Numerical Solution of Nonlinear Singular Ordinary Differential Equations Arising in Biology Via Operational Matrix of Shifted Legendre Polynomials

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1 Aercan Journal of Coputatonal and Appled Matheatcs: ; (): -9 DOI:.93/.aca..4 Nuercal Soluton of Nonlnear Sngular Ordnary Dfferental Equatons Arsng n Bology Va Operatonal Matr of Shfted Legendre Polynoals K. Maleknead * E. Hashezadeh Departent of Matheatcs Kara Branch Islac Azad Unversty Kara Iran Abstract hs paper proposed a nuercal ethod for nonlnear sngular ordnary dfferental equatons that arses n bology and soe dseases. We solved these nonlnear probles by a new ethod based on shfted Legendre polynoals. Operatonal atrces of dervatves for ths functon are presented to reduce the nonlnear sngular boundary value probles to a syste of nonlnear algebrac equatons. he ethod s coputatonally very sple and attractve and applcatons are deonstrated through llustratve eaples. he results obtaned are copared by the known results. Keywords Nonlnear Sngular Boundary Value Proble Legendre Polynoals Operatonal Matr of Dervatve Collocaton Method Bology. Introducton he a of ths paper s to ntroduce a new ethod for the nuercal soluton of the followng class of sngular boundary value probles y ( ( a ) y f ( () ) ) () ) ) (3) whch arsng n bology and physology probles. We assue that f ( s contnuous f / y ests and s contnuous and also f / y. he boundary value proble ()-(3) wth and a arse n the study of varous tuor growth probles see ([-6]) wth lnear f ( and wth nonlnear f ( of the for ny f ( f ( y n > > (4) A atheatcal odel of tuor growth s a atheatcal epresson of the dependence of tuor sze on te. And when a n the study of oygen dffuson proble n a sphercal cell wth Mchaels-Menten Knetcs see ([7-9]). A slar proble arse wth * Correspondng author: aleknead@ust.ac.r (K. Maleknead) Publshed onlne at Copyrght Scentfc & Acadec Publshng. All Rghts Reserved and a n odellng of heat conducton n huan head see[-3] wth f ( of the for f ( f ( e y > >. () Estence-unqueness results for such probles have been establshed by several researchers[4-6]. In recent years fndng nuercal solutons of sngular dfferental equatons partcularly those arsng n physology has been the focus of a nuber of authors whch you can see soe of the n[7-]. he purpose of ths paper s to ntroduce a novel ethod based on operatonal atrces of dervatves of shfted Legandre polynoals that have been ntroduced recently n Saadatand and Dehghan work's[] for the nuercal soluton of the class of sngular second-order boundary value probles gven n the (-3) that arse n physology. In ths work by use of shfted Legendre polynoals as bass and operatonal atrces of dervatves of the we convert these knds of equatons to algebrac equatons. he advantage of ths ethod analogy to other ested ethod for these probles s ts trusty and sply n pleentaton we copared our results wth soe ested results to prove ths cla. hs paper s organzed as follows: Secton represents prelnares n ths secton we ntroduced shfted Legendre polynoals and soe propertes of the specally the operatonal atrces of dervatves n Secton 3 we pleented the on physology probles. In Secton 4 a nuber of appled odels n physology are dscussed to show the effcency and accuracy of the proposed ethod the results obtaned are copared by the known results. Fnally Secton ncludes a concluson for the paper.

2 6 K. Maleknead et al.: Nuercal Soluton of Nonlnear Sngular Ordnary Dfferental Equatons Arsng n Bology Va Operatonal Matr of Shfted Legendre Polynoals. Defntons and Propertes of Shfted Legendre Polynoals.. Shfted Legendre Polynoals Consder the Legendre polynoals L (z) on the nterval [ ] L ( z) L ( z) z the set { L ( z):...} n Hlbert space L [ ] s a coplete orthogonal set[3]. In order to use these polynoals on the nterval [] we defne the so-called shfted Legendre polynoals by ntroducng the change of varable z. Let the shfted Legendre polynoals L ( ) be denoted by P (. hen P ( can be obtaned as follows: ( )( ) P ( P ( P ( (6) ( ) where P ( and P (. he analytc for of the shfted Legendre polynoals P ( of degree gven by k k ( k)! P ( ( ). (7) k ( k)! ( k!) Note that P () ( ) and P (). he orthogonalty condton s P ( P ( d.. Functon Approaton for for. Any functon L [] can be epanded n ters of shfted Legendre polynoals as where the coeffcents c ( ) c P ( c are gven by In practce only the frst ( ) -ters shfted Legendre polynoals are consdered. hen we have P ( d c P ( C B( (9) where the shfted Legendre coeffcent vector C and the shfted Legendre vector B are gven by: C [ c c... c ] () B [ P ( P (... P ( ]. () (.3. Operatonal Matr of Dervatve. he dervatve of the vector B ( can be epressed by d B ( () D B( () d () where D s the ( ) ( ) operatonal atr of dervatve gven by[] ( ) for D () ( d ) k otherwse as f s odd k 3... and f s even k For eaple for even we have 3 () D By usng Eq. () t s clear that n d B ( () n ( D ) B( (3) n d where n and the superscrpt n () D denote atr powers. hus ( n) () n D ( D ) n. (4) 3. Ipleentaton of Shfted Legendre Polynoals Method on Physology Probles In ths secton we solve nonlnear sngular boundary value proble of the for Eq.() wth the ed condtons () and (3) by usng shfted Legendre polynoals. Fro Eq. (9) we can approate our unknown as C B( () where B ( and C are defned n Eqs.() and (). By usng Eqs.() and (3) we have () y ( C B'( C D B( (6) and () y ( C B' ( C ( D ) B(. (7) By substtutng Eqs.() (6) and (7) n Eq. () we have C ( D () () ) B( ( a ) C D B( f ( C B( ). (8) Also by usng Eqs.() (3) () and (6) we have () C B() C D B() (9) () C B() C D B(). () Eqs.(9) and () gve two lnear equatons. Snce the total unknowns for vector C n Eq.() s ( ) we collocate Eq.(8) n ( ) ponts n the nterval [] that are roots of shfted Legendre polynoal P then we have () () C ( D ) B( ) ( a ) C D B( f ( C B( )) () for.... Now the resultng Eqs. (9) () and () generate a syste of ( ) nonlnear equatons whch can be solved usng Newton's teratve ethod[4]. We used the Matheatca 7 software to solve ths nonlnear syste.

3 Aercan Journal of Coputatonal and Appled Matheatcs: ; (): Illustratve Eaples and Appled Models o show the effcency of the proposed nuercal ethod we pleent t on three nonlnear sngular boundary probles that arse n real physology applcatons. Our results are copared wth result n Refs.[7-]. he austerty of our ethod n pleentaton n analogy to other ested ethods and ts trusty answers s consderable. 4.. Eaple Consder the followng oygen dffuson proble.769 y y.39 wth the boundary condtons: y () ) ). able shows the nuercal results for varous nuber of eshes and present ethod solutons are copared wth results n Refs.[7] and[8]. able. Approate solutons for Eaple. Present ethod wth Method n[7] Wth n Method n[8] wth n Eaple able. Nuercal errors for Eaple. Present ethod wth Present ethod wth Approach II[7] wth n Consder the followng sngular two pont boundary value proble: y e ) ) wth the eact soluton ( c y ln ( ) c where c 3. able shows nuercal errors of ths eaple n analogy to errors for ths eaple n[7] Eaple 3 Consder ths proble that s concde by heat conducton odel of the huan head y y ( e we consder the soluton of ths proble wth condtons as follows: y () ) ). able 3 llustrates results for ths eaple by proposed ethod alongsde nuercal solutons for ths eaple that have been gven n Refs[9-]. able 3. Approate solutons for Eaple 3. Present ethod Method n[9] Method wth wth forth-order n[] able 4. he au absolute errors n soluton of Eaple 4 for h. h.7. Case () h Case () h Case () h Case () h able. he au absolute errors n soluton of Eaple 4 for h h. Case () h Eaple 4 Case () h Case () h Case () h Consder the followng sngular two pont boundary value proble: 3 y h ( e h 4 ( ) 4 for the followng two cases:

4 8 K. Maleknead et al.: Nuercal Soluton of Nonlnear Sngular Ordnary Dfferental Equatons Arsng n Bology Va Operatonal Matr of Shfted Legendre Polynoals ( ) ) ln ( ) ) ) ln ( ) 4 ( ) ) ) ) ln ( ) wth the eact soluton ln ( ). 4 Mau absolute errors for ths proble have been dsplayed for h < n able 4 and for h n able whch show the accuracy of proposed ethod and these results n analogy to ehbted results for ths eaple n[9-] show advantage of ths ethod.. Conclusons hs paper present a new approach based on shfted Legendre polynoals for the nuercal soluton of a class of sngular boundary value probles arsng n bology and physology probles. By use of shfted Legendre polynoals as bass and operatonal atrces of dervatves of these functons we convert such probles to an algebrac syste. he pleentaton of current approach n analogy to ested ethods s ore convenent and the accuracy s hgh and we can eecute ths ethod n a coputer speedy wth nu CPU te used. he nuercal appled odels that have been presented n the paper and the copared results support our cla. ACKNOWLEDGEMENS he authors would lke to thank Islac Azad Unversty of Kara Branch for partally fnancally supportng ths re-search and provdng facltes and encouragng ths work. REFERENCES [] J.A. Ada A splfed atheatcal odel of tuor growth Math. Bosc. vol.8 pp [] J.A. Ada A atheatcal odel of tuor growth II: effect of geoetry and spatal non-unforty on stablty Math. Bosc. vol.86 pp [3] J.A. Ada S.A. Maggelaks Matheatcal odel of tuor growth IV: effect of necrotc core Math. Bosc. vol. 97 pp [4] A.C. Burton Rate of growth of sold tuor as a prob-le of dffuson Growth vol.3 pp [] H.P. Greenspan Models for the growth of sold tuor as a proble by dffuson Stud. Appl. Math. vol. pp [6] N.S. Asathab J.B. Goodan Pont wse bounds for a class of sngular dffuson probles n physology Appl. Math. Coput. vol. 3 pp. 989 [7] H.S. Ln Oygen dffuson n a sphercal cell wth non-lnear oygen uptake knetcs J. heor. Bol. vol. 6 pp [8] D.L.S. McElwan A re-eanaton of oygen dffuson n a sphercal cell wth MchaelsMenten oygen uptake knetcs J. heor. Bol. vol. 7 pp [9] N. Rashevsky Matheatcal Bophyscs vol. Dover New York 96 [] U. Flesch he dstrbuton of heat sources n the hu-an head: a theoretcal consderaton J. heor. Bol. vol. 4 pp [] J.B. Garner R. Shva Dffuson probles wth ed nonlnear boundary condton J. Math. Anal. Appl. vol. 48 pp [] B.F. Gray he dstrbuton of heat sources n the hu-an head: a theoretcal consderaton J. heor. Bol. vol. 8 pp [3] R.C. Duggan A.M. Goodan Pont wse bounds for nonlnear heat conducton odel for the huan head Bull. Math. Bol. vol. 48 () pp [4] R.K. Pandy On a class of weakly regular sngular two pont boundary value probles II J Dfferental Equa-tons vol. 7 pp [] M.M. Chawla P.N. Shvkuar On the estence of so-luton of a class of sngular two--pont nonlnear boun-dary value probles J. Coput. Appl. Math. vol. 9 pp [6] R.D. Russell L.F. Shapne Nuercal ethods for sngular boundary value probles SIAM J. Nuer. Anal. vol. pp [7] S.A. Khur A. Sayfy A novel approach for the soluton of a class of sngular boundary value probles arsng n physology J. Math. Coput. Model. vol. pp [8] Hket Caglar Nazan Caglar Mehet Ozer B-splne soluton of non-lnear sngular boundary value probles arsng n physology Chaos Soltons Fractals vol. 39 pp [9] J. Rashdna R. Mohaad R. Jallan he nuercal soluton of non-lnear sngular boundary value probles arsng n physology J. Appl. Math. Coput. vol. 8 pp [] R.K. Pandey Arvnd K. Sngh On the convergence of a fnte dfference ethod for a class of sngular boundary value probles arsng n physology J. Coput. Appl. Math. vol. 66 pp [] A. Saadatand M. Dehghan A new operatonal a-tr for solvng fractonal-order dfferental equatons J. Coput. Math. Appl. vol. 9 pp [] K. Maleknead B. Basrat E. Hashezadeh Hybrd Le-gendre polynoals and Block-Pulse functons approach for nonlnear Volterra Fredhol ntegro-dfferental eq-uatons Coput. Math. Appl.vol. 6 pp [3] K. Maleknead E. Hashezadeh A nuercal approach for Haersten ntegral equatons of ed type usng operatonal atrces of hybrd functons Scentfc Bulle-tn Seres A: Appled Matheatcs and Physcs vol.73(3) pp. 9-4

5 Aercan Journal of Coputatonal and Appled Matheatcs: ; (): -9 9 [4] K. Maleknead S. Sohrab H. Derl A new coputa-tonal ethod for soluton of non-lnear Volter-ra-Fredhol ntegro-dfferental equatons Int. J. Appl. Math. vol. 87() pp ethod based on Bernesten operatonal atrces for nonlnear Volterra-Fredhol-Haersten ntegral equatons Coun. Nonlnear. Sc. Nuer. Sulat. vol. 7 pp. 6 [] K. Maleknead E. Hashezadeh B. Basrat Coputatonal