A Numerical Technique of Initial and Boundary Value Problems by Galerkin s Weighted Method and Comparison of the Other Approximate Numerical Methods

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1 Internatonal Journal of Engneerng Research & Technolog (IJERT) ISS: 78-8 Vol 3 Issue, Februar - 4 A umercal Technque of Intal and Boundar Value Problems b Galerkn s Weghted Method and Comparson of the Other Appromate umercal Methods Dpankar Kumar, Gour Chandra Paul, Ashabul Hoque 3 Department of Mathematcs, Bangabandhu Shekh Mubur Rahman Scence & Technolog Unverst, Gopalgan-8, Bangladesh Department of Mathematcs, Unverst of Rashah, Rashah-65, Bangladesh 3 Department of Mathematcs, Unverst of Rashah, Rashah-65, Bangladesh Abstract In ths paper, the ntal and boundar value problems are solved b Galerkn weghted resdual method In the case of ntal value problem, accurac of Galerkn method s shown over eact soluton Accurac s also contnued to mprove over the solutons b some standard numercal methods It s shown that there s an astonshng accurac of the Galerkn s appromaton method wth even two terms n the case of ntal value problem Agan In the cases of boundar value problem, some aspects of boundar problem are shown n solvng them b Galerkn weghted resdual appromaton method In ths stuaton, the result of our calculaton shows that bass functons are ver dense n a space contanng the actual soluton Galerkn fnte element method s also ntroduced n solvng boundar value problem Resultng accurac s also tested Galerkn fnte element method s found to be so effectve that n ths method an etraordnar accurac s acheved wth modest effort Kewords Galerkn weghted resdual, Galerkn fnte element method, ntal value problem, boundar value problem I ITRODUCTIO In Mathematcs, Engneerng and other branches of scence, dfferental equatons are used to model problems Most of the problems requre the soluton to an ntal value problem that s the soluton to a dfferental equaton that that satsfes a gven condton But n some cases arse n real lfe stuatons; the dfferental equaton that models the problem s so complcated that, there s rarel a soluton In such cases, where an analtc soluton s not possble, one must have to adopt one of the two was, namel ( ) to smplf the dfferental equaton to one that that can be solved eactl and () to use methods for appromatng the soluton of the orgnal problem There are varous procedures for obtanng a numercal soluton to a dfferental equaton These methods can be separated nto three basc groupngs, namel () the fnte dfference method, () the varatonal method and ) the method that weght a resdual Our am s to solve an ntal value problem b Galerkn weghted resdual appromaton method Several problems arsng n scence and engneerng are modeled b dfferental equatons that nvolve condtons that are specfed at more than one pont Such tpes of problems are called boundar-value problems The crucal dstncton between ntal value problems and boundar value problems s that n the former case we are able to start an acceptable soluton at ts begnnng (ntal values) and ust march t along b numercal ntegraton to ts end (fnal values), whle n the case of boundar value problem, boundar condtons at the startng pont do not determne a unque soluton to start wth a random choce among the solutons that satsf these ncomplete startng boundar condtons s almost certan not to satsf the boundar condtons at the other specfed ponts There are three standard methods for solvng two pont boundar value problems, namel shootng method, fnte dfference method and proecton method Among these, fnte dfference method s popular one Our assumpton s that the dfferental equaton s lnear The fnte dfference method appromates the dervatves n the governng dfferental equaton usng dfference equaton The varatonal approach nvolves the ntegral of a functon that produces a number Each functon produces a new number The functon that produces the lowest number has the addtonal propert of satsfng a specfc dfferental equaton The weghted resdual methods also nvolve an ntegral In these methods, an appromate soluton s substtuted nto the dfferental equaton Snce the appromate soluton does not satsf the dfferental equaton, therefore an error term or a resdual results Weghted resdual method requres that the nner product of the resdual and each of the weghted functons must be zero There are several processes to choce weghted functons Galerkn s method s one of them In order to solve to solve ordnar dfferental equatons b Galerkn method the followng terms are ver essental to descrbe The fnte element method s another numercal technque that gves appromate solutons to dfferental equatons that model problems arsng n phscs and engneerng As n smple fnte dfference schemes, the fnte element method requres a problem defned n geometrcal space (or doman), to be subdvded nto a fnte number of smaller regons The earl work on numercal soluton of IJERTV3IS565 wwwertorg 334

2 Internatonal Journal of Engneerng Research & Technolog (IJERT) ISS: 78-8 Vol 3 Issue, Februar - 4 boundar-valued problems can be traced to the use of fnte dfference schemes The begnnngs of the fnte element method actuall stem from these earl numercal methods and the frustraton assocated wth attemptng to use fnte dfference methods on more dffcult, geometrcall rregular problems [7] The frst publcatons n fnte element method appeared n 95 s wth the works wrtten b [, 6, and ] These were used to solve problems n structural analss Some decades later, Zenkewcz and Cheung, Oden and Wellford, Chung and Baker among other publcatons, treated the heat transfer and flud flow problems solutons nvolvng soluton of Laplace and Posson equatons [,, 5 and ] A vgorous mathematcal dscusson s gven b Johnson [], and programmng the fnte element method s descrbed b Smth [8] The mathematcal bass of the fnte element method frst les wth the classcal Ralegh-Rtz and varatonal calculus procedures ntroduced b Ralegh [4] and Rtz [6] Recent descrptons of the method are dscussed n [4,, 8, 9, 5, 9, 3 and 7] Most practtoners of the fnte element method now emplo Galerkn's method to establsh the appromatons to the governng equatons Instead of gong nto rgorous treatment about ths Galerkn fnte element, we onl ntend to show n ths artcle s that wh ths method s so effectve II GOVERIG EQUATIOS A Intal Value Problems We consder the general equaton of ntal value problems of the frst ordered as d f (, ) subect to the ntal condton d ( ), where n To keep the dscusson smple whle meanngful a general formulaton, we consder the followng ntal value problems of the frst ordered as d () d subect to the ntal condton ( ) () where B Boundar value problems We consder the general equaton of boundar value problems of the frst ordered as d d ) f (,, ) subect to the boundar condton ( n ( ), where n Agan to keep the dscusson smple whle meanngful a general formulaton, we agan consder the followng ntal value problems of the frst ordered as d d subect to the boundar condton ( ) () (4) (3) III A Galerkn s weghted Method METHODS In ths method, an appromatng functon called the tral functon s substtuted n the gven dfferental equaton and the result s called the resdual It s mentoned that the result wll not be zero snce an appromaton functon s substtuted The resdual s then weghted and the ntegral of the product, taken over the doman, s set to zero An advantage of ths method s that t works wth the governng equatons of the problem and does not requre a functonal Galerkn s Requrements Let us solve the lnear dfferental equaton L( u) f b choosng bass functon Then appromatng the actual soluton u ~ b a lnear combnaton of these functons c u~ for all values of appromate u ~ satsfes The resdual R L u~ ) f bass element,, appromaton e,, R c the ( must be orthogonal to the, 3 used n the, where R L c f Hence, R c,, f B Galerkn fnte element method The fnte element method s a numercal technque that gves appromate solutons to dfferental equatons that model problems arsng n phscs and engneerng As n smple fnte dfference schemes, the fnte element method requres a problem defned n geometrcal space (or doman), to be subdvded nto a fnte number of smaller regons (a mesh)ths method s based on the dea of buldng a complcated obect wth smple blocks, or dvdng a complcated obect nto small and manageable peces It s provdes a greater fleblt to model comple geometres than fnte dfference and fnte volume methods do It has been wdel used n solvng structural, mechancal, heat transfer, and flud dnamcs problems as well as problems of other dscplnes The fnte element method has grown out of Galerkn s method, emergng as a unversal method for the soluton of dfferental equatons Much of the success of the fnte element method can be contrbuted to ts generalt and smplct, allowng a wde range of dfferental equatons from all areas of scence to be analzed and solved wthn a common framework Another contrbutng factor to the success of the fnte element method s the fleblt of formulaton, allowng the propertes of the dscretzaton to be controlled b the choce of fnte element appromatng spaces Hstorcall, all maor practcal advances of the fnte element method have taken place snce the earl 95s n conuncton wth the development of dgtal computers However, nterest n appromate solutons of feld equatons IJERTV3IS565 wwwertorg 335

3 Internatonal Journal of Engneerng Research & Technolog (IJERT) ISS: 78-8 Vol 3 Issue, Februar - 4 dates as far back n tme as the development of the classcal feld theores themselves The work of Ralegh [4] and Rtz [6] on vbratonal methods and the weghted-resdual approach taken b B G Galerkn and others form the theoretcal framework to the fnte element method C Fnte dfference method The fnte dfference method for the soluton of a two pont boundar value problem conssts n replacng the dervatves occurrng n the dfferental equaton b means of ther fnte dfference appromatons and then solvng the resultng lnear sstem of equatons b a standard procedure D Dscontnuous Galerkn weghted method The Dscontnuous Galerkn weghted method s somewhere between a fnte element and a fnte volume method and has man good features of both It provdes a practcal framework for the development of hgh-order accurate methods usng unstructured grds The method s well suted for large-scale tme-dependent computatons n whch hgh accurac s requred [3] An mportant dstncton between the Dscontnuous Galerkn weghted (DGW) method and the usual fnte-element method s that n the Dscontnuous weghted Galerkn (DGW) method the resultng equatons are local to the generatng element The soluton wthn each element s not reconstructed b lookng to neghborng elements Its compact formulaton can be appled near boundares wthout specal treatment, whch greatl ncreases the robustness and accurac of an boundar condton mplementaton IV SOLUTIOS A Analtcal soluton of ntal value problem The analtc soluton of the Eq () subect to the condton () s gven b ( ) e (5) B Soluton of ntal value problem b Galerkn weghted method Let us use the basc functons,, 3, 4, ( ) Each of whch satsfes the condton Let the tral soluton be ~ c The resdual for ths tral soluton s R (6) c ( ) (7) Imposng Galerkn s requrement, we have, c (, Ths equaton elds equatons k c ( ) ), (8) where,,, 3,, We have solved the equatons Ac b for the unknown c wth the help of the MATLAB routne Accurac wll contnue to mprove over the solutons of the dfferental equaton b two standard numercal methods, namel Euler s method and Range-Kutta method Fg Soluton curves b Galerkn s, Euler and Range- Kutta method for = Fg Soluton curves b Galerkn s, Euler and Range-Kutta method for = Fg3 Soluton curves b Galerkn s, Euler and Range-Kutta method for =3 IJERTV3IS565 wwwertorg 336

4 Internatonal Journal of Engneerng Research & Technolog (IJERT) ISS: 78-8 Vol 3 Issue, Februar - 4 Snce our am s estmate a comparson and to show accurac of our problem b Galerkn weghted resdual method wth the eact soluton and those obtaned b the two standard methods mentoned above, so nstead of gong nto gvng detal descrpton about these methods, onl MATLAB routne n the respectve cases are gven C Analtcal soluton of boundar value problem Analtc soluton of boundar value problem of Eq (3) mposng the condton eq (4), we get sn ( ) cos ( cos) (9) sn D Soluton of boundar value problem b Galerkn weghted method We emplo as usual the bass functons, where,, 3,, Fg4 Soluton curves b Galerkn s, Euler and Range- Kutta method for =5 Fg5 Soluton curves b Galerkn s, Euler and Range- Kutta method for = Each of the bass functon satsfes both the boundar condtons gven b Eq (3) We assume the tral soluton of the problem prescrbed b Eq (3) as c The resdual for ths tral soluton s gven b R () c c () Imposng Galerkn s requrement, we have c, c, As Sobolev matr Eq () then can be wrtten as c d S,, c d d or, ( )( ) c 3 where,,, 3, To obtan Galerkn appromate soluton of the gven boundar value problem, the unknown c must have to be determned The values of c are elds MATLAB routne If we set, then the value of the unknown wth the help of the above MATLAB routne s obtaned as c 778 and then the Galerkn appromate soluton s obtaned as ~ 778( ) For 5, we have frst appromate soluton as ~ 6945, whereas the eact soluton gven b Eq (8) of the gven boundar value problem for that pont elds Comparson wth the eact soluton the error n the computed soluton b Galerkn method s 74 Agan f we set, then the values of the unknowns wth the help of the above MATLAB routne are obtaned as c 94 and c 77, and then the Galerkn appromate ~ soluton s obtaned as 94( ) 77( 3 ) For 5, the second appromate soluton s gven b ~ In ths case the error s 34 Smlarl the other appromate solutons are obtaned some of them are gven n Table IJERTV3IS565 wwwertorg 337

5 Internatonal Journal of Engneerng Research & Technolog (IJERT) ISS: 78-8 Vol 3 Issue, Februar - 4 E Soluton of boundar value problem b fnte dfference method Wth h 5, use the fnte-dfference method to determne the value of 5 It s shown that ts eact soluton of Eq (3) s gven b cos cos sn, sn from whch, we obtan Here nh Then the gven boundar value problem dscretzed as fnte dfference method can be wrtten as () h and the Eq () after smplfcaton gves h h, (3) where,,3,, n whch together wth the boundar condtons and, comprses a sstem of n n equatons for the n unknowns,,,, n Choosng h (e n ), the above sstem becomes 4 4 Wth, ths gves Comparson wth the eact soluton gven above shows that the error n the computed soluton s 336 On the other hand, f we choose h (e n 4 ), we 4 obtan the three equatons: , 6 6 where Solvng the sstem we obtan , 449 the error n whch s 8 Snce the rato of the two errors s about 4, t follows that the order of convergence s h The results of our calculaton n respectve cases are tabulated n shown Table Table : Comparson of our calculated value wth eact value Computed value of b Galerkn weghted method 5 Eact value of 395 st appro nd appro rd appro Table : Comparson of our calculated value wth eact value Computed value of b Fnte dfference method 5 Eact value of 395 n n n F Soluton of boundar value problem b Galerkn fnte element method Dvde the nterval [, ] nto equal subntervals, each of,, 3,, length For, we take the pecewse lnear bass functon that s zero off the open nterval (( ),( ) ) but has value at as shown n the Fg 6 Fg6 Bass functons wth small support Fg6 Bass functons wth small support Let us assume the pecewse lnear tral soluton c (4) Ths tral soluton gves the resdual The resdual for ths tral soluton s gven b R c c (5) Imposng Galerkn s requrement, we have IJERTV3IS565 wwwertorg 338

6 Internatonal Journal of Engneerng Research & Technolog (IJERT) ISS: 78-8 Vol 3 Issue, Februar - 4 c, c, (6) The pecewse lnear nterpolaton (4) has zero second dervatve almost everwhere, therefore, our Eq (6) wll then be reduced to the smple form as Gc,, (7) where G s the Graman and s gven b G, (8) Equaton (7) fnds the best ft to the horzontal lne It s not solvng the gven boundar value problem prescrbed b the Eqs () and () Ths same falure wll alwas occur when usng pecewse lnear tral functons to solve second order problems One wa to vew ths falure s that the approach does not take nto account the Drac delta functons that should arse when dfferentatng these bass functons twce Another vew s that we must not ask so much of solutons-the need not be so dfferentable Rather than requrng the soluton satsfes the classcal statement of the problem gven b Eq () We onl requre that the soluton holds when proected nto fnte dmensonal subspaces, e satsfes the weak condton,,, (9) for an test functon wth square-ntegrable dervatve and zero boundar values B ntegratng b parts, we throw one dervatve onto the test functon Thus the weak restatement of our Galerkn problem s gven b Eqs () and () s c, c,,, () where,, 3,, The Eq () can be wrtten as G S) c (, ), () ( where S represent Sobolev matr and s gven b S,, () Because all the bass functons are translates of one another, G and S are eas to compute A framework so of such fnte element problems s gven below f 3, f mod( ) 6 otherwse f, f otherwse, (3) (4) and, (5) Thus equaton () becomes the tr-dagonal sstem 4 c 4 c 6 4 c c c c (6) Implementaton of the above sstem s gven b the MATLAB routne V DISCUSSIO OF THE RESULT AD COCLUSIO Galerkn s appromaton weghted resdual method for the soluton of ntal value problem s nvestgated In each case the result of our calculaton s shown graphcall Fgs -5 ndcate that there s an astonshng accurac of the Galerkn s appromaton method wth that of eact method After two terms, Fgs 3-5 show that each of the soluton curve obtaned b eact soluton overlaps on the curves that are etracted b our calculaton n respectve cases of nterest Accurac s also tested over some standard numercal appromaton methods Fg8 Soluton curves b Galerkn s and fnte dfference methods for =3 IJERTV3IS565 wwwertorg 339

7 Internatonal Journal of Engneerng Research & Technolog (IJERT) ISS: 78-8 Vol 3 Issue, Februar - 4 Fg9 Soluton curves b Galerkn s and fnte dfference methods for =5 Fg Soluton curves b Galerkn s and fnte dfference methods for =7 Galerkn s fnte element soluton of boundar value problem s nvestgated In each case the result of our calculaton s shown graphcall Fgs 7-8 ndcate that there s an astonshng accurac of the Galerkn s appromaton method wth that of eact method Fgures 9- show that each of the soluton curve obtaned b eact soluton overlaps on the curves that are etracted b our calculaton n respectve cases of nterest Accurac s also tested over some standard numercal appromaton methods Therefore, we ma conclude that the bass functons are dense n a space contanng the actual soluton Two pont Boundar value problems are solved b Galerkn method The accurac of our calculated values s compared wth the results obtaned b eact soluton and fnte dfference method The errors are also estmated n the respectve cases The are gven n Tables and The results show that the accurac obtaned b the fnte-dfference method depends upon the wdth of the subnterval chosen and also on the order of the appromatons As h s reduced, the accurac ncreases but the number of equatons to be solved also ncreases Whereas, n the case of Galerkn method, the accurac depends upon the number of bass functons chosen It s shown that even thrd appromaton elds an astonshng accurac There s onl one problem of ths s method that more computatons are needed for more choce of bass functons The same concluson can be drawn here n the case of boundar value problem as s drawn n the case of ntal value problem Fnall the concluson can be drawn that the Galerkn fnte element method s so effectve that, n ths method such an etraordnar accurac s acheved wth modest effort REFERECES [] Argrs, J H, 963, Recent Advances n Matr Methods of Structural Analss, Pergamon Press, Elmsford, ew York [] Baker, A J, 983, Fnte Element Computatonal Flud Mechancs, ew York: Hemsphere, McGraw-Hll [3] Becker, A A, 4 An ntroductor gude to fnte element analss, ASME Press, Y [4] Chandrupatla, T R and Belegundu, A D, Introducton to fnte elements n engneerng, Prentce Hall, Upper Saddle Rver, J [5] Chung, T J, 978, Fnte Element Analss n Flud Dnamcs, ew York: McGraw-Hll [6] Clough, R W, 96, The Fnte Element Method n Plane Stress Analss, Proceedngs of nd Conf on Electronc Computaton, Amercan Socet of Cvl Engneers, Pttsburgh, Penn, pp [7] Ern, A and Guermond, J L, 4 Theor practce of fnte elements, Sprnger-Verlag, Y [8] Hollg, K, 3 Fnte elements wth B-splnes, Socet of Industral and Appled Mathematcs, Phladelpha, PA [9] Hutton, D V, 4 Fundamentals of Fnte Element Analss, McGraw-Hll, Boston, MA [] Johnson, C, 987 umercal Soluton of Partal Dfferental Equatons b the Fnte Element Method, Cambrdge Unverst Press, Cambrdge, UK [] Lu, G R and Quek, S S, 3 The fnte element method: A practcal course, Butterworth- Henemann, Boston, MA [] Oden, J T, and Wellford Jr, L C, 97, Analss of vscous flow b the fnte element method, AIAA J, Vol, pp [3] Qu JX, Khoo BC, Shu CW, 6, A umercal Stud for the Performance of the Runge-Kutta Dscontnuous Galerkn Method Based on Dfferent umercal Flues, Journal of Computatonal Phscs Vol, o, pp [4] Ralegh, J W S, 877 Theor of Sound, st Revsed edton, Dover Publshers, Y [5] Redd, J, 4 An ntroducton to nonlnear fnte element analss, Oford Unverst Press, Oford, UK [6] Rtz, W, 99 "Uber ene eue Methode zur Lïsung Gewsses Varatons-Problem der Mathematschen Phsk," J Rene Angew Math, Vol 35, pp -6 [7] Roache, P J, 97 Computatonal Flud Mechancs, Hermosa Publshers, Albuquerque, M [8] Smth, I M, 98 Programmng the Fnte Element Method, John Wle & Sons, Y [9] Soln, P, Segeth, K and Dolezel, I, 4 Hgher-order fnte element methods, Chapman and Hall/CRC, Boca Raton, FL [] Turner, M J, Clough, R W, Martn, H C, and Topp, L P, 956, Stffness and Deflecton Analss of Comple Structures, J Aeron Sc, Vol 3, o 9, pp [] Zenkewcz, O C and Cheung, Y K, 965, Fnte elements n the soluton of feld problems The Engneer, Vol, pp 57-5 IJERTV3IS565 wwwertorg 34

Mathematics 256 a course in differential equations for engineering students

Mathematics 256 a course in differential equations for engineering students Mathematcs 56 a course n dfferental equatons for engneerng students Chapter 5. More effcent methods of numercal soluton Euler s method s qute neffcent. Because the error s essentally proportonal to the