Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed Mathematics Departmet, COMSATS Istitute of Iformatio Techology, Park Road, Islamabad, Pakista Mathematics Departmet, HITEC Uiversity, Teila Catt, Islamabad, Pakista oormaslam@hotmail.com, khalidaoor@hotmail.com, waheedasif@hotmail.com Abstract- I this paper, we use the modified decompositio method for solvig the system of third-order oliear boudary value problems associated with obstacle problems. Some eamples of system of third-order oliear boudary value problems are give. The compariso of the results obtaied by modified decompositio method with modified variatio of parameters method is also give. Results are also show graphically that demostrate the ature of the obstacles i particular problems. The results are very ecouragig idicatig the reliability ad efficiecy of the techique. Keywords- Modified decompositio method, Adomia s polyomials, system of thirdorder oliear boudary value problem.. INTROUCTION I recet years, much attetio has bee give to solve a system of third-order obstacle boudary value problems, see [-4, 5, 6, 7]. To be more precise, we cosider the followig systems of third-order oliear boudary value problems: f, u, a c, u f, u u g r, c d, f, u, d b, ad f, u u g r, a c, u f, u, c d, f, u u g r, d b, (a) (b) with boudary coditios u a, u a, u b, 3 ad cotiuity coditios of u, u ad u iterval ab,. Here rad i, i 3 are real ad fiite costats ad ab For simplicity, we will cosider f, u at iteral poits c ad d of the g is a cotiuous fuctio o,. f u. Systems (a) ad (b) arise i obstacle, cotact, uilateral ad equilibrium problems.
M.A. Noor, K.I. Noor ad A. Waheed 3 Most of the problems i oceaography, ecoomics, trasportatio, oliear optimizatio, ocea wave egieerig, fluid flow through porous media ad some other braches of sciece ad egieerig are modeled by these systems, see [-] ad refereces therei. Several techiques have bee used to solve system of liear thirdorder boudary value problems associated with obstacle, cotact ad uilateral problems. All the above metioed methods are proposed to solve liear system of boudary value problems associated with obstacle, cotact ad uilateral problems. Noor et al. [5] have proposed modified variatio of parameters method for solvig system of oliear obstacle boudary value problems of eve ad odd orders. Ispired ad motivated by the recet research activities i this field, we cosider the systems of oliear boudary value problems. Adomia [] developed a ovel techique kow as Adomia decompositio method for solvig boudary value problems. Further Wazwaz [3] has modified this techique kow as modified decompositio method. Modified decompositio method has bee etesively used to solve a class of problems of diversified ature. I this paper, we agai use the modified decompositio method to solve systems of thirdorder oliear obstacle boudary value problems. The modified decompositio method provides the rapidly coverget series solutio with easily computable compoets. This techique makes the solutio procedure simple while still maitaiig the higher level of accuracy. I the preset study, we implemet this techique for solvig systems of third-order oliear boudary value problems. The idea ad techique of this paper may lead to ovel applicatios of these problems i pure ad applied scieces.. MODIFIED DECOMPOSITION METHOD Cosider the followig differetial equatio: L( u) R( u) N( u) g, () where L is the highest-order derivative which is assumed to be ivertible, R is a liear differetial operator of order lesser tha L, N u represets the oliear terms ad g is the source term. Applyig the iverse operator L to both sides of () ad usig the give coditios, we obtai u f L ( Ru) L ( N u), where the fuctio f represets the terms arisig from itegratig the source term g ad by usig the give coditios. I modified decompositio method [3], the fuctio f ca be set as the sum of two partial fuctios amely f ad f ad defies the solutio u () by the series 0 u( ) u ( ), (3) where the compoets u () are usually determied recurretly by usig the relatio u f, u f L ( Ru ) L ( N u ), 0 0 0 u f L ( Ru ) L ( N u ), k. k k k The oliear operator N (u) ca be decomposed ito a ifiite series of polyomials give by
3 Decompositio Method for Solvig a System of Boudary Value Problems N ( u) 0 A where A are the so-called Adomia s polyomials. These Adomia s polyomials ca be geerated for various classes of oliearities accordig to the specific algorithm developed i [3] which yields d i A, 0,,,.! N ui (4) d i 0 0 3. NUMERICAL RESULTS, I this sectio, we apply the modified decompositio method [3] for solvig a system of third-order oliear obstacle boudary value problems. The solutios are compared with modified variatio of parameters method ad are also graphically represeted. The results are very ecouragig idicatig the reliability ad efficiecy of the techique. Eample 3.. Cosider the system of third-order oliear boudary value problems 3 u u equivalet to system (a), f u u, g, r ad ab,,, as 3!! follows: 3 u u u, for, 3!! 3 u u u u, for, (5) 3!! 3 u u u, for, 3!! with boudary coditios u u u 0,. Case :. I this case, we implemet modified decompositio method as follows 3 u u Lu u. Applyig L o both the sides, we have 3!! 3 u u u c3 c c L L u, (6)! 3!! where the operator L is defied as follows: L. ddd, ad c, c ad c3 will be determied further by usig boudary coditios ad cotiuity coditios. 000
M.A. Noor, K.I. Noor ad A. Waheed 33 Substitutig the decompositio series (3) for u ad the series of polyomials for oliearity (4) ito (6), we have u c3 c c L A L u, 0 0 0 Usig the modified decompositio method [3], we have followig approimatios u c, 0 3 4 5 3 6 u c3 c c c c, 6 4 0 70 u c c c c c c + c c 6 4 0 0 70 40 7 8 3 9 + c cc c + c c..., 008 840 50 8440 3 4 5 6 3 3 3 3 u u Case.:. I this case, we have Lu. u Applyig 3!! the sides, we have 3 u u u c6 c5 c4 L L u, 3!! Substitutig the decompositio series for oliearity ito (7), we have followig approimatios u c, 0 5 L o both (7) u ad the series of polyomials for 4 5 3 6 u c6 c4 c5 c5 c5, 0 70 u c c c c c c + c c + c c c c 3 4 60 0 40 60 840 8 9 3 9 4 0 5 + c5 c5 c5 c5, 3360 8440 9600 45600 3 4 5 6 7 6 5 6 4 5 6 4 5 5 5 4 Case..: 3 u u. I this case, we have Lu u. Applyig L o 3!! both the sides, we have 3 u u u c9 c8 c7 L L u, (8) 3!!
34 Decompositio Method for Solvig a System of Boudary Value Problems Substitutig the decompositio series for oliearity ito (8), we have followig approimatios u c, 0 8 u ad the series of polyomials for 3 4 5 3 6 u c9 c7 c8 c8 c8, 6 4 0 70 3 4 5 u c9 c8c9 c7 c8 c9 6 4 0 0 6 7 + c7c8 + c7c8 c8 70 40 008 840 8 3 9 4 0 5 + c8 c8 c8 c8, 50 8440 9600 45600 By usig modified decompositio method, we have followig formula for gettig series solutio i the whole domai from the above cases uk for, k 0 u uk for, k 0 uk for. k 0 Hece, we have the followig series solutio after two iteratios
M.A. Noor, K.I. Noor ad A. Waheed 35 u 3 4 5 c3 c c c3 c3 c c c c c3 6 6 4 4 0 0 0 3 6 7 8 3 9 + cc c + cc c + c c 70 40 70 008 840 50 8440 4 0 5 c c, for, 9600 45600 3 4 5 c6 c5 c4 c6 c6 c5 c5 c4 c5 c6 3 4 0 60 0 6 7 8 c5 + c4 c5 + c5 c5 c4 + c5 70 40 60 840 3360 9 3 9 4 0 5 c5 c5 c5, for, 8440 9600 45600 3 4 5 c9 c8 c7 c9 c9 c8 c7 c8 c9 c8 6 6 4 4 0 0 0 3 6 7 8 3 9 + c7c8 c8 + c7c8 c8 + c8 c8 70 40 70 008 840 50 8440 4 0 5 c8 c8, for. 9600 45600 (9) Now we use boudary coditios ad cotiuity coditios at ad, ad we obtai system of oliear equatios. I order to solve system of oliear equatios, we use Newto s method. Hece we have the followig values of ukow costats c.607470034,c.04437, c 3.0084536, c4.08606866, c.00867458, c.04384406, c.4306744635, c.074506, (0) 5 6 7 8 c.06394470. 9 By usig values of ukows from (0) ito (9), we have followig aalytic solutio of system of secod-order oliear boudary value problems (5)
36 Decompositio Method for Solvig a System of Boudary Value Problems 3 4.0084536.04437 +.3037350067 +.63959578 +.00493848330 5 6 7-6 8 +.005880507 +.0069773077 +.000306793633 +5.8469060 0-7 9-9 0 - +.0750665 0 +.65643530 0 +.87394 0, for, 3 4.04384406.00867458.04303430580.0483687.0007090999 5-6 6-6 7-8 8 u.00435069469 3.0974860 6.8364355 0 +.036430-9 -4 0-7 6.7085560 4.559099 0 3.33348776 0, for, 3 4.06394470+.074506.533738 +.560093050.004709340587 5 6 7-6 8 -.0034757883.007465954.00038636 +5.785469900-7 9-9 0 -.06755537 0.6404370.8003300, for. Table 3. shows the compariso of the aalytic solutio of problems (5) betwee modified decompositio method (MDM) ad modified variatio of parameters method(mvpm) Sr. No MDM MVPM Error - 0 0 0 -.8 -.0060830 -.0060830 8.00E-0 -.6 -.090437794 -.090437793.40E-0 -.4 -.0380487 -.0380486.0E-0 -. -.03888058 -.03888056.00E-0 0 -.04384409 -.04384406 3.00E-0. -.04500 -.0450008 4.0E-0.4 -.039853493 -.039853487 6.0E-0.6 -.0349348 -.03493473 8.00E-0.8 -.0464487 -.04644860.0E-0 0 0 0
M.A. Noor, K.I. Noor ad A. Waheed 37 Figure 3. depicts the graphical represetatio of aalytical solutio of (5) by usig modified decompositio method Eample 3.. Cosider the system of third-order oliear boudary value problems 3 equivalet to system (b), f u u, g, r ab,,, as follows: ad 3 u u, for, 3 u u, for, u u 0, u. 3 u u, for, with boudary coditios Usig the procedure of Eample 3., we have the followig series solutio after two iteratios ()
38 Decompositio Method for Solvig a System of Boudary Value Problems 3 4 5 3 6 c3 c c c3 c c + c c3 c 6 6 4 0 0 60 70 7 8 3 9 5 cc c c c c, for, 70 5040 336 890 9900 u 5 3 6 7 5 c6 c5 c4 c5 c6 c5 c5 c4 c5, for, 0 60 70 9900 3 4 5 3 6 c9 c8 c7 c9 c8 c7 + c8 c9 c8 6 6 4 0 0 60 70 7 8 3 9 5 c8c7 c8 c8 c8 c8, for. 70 5040 336 890 9900 (5) Now we use boudary coditios ad cotiuity coditios at ad, ad we obtai system of oliear equatios. Hece we have the followig values of ukow costats by usig the Newto method. c.37759790,c.0533738, c.5496743, c.07994040, 3 4 c.04095360, c.53693340, c.8989496, c.059640849, 5 6 7 8 c.547604. 9 (6) By usig values of ukows from (6) ito (5), we have followig aalytic solutio of system of secod-order oliear boudary value problems () 3 4.5496743.0533738.6638796395.0797435.008077658 5 6-5 7-6 8.00908645.00397845 6.340570 8.5336950-8 9-7.585559840 3.966598440, for, -5 5.53693340+.04095360.53599700.06550374 0 u -8 6-6 7-4 +4.66744350 3.0463660 +5.60460, for, 3 4.547604.059640849.409947463.075399630.00657004 5 6-5 7-6 8.00668459058.00397503 4.937969750 8.0365063070-8 9-7.447376 0 3.8700579 0, for.
M.A. Noor, K.I. Noor ad A. Waheed 39 Table 3. represets the compariso of the aalytic solutio of problems () betwee modified decompositio method (MDM) [] ad modified variatio of parameters method (MVPM) [5] Sr. No MDM MVPM Error - 0 0 0 -.8.848457.848457 0 -.6.3349777.3349777.00E-0 -.4.4448955397.4448955397 0 -..50343645.50343646.00E-0 0.53693340.5369334.00E-0..5767553.5767556 3.00E-0.4.4567036.45670363.00E-0.6.35689738.35689738.00E-0.8.06078444.06078444 0 0 0 0 Figure 3. is a graphical represetatio of aalytical solutio of system of third-order oliear boudary value problem () by usig modified decompositio method 4. CONCLUDING REMARKS I this paper, we have used the modified decompositio method for solvig system of third-order oliear boudary value problems. We have cosidered two eamples for both the systems. The compariso of the results obtaied by this method are also give with modified variatio of parameters method. The results are also represeted graphically which demostrate the ature of the obstacle i each problem. It
40 Decompositio Method for Solvig a System of Boudary Value Problems is a iterestig problem to compare these methods with other methods such as variatioal iteratio methods ad its various modificatios. Acklowdegemet- The authors are grateful to Dr. S. M. Juaid Zaidi, Rector, COMSATS Istitute of Iformatio Techology, Pakista for providig ecellet research ad academic eviromets. This research is partially supported by HEC NRPU project No: 0-966/R\&D/-553, titled: Research uit of Academic Ecellece i Geometric Fuctio Theory ad Applicatios.. The authors would like to thak the referees for their valuable ad costructive commets. 6. REFERENCES. F. Gao ad C. M. Chi, Solvig third-order obstacle problems with quartic B-splies, Applied Mathematics ad Computatio 80, 70-70, 006... A. K. Khalifa ad M. A. Noor, Quitic splies solutios of a class of cotact problems, Mathematical ad Computer Modellig 3, 5-58, 990.. 3. S. Momai, K. Moadi ad M. A. Noor, Decompositio method for solvig system of forth-order obstacle boudary value problems, Applied Mathematics ad Computatio 75, 83-93, 006. 4. M. A. Noor, Variatioal iequalities i physical oceaography, i: Ocea Wave Egieerig (edit. M. Rahma), Computatioal Mechaics Publicatios, Southampto, UK, 0 6, 994. 5. M. A. Noor, K. I. Noor, A. Waheed ad E. A. Al-Said, Modified variatio of parameters method for solvig system of secod-order oliear boudary value problem, Iteratioal Joural of Physical Scieces 5, 46-43, 00. 6. M. A. Noor ad S. I. Tirmizi, Numerical methods for uilateral problems, Joural of Computatioal ad Applied Mathematics 6, 387-395, 986. 7. M. A. Noor ad S. I. Trimizi, Fiite differece techiques for solvig obstacle problems, Applied Mathematics Letters, 67-7, 988. 8. M. A. Noor, Some developmets i geeral variatioal iequalities, Applied Mathematics ad Computatio 5, 99-77, 004. 9. M. A. Noor ad A. K. Khalifa, A umerical approach for odd-order obstacle problems, Iteratioal Joural of Computer Mathematics 54, 09-6, 994. 0. M. A. Noor, K. I. Noor ad E. Al-Said, Iterative methods for solvig ocove equilibrium variatioal iequalities, Applied Mathematics & Iformatio Scieces 6(), 65-69, 0.. S. Islam, M. A. Kha, I. A. Tirmizi, ad E. H. Twizell, No-polyomial splie approach to the solutio of a system of third-order boudary-value problems, Applied Mathematics ad Computatio 68, 5-63, 005.. G. Adomai, A review of the decompositio method i applied mathematics, Joural of Mathematical Aalysis ad Applicatios 3, 50-544, 988. 3. A. M. Wazwaz, Approimate solutios to boudary value problems of higher-order by the modified decompositio method, Computers & Mathematics with Applicatios 40, 679-69, 000.