Numerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method

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1 Mathematcs ad Computer Scece 7; (5: do:.648/.mcs.75. Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method Aalu Abrham Aulo,, Alemayehu Shferaw Kbret, Geaew Gofe Gofa, Ayaa Deressa Negassa Departmet of Mathematcs, Isttute of Techology, Dre Dawa Uversty, Dre Dawa, Ethopa Departmet of Mathematcs, Jmma Uversty, Jmma, Ethopa Emal address: (A. A. Aulo, (A. S. Kbret, (G. G. Gofa, (A. D. Negassa Correspodg author To cte ths artcle: Aalu Abrham Aulo, Alemayehu Shferaw Kbret, Geaew Gofe Gofa, Ayaa Deressa Negassa. Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method. Mathematcs ad Computer Scece. Vol., No. 5, 7, pp do:.648/.mcs.75. Receved: Aprl 8, 7; Accepted: Jue 6, 7; Publshed: September 8, 7 Abstract: I ths paper, the Galer method s appled to secod order ordary dfferetal equato wth mxed boudary after covertg the gve lear secod order ordary dfferetal equato to equvalet boudary value problem by cosderg a vald assumpto for the depedet varable ad also covertg mxed boudary codto to Neuma type by usg secat ad Ruge-Kutta methods. The resultg system of equato s solved by drect method. I order to chec to what extet the method approxmates the exact soluto, a test example wth ow exact soluto s solved ad compared wth the exact soluto graphcally as well as umercally. Keywords: Secod Order Ordary Dfferetal Equato, Mxed Boudary Codtos, Ruge-Kutta, Secat Method, Galer Method, Chebyshev Polyomals. Itroducto The goal of umercal aalyss s to fd the approxmate umercal soluto to some real physcal problems by usg dfferet umercal techques, especally whe aalytcal solutos are ot avalable or very dffcult to obta. Sce most of mathematcal models of physcal pheomea are expressed terms of ordary dfferetal equatos, ad these equatos due to ther ature ad further applcatos to use computers, t eeds to establsh approprate umercal methods correspodg to the type of the dfferetal equato ad codtos that gover the mathematcal model of the physcal pheomea. The codtos may be specfed as a tal Value (IVP or at the boudares of the system, Boudary Value (BVP []. May problems egeerg ad scece ca be formulated as two-pot BVPs, le mechacal vbrato aalyss, vbrato of sprg, electrc crcut aalyss ad may others. Ths shows that the umercal methods used to approxmate the solutos of two-pot boudary value problems play a vtal role all braches of sceces ad egeerg []. Amog dfferet umercal methods used to approxmate two-pot boudary value problems terms of dfferetal equatos are shootg method, fte dfferece methods, fte elemet methods (FEM, Varatoal methods (Weghted resdual methods, Rtz method ad others have bee used to solve the two-pot boudary value problems [3]. Both FEM ad Varatoal methods the ma attempts were to loo a approxmato soluto the form of a lear combato of sutable approxmato fucto ad udetermed coeffcets [4]. For a vector space of fuctos S = φ ( x be bass of V, a set of learly V, f { } = depedet fuctos, ay fucto f ( x V could be uquely wrtte as a lear combato of the bass as:

2 Mathematcs ad Computer Scece 7; (5: f ( x = c φ ( x ( = The weghted resdual methods use a fte umber of φ ( x as tral fucto. learly depedet fuctos { } = Suppose that the approxmato soluto of the dfferetal equato, D( u = L( u( x + f ( x =, o the boudary B( u = [ a, b] s the form: N φ φ ( = u( x U ( x = c ( x + ( x N Where UN ( x s the approxmate soluto, u( x s the exact soluto L s a dfferetal operator, f s a gve fucto, φ ( x' s are fte umber of bass fuctos ad uow coeffcets for =,,..., N. The resdual R( x, c s defed as: R( x, c = D( U ( x ( L( U ( x + f ( x. Now determe N N c by requrg R to vash a weghted-resdual sese: b w ( x R( x, c = ( =,,..., N (3 a Where w ( x are a set of learly depedet fuctos, called weght fuctos, whch geeral ca be dfferet from the approxmato fuctos φ ( x, ths method s ow as the weghted-resdual method.. Galer Method If φ ( x = w ( x equato (3, the the specal ame of the weghted-resdual method s ow as the Galer method. Thus Galer method s oe of the weghted resdual methods whch the approxmato fucto s the same as the weght fucto ad hece t s also used to fd the approxmate soluto of two-pot boudary value problems [4]. The Galer method was veted 95 by Russa mathematca Bors Grgoryevch Galer ad the org of the method s geerally assocated wth a paper publshed by Galer 95 o the elastc equlbrum of rods ad th plates. He publshed hs fte elemet method 95. The Galer method ca be used to approxmate the soluto to ordary dfferetal equatos, partal dfferetal equatos ad tegral equatos [5]. May authors have bee used the Galer method to fd approxmate soluto of ordary dfferetal equatos wth boudary codto. Amog ths, a sple soluto of two pot boudary value problems troduced [6], a method for solutos of olear secod order mult-pot boudary value problems produced [7], [8] lear ad o-lear c dfferetal equatos were solved umercally by Galer method usg a Berste polyomals bass, [9] a umercal method s establshed to solved secod order ordary dfferetal equato wth Neuma ad Cauchy boudary codtos usg Hermte polyomals, [] a parametrc cubc sple soluto of two pot boudary value problems were obtaed, a secod-order Neuma boudary value problem wth sgular olearty for exact three postve solutos were solved [], a Numercal soluto of a sgular boudary-value problem o-newtoa flud mechacs were establshed [], a Fourth Order Boudary Value Problems by Galer Method wth Cubc B-sples were solved by cosderg dfferet cases o the boudary codto [3] ad a specal successve approxmatos method for solvg boudary value problems cludg ordary dfferetal equatos were proposed.[4] I ths paper Galer method wll be appled to the lear secod order ordary dfferetal equato of the form d y dy α( x + β ( x + δ ( x y = g( x ; a x b y( a = µ wth boudary codto ad y '( b = µ d y dy α( x + β( x + δ ( x y = g( x ; a x b wth y '( a = β boudary codto y '( b = β 3. Chebyshev Polyomal The polyomals whose propertes ad applcatos are dscussed ths paper were 'dscovered' almost a cetury ago by the Russa mathematca Chebyshev. It s a fucto defed usg trgoometrc fuctos cosθ ad sθ for x [,]. Chebyshev polyomals of frst d wth degree for x [, ] defed as: T ( x = cos θ, such that cos θ = x, for x ad Thus ( = T x cos ( cos x, T ( x = cos( = ad T ( x = cos(cos x = x From the trgoometrc detty, ( l θ ( l cos + + cos θ = cos θ cosθ s ( θ s( θ + s ( θ s( θ ( ( ( ( ( cos + l θ + cos l θ = cos θ cosθ T x = xt x - T x. Thus usg the recursve + - relato above for =, there s a seres of Chebyshev polyomal T ( x = x

3 68 Aalu Abrham Aulo et al.: Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method T ( x = x - 3 T 3 ( x = 4x - 3x 4 T 4 ( x = 8x -8x + Here the coeffcet of T ( x = 6 x x + 5 x etc. x T ( x s. The fgure below shows the graph of the frst seve Chebyshev polyomals for x [,]. Y-axs.5 The Graph of the frst eght Chebyshev Polyomals T(x T(x T3(x T4(x T5(x T6(x T7(x T8(x X-axs Fgure. The graph of the frst eght Chebyshev polyomals for x [,]. 4. Ruge-Kutta Method for Secod Order ODE Ruge-Kutta method s a umercal method used to fd approxmate soluto for tal value problems. I order to use Ruge-Kutta method to fd a approxmate soluto of secod order ODE, t eeds to covert to a system of two frst order ODEs. For two evaluato of f the method s gve by Where y+ = y + hy + ( K + K y + = y + hy + ( K + 3 K h h K = f ( x, y h 4 K = f ( x + h, y + hy + K Secat Method Ths method approxmates the graph of the fucto y = f ( x the eghborhood of the root by a straght le (secat passg through the pots ( x, f ( x ( x, f ( x, where f f ( x ad = ad tae the pot of tersecto of ths le wth the x-axs as the ext terate. Hece Or x x x x f + = f f x + x f x f = f f, =,,..., =,,... Where x ad x are two cosecutve terates. I ths method there are two tal approxmatos x ad x. Ths method s also called Secat method. 6. Mathematcal Formulato of the Method Cosder a geeral lear secod order dfferetal equato wth two type boudary codtos d y dy Type I: α( x + β( x + δ ( x y = g( x ; a x b wth boudary codto y( a = µ (Mxed type (4 y '( b = µ Type II: d y dy α( x + β( x + δ ( x y = g( x ; a x b wth boudary codto where fuctos for α( x, β ( x, δ ( x, y '( a = β y '( b = β (Neuma type (5 g( x are gve cotuous a x b where β, β, µ ad µ are gve

4 Mathematcs ad Computer Scece 7; (5: costats ad y( x s uow fucto or exact soluto of the boudary value problem whch s to be determed. I a BVP wth mxed boudary codto, the soluto s requred to satsfy a Drchlet or a Neuma boudary codto a mutually exclusve way o dsot parts of the boudary. Now Cosder the BVP of type II (Neuma type. To use a approxmatg polyomal defed for x [,] the gve BVP defed o arbtrary terval [a, b] must be coverted to a equvalet BVP defed o [-, ]. So that the approxmatg polyomal should be defed o [-, ]. Sce Chebyshev polyomal s defed o [-, ], t s possble to use Chebyshev polyomal after covertg the BVP defed o arbtrary terval [a, b] to a equvalet BVP defed o [-, ]. 6.. Coverso of the Doma of the BVP The dfferetal equato (5 together wth the Neuma boudary codto ca be coverted to a equvalet problem o [-, ] by lettg b a b + a x = x +, for x ad a x b The equato (4 wth boudary codto s equvalet to the BVP gve by ( d y dy ɶ α x + ɶ( x ( x y g( x; - x β + ɶ δ = ɶ Subect to the boudary codto, where y '( = d y '( = d 4 b a b + a b a b + a ɶ α( x = α x sce for x x, + = + ( b a Ths mples that dy dy d y 4 d y ad = ( b a = ( b a Now equatg the D. E (4 wth (6, d y dy α( x + β( x + δ ( x y g( x = d y dy ɶ α( x + ɶ β( x + ɶ δ ( x y gɶ ( x d y α x d y ɶ α( x = (, ɶ dy dy β ( x = β( x, ɶ δ ( x = δ ( x (6 (7 ad gɶ ( x = g( x (8 Therefore, the DE (5 wth Neuma boudary codto s a equvalet BVP wth the BVP (6. Up o substtuto of (7 to (4, equato (4 yelds d y d y α x = ɶ α x 4 ( (, ( b a dy dy β ( x = ɶ β( x, ( b a ɶ δ ( x y = δ ( x y ad gɶ ( x = g( x 4 b a b + a ɶ α( x = α( x +, ( b a ɶ b a b + a β ( x = β ( x +, ( b a ɶ b a b + a δ ( x= δ ( x + ad b a b + a gɶ ( x=g( x Applyg Galer Method To apply the techque of Galer method to fd a approxmate soluto of (4, say y( x, wrtte as a lear combato of base fuctos ad uow costats. That s; y( x = ct ( x (9 = where T ( x are pecewse polyomal, amely Chebyshev polyomals of degree ad c 's are uow parameters, to be determed. Now applyg Galer method wth the bass fucto T ( x gves, d y dy [ ɶ α ( x + ɶ β ( x + ɶ δ ( x y] T ( x = gɶ ( x T ( x ( Itegratg the frst term by parts o the left had sde of (, that s d y d y T ( x ɶ α( x, u = T ( ( ad x ɶ α x dv = d dy du = T ( x ɶ α( x ad v = d y ɶ α( x = uv vdu dy dy d = T( x ( x T( x ( x ɶ α α ɶ ( Upo substtuto of ( to (, yelds

5 7 Aalu Abrham Aulo et al.: Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method - - ( ɶ α dy d dy [- ( x T ( x + ɶ β ( x T ( x + ɶ δ ( x y( x T ( x] = gɶ ( x T ( x + ɶ α(- y '(- T (- - ɶ α( y '( T ( ( But from equato (9 the approxmate soluto s gve by y( x = ct ( x = Substtutg ths to equato ( yelds, - = ( ɶ α x T x d [- ct ( x ( ( + ɶ β( x ct ( x T ( x + ɶ δ ( x ct ( x T ( x] = = - = gɶ ( x T ( x + ɶ α(- uɶ (- T (- - ɶ α( uɶ ( T ( ( ɶ α d c [- T ( x ( x T ( x + ɶ β( x T ( x T ( x = - + ɶ δ ( x T ( x T ( x] = gɶ ( x T ( x + ɶ α(- y '(- T (- - - ɶ α( y '( T ( (3 I the left had sde of equato (3 above t eeds to ow the values of y (- ad y ( whch approxmately equal to y (- ad y ( respectvely, where y s the exact soluto of the BVP equato ( The Resultg System of Equato Sce the values of y ( ad y ( are ow from the boudary codto, substtutg these values to (3, ad equato (3 gves a system of equatos to solve the parameters c 's thus equato (3 matrx form becomes: ck = F (4 = ( d = ɶ ( [- T ( x ( x T ( x α ( = ɶ β (3 = ɶ ( δ ( x T ( x T ( x f - ( x T ( x T ( x = g ɶ ( x T ( x ( =ɶ α ɶ α f (- y (- T (-- ( y ( T ( Now, the uow parameters c 's are determed by solvg the system of equato (4 by drect method ad substtutg these values to (9 yelds the approxmate soluto y( x of the DE (4 satsfyg the gve boudary codtos (6. Cosder a BVP of type I. I ths case t s mpossble to use the above method drectly; sce y ( a s ot gve ad hece stead t eeds to covert the BVP to type II. The coverso s made by usg dfferet umercal methods. Cosder to solve the followg boudary-value problem: α d y dy ( x + ( x ( x y g( x ; a x b β + δ = (5 Wth boudary codto y( a = µ y '( b = µ The dea of shootg method for (5 s to solve for y ( a hopg that y ( b = µ. I order to fd y ( a such that y ( b = µ, guess y ( a = z ad solve for y ( b usg Ruge-Kutta method for secod order ODE, after havg a value usg the guess, deote ths approxmate soluto y z ad hope y z ( b = µ. If ot, use aother guess for y ( a, ad try to solve usg the Ruge-Kutta method. Ths process s repeated ad ca be doe systematcally utl ths choce satsfy y ( b. To do ths, follow the steps below. Step:- select z so that y z ( b = µ, letψ ( z = y z( b µ. The guess for z Step :- Now the obectve s smply to solve for ψ ( z =, hece secat method ca be used. Step 3:- How to compute z Suppose that the solutos yz ( b ad y z ( b obtaed from guesses z ad z respectvely. Step 4:- Now usg secat method to fd z gve by; z z z z y, =,,... + = z yz y z Followg ths sequece of terato there exsts z such that Where ( ( (3 ( ( ad K = + + F = f + f such that y ( b = y ( b z Thus, the Neuma boudary codto for the DE (5

6 Mathematcs ad Computer Scece 7; (5: s gve by y ( a = z y ( b = µ (6 Now to solve the DE wth boudary codto (6 t s coveet to use equato (3. Example: - Cosder the lear boudary value problem d y x + y = x e ; x, subect to the boudary codto y ( =.678, y ( = Whose exact soluto s:- y = / 5 s( x(349 s( e 5 / cos( / e ( ( x 3839 / 5 cos( x + / ( x + e Soluto: -The above problem s a mxed boudary codto or (type II; to apply the above method t eeds to covert the gve boudary codto to Neuma boudary codto. Now assume a guess depedg o the value of y ( =, let y ( = be the frst guess ad hopg that y ( =. The ext step s usg Ruge-Kutta method for secod order dfferetal equato, where y ( x = f ( x, y, y. But for ths problem, y ( x = f ( x, y sce f s depedet of y x =, ad x = ad tae step sze h=.5, ed ( y = y( x = for x = y =.678 y ( x = for x = y = y+ = y + hy + ( K + K y + = y + hy + ( K + 3 K h h K = f ( x, y h 4 K = f ( x + h, y + hy + K for =,, Ths gves the result table for the frst terato, where the th step x = x, y = y( x ad y = y ( x Referrg to table, tae y z =.588. But, y ( y (, thus t eeds to guess aother value for z y (. Let y ( = hopg that y ( =. Usg Ruge- Kutta method for x = to x = ad tag h =.5, y = y( x = for x = ad y = y ( x = for x =, ths yelds the followg result, where the y = y( x ad y = y ( x th step x = x, st terato d terato Now Table. Shows the result of y ad y o the frst terato. x y y Table. Shows the result of y ad y o secod terato. x y y y ( =.548 where z = ad sce z ad z gve the fd z z z z ( z = z yz = (.548 =.99 y y z z Usg z =.99 applyg Ruge-Kutta method where h=.5, y = y( x = for x = ad y = y ( x =.99 for x =

7 7 Aalu Abrham Aulo et al.: Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method gves the followg result for the 3 rd terato. 3 rd terato Table 3. Shows the result of y ad y o the thrd terato. x y y Now calculate the ext guess z z.99 z3 = z yz =.99 (.99 =.3 y y z z Ruge-Kutta method for y =.3 yelds the followg result for the 4 th terato. Table 4. Shows the result of y ad y o the fourth terato. 4 th terato x y y As table 4 shows the guess for y (.3. Thus, the Neuma boudary value problem gve by d y x + y = x e ; x y ( =.3, ad y ( = The ext step s covertg the BVP to equvalet BVP defed for x by lettg ( b a ( b + a x = x + = 5x + 5, sce a = ad b =. The equvalet BVP for the above problem o x becomes, d y ( 5x+ 5 + y = ( 5x + 5 e, for x 5 wth boudary codto (7 y'( =.3 y'( = (7 Now, suppose that y be the approxmate soluto of (7, gve by a lear combato of costats c ' s ad a approxmatg polyomal, called Chebyshev polyomal, thus y = ct ( x (8 = Upo substtuto of y, the approxmate soluto, to the dfferetal equato (7 gves a equato called resdue gve by: d y ( R( x, c y 5x 5 e 5 5x+ 5 = + ( +, for x (9 Applyg Galer method; R( c, x T ( x = d y ( 5x+ 5 [ + y ( 5x + 5 e ] T ( 5 x + = + 5 ( 5x+ 5 [ y '' T ( x yt ( x] ( 5x 5 e T ( x ( Usg tegrato by parts to smplfy the frst term the rght had sde, let u = T ( x du = T ( x ad dv = y ( x v = y ( x, hece t gves y T ( x y ( x T ( x y T ( x 5 5 = Therefore equato ( becomes

8 Mathematcs ad Computer Scece 7; (5: ( 5x+ 5 yt ( x y ' T '( x = y '( x T ( x + ( 5x + 5 e T ( x ( Substtutg the approxmate soluto y = ct ( x to ( yelds, = ( 5x+ 5 c ( ( '( '( '( ( T x T x T x T x = y x T x + ( 5x + 5 e T ( x 5 5 = = ( 5x x + 5 e T ( x c T ( x T ( x T ( x T ( x y ( T ( y = ( T ( ( ( Now, usg the gve boudary codto to (, equato ( becomes ( 5x+ 5 c T ( x T ( x T ( x T ( x = ( 5x + 5 e T ( x + (.3 T ( (3 5 5 = For equato (3 there s a system of equato gve by: ck = F (4 = where ( ( ( ( = + ad = + K F f f ( = such that T ( x T ( x ( = 5 T ( x T ( x I order to fd the value of fucto. For = 6 : x x 3 4x 3x T ' = 4 8x 8x x x + 5x 6 4 3x 48x + 8x ( f = (.3 T ( 5 ( ( 5x+ 5 f = 5x + 5 e T ( x ( c ' s tae tral fuctos defed for x [,], usg Chebyshev polyomals as tral, where T = [ T T T3 T4 T5 T6 ] ad T ' s the traspose of T.

9 74 Aalu Abrham Aulo et al.: Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method /3 / 5 / 4 /5 38 /5 6 / 35 / 5 34 / 35 / 63 = T ( x T ( x = 38 /5 6 / / 99 / / / 99 6 / / 99 4 /43 ( /5 /5 /5 3/75 8/375 88/875 /5 38/5 4/75 = T ( x T ( x = 5 8/ / /65 /5 4/75 6/35 88/ /65 564/965 ( To fd the value of the coeffcet matrx equato (4 use ( ( = + K So, by expressg the coeffcet matrx K, ad the uow coeffcet c ' s a system of equato matrx form: ( ( = + K 44/75 / 5 9 / 55 c 38 / / / 7875 c / 5 6 / /575 c3 = [ f ] 846 / / / 8665 c4 9 / / / 3465 c5 34 / / / 55 c6 (5 Where f s 6 a colum vector gve by:- (- - 4 / / 5e (- - / /5e (- 8 /5-478 /5e f = (- -38 / / 65e 8 / / 35e 88 / /565e (- (- Now a 6 6 coeffcet matrx whch s symmetrc, 6 uow colum vector that represet c ' s ad 6 colum vector that represets f. So there are sx equatos wth sx uows. Usg ( values of the sx uows are:, c = K f to solve (5, the c =.3655 c = c 3 = c 4 =.3697 c 5 = c 6 =.9 Now t s possble to express the approxmate soluto as a lear combato of costats c ' s ad a approxmatg polyomal. So substtutg c ' s ad polyomals for =6 the approxmate soluto s: Chebyshev 3 4 y =.3655x.596(x (4x 3 x (8x 8x (6 x x + 5 x.9(3x 48x + 8x The graph of the exact ad approxmate soluto, to loo the covergece of the approxmate soluto to the graph of the exact soluto, loos le fgure below.

10 Mathematcs ad Computer Scece 7; (5: The Graph of Exact ad Approxmate Solutos for =4 ad =6 The exact soluto Approx. soluto for =4 Approx. soluto for =6.5 Y-axs X-axs -<x< Fgure. The graph of the exact ad approxmate soluto for =4 ad =6. The above graph shows that the graph of the approxmate soluto approaches the graph of the exact soluto for the dfferetal equato wth boudary codto problem. Cosderg =8, the the approxmate soluto ad the Chebyshev polyomals are gve by: 8 y = ct ( x (6 x x 3 4x 3x 4 8x 8x + T ' =, where T = [ T 5 3 6x x 5x T T3 T4 T5 T6 T7 T8 ] ad T ' s the traspose of T x 48x + 8x x x + 56x 7x x + 6x 56x + 8x Usg MATLAB code; = ( K = /3 -/5 -/ -/45 4/5-38/5-6/35-34/3465 -/5 34/35 -/63-38/495 ] -38/5 6/63-34/99-58/45 -/ -/63 98/99-46/49-6/35-34/99 4/43 -/65 -/45-38/495-46/49 94/95-34/ /45 -/65 54/55

11 76 Aalu Abrham Aulo et al.: Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method ( K = /5 /5 /5 /5 3/75 8/375 88/875 5/575 /5 38/5 4/75 86/375 8/ / /65 83/8665 /5 4/75 6/35 686/475 88/ /65 564/ / /5 86/ / /6875 5/575 83/ / /65 f = (- - 4 / / 5e (- - / /5e (- 8 /5-478 /5e (- -38 / / 65e (- 8 / / 35e 88 / /565e 3544 / / 785e 544 / / 785e (- (- (- Thus the uow parameters c ' s are c = c = c 3 = c 4 =.8693 c 5 =.4546 c 6 =.79 c 7 =.579 c 8 =.9963 Now substtutg the uow parameters ad eght Chebyshev polyomals to (6, the approxmate soluto s: y = x x + x x + x x (.95956( ( (6x x 5 x.79(3x 48x 8x (64x x 56x 7 x.9963( 3x 6x 56x 8 x The graph of the exact ad approxmate soluto, to loo the covergece of the approxmate soluto to the graph of the exact soluto, loos le fgure 3 below.

12 Mathematcs ad Computer Scece 7; (5: The graph of the exact ad approxmate souluto for =6 ad =8 Exact soluto Approxmate soluto for =6 Approxmate soluto for =8.5 Y-axs X-axs -<=X<= Fgure 3. The graph of the exact soluto ad approxmate soluto for =6 ad =8. From the graph above the approxmate soluto approaches the graph of the exact soluto whe the umber of the tral fuctos creases from 6 to 8. Now compare the absolute error whch s gve by error = y y exact approx Table 5. Shows the computed the absolute error for problem whe =6 ad =8. x Exact soluto Approxmate soluto for =6 Approxmate soluto for =8 Absolute error for =6 Absolute error for = As observed from table 5 the approxmate soluto s approachg to the exact value as the value of creases. Tae =, the graph of the correspodg approxmate soluto together wth the graph of the approxmate soluto for =6 ad =8 the same plae wth the exact soluto s show below.

13 78 Aalu Abrham Aulo et al.: Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method.5.5 The Graph of Exact ad Approxmate solutos for =6,=8 ad = The exact soluto Approx. soluto for =6 Approx. soluto for =8 Approx. soluto for = Y-axs X-axs -<x< Fgure 4. The graph of the exact soluto ad approxmate soluto for =6, =8 ad =. As the umber of Chebyshev polyomal creases the correspodg approxmate soluto of the dfferetal equato wth mxed boudary codto problem approaches the graph of the exact soluto. 7. Cocluso Ths study troduces that, by applyg Galer method to lear secod order ordary dfferetal equatos wth mxed ad Neuma boudary codtos, t s possble to fd ther approxmate solutos. The umercal results that are obtaed usg ths method coverges to the exact soluto as the umber of Chebyshev polyomal creases, that wll be used as a tral fucto; ad also usg small step sze h, whle covertg the gve lear secod order ordary dfferetal equato from mxed type to Neuma boudary codto, creases the accuracy of the approxmate soluto. So that usg ths method better results wll be obtaed as the umber of Chebyshev polyomal creases ad usg small step sze whle usg Rug-Kutta method. 8. Future Scope Ths study has led to a attetveess of several topcs that requre further vestgato, for stace lear ordary dfferetal equatos wth dfferet order, o-lear ordary dfferetal equatos. I order to fll the gap terms of accuracy t s mportat to aalyze the error of the method. Thus aalyzg the error ad creasg the accuracy of ths method s left for future vestgato. Refereces [] Yatteder Rsh Dubey: A approxmate soluto to buclg of plates by the Galer method, (August 5. [] Ta-Ra Hsu: Mechacal Egeerg 3 Appled Egeerg aalyss, Sa Jose State Uversty, (Sept 9. [3] E. Sul: Numercal Soluto of Ordary Dfferetal Equatos, (Aprl 3. [4] J. N. Reddy: A Itroducto to the fte elemet method, 3rd edto, McGraw-Hll, (Ja [5] Marcos Cesar Rugger: Theory of Galer method ad explaato of MATLAB code, (6. [6] Jall Rashda ad Reza Jalla: Sple soluto of two pot boudary value problems, Appl. Comput. Math 9 ( [7] S. Das, Sul Kumar ad O. P. Sgh: Solutos of olear secod order multpot boudary value problems by Homotopy perturbato method, Appl. Appl. Math. 5 ( [8] M. Idress Bhatt ad P. Brace: Solutos of dfferetal equatos a Berste polyomals bass, J. Comput. Appl. Math. 5 ( [9] M. M. Rahma. et.al: Numercal Solutos of Secod Order Boudary Value Problems by Galer Method wth Hermte Polyomals, (. [] Arshad Kha: Parametrc cubc sple soluto of two pot boudary value problems, Appl. Math. Comput. 54 ( [] Yuqag Feg ad Guagu L: Exact three postve solutos to a secod-order Neuma boudary value problem wth sgular olearty, Araba J. Sc. Eg. 35 ( [] P. M. Lma ad M. Carpeter: Numercal soluto of a sgular boudary-value problem o-newtoa flud mechacs, Computer Phys. Commuca. 6( 4-. [3] K. N. S. Kas Vswaadham ad Sreevasulu Ballem: Fourth Order Boudary Value Problems by Galer Method wth Cubc B-sples, (May 3. [4] Jahashah et al.: A specal successve approxmatos method for solvg boudary value problems cludg ordary dfferetal equatos, (August 3. [5] L. Fox ad I. B. Parer: Chebyshev Polyomals Numercal Aalyss, Oxford Uversty Press, ( May, 967.