Numerical Solution of Non-Linear Ordinary Differential Equations via Collocation Method (Finite Elements) and Genetic Algorithms

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1 Proceedigs of te 6t WSEAS It. Cof. o EVOLUTIONARY COPUTING Lisbo Portugal Jue pp6- Numerical Solutio of No-Liear Ordiary Differetial Equatios via Collocatio etod Fiite Elemets ad Geetic Algoritms Nikos E. astorakis ilitary Istitutes of Uiversity Educatio ASEI Helleic Naval Academy Terma Hatzikyriakou 859 Piraeus GREECE ttp:// Abstract: - I tis paper a ew metod for solvig o-liear ordiary differetial equatios is proposed. Te metod is based o fiite elemets collocatio metod as well as o geetic algoritms. Te metod seems to ave some advatages i compariso wit te typical sequetial oe step ad multi step metods. Key-Words: - Ordiary Differetial Equatios Fiite Elemets Geetic Algoritms Evolutioary Computig Collocatio Itroductio Researc i umerical solutio of Ordiary Differetial Equatios ODEs is a ope field durig te last ceturies ad may umerical metods ave bee adopted to solve iitial value problems. Te importace of ODE is great because matematical models tat occur i sciece pysics cemistry biology ecoomy geology space sciece etc ad egieerig electrical mecaical civil etc are usually ordiary differetial equatios. Euler s metod Heu s metod Polygo metod Ruge Kutta metods are te most usual oe step metods wile ile Simpso Adams Basfort oulto metods are te most usual multi step metod. Suppose tat a first order dy ODE is epressed i te form: f y d Commo feature i oe step metods is te discovery ad te usage of a discrete sequece y i yi F i yi were F is a fuctio or procedure or algoritm i wic i ad y i are ivolved. Commo feature i te so called multistep metods is te discovery ad te usage of a discrete sequece yi yi F i yi i yi i yi L ik yik Te previous formula describes a k step metod. ore details ca be foud i may tetbooks ad surveys like [] []. A disadvatage of all tese metods is tat te umerical errors are accumulated ad so te solutio of te iitial value problems is affected by tese errors. Especially we te iterval of solutio is big we ave to use small step of icrease wic affects te compleity of te metod. Terefore to icrease accuracy we must decrease te step of te metod wic uavoidably leads to big computatioal load ad compleity. I tis paper a attempt is made to solve tis problem by usig fiite elemets collocatio metod ad geetic algoritms. Fiite elemets metod yields a set of o liear algebraic equatios. Tese o liear equatios ca be solved via Geetic Algoritm GA []. Te solutio of te o liear algebraic equatios via GAs is eamied i [] wit more details ad specific eamples. I sectio we preset te mai fiite elemets. I sectio a formulatio of te geeral iitial value problem is give. I sectio a formulatio of te geeral boudary value problem is preseted. Fially some cocludig remarks are provided i 5. Overview of te ai Fiite Elemets Fiite Elemets are bases of fuctios suc tat te solutio of a iitial or boudary value problem to be able to be epressed as a liear combiatio of tese fuctios. Below we remid te most commo fiite elemets. Fiite elemets are basis fuctios tat are ozero oly i small regios te elemets.

2 Suppose tat we ave to solve te followig iitial value problem over te iterval ] [ y f d dy wit te iitial coditio y y.a. Fiite Elemets: Piecewise Liear Fuctios at fuctios I te iterval ] [ we cosider te poits: L as well as te fuctios... It is proved tat te fuctios compose a basis of te space of fuctios tat are cotiuous i te iterval ] [. I applicatios we cosider te solutio of our No-Liear Ordiary Differetial Equatios as a liear combiatio of te fiite subset L of tis basis. Usually te poits L are equispaced. So Fiite Elemets: Piecewise Square Fuctios I te iterval ] [ we cosider te poits: L We defie We also cosider Proceedigs of te 6t WSEAS It. Cof. o EVOLUTIONARY COPUTING Lisbo Portugal Jue pp6-

3 6 ad 7 Especially ad. It is proved tat te fuctios... L compose a basis of te space of fuctios tat are cotiuous i te iterval ] [. I applicatios we cosider te solutio of our No-Liear Ordiary Differetial Equatios as a liear combiatio of te fiite subset L of tis basis.. Fiite Elemets: Hermite Fuctios Te basic Hermite fuctios are defied as follows 8 9 For L we defie as well as... Proceedigs of te 6t WSEAS It. Cof. o EVOLUTIONARY COPUTING Lisbo Portugal Jue pp6-

4 . We cosider agai te solutio of te differetial equatio as a liear combiatio of te fiite subset L of tis basis.. Fiite Elemets: Splies Fuctios Te basic procedure splie fuctio is defied as follows ] [ ] [ So te Splie basis is L were L. Obviously a sligt modificatio is eeded for Hece... We ca cosider agai te solutio of te differetial equatio as a liear combiatio of te fiite subset... of tis basis. Formulatio ad Solutio of te Geeral Iitial Value Problem via Collocatio ad Geetic Algoritm Cosider te geeral first order iitial problem. y f d dy y y.a Suppose tat { } is a subset of te basis of fuctios like. or. or. or. o wic our solutio is epaded. Te our solutio ca be epressed as a liear combiatio c y Proceedigs of te 6t WSEAS It. Cof. o EVOLUTIONARY COPUTING Lisbo Portugal Jue pp6-

5 Proceedigs of te 6t WSEAS It. Cof. o EVOLUTIONARY COPUTING Lisbo Portugal Jue pp6- I te metod of collocatio [] we select poits i te iterval ] L. [ Tese poits ca be equispaced or o equispaced. Te we ave te equatios y c 5. c 5. f c 5. f We ca solve te Equatio 5. wit respect to oe c say c l were l {... }. Substitutig tis c l ito we obtai a system of o liear equatios i ukows: c c... cl cl... c. Tese equatios are umbered as 5a. 5a.. Please ote tat tese equatios do ot cotai c l c 5a. f c 5a. f Te solutio of 5a. 5a. ca be obtaied via a Geetic Algoritm followig te metod of [6]. Te Geetic Algoritm is used to elimiate a error i 5a. 5a.. Te reaso tat we solve 5. wit respect to c l as well as te reaso for eecutig te Geetic Algoritm o 5a. 5a. istead of is tat we must satisfy te iitial coditio.a wit zero error. A brief overview of te GAs metodology could be te followig: Suppose tat we ave to maimize miimize te fuctio Q wic is ot ecessary cotiuous or differetiable. GAs are searc algoritms wic iitially were isiped by te process of atural geetics reproductio of a origial populatio performace of crossover ad mutatio selectio of te best. Te mai idea for a optimizatio problem is to start our searc o wit oe iitial poit but wit a populatio of iitial poits. Te umbers poits of tis iitial set called populatio quite aalogously to biological systems are coverted to te biary system. I te sequel tey are cosidered as cromosomes actually sequeces of ad. Te et step is to form pairs of tese poits wo will be cosidered as parets for a "reproductio" see te followig figure parets... }... cildre "Parets" come to "reproductio" were tey itercage parts of teir "geetic material". Tis is acieved by te so-called crossover see te previous figure wereas always a very small probability for a utatio eists. utatio is te peomeo were quite radomly - wit a very small probability toug - a becomes or a becomes. Assume tat every pair of "parets" gives k cildre. By te reproductio te populatio of te "parets" are eaced by te "cildre" ad we ave a icreasemet of te origial populatio because ew members were added parets always belog to te cosidered populatio. Te ew populatio as ow k members. Te te process of atural selectio is applied. Accordig te cocept of atural selectio from te k members oly survive. Tese members are selected as te members wit te iger values of QQ if we attempt to acieve maimizatio of Q or wit te lower values of Q if we attempt to acieve miimizatio of QQ. By repeated iteratios of reproductio uder crossover ad mutatio ad atural selectio we ca fid te miimum or maimum of Q as te poit to wic te best values of our populatio coverge. Te termiatio criterio is fulfilled if te mea value of Q i te -members populatio is o loger improved maimized or miimized. ore detailed overviews of GAs ca be foud i [] [] [] ad [].

6 Proceedigs of te 6t WSEAS It. Cof. o EVOLUTIONARY COPUTING Lisbo Portugal Jue pp6- So we we ave to solve a system of equatios i ukow variables. f... f f Te square fuctio Q f f L f or te absolute value fuctio Q f f L f are defied or i r geeral ay suitable orm of f f f... f ad our problem is mi Q If te global miimum of Q is at te poit * * * * * * L te L is a solutio of te aforemetioed systems of o-liear equatios. Te metod as also bee used i []. Formulatio ad Solutio of te Geeral Boudary Value Problem via Collocatio ad Geetic Algoritm Cosider te geeral secod order iitial value problem. d y f y y 6 d dy d dy d ya 6.a y b 6.b Suppose tat { } is a subset of te basis of fuctios like. or. or. or. o wic our solutio is epaded as follows: y c. We ca use te metod of collocatio agai i te iterval ]. [ Te followig system of o liear equatios is obtaied y c 7. c f c 7. f c c 7. y 7. We ca solve te systems of te Equatio wit respect to two variables say c l c l were l l {... }. Substitutig tese epressios of c l c l ito we obtai a system of o liear equatios i ukows: c c... cl cl cl cl c. Tese equatios ew equatios after te elimiatio of c l c l are umbered as 7a. 7a.. Please ote tat tese equatios do ot cotai c c l l c f c 7a. f c 7a. Te solutio of 7a. 7a. ca be obtaied via a Geetic Algoritm.. Te Geetic Algoritm is used to elimiate a error i 7a. 7a.. Te reaso tat we solve te 7. ad 7. wit respect to c l c l as well as te reaso for eecutig te Geetic Algoritm i 7a. 7a. istead of is tat we must

7 Proceedigs of te 6t WSEAS It. Cof. o EVOLUTIONARY COPUTING Lisbo Portugal Jue pp6- satisfy te boudary value coditios 6.a ad 6.b wit zero error. 5 Cocludig Remarks ad Future Researc As GAs is a powerful tool for te solutio of systems of o liear equatios tey ca fid applicatios i te solutio of o liear Ordiary Differetial Equatios. Te collocatio metod is used ad a system of o liear equatios is obtaied. Tis system is solved by GAs. Numerical eamples ca outlie te validity ad efficiecy of our proposed metod. usig Geetic Algoritms IEEE Trasactios o Circuits ad Systems I: Fudametal Teory ad Applicatios Part I Vol. 5 No. 5 pp ay. [] astorakis N.E. Goos I.F. Swamy.N.S.: Stability of ultidimesioal Systems usig Geetic Algoritms IEEE Trasactios o Circuits ad Systems Part I Vol. 5 No. 7 pp July. [] Neil Gersefeld 999 Te Nature of atematical odelig Cambridge Uiversity Press. Refereces: [] Goldberg D.E. 989 Geetic Algoritms i Searc Optimizatio ad acie Learig Addiso-Wesley Secod Editio 989 [] Grefestette J.J. Optimizatio of cotrol parameters for Geetic Algoritms IEEE Tras. Systems a ad Cyberetics SC 6 Ja/Feb 986 pp. 8 [] Eberart R. Simpso P. ad Dobbis R. 996 Computatioal Itelligece PC Tools AP Professioals. [] Kosters W.A. Kok J.N. ad Floree P. Fourier Aalysis of Geetic Algoritms Teoretical Computer Sciece Elsevier 9 99 pp [5] E. Balagusuramy Numerical etods Tata cgraw Hill New Deli 999 [6] Nikos E. astorakis Solvig No-liear Equatios via Geetic Algoritms Proceedigs of te 6t WSEAS Iteratioal Coferece o Evolutioary Computig Lisbo Portugal Jue [7] Ioais F. Goos Lefteris I. Virirakis Nikos E. astorakis.n.s. Swamy "Evolutioary Desig of -Dimesioal Recursive Filters via te Computer Laguage GENETICA" to appear i IEEE Trasactios o Circuits ad Systems I: Fudametal Teory ad Applicatios. 5 [8] Goos I.F. astorakis N.E. Swamy.N.S.: A Geetic Algoritm Approac to te Problem of Factorizatio of Geeral ultidimesioal Polyomials IEEE Trasactios o Circuits ad Systems I: Fudametal Teory ad Applicatios Part I Vol. 5 No. pp. 6- Jauary. [9] astorakis N.E. Goos I.F. Swamy.N.S.: Desig of -Dimesioal Recursive Filters

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat