Partial preview of the text
Download Numerical Solution of ODEs Lecture Notes and more Study notes Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
tSLatre duction Mon problems im science and en incevury Coymot be “solved analytically. So Ww ordev to et the sob of Such type of provlem<, numerical — methods |tech- ove used - v= ore “he. deviation 1 analysis and implemen takior of numerical qethods per compiler mathemakicak provlens We within the disciptine oS — Nomericak A “Since nunericod method jv appwOF. Sod So Shere ave. essentialy chones sf — error: ExtoY “w the dah bho enact value and approximate vasve 5) a a Envoy = = ak at = exack val. MK = appro valk tommmnw Soukcts Or i Yoss- Late Often ov'isey due fo homan mistake! e 4 inaccvvate emdaka suse of “of one Serie» another with ewer terms. The _errov... troduced by thy Yeplaumenk.. index med arty unc hon _ -erroy: Exomple The nfyrite Ca IANNCARNERPANEINS of LK Ok Asbo 16: Coo aL ab ya ee 21 Wy 6! an ok a) be Sa ir joovse f= 2 £ 7 key m ae \y clan yh ONY oil COSA = leat + ae at phe tete caer IN Some “compolchon & ; tense TD ngy Or Errigamesssoeo BGrety ae “late ot porcieel “ofp Abanlle Evvore A-E =] * Pl Ik is defined ag the vation of absolute eyror to tke er valve Relative €xror aan 3 P% sho - |p¥-P] \P*! Relative €xvor ghwes hebler wtp d Oxo Whormakion about error. Ee Pescentage E caer Relate ercor expressed interme of Perceniag®. ic _krowon ay percentage. error canner’ Enor= FEe REXI00 aa AE = |PF-PI = 0.04166343 MB Relative Exyor RE = ME = 0-041 bb 343 }P*| logy 41099] RE = 9-044 $1321 wee? Percentade Evror PEs RE Xto0 = 0-049 (321 ¥ 100 = 44g} 321°/- 1 { } ____| Prot opogation “oF Exvoy |Evror Accwmolation ————————— = SS al a an — 2.36 eee “rabeecte ie opmmow 2 Sore a Lek. pt y bo ¥ values. with o> “v “le values ie Ton a t= eee oy Sen 20 a ——. Ao we be a. | a pts ot = Gp 4p + eave yy - | | Pet —mepcay PA | Pleas -(pea)—aeptee — | _S¥ -S = Op +epy, a S= poy eg 2 epteq). = Ccagk~c less fep ey] Jepl sey! steals sat ERIE SEE ain Sent eee eee _ he_absoldteOrrorin som 2 no. Cannth neces _Bea_som fF abSo lute ertors 4 No. 9 Sn. general lesje Jel + jerltless ten! gy fegy. Bt “JE Peay 7 - Cpa YeptPey,. ee eH t P ¥y lee) el ex Phat pay aeaep vep it Req — Relative. extor my _groo h 2 No: ‘y “the, Sum _d) Cehakive er vOr$- ~ Ah taps Aoggnot excead 42 S211 Ft]s* = eyrautabional constawt - @- angle with. vorticod. WMC cassieication Or Drrecennian Evcunrionssh ent fa Orpiwary Diererentiar Eovanow The ai pevential vation which has only one indeperdart enn) is walled 4 ODE J Exomple 4 = B+ SNH BY Partick Dirrerentiat Eourton Gt The. dif) ey hid hor moe fon 4 indep Vow is “now os PDE Cuample - coy indep voy XAG: On 4 cm yy Te Or ODEsp, ATEGORIES IF S The ODEs can be categorized accord ing to they 4s B21 $efs + ayouitetional _carstans | &- angle wilh _vorliced. 7 WOMAN vasseicaion Or Drrecenrinn Eouanons BB Ceoiwney Dicrerentiar Equanon Md Fre di pevential vation which has only one_ independent neal is walled ODE J Example ue = 3B + oN | BE Portiol Diererentiat Eovrvon The. dif ea. whidr har moe thon 4 indeyp var: is non od PDE _ Example ay = coy indep Voy XAG: Ok On Bo nrecones Ot ODED, he ODEs can be categarined according to. nev Rf Db Lope wilh pe _ _Uneay.. OO. o_inear _ fon onl on. ot _voriahie, ben, ie Fy {Yury af ly) ii. tranccedenta). AG non - Linear _ he. _contaning, “erm BS e ey vo ee coer a TBOUNDARY CONDITIONS ia Z _Exasple — ——MiBowrioy Conkion Sa rete -—HBIINTIAL_CONDITION = nk gti pe Pyle i PE Ey Methods —Sovve. : Ove Pa i IR the dep wow. ib defined od 4. pont the _endth ill be) ¢ The Ai “along with ee WP - ‘= ak ey WP ave _also_termed as. are _p-toblems or Or than problems ae ees eae tee ae OF te__probs ore. Heb ghes, of _more_pks_. such _icondrtions wi be Bc and the. associated problem “VV -* Analutica) Methods _ © Crophical Methods _ 6 Nl Methods GBTYPES OF NUMERICAL METHODS ER. # class. of methods Lohidh produces. “ex pressims by ty! interms of | hunctions of "at pvor whic _ Volues of "y" Can be determined by diveck — Substitution ; @ PICARD's METHOS @ TAYLOR'S SERIES Method 6 4 ie conkinvors on [ajb] and OX (ayb) IReA a number ) “a dif heventiable Ew _layb) exists cuch (Ce) = fb) fla), cope ot b- a uve feC lad] x F'n) exists yw xe (ais) tren, 4 Ano. c— wilh Cela) enor mon value Ctohtr thar Ue is adteayr 4 no. CElab) SE lhe slope_of The bongent tine, to the Qvapn Coy Flgy = te elope of te i pols (a. fil) & (vf (by) | 0 Y=fu ot the powt Secant ine Hevougin Te ry, OX Sosa tinas defy _ b-a_ - t . b aa