A Galerkin Finite Element Method for Two-Point Boundary Value Problems of Ordinary Differential Equations

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1 Applied ad Computatioal Matematics 5; 4(: Publised olie Mac 9, 5 (ttp:// doi:.648/j.acm.54.5 ISS: (Pit; ISS: (Olie A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios Getia Zavalai Faculty of Matematics ad Pysics Egieeig Polytecic Uivesity of Tiaa, Albaia addess: zavalaigetia@otmail.com To cite tis aticle: Getia Zavalai. A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios. Applied ad Computatioal Matematics. Vol. 4, o., 5, pp doi:.648/j.acm.54.5 Abstact: I tis pape, we peset a ew metod fo solvig two-poit bouday value poblem fo cetai odiay diffeetial equatio. Te two poit bouday value poblems ave geat impotace i cemical egieeig, deflectio of beams etc. I tis study, Galeki fiite elemet metod is developed fo iomogeeous secod-ode odiay diffeetial equatios. Seveal examples ae solved to demostate te applicatio of te fiite elemet metod. It is sow tat te fiite elemet metod is simple, accuate ad well beaved i te pesece of sigulaities. Keywods: Exact Solutio, Two-Poit Value Bouday Poblem, Fiite Elemet Metod. Itoductio Two-poit bouday-value poblems i odiay diffeetial equatios occu i may baces of pysics; examples iclude te two-dimesioal, icompessible, oedimesioal eat tasfe, bouday laye equatios, etc. Te coespodig odiay diffeetial equatios ca be oliea o liea but wit complex coefficiets. If te diffeetial equatio is oliea o liea but wit complex coefficiets, a closed fom aalytic solutio is, i geeal, difficult to obtai, if ot possible. Teefoe, a umeical solutio is sougt. May eseaces ave developed umeical tecique to study te umeical solutio of two poit bouday value poblems. Villadse ad Stewat [5] poposed solutio of bouday value poblem by otogoal collocatio metod. Jag [6] poposed te solutio of twopoit bouday value poblem by te exteded Adomia decompositio metod. Te Galeki-fiite elemet metod is well kow umeical tecique fo te umeical solutio of diffeetial equatios. Doga [7] poposed te Galekifiite elemet appoac fo te umeical solutios of Buges equatio. Segupta et al. [8] caied out Gakeki fiite elemet metods fo wave poblems. Kaeko et al. [9] discussed te Discotiuous Galeki-fiite elemet metod fo paabolic poblems. EI-Gebeily et al. [] studied te fiite elemet- Galeki metod fo sigula self-adjoit diffeetial equatios. Sama et al. [] poposed Galekifiite Elemet Metods fo umeical solutio of advectio- diffusio equatio. Oa [] poved te asymptotic covegece of te solutio of a paabolic equatio by usig two metods amely, te Galeki metod expessed i tems of liea splies ad te Fiite Elemet Collocatio metod expessed by cubic splie basis fuctios. I tis pape, we coside te followig iomogeeous secod ode diffeetial equatio wee u ( x + p( xu ( x + q( xu ( x = f( x, < x < β u( = u( β = p x ( [, β] (. =C p( x λ > i [, β ], q( x =C [, β], q( x o [, β] ad f x ( =C [, β] We assume tat poblem (. as a uique solutio u( x I te peset wok, we use Galeki-fiite elemet metod fo te umeical solutio of two poit bouday value poblems. Te appoac is simple ad effective. Te emaiig pat of te aticle is ogaised as follows. I Sectio, we sall fist efomulate (. as a vaiatioal poblem i te space vaiablesx.we sall te defie a Galeki appoximatio u( x to te solutio of (. by equiig tat ulie i a fiite-dimesioal space of fuctios, also a eo estimate is give. Te Full Discetized system aisig fom eite of te spatial discetisatios is give i Sectio 3. I sectio 4 of tis pape, we sall make some diect applicatios of appoximatio teoy to some test

2 65 Getia Zavalai: A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios poblems. Fially, Sectio 5 cocludes te aticle wit fial emaks. Fomulatio of te Vaiatioal Poblem ad Galeki Appoximatios Tis poblem may also be stated i weak fom: fid u H ( [, β] suc tat wee β Θ ( u, = ( f, fo w H ([, β ] (. dw du du Θ ( u, = β + wp( x + wq( x u dx dx dx dx, u, w = uwdx (.* Te (stadad Galeki metod fo appoximatig te solutio u of (. amouts to costuctig a family of fiite dimesioal subspaces { S } < <, ad seekig u S satisfyig te liea system of equatios Θ ( u, χ = ( f, χ fo χ S (. We sall assume tat te data ae suc tat te uique solutio u of (. belogs to u ( H ( H ( Ω Ω ad satisfies te elliptic egulaity estimate tat fo some C >, idepedet of f ad uwe ave u C f (.3 Ude ou ypoteses, a uique solutio u of (. exists ad satisfies if u u C u χ x S (.4 fo some costat C idepedet of. Te existeceuiquess of u S is guaateed by Lax-Milgam teoem applied to te Hilbet space ( S, te poof of Lax- Milgam teoem is give i Appedix A. Assumig tat if { ϕ χ + ϕ χ } C ϕ ( H ( H ( x S we obtai fom (.4 te optimal ate ϕ Ω Ω (.5 H ( Ω eo estimate u u C u (.6 Te L eo estimate is obtaied by te itsce tick, by lettig e = u u ad cosideig w ( H ( Ω H ( Ω te solutio of te poblem Θ ( w, ϕ = ( e, ϕ fo ϕ H ( Ω (.7 Te χ S, e = ( ee, = Θ ( we, = Θ ( e, = Θ( e, w χ fo ay By te cotiuity of L i H ( Ω H ( Ω we ave te e = C e w (.8 χ C e w C e e. Hece e C e C u I geeal, assumig tat fo some itege if { w χ + w χ } C w fo ( ( w H ( H x S Ω Ω (.9 s wee. deotes te om i H ( Ω,. =. ad te itsce agumet give et ( + et ( C u ( ( ( H u H Ω Ω 3. Fully Discetized Fiite Elemet Models We sall appoximate te solutio of (. by equiig tat u ad χ lie i { S } < <. Let χκ S fo κ =,,...,. Assume tat te set χ,..., χ is liealy idepedet. Deote by ϒ te subspace spaed by χ,..., χ,let { χ κ } κ = be a basis of S wee = dims.we sall appoximate u of (. by a fuctio u ( x = c χ ( x (3. κ κ κ = Θ ( u, χ = ( f, χ fo χ H ( Ω χ S (3. Substitutig tis expessio fo u i (. ad takig χ = χ, κ =,..., we see tat κ Gc = f (3.3 Wee G is te x matix defied by β ( G = Gκ j = Θ ( χκ, χj = χκ χ + p( x χκ χj + q( x χκ χj dx, κ,j C [ c,, c ] T f (,,,(, T = f χ f χ. Wee G is a positive defiite matice. = ad [ ] 4. umeical Example I tis sectio, some umeical examples ae studied to demostate te accuacy of te peset metod. Te examples ae computed usig MatlabRb. Te vesatility ad accuacy of poposed metod is measued usigl. L = u U = max u ( U j j j Example. Cosideig equatio

3 Applied ad Computatioal Matematics 5; 4(: u ( x + p( x u ( x + q( x u( x = f( x, = x β = (4. wit bouday coditios u( = u( = wee te fuctio p( x ad q( x ae assumed costat, 3, espectively, wile te fuctio f( x is assumed. Te tue solutio of tis poblem is wee / c =, c exp( = + exp( Table. cocetatio eos. Liea elemets Elemets L E E E-6 x x u( x = ce + ce +, Example. Let's coside te same example wit mixed bouday coditios as below u( = u ( = Te tue solutio of tis poblem is x x u( x = ce + ce +, wee c = + exp(, ( + exp( c = ( exp( exp( ( exp( exp( Solutio at ode odes Fig.. Compaiso of umeical ad exact solutio of Example. Liea elemets Example 3. Cosideig equatio x u ( x xu ( x + 4 u( x = x x wit bouday coditios u( = u( = Te tue solutio of tis poblem is.x.6667x x 4 Appoximate solutio Tue solutio Elemets Table. cocetatio eos. Liea elemets L E-4 4.6E E-6 Solutio at ode Appoximate solutio Tue solutio Solutio at ode Appoximate solutio Tue solutio odes Fig. 3. Compaiso of umeical ad exact solutio of Example 3. Liea elemets Table 3. cocetatio eos. Liea elemets odes Fig.. Compaiso of umeical ad exact solutio of Example. Liea elemets Elemets L 6.8E- 4 5.E-.E-

4 67 Getia Zavalai: A Galeki Fiite Elemet Metod fo Two-Poit Bouday Value Poblems of Odiay Diffeetial Equatios 5. Cocludig Remaks I tis aticle, Galeki-fiite elemet metod is poposed to fid te appoximate solutios of two poit bouday value poblems. I te solutio pocedue, te fist step is to make weak fomulatio ad te develop fiite elemet fomulatio. Lastly, weigted aveage is used fo fully discetizatio. As test poblem, tee diffeet solutios of tee poit bouday value poblems ae cose. Also, a compaiso of umeical ad aalytical solutios is made ad foud tat te poposed sceme as good accuacy. Appedix A Teoem (Lax Milgam Teoem. Let H be a (eal Hilbet space ad let Θ(, :H H R be a biliea fom o H wic satisfies: Θ φ, ψ c φ ψ φ, ψ H. (. Θ( φ, φ c φ φ H wee c, c ae positive costats idepedet of φ, ψ H. Let F : H R be a (eal valued liea fuctioal o H suc tat. 3. c3 > ψ H F( ψ < c3 ψ Te tee exists a uique u H satisfyig Moeove, Poof. Let evey w H ( Θ u, w = F( w H u F c φ H be fixed. Te Φ:H R defied fo by ( ( φ, Φ = Θ defies a cotiuous liea fuctioal o H. Fo boudedess obseve tat fo eac w H ( φ c Φ ( = Θ, φ ψ Hece Φ c φ < By te Riesz Repesetatio Teoem teefoe, tee exists a uique elemet φ suc tat Φ ( = Θ φ, w = w, φ w H (A. ( ( Hece fo evey φ H we defie a φ H by (A. ad deote te coespodece ϕ φ by φ = Λφ ( φ, ( w, φ Θ = Λ w H φ H (A. ow Λ is a liea opeato defied o H. We claim ow tat Λ, defied by (A. as a age Ra( Λ wic is a closed subspace of H. Let φ = Λφ be a coveget sequece, suc tat φ φ. ow, sice Θ ( φ, = ( w, Λφ w H Θ( φ φ, = ( Λφ Λ φ, w H. Coose m m m w = φ φ ad usig ( get Hece { } φ φ Λφ Λ φ. m m c φ is a Caucy sequece i H,tee exist φ H suc tat φ φ.we ow sow tat φ = Λφ tus sowig tat φ Ra( Λ, tat Ra( Λ is closed. tat Θ φ, w Θ φ, w C φ φ w w H gives ow ( ( ( φ ( φ lim Θ, = Θ, w H Also Λ ( φ, = ( φ, ( φ, ( φ, ( φ, φ φ w Θ φ sice. Sice (, = Λ ( φ, w H Θ ( φ, = ( φ,. Hece Ra( Λ is closed. Also we claim tat Ra( Λ = H Give F o H, by Riesz epesetatio! ξ H suc tat F( = ( ξ, v H tat Λ u = ξ.hece u suc tat.sice Ra( Λ = H F( = ( Λ u, = Θ( u, w H u H suc Fo uiqueess, suppose tat u u suc tat Θ ( u, = F( = Θ( u, w H. Hece Θ( u u, = w H Θ( u u, u u c u u u = u Sice Θ ( u, u = F( u,(,( give tat u c u F( u fom wic u Refeeces F( u c u. Hece F( u sup = F w c w c [] C.w. Cye, Te umeical solutio of bouday value poblems fo secod ode fuctioal diffeetial equatios by fiite diffeeces, ume. Mat. ( [] Y.F. Holt, umeical solutio of oliea two-poit bouday value poblems by fiite diffeece metod, Comm. ACM 7 ( [3] P.G. Cialet, M.H. Scultz ad R.S. Vaga, umeical metods of ig ode accuacy fo oliea bouday value poblems, ume. Mat. 3 ( [4] S Aoa, S S Daliwal, V K Kukeja. Solutio of two poit bouday value poblems usig otogoal collocatio o fiite elemets. Appl. Mat. Comput., 7(5: [5] J Villadse, W E Stewat. Solutio of bouday value poblems by otogoal collocatio. Cem. Eg. Sci., (967:

5 Applied ad Computatioal Matematics 5; 4(: [6] B Jag. Two-poit bouday value poblems by te exteded Adomia decompositio metod. J. Comput. Appl.Mat., 9 ((8: [7] A Doga. A Galeki fiite elemet appoac of Buges equatio. Appl. Mat. Comput., 57 ((4: [8] T K Segupta, S B Talla, S C Pada. Galeki fiite elemet metods fo wave poblems. Sadaa, 3 (5(5: [9] H Kaeko, K S Bey, G J W Hou. Discotiuous Galeki fiite elemet metod fo paabolic poblems. Appl. Mat. Comput., 8 ((6: [] D Sama, R Jiwai, S Kuma. Galeki-fiite Elemet Metods fo umeical Solutio of Advectio- Diffusio Equatio. It. J. Pue ad Appl. Mat., 7 (3(: [] S E Oat. Asymptotic beavio of te Galeki ad te fiite elemet collocatio metods fo a paabolic equatio. Appl. Mat. Comput., 7(:7-3. [3] T Jagveladze, Z Kiguadze, B eta. Galeki fiite elemet metod fo oe oliea itego-diffeetial model. Appl. Mat. Comput., 7 (6(: [] M A EI-Gebeily, K M Fuati, D O Rega. Te fiite elemet- Galeki metod fo sigula self-adjoit diffeetial equatios. J. Comput. Appl. Mat., 3 ((9:

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special