Solving Second Order Linear Dirichlet and Neumann Boundary Value Problems by Block Method

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1 IAENG International Journal o Applied Matematics 43: IJAM_43 04 Solving Second Order Linear Diriclet and Neumann Boundar Value Problems b Block Metod Zanaria Abdul Majid Mod Mugti Hasni and Norazak Senu Abstract In tis paper te direct tree-point block one-step metods are considered or solving linear boundar value problems (BVPs) wit two dierent tpes o boundar conditions wic is te Diriclet and Neumann boundar conditions. Tis metod will solve te second order linear BVPs directl witout reducing it to te sstem o irst order equations. Te direct solution o tese two tpes o BVPs will be calculated at tree points simultaneousl using constant step size. Tis metod will be used togeter wit te linear sooting tecnique to construct te numerical solution. Te implementation is based on te predictor and corrector ormulas in te PE(CE) r mode. Numerical results are given to sow te perormance o tis metod compared to te eisting metods. Inde Terms diriclet boundar value problems neumann boundar value problems block metod M I. INTRODUCTION ANY problems in science and tecnolog are ormulated in boundar value problems as in diusion eat transer delection in cables and te modeling o cemical reaction. Tere are several tpes o boundar value problems (BVPs) and some o tem depend on te boundar condition itsel. In tis paper we consider te second order linear two-point BVPs wic as ollows: p q r q 0 ab wit te Diriclet boundar conditions: a and (b) () and wit te Neumann boundar conditions: a and b. () (3) Manuscript received Marc 3 03; revised April 03. Tis work was supported in part b te Graduate Researc Fellowsip (GRF) rom Universit Putra Malasia and MMaster rom te Ministr o Higer Education. Zanaria Abdul Majid is a lecturer at te Matematics Department Facult o Science and also as associate researcer at te Institute or Matematical Researc Universiti Putra Malasia Serdang Selangor DE Malasia ( am_zana@upm.edu.m). Mod Mugti Hasni is a Master student at te Institute or Matematical Researc Universiti Putra Malasia Serdang Selangor DE Malasia ( ModMugti@gmail.com). Norazak Senu is a lecturer at te Matematics Department Facult o Science Universiti Putra Malasia Serdang Selangor DE Malasia. ( norazak@science.upm.edu.m). Teorem: Suppose te unction in te BVPs in () is continuous on te set D a b and te partial derivatives and are also continuous on D. I (i) 0 or all D and (ii) A constant M eists wit M or all D Ten te BVPs as a unique solution. Corollar: In linear BVPs in () satisies (i) p()q() and r() are continuous on a b (ii) q() > 0 on a b ten te problem as unique solution. In literature tere are several researcer tat as been conducted te researc or solving te BVPs suc as Wang et al. [4] Emad et al. [9] and Bongsoo [3]. Adomian Decomposition Metod (ADM) as been widel used b man researcers or solving dierential and integral problems. However tis metod as dealt wit some diiculties or solving te problem involving wit te boundar conditions. Tus Bongsoo [3] in 00 ave proposed a metod called te Etended Adomian Decomposition Metod (EADM) or solving te two-point linear and nonlinear second order boundar value problems. Bongsoo [3] managed to overcome te problems inibit in te ADM b creating a new canonical orm containing all boundar conditions make it suitable or solving te BVPs. Finite Dierence (FDM) Finite Element (FEM) and Finite Volume (FVM) metods ave been proposed b Fang et al. [] in 00 to solve te two-point BVPs. Te comparison in terms o te accurac as been made between tese tree metods and te results sown tat te are comparable to eac oter wit no remarkable dierences. Later in 00 Caglar et al. [] ave proposed a new metod called te B- spline Interpolation (CBIM) to compare te accurac o te results obtained wit te metods proposed b Fang et al. [] in 00. Clearl te CBIM managed to give better results. A ew ears later Hamid et al. [3] ave proposed a new metod called te Etended Cubic B-Spline metod (ECBIM) or solving te linear two-point BVPs b appling te same procedure in CBIM in [] but using te etended version cubic (Advance online publication: Ma 03)

2 IAENG International Journal o Applied Matematics 43: IJAM_43 04 B-spline. Hamid et al. [3] ave compared te results obtained b ECBIM wit te CBIM in [] FDM in [] FEM in [] and FVM in []. Te results ave sown tat tis metod wic is te ECBIM was better in terms o teir accurac. Man works as been done to solve te BVPs wit te Diriclet boundar condition but tere is not muc attention pa to te BVPs wit te Neumann boundar condition. However tere are several researcer tat ave sown teir interest or solving te BVPs wit te Neumann boundar conditions suc as Li-Bin et al. [] Ramadan et al. [] Emad et al. [9] and Siraj-ul-Islam et al. []. Ramadan et al. [] in 00 ave proposed te polnomial and nonpolnomial spline approaces to te numerical solution o second order BVPs subjected to Neumann boundar conditions. Te numerical results ave sown tat te accurac o te nonpolnomial spline metod was better tan te two polnomial spline metods wic are te quadratic and cubic splines. Recentl in 0 Li-Bin et al. [] managed to come out wit a new numerical based on te interpolation b quartic spline unctions to solve te second order BVPs wit Neumann conditions. Li-Bin et al. [] ave compared teir numerical result wit Ramadan et al. []. Te results ave sown tat te metod proposed b Li-Bin et al. [] was better. Several researcers suc as Fatunla [4] Majid et al. [5] Rosser [] and Sampine et al. [] ave proposed a block metod wic computes simultaneousl te solution values at dierent points along te interval. One-step block metod being reerred as one previous point to obtain te solution. Recentl in 0 Muktar et al. [5] ave derived te treepoint block one-step metod in teir paper or solving general second order ordinar dierential equations (ODEs) directl witout reducing it into te sstem o irst order equations. Te results sowed tat it is better in terms o te accurac and te computed times. Tus we are motivated to implement tis metod or solving te two tpes o linear BVPs subjected to te Diriclet and Neumann boundar conditions respectivel. In tis researc we will etend te idea in [5] or solving () wit bot condition () and (3) respectivel witout reducing it into te sstem o irst-order ODEs b using te direct tree-point block one-step metods wit te linear sooting tecniques or solving second order linear Diriclet and Neumann BVPs. II. FORMULATION Te ormulation o te direct tree-point block one-step metod or solving second order linear BVPs will be based in Muktar et al. [5]. Tis metod will compute tree approimation values wic is n at n at n and n3 at n3 simultaneousl. In Fig. te interval o a b need to be divided into a series o block wit eac block containing tree points wit te step size 3. Ten te solution obtained in te last point witin te k t block will be restored as te initial value or te net block. Te same procedure will be repeated to compute te solutions until te end o te interval. Tis metod possesses te desirable eature o one-step metod in wic being reerred to onl one previous point to obtain te solutions. n n n n3 n4 n5 Fig.. Tree-Point Block One-Step Metod B letting te second order dierential equation as ollows:. (4) Te approimating values or n n and n3 was obtained b integrating once and twice over (4) wit respect to. Te evaluation o te irst point will be approimated b integrating (4) once and twice at bot sides over te interval n as ollows: n d d (5) Tereore n dd dd. n n n d () n n n n d () Tus Lagrange interpolating polnomial will replace te in () and (). Te interpolation points involved are n n n n n and n3 n3 witin te block. B doing tat we will obtain te Lagrange interpolating polnomial wic as ollows: n n n3 P n n n n n n n3 n n n3 n n n n n n3 n n n3 n n n n n n3 n n n n3 n n3 n n3 n 3 n n n 3. () (9) (Advance online publication: Ma 03)

3 IAENG International Journal o Applied Matematics 43: IJAM_43 04 Now b taking n 3 s d ds and replace it into () and (). Ten taking -3 to - as te limit o integration in () and () and te corrector ormulae or te irst point will be obtained as ollows: n 4 9 n 4n n n n n3 n n 9 n 9 n 5 n n3 () Net te same process will be repeated similar as te corrector ormulae or te irst point but te integration point is witin te interval and te limit o integration is n n rom - to - in order or us to obtain te corrector ormula or te second point. Tus te corrector ormulae wic is te approimate value o n and n obtained would be as ollows: 4 n n n 3n 3 n n n n n 9 n 30 n n 3 n3 () Te corrector ormulae or te tird point would be obtained b using te same process to obtain te corrector ormulae or te irst and second point ecept teir integration points. Te integration points would be witin te interval. B n n3 taking te limit o integration rom - to 0 te corrector ormulae obtained would be as ollows: 4 n3 n n 5 n 9 n 9 n3 n n n3 n 3 n 30n 3n III. IMPLEMENTATION 3 () Te evaluation o te approimation points n n and n3 will be based on te PE ( CE) mode were P E and C stands or predictor evaluation and corrector respectivel. For eac step r unctions evaluation will be used until te convergence is satisied. Te Modiied Euler metod will be used as te predictor in tis algoritm. Tis metod will act as te initial starting point beore te corrector ormulae take place to compute te approimation values or n n and n3. Te same process will be used or eac block along te interval until te end o it. Tere are two tpes o BVPs tat we need to solve in tis researc wic is te BVPs wit te Diriclet boundar r conditions and BVPs wit te Neumann boundar conditions. Te implementation or solving tese two tpes o BVPs was basicall te same but tere are sligtl dierences wic can be seen as sown below: To solve te BVPs te linear sooting tecnique will be used togeter wit te direct tree-point block one-step metod. Te BVPs () wit te Diriclet boundar conditions will be replaced into two initial value problems (IVP) wic as ollows: p q r a a 0 p q a a. 0 (3) Ten b solving te two IVP wic is te nonomogeneous and omogeneous equation in (3) te linear sooting metodwas obtained wic as ollows: ( b) were w (4). ( b) w Te same procedure will be used or solving te BVPs wit te Neumann boundar conditions wit a sligtl modiications in terms o te initial conditions and te linear sooting metod. First te BVPs () wit te Neumann boundar conditions will be replaced into two IVP wit teir initial conditions wic as ollows: p q r a 0 a p q a a. 0 (5) Ten b perorming te linear combination between tis two IVP in (5) te linear sooting metod will be obtained wic as ollows: w b were w. () b Tis metod will be implemented wit te constant step size. Te convergence test will be used during te calculation o te approimated solution in te corrector ormulae to obtain better accurac. Te convergence test: 3 r 3 r 0. TOL () were r is te number o iterations and TOL is te tolerance. All problems were tested using te absolute error test. Te iterations in te corrector ormulae will be repeated until te convergence test was satisied. IV. NUMERICAL RESUL In tis section siumerical eamples are presented. From te siumerical eamples tat as been tested tere are tree (Advance online publication: Ma 03)

4 IAENG International Journal o Applied Matematics 43: IJAM_43 04 o tem come rom te BVPs wit te Diriclet boundar condition and anoter tree rom te BVPs wit te Neumann boundar conditions. Te problems will be tested using direct tree-point block one-step metod (). A. Diriclet Boundar Conditions Problem ( ) cos( ) Eact Solution 3cos() 3sin() cos() e 4sin() 3cos() 3sin() cos() e 4sin() Source: Bongsoo [3] Problem ( ) ep( ) Eact Solution ( ep( )). Source: Hamid et al. [3] NOTATIONS cos( ). MAXE : Maimum Error o te Computed Solution. ECBIM(N) : Etended Cubic B-Spline Metod Minimizing Using Newton s Metod in Hamid et al. [3]. ECBIM(B) : Etended Cubic B-Spline Metod Minimizing Using Built-In Function in Hamid et al. [3]. EAD : Etended Adomian Decomposition Metod in Bongsoo [3]. : Implementation o te Direct Tree-Point Block One-Step Metod For Solving Te Linear Diriclet and Neumann BVPs. : Maimum Error o te Computed Solution. COLHW : Collocation metod wit te Haar Wavelets in Siraj et al. []. SPLINE : Polnomial spline metod in Li-Bin et al. []. : Step size. : Total steps. TABLE I NUMERICAL RESUL FOR SOLVING PROBLEM WHEN 0. 5 MAXE EAD [3] EAD [3] MAXE * * Eisting Metod = 0. ECBIM(N) [3] ECBIM(B) [3] TABLE II NUMERICAL RESUL FOR SOLVING PROBLEM WHEN 0. MAXE * TABLE III NUMERICAL RESUL FOR SOLVING PROBLEM WHEN 0. 0 Eisting Metod = 0.0 EAD [3] MAXE EAD [3].05 0 = * In Table I te comparison as been made between two tpes o metods wic is EAD and. Te numerical results ave been listed and te maimum error or bot 34 (Advance online publication: Ma 03)

5 IAENG International Journal o Applied Matematics 43: IJAM_43 04 metods as been noted b te asterisk sign. It can be seen tat te maimum error or was 3.0 and it was better tan te EAD wic is onl.4. Moreover te total steps or was lesser compared to te EAD. Table II and Table III sow te numerical results or problem based on te step size = 0. and = 0.0 respectivel. In table II te comparison as been made between te ECBIM(N) and ECBIM(B) and te results obtained sow tat te maimum error or was better wit lesser total steps. Te same ting goes in table III wen te step size was 0.0. In tis table gave better results compared to te EAD wit a lesser total steps. B. Neumann Boundar Conditions Problem Eact Solution e. Source: Siraj et al. [] Problem 4 e 3 3 sin 4 cos ( ) 0 0 sin Eact Solution sin. Source: Liu et al. [] It s clear to see tat or eac dierent step size te could provide a better result in terms o accurac at teir point respectivel. For te step size te local error was taken at te point number 3 tus giving us te local error or bot metods wic is te COLHW and was 4.90 and.5 respectivel. Furtermore te total steps taken b te also were nearl al compared to te COLHW. From tis result it as been sown tat te could provide a better result or eac dierent step size wit lesser total steps. TABLE V NUMERICAL RESUL FOR PROBLEM 4 MAXE MAXE SPLINE SPLINE In Table V te comparison as been made or eac dierent step size tested at problem 5. Te maimum error or bot metods wic is SPLINE and were comparable to eac oter. In addition to tat te total steps or te were lesser as compared to te total steps taken or te SPLINE. V. CONCLUSIONS t j TABLE IV NUMERICAL RESUL FOR PROBLEM 3 j MAXE COLHW COLHW MAXE In tis researc we ave implemented te direct treepoint block one-step metod togeter wit te linear sooting tecnique wit constant step size wic is eicient and suitable or solving te linear Diriclet and Neumann BVPs directl. ACKNOWLEDGMENT Te autor grateull acknowledged te inancial support o Fundamental Researc Grant Sceme (FRGS) and MBrain scolarsip rom te Ministr o Higer Education Malasia REFERENCES From Table IV te local error was computed at teir collocation point j were t j j... N were N is te last number on te interval. Te comparison as been made between te two metods wic is COLHW and. [] H. Caglar N. Caglar and K. Elaituri B-spline interpolation compared wit inite dierence inite element and inite volume metods wic applied to two-point boundar value problems. Applied Matematics and Computation 5-9 (00). [] Q. Fang T. Tsucia and T. Yamamoto Finite dierence inite element and inite volume metods applied to two-point boundar value problems.journal o Computational and Applied Matematics39() 9-9 (00) (Advance online publication: Ma 03)

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