A New Iterative Scheme for the Solution of Tenth Order Boundary Value Problems Using First-Kind Chebychev Polynomials

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1 Fll Length Research Artcle Avalable onlne at Ngeran Jornal of Basc and Appled Scence (Jne, 6), (): 76-8 DOI: ISSN A New Iteratve Scheme for the Solton of Tenth Order Bondary Vale Problems Usng Frst-Knd Chebychev Polynomals E.J. Mamad and * I.N. Njoseh Department of Mathematcs, Unversty of Ilorn, P.M.B 55, Ilorn, Ngera Department of Mathematcs, Delta State Unversty, Abraka, Ngera [ * Correspondng athor: e-mal:njoseh@dels.ed.ng;: ] ABSTRACT A new teratve scheme for the solton of tenth order bondary vale problems has been mplemented sng frst-knd Chebychev polynomals as tral fnctons. The method nvolves transformng tenth order bondary vale problems nto a system of ordnary dfferental eqatons (ODEs). The tral solton s ntrodced nto the ODEs, and s evalated at the bondares to obtan the approxmate solton. The method avods qas-lnearzaton, lnearzaton, dscretzaton or pertrbaton. Also, the method s comptatonally smple wth rond-off and trncaton errors avoded. Nmercal reslts obtaned wth the method show that the method s hghly relable and accrate n obtanng the approxmate solton of tenth order bondary vale problems as compared wth the reslts generated from the Galerkn method avalable n lteratre. All comptatons were performed wth maple 8 software. Keywords: Bondary Vale Problem, Frst-Knd Chebychev Polynomals, Tral solton, Approxmate Solton INTRODUCTION Tenth order bondary vale problems arse n the mathematcal modellng of vscoelastc flows, hydro-magnetc and hydrodynamcs stablty and other areas of appled mathematcs, scences and engneerng. When nstable ordnary convecton spersedes stable convecton, the eght or tenth bondary vale problem s appled to model the nstablty (Vswanadham and Ballem, 5). Analytcal methods for these problems prove nadeqate. Ths, several nmercal schemes have been developed by many athors for the solton of bondary vale problems de to ther mathematcal sgnfcance n dversfed applcatons n scence and engneerng. For nstance, Njoseh and Mamad (6), sed the power seres approxmaton method for the nmercal solton of generalzed nth order bondary vale problems. Smlarly, Mamad and Njoseh (6) appled the Ta-collocaton approxmaton approach for solvng frst and second order ordnary dfferental eqatons. Olagnj and Joseph (3) solved the bondary vale problems n a collocaton method sng thrd-knd Chebychev method. Grover and Tomer () appled the homotopy pertrbaton method for evalatng twelfth order bondary vale problems. Islam et al. (9) employed the dfferental transform method for a specal twelfth order bondary vale problem. Frthermore, Vswanadham and Ballem (5) appled the Galerkn method wth septc B- splne for the nmercal solton of tenth order bondary vale problems. The weghted resdal method was adopted by Odern () on the nmercal solton of tenth and twelfth order bondary vale problems. Also, Al et al. () sed the optmal homotopy asymptotc method for the solton of specal twelfth order bondary vale problems. Rashdna et al. () employed the non polynomal splne soltons for specal lnear tenth-order bondary vale problems. The exstence and nqeness solton of the bondary vale problems has been explored by Agarwa (986). Ths stdy develops a new method for solvng the tenth order bondary vale problems. The method adopts the frst-knd Chebychev polynomals as tral fnctons. The method nvolves transformng the tenth order bondary vale problems nto a system of ordnary 76

2 Mamad and Njoseh: A New Iteratve Scheme for the Solton of Tenth Order dfferental eqatons (ODEs). Thereafter, the tral solton s ntrodced nto the ODEs, and s evalated at the bondares to obtan the approxmate solton. The method avods qas-lnearzaton, lnearzaton, dscretzaton or pertrbaton. Also, the method s comptatonally smple wth rond-off and trncaton errors avoded. Nmercal reslts obtaned wth the proposed method show that the method s hghly relable and accrate n obtanng the approxmate solton of tenth order bondary vale problems. PROPOSED METHODOLOGY In ths secton we consder the th order bondary vale problem of the form (Njoseh and Mamad, 6; Vswanadham and Ballem, 5), () Sbject to the bondary condtons, (), (3) where and are assmed real and contnos on [ ] are fnte real constants. Transformng (), () and (3) to a system of ODEs, we have d d, 3 dx dx 3 dx dx () wth the bondary condtons ( x), ( x), 3( x),, n n (5) ( x ), x ), x ),, n n (6) ( 3( Let the approxmate solton of (), () and (3) be gven as 9 y( x) a T ( x), x [,], (7) where a,,,,,( n ) are nknown parameter to be determned,, are the frst-knd shfted Chebychev polynomals n the nterval evalated sng: wth the ntal condtons, and. (8) Usng Eqaton (7) n (), (5) and (6) n the nterval [ Where the coeffcent matrx A s gven as: ], we obtan the matrx eqaton (9) 77

3 Ngeran Jornal of Basc and Appled Scence (Jne, 6), (): x a a a a a a a a a a T b T 3 3 The constants a,,,,...,( n ), are determned n (9) wth a matrx solver, and sbstttng these parameters nto (7) yelds the approxmate solton of () n the nterval [,]. Smlarly, sng Eqaton (7) n (), (5) and (6) n the nterval [ ], the coeffcent matrx eqaton A s obtaned sng the standard frstknd shfted Chebychev polynomals, whch can be evalated sng the relaton n eqaton (8) wth the ntal condtons, and as: x a a a a a a a a a a T b T 3 3 The constants a,,,,,( n ), are determned n (9) wth a matrx solver, and sbstttng these parameters nto (7) yelds the approxmate solton of () n the nterval [ ] The error formlaton for ths problem s gven as ( x) ( x), where (x) s the exact n and n (x) s the approxmate solton. Nmercal Examples Example 3. (Vswanadham and Ballem, 5): Consder the gven problem below () Sbject to the bondary condtons 78

4 Mamad and Njoseh: A New Iteratve Scheme for the Solton of Tenth Order The exact solton s Usng the proposed methodology, we have: Nmercal reslts obtaned wth the proposed method are gven n Table. Table : Shows the nmercal reslts of the proposed method and error obtaned for example 3. compared wth that obtaned from the Galerkn method wth Septc B-splne n Vswanadham and Ballem (5). Exact Solton Approxmate Solton by proposed method Proposed Method Error Error n Galerkn method e-.5378e e-.567e e E E E E E E E E E E E E+.33E-5 Example 3. (Vswanadham and Ballem, 5) Gven () Sbject to the bondary condtons The exact solton s Usng the proposed methodology, we have Nmercal reslts obtaned wth the proposed method are gven n Table. 79

5 Ngeran Jornal of Basc and Appled Scence (Jne, 6), (): 76-8 Table : Shows the nmercal reslts of the proposed method and error obtaned for example 3. compared wth that obtaned from the Galerkn method wth septc B-splne n Vswanadham and Ballem (5) Exact Solton Approxmate Solton by proposed method Proposed Method Error Error n Galerkn method E-8.696E E-7.398E E-6.5E E E E E E-6.588E E-6.E E E E E-5 Example 3.3 (Njoseh and Mamad, 6; Vswanadham and Ballem 5) (3) Sbject to the bondary condtons ( ) The exact solton s Usng the proposed methodology, we have Nmercal reslts obtaned wth the proposed method are gven n Table 3. Table 3: Shows the nmercal reslts of the proposed method and error obtaned for example 3.3 compared wth Galerkn method wth Septc B-splne n Vswanadham and Ballem (5). X Exact Solton Approxmate Solton by proposed method Proposed Method Error Error n Galerkn method e E E+.7E E E E E E-.993E E E E E E E E E-6 DISCUSSION In ths stdy, the approxmate solton of tenth order bondary vale problems have been obtaned sng frst-knd Chebychev polynomals as tral fnctons. The maxmm errors obtaned by the proposed method are.e-,.6e-8 and.e- n examples, and 3 respectvely. These vales are by far speror to those obtaned from the Galerkn method wth septc B-splne n Vswanadham and Ballem (5) as shown n Tables, and 3. 8

6 Mamad and Njoseh: A New Iteratve Scheme for the Solton of Tenth Order CONCLUSION The method avods qas-lnearzaton, lnearzaton, dscretzaton or pertrbaton. Also, the method s comptatonally smple wth rond-off and trncaton errors avoded. Nmercal reslts obtaned wth the proposed method show that the method s hghly relable and accrate n obtanng the approxmate solton of tenth order bondary vale problems. The method can be appled to hgher order bondary vale problems. REFERENCES Agarwa, R.P. (986). Bondary Vale Problems for Hgher Order Dfferental Eqatons. World Scentfc, Sngapore. Al, J., Islam, S., Rahm, T.M. and Zaman, G. (). The Solton of Specal Twelfth Order Bondary Vale Problems by the Optmal Homotopy Asymptotc Method. World Appled Scences Jornal, (3): Grover, M. and Tomer, A.K. (). A New Approach to Evalate th Order Bondary Vale Problems wth HPM. Internatonal Jornal of Comptatonal Mathematcs and Nmercal Smlaton, (): 99-. Islam, S., Haq, S. and Al, J. (9). Nmercal Solton of Specal th -Order Bondary Vale Problems sng Dfferental Transform. Commn Nonlnear Sc Nmer Smlat, : Mamad, E. J. and Njoseh, I.N. (6). Ta- Collocaton Approxmaton Approach for Solvng Frst and Second Order Ordnary Dfferental Eqatons. Jornal of Appled Mathematcs and Physcs, : Njoseh, I.N. and Mamad, E.J. (6). Nmercal Soltons of Generalzed Nth Order Bondary Vale Problems sng Power Seres Approxmaton Method. Appled Mathematcs, 7: 5-. Odern, R.A. (). On the Nmercal Solton of Tenth and Twelfth Order Bondary Vale Problems sng Weghted Resdal Method (WRM). Gen. Math. Notes, (): 7-. Olagnj, A.S. and Joseph, F.L. (3). Thrdknd Chebychev Polynomals n Collocaton Methods of Solvng Bondary Vale Problems. IOSR Jornal of Mathematcs, 7(): -7. Rashdna, J., Jallan, R., and Farajeyan, K. (). Non Polynomal Splne Soltons for Specal Lnear Tenth-Order Bondary Vale Problems. World Jornal of Modellng and Smlaton, 7(): -5 Vswanadham, K.N.S and Ballem, S. (5). Nmercal solton of tenth order bondary vale problems by Galerkn method wth septc B-splne. Internatonal Jornal of Appled Scence and Engneerng, 3(3):