scence and Technology ISSN - Avalable onlne at www.mjst.mju.ac.th Fourth-order method for sngularly perturbed sngular boundary value
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationTR/28. OCTOBER CUBIC SPLINE INTERPOLATION OF HARMONIC FUNCTIONS BY N. PAPAMICHAEL and J.R. WHITEMAN.
TR/8. OCTOBER 97. CUBIC SPLINE INTERPOLATION OF HARMONIC FUNCTIONS BY N. PAPAMICHAEL and J.R. WHITEMAN. W960748 ABSTRACT It s sown tat for te two dmensonal Laplace equaton a unvarate cubc splne approxmaton
More informationChapter 6 The Effect of the GPS Systematic Errors on Deformation Parameters
Chapter 6 The Effect of the GPS Sytematc Error on Deformaton Parameter 6.. General Beutler et al., (988) dd the frt comprehenve tudy on the GPS ytematc error. Baed on a geometrc approach and aumng a unform
More informationCubic Trigonometric B-Spline Applied to Linear Two-Point Boundary Value Problems of Order Two
World Academy of Scence Engneerng and echnology Internatonal Journal of Mathematcal and omputatonal Scences Vol: No:0 00 ubc rgonometrc B-Splne Appled to Lnear wo-pont Boundary Value Problems of Order
More informationPositivity Preserving Interpolation by Using
Appled Matematcal Scences, Vol. 8, 04, no. 4, 053-065 HIKARI Ltd, www.m-kar.com ttp://dx.do.org/0.988/ams.04.38449 Postvty Preservng Interpolaton by Usng GC Ratonal Cubc Splne Samsul Arffn Abdul Karm Department
More informationStanford University CS254: Computational Complexity Notes 7 Luca Trevisan January 29, Notes for Lecture 7
Stanford Unversty CS54: Computatonal Complexty Notes 7 Luca Trevsan January 9, 014 Notes for Lecture 7 1 Approxmate Countng wt an N oracle We complete te proof of te followng result: Teorem 1 For every
More informationHarmonic oscillator approximation
armonc ocllator approxmaton armonc ocllator approxmaton Euaton to be olved We are fndng a mnmum of the functon under the retrcton where W P, P,..., P, Q, Q,..., Q P, P,..., P, Q, Q,..., Q lnwgner functon
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationImprovements on Waring s Problem
Imrovement on Warng Problem L An-Png Bejng 85, PR Chna al@nacom Abtract By a new recurve algorthm for the auxlary equaton, n th aer, we wll gve ome mrovement for Warng roblem Keyword: Warng Problem, Hardy-Lttlewood
More informationNumerical Solution of Singular Perturbation Problems Via Deviating Argument and Exponential Fitting
Amercan Journal of Computatonal and Appled Matematcs 0, (): 49-54 DOI: 0.593/j.ajcam.000.09 umercal Soluton of Sngular Perturbaton Problems Va Devatng Argument and Eponental Fttng GBSL. Soujanya, Y.. Reddy,
More information1 Introduction We consider a class of singularly perturbed two point singular boundary value problems of the form: k x with boundary conditions
Lakshm Sreesha Ch. Non Standard Fnte Dfference Method for Sngularly Perturbed Sngular wo Pont Boundary Value Problem usng Non Polynomal Splne LAKSHMI SIREESHA CH Department of Mathematcs Unversty College
More informationChapter 4: Root Finding
Chapter 4: Root Fndng Startng values Closed nterval methods (roots are search wthn an nterval o Bsecton Open methods (no nterval o Fxed Pont o Newton-Raphson o Secant Method Repeated roots Zeros of Hgher-Dmensonal
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationSpecification -- Assumptions of the Simple Classical Linear Regression Model (CLRM) 1. Introduction
ECONOMICS 35* -- NOTE ECON 35* -- NOTE Specfcaton -- Aumpton of the Smple Clacal Lnear Regreon Model (CLRM). Introducton CLRM tand for the Clacal Lnear Regreon Model. The CLRM alo known a the tandard lnear
More informationUNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 2017/2018 FINITE ELEMENT AND DIFFERENCE SOLUTIONS
OCD0 UNIVERSITY OF BOLTON RAK ACADEMIC CENTRE BENG(HONS) MECHANICAL ENGINEERING SEMESTER TWO EXAMINATION 07/08 FINITE ELEMENT AND DIFFERENCE SOLUTIONS MODULE NO. AME6006 Date: Wednesda 0 Ma 08 Tme: 0:00
More informationM. Mechee, 1,2 N. Senu, 3 F. Ismail, 3 B. Nikouravan, 4 and Z. Siri Introduction
Hndaw Publhng Corporaton Mathematcal Problem n Engneerng Volume 23, Artcle ID 795397, 7 page http://dx.do.org/.55/23/795397 Reearch Artcle A Three-Stage Ffth-Order Runge-Kutta Method for Drectly Solvng
More informationOn Pfaff s solution of the Pfaff problem
Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of
More informationThe finite element method explicit scheme for a solution of one problem of surface and ground water combined movement
IOP Conference Seres: Materals Scence and Engneerng PAPER OPEN ACCESS e fnte element metod explct sceme for a soluton of one problem of surface and ground water combned movement o cte ts artcle: L L Glazyrna
More informationVariable Structure Control ~ Basics
Varable Structure Control ~ Bac Harry G. Kwatny Department of Mechancal Engneerng & Mechanc Drexel Unverty Outlne A prelmnary example VS ytem, ldng mode, reachng Bac of dcontnuou ytem Example: underea
More informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationScattering of two identical particles in the center-of. of-mass frame. (b)
Lecture # November 5 Scatterng of two dentcal partcle Relatvtc Quantum Mechanc: The Klen-Gordon equaton Interpretaton of the Klen-Gordon equaton The Drac equaton Drac repreentaton for the matrce α and
More information2nd International Conference on Electronics, Network and Computer Engineering (ICENCE 2016)
nd Internatonal Conference on Electroncs, Network and Computer Engneerng (ICENCE 6) Postve solutons of the fourth-order boundary value problem wth dependence on the frst order dervatve YuanJan Ln, a, Fe
More informationResearch Article Numerov s Method for a Class of Nonlinear Multipoint Boundary Value Problems
Hndaw Publsng Corporaton Matematcal Problems n Engneerng Volume 2012, Artcle ID 316852, 29 pages do:10.1155/2012/316852 Researc Artcle Numerov s Metod for a Class of Nonlnear Multpont Boundary Value Problems
More informationIntroduction to Interfacial Segregation. Xiaozhe Zhang 10/02/2015
Introducton to Interfacal Segregaton Xaozhe Zhang 10/02/2015 Interfacal egregaton Segregaton n materal refer to the enrchment of a materal conttuent at a free urface or an nternal nterface of a materal.
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationResearch Article Runge-Kutta Type Methods for Directly Solving Special Fourth-Order Ordinary Differential Equations
Hndaw Publhng Corporaton Mathematcal Problem n Engneerng Volume 205, Artcle ID 893763, page http://dx.do.org/0.55/205/893763 Reearch Artcle Runge-Kutta Type Method for Drectly Solvng Specal Fourth-Order
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationSmall signal analysis
Small gnal analy. ntroducton Let u conder the crcut hown n Fg., where the nonlnear retor decrbed by the equaton g v havng graphcal repreentaton hown n Fg.. ( G (t G v(t v Fg. Fg. a D current ource wherea
More informationResearch Article Cubic B-Spline Collocation Method for One-Dimensional Heat and Advection-Diffusion Equations
Appled Mathematcs Volume 22, Artcle ID 4587, 8 pages do:.55/22/4587 Research Artcle Cubc B-Splne Collocaton Method for One-Dmensonal Heat and Advecton-Dffuson Equatons Joan Goh, Ahmad Abd. Majd, and Ahmad
More informationPathan Mahabub Basha *, Vembu Shanthi
Amercan Journal of umercal Analyss 5 Vol 3 o 39-48 Avalable onlne at ttp://pubsscepubcom/ajna/3// Scence Educaton Publsng DOI:69/ajna-3-- A Unformly Convergent Sceme for A System of wo Coupled Sngularly
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationNumerical Solutions of a Generalized Nth Order Boundary Value Problems Using Power Series Approximation Method
Appled Mathematcs, 6, 7, 5-4 Publshed Onlne Jul 6 n ScRes. http://www.scrp.org/journal/am http://.do.org/.436/am.6.77 umercal Solutons of a Generalzed th Order Boundar Value Problems Usng Power Seres Approxmaton
More informationLecture 13 APPROXIMATION OF SECOMD ORDER DERIVATIVES
COMPUTATIONAL FLUID DYNAMICS: FDM: Appromaton of Second Order Dervatves Lecture APPROXIMATION OF SECOMD ORDER DERIVATIVES. APPROXIMATION OF SECOND ORDER DERIVATIVES Second order dervatves appear n dffusve
More informationComputers and Mathematics with Applications
Computers and Mathematcs wth Applcatons 56 (2008 3204 3220 Contents lsts avalable at ScenceDrect Computers and Mathematcs wth Applcatons journal homepage: www.elsever.com/locate/camwa An nnovatve egenvalue
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More informationChapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder
S-. The Method of Steepet cent Chapter. Supplemental Text Materal The method of teepet acent can be derved a follow. Suppoe that we have ft a frtorder model y = β + β x and we wh to ue th model to determne
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationTR/01/89 February An O(h 6 ) cubic spline interpolating procedure for harmonic functions. N. Papamichael and Maria Joana Soares*
TR/0/89 February 989 An O( 6 cubc splne nterpolatng procedure for armonc functons N. Papamcael Mara Joana Soares* *Área de Matematca, Unversdade do Mno, 4700 Braga, Portugal. z 6393 ABSTRACT An O( 6 metod
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationHaar wavelet collocation method to solve problems arising in induction motor
ISSN 746-7659, England, UK Journal of Informaton and Computng Scence Vol., No., 07, pp.096-06 Haar wavelet collocaton method to solve problems arsng n nducton motor A. Padmanabha Reddy *, C. Sateesha,
More information5 The Laplace Equation in a convex polygon
5 Te Laplace Equaton n a convex polygon Te most mportant ellptc PDEs are te Laplace, te modfed Helmoltz and te Helmoltz equatons. Te Laplace equaton s u xx + u yy =. (5.) Te real and magnary parts of an
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationOn the SO 2 Problem in Thermal Power Plants. 2.Two-steps chemical absorption modeling
Internatonal Journal of Engneerng Reearch ISSN:39-689)(onlne),347-53(prnt) Volume No4, Iue No, pp : 557-56 Oct 5 On the SO Problem n Thermal Power Plant Two-tep chemcal aborpton modelng hr Boyadjev, P
More informationModule 5. Cables and Arches. Version 2 CE IIT, Kharagpur
odule 5 Cable and Arche Veron CE IIT, Kharagpur Leon 33 Two-nged Arch Veron CE IIT, Kharagpur Intructonal Objectve: After readng th chapter the tudent wll be able to 1. Compute horzontal reacton n two-hnged
More informationBézier curves. Michael S. Floater. September 10, These notes provide an introduction to Bézier curves. i=0
Bézer curves Mchael S. Floater September 1, 215 These notes provde an ntroducton to Bézer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of
More informationAdditional File 1 - Detailed explanation of the expression level CPD
Addtonal Fle - Detaled explanaton of the expreon level CPD A mentoned n the man text, the man CPD for the uterng model cont of two ndvdual factor: P( level gen P( level gen P ( level gen 2 (.).. CPD factor
More informationA Discrete Approach to Continuous Second-Order Boundary Value Problems via Monotone Iterative Techniques
Internatonal Journal of Dfference Equatons ISSN 0973-6069, Volume 12, Number 1, pp. 145 160 2017) ttp://campus.mst.edu/jde A Dscrete Approac to Contnuous Second-Order Boundary Value Problems va Monotone
More informationModelli Clamfim Equazioni differenziali 22 settembre 2016
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 22 settembre 2016 professor Danele Rtell danele.rtell@unbo.t 1/22? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationFINITE DIFFERENCE SOLUTION OF MIXED BOUNDARY-VALUE ELASTIC PROBLEMS
4 t Internatonal Conference on Mecancal Engneerng, December 6-8, 1, Daa, Banglades/pp. V 171-175 FINITE DIFFERENCE SOLUTION OF MIXED BOUNDARY-VALUE ELASTIC PROBLEMS S. Reaz Amed, Noor Al Quddus and M.
More informationFormal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
More informationThe Analytical Solution of a System of Nonlinear Differential Equations
Int. Journal of Math. Analyss, Vol. 1, 007, no. 10, 451-46 The Analytcal Soluton of a System of Nonlnear Dfferental Equatons Yunhu L a, Fazhan Geng b and Mnggen Cu b1 a Dept. of Math., Harbn Unversty Harbn,
More informationChapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods
Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom ------ Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary
More informationTwo-Layered Model of Blood Flow through Composite Stenosed Artery
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 4, Iue (December 9), pp. 343 354 (Prevouly, Vol. 4, No.) Applcaton Appled Mathematc: An Internatonal Journal (AAM) Two-ayered Model
More informationThe Finite Element Method
The Fnte Element Method GENERAL INTRODUCTION Read: Chapters 1 and 2 CONTENTS Engneerng and analyss Smulaton of a physcal process Examples mathematcal model development Approxmate solutons and methods of
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationSTATIC ANALYSIS OF TWO-LAYERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION
STATIC ANALYSIS OF TWO-LERED PIEZOELECTRIC BEAMS WITH IMPERFECT SHEAR CONNECTION Ákos József Lengyel István Ecsed Assstant Lecturer Emertus Professor Insttute of Appled Mechancs Unversty of Mskolc Mskolc-Egyetemváros
More informationThe multivariate Gaussian probability density function for random vector X (X 1,,X ) T. diagonal term of, denoted
Appendx Proof of heorem he multvarate Gauan probablty denty functon for random vector X (X,,X ) px exp / / x x mean and varance equal to the th dagonal term of, denoted he margnal dtrbuton of X Gauan wth
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationORDINARY DIFFERENTIAL EQUATIONS EULER S METHOD
Numercal Analss or Engneers German Jordanan Unverst ORDINARY DIFFERENTIAL EQUATIONS We wll eplore several metods o solvng rst order ordnar derental equatons (ODEs and we wll sow ow tese metods can be appled
More informationLecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
More informationA New Recursive Method for Solving State Equations Using Taylor Series
I J E E E C Internatonal Journal of Electrcal, Electroncs ISSN No. (Onlne) : 77-66 and Computer Engneerng 1(): -7(01) Specal Edton for Best Papers of Mcael Faraday IET Inda Summt-01, MFIIS-1 A New Recursve
More informationCALCULUS CLASSROOM CAPSULES
CALCULUS CLASSROOM CAPSULES SESSION S86 Dr. Sham Alfred Rartan Valley Communty College salfred@rartanval.edu 38th AMATYC Annual Conference Jacksonvlle, Florda November 8-, 202 2 Calculus Classroom Capsules
More informationMultivariate Ratio Estimator of the Population Total under Stratified Random Sampling
Open Journal of Statstcs, 0,, 300-304 ttp://dx.do.org/0.436/ojs.0.3036 Publsed Onlne July 0 (ttp://www.scrp.org/journal/ojs) Multvarate Rato Estmator of te Populaton Total under Stratfed Random Samplng
More informationShuai Dong. Isaac Newton. Gottfried Leibniz
Computatonal pyscs Sua Dong Isaac Newton Gottred Lebnz Numercal calculus poston dervatve ntegral v velocty dervatve ntegral a acceleraton Numercal calculus Numercal derentaton Numercal ntegraton Roots
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationChapter 8: Fast Convolution. Keshab K. Parhi
Cater 8: Fat Convoluton Keab K. Par Cater 8 Fat Convoluton Introducton Cook-Too Algort and Modfed Cook-Too Algort Wnograd Algort and Modfed Wnograd Algort Iterated Convoluton Cyclc Convoluton Degn of Fat
More informationA MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS
Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ISSN 1549-644 1 Scence Publcatons do:1.844/jmssp.1.4.8 Publshed Onlne 9 (1) 1 (http://www.thescpub.com/jmss.toc) A MODIFIED METHOD FOR SOLVING SYSTEM OF
More informationSTABILITY OF DGC S QUADRATIC FUNCTIONAL EQUATION IN QUASI-BANACH ALGEBRAS: DIRECT AND FIXED POINT METHODS
Internatonal Journal of Pure and ppled Mathematcal Scence. ISSN 097-988 Volume 0, Number (07), pp. 4-67 Reearch Inda Publcaton http://www.rpublcaton.com STILITY OF DGC S QUDRTIC FUNCTIONL EQUTION IN QUSI-NCH
More informationNew Method for Solving Poisson Equation. on Irregular Domains
Appled Mathematcal Scences Vol. 6 01 no. 8 369 380 New Method for Solvng Posson Equaton on Irregular Domans J. Izadan and N. Karamooz Department of Mathematcs Facult of Scences Mashhad BranchIslamc Azad
More informationTHE STURM-LIOUVILLE EIGENVALUE PROBLEM - A NUMERICAL SOLUTION USING THE CONTROL VOLUME METHOD
Journal of Appled Mathematcs and Computatonal Mechancs 06, 5(), 7-36 www.amcm.pcz.pl p-iss 99-9965 DOI: 0.75/jamcm.06..4 e-iss 353-0588 THE STURM-LIOUVILLE EIGEVALUE PROBLEM - A UMERICAL SOLUTIO USIG THE
More information9.2 Seismic Loads Using ASCE Standard 7-93
CHAPER 9: Wnd and Sesmc Loads on Buldngs 9.2 Sesmc Loads Usng ASCE Standard 7-93 Descrpton A major porton of the Unted States s beleved to be subject to sesmc actvty suffcent to cause sgnfcant structural
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 13
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 13 GENE H GOLUB 1 Iteratve Methods Very large problems (naturally sparse, from applcatons): teratve methods Structured matrces (even sometmes dense,
More information( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.
3 Interpolaton {( y } Gven:,,,,,, [ ] Fnd: y for some Mn, Ma Polynomal Appromaton Theorem (Weerstrass Appromaton Theorem --- estence ε [ ab] f( P( , then there ests a polynomal
More informationBezier curves. Michael S. Floater. August 25, These notes provide an introduction to Bezier curves. i=0
Bezer curves Mchael S. Floater August 25, 211 These notes provde an ntroducton to Bezer curves. 1 Bernsten polynomals Recall that a real polynomal of a real varable x R, wth degree n, s a functon of the
More informationModule 3: Element Properties Lecture 1: Natural Coordinates
Module 3: Element Propertes Lecture : Natural Coordnates Natural coordnate system s bascally a local coordnate system whch allows the specfcaton of a pont wthn the element by a set of dmensonless numbers
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationMultigrid Methods and Applications in CFD
Multgrd Metods and Applcatons n CFD Mcael Wurst 0 May 009 Contents Introducton Typcal desgn of CFD solvers 3 Basc metods and ter propertes for solvng lnear systems of equatons 4 Geometrc Multgrd 3 5 Algebrac
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationNorms, Condition Numbers, Eigenvalues and Eigenvectors
Norms, Condton Numbers, Egenvalues and Egenvectors 1 Norms A norm s a measure of the sze of a matrx or a vector For vectors the common norms are: N a 2 = ( x 2 1/2 the Eucldean Norm (1a b 1 = =1 N x (1b
More informationInexact Newton Methods for Inverse Eigenvalue Problems
Inexact Newton Methods for Inverse Egenvalue Problems Zheng-jan Ba Abstract In ths paper, we survey some of the latest development n usng nexact Newton-lke methods for solvng nverse egenvalue problems.
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationChapter 3 Differentiation and Integration
MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 12
REVIEW Lecture 11: 2.29 Numercal Flud Mechancs Fall 2011 Lecture 12 End of (Lnear) Algebrac Systems Gradent Methods Krylov Subspace Methods Precondtonng of Ax=b FINITE DIFFERENCES Classfcaton of Partal
More informationSeparation Axioms of Fuzzy Bitopological Spaces
IJCSNS Internatonal Journal of Computer Scence and Network Securty VOL3 No October 3 Separaton Axom of Fuzzy Btopologcal Space Hong Wang College of Scence Southwet Unverty of Scence and Technology Manyang
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationENTROPY BOUNDS USING ARITHMETIC- GEOMETRIC-HARMONIC MEAN INEQUALITY. Guru Nanak Dev University Amritsar, , INDIA
Internatonal Journal of Pure and Appled Mathematc Volume 89 No. 5 2013, 719-730 ISSN: 1311-8080 prnted veron; ISSN: 1314-3395 on-lne veron url: http://.jpam.eu do: http://dx.do.org/10.12732/jpam.v895.8
More information