Numerical Solution of Nonlinear Integro-Differential Equations with Initial Conditions by Bernstein Operational Matrix of Derivative

Transcription

1 Iteratioal Joural of Moder Noliear heory ad Applicatio Published Olie Jue 3 ( Nuerical Solutio of Noliear Itegro-Differetial Equatios with Iitial Coditios by Berstei Operatioal Matri of Derivative Behrooz Basirat Mohaad Ai Shahdadi Departet of Matheatics Islaic Azad Uiversity of Birjad Birjad Ira Eail: behroozbasirat@iaubiracir Received Jauary 8 3; revised February 3 3; accepted March 5 3 Copyright 3 Behrooz Basirat Mohaad Ai Shahdadi his is a ope access article distributed uder the Creative Coos Attributio Licese which perits urestricted use distributio ad reproductio i ay ediu provided the origial work is properly cited ABSRAC I this paper we preset a practical atri ethod for solvig oliear Volterra-Fredhol itegro-differetial equatios uder iitial coditios i ters of Berstei polyoials o the iterval [] he oliear part is approiated i the for of atrices equatios by operatioal atrices of Berstei polyoials ad the differetial part is approiated i the for of atrices equatios by derivative operatioal atri of Berstei polyoials Fially the ai equatio is trasfored ito a oliear equatios syste ad the ukow of the ai equatio is the approiated We also give soe uerical eaples to show the applicability of the operatioal atrices for solvig oliear Volterra-Fredhol itegro-differetial equatios (NVFIDEs) Keywords: Berstei Polyoial; Operatioal Matri; Itegro-Differetial Equatios Itroductio Soe of the pheoea i physics electroics biology ad other applied scieces lead to oliear Volterra- Fredhol itegro-differetial equatios Of course these equatios ca also appear whe trasforig a differetial equatio ito a itegral equatio [-7] I these kids of equatios the ukow fuctio u appears o both sides; it appears uder the itegral sig o oe side while it appears as a ordiary derivative o the other side Ivestigatig the results of oliear itegrodifferetial equatios we observe that o obvious algebraic ethod has bee recogized to solve these equatios ad thus approiatio ethods are used to solve such equatios [8-] Berstei polyoials (B-plyoials) have ay useful properties [] I this paper we use Berstei polyoials ad their resultig operatioal atrices to solve the followig oliear Volterra- Fredhol itegro-differetial equatios with the iitial coditios idicated as: u q u r u d d k s u s s k s u s s f () u t u t the fuctios u q r k s k s ad f are kow as is ukow We approiate phrases icludig derivative of ukow fuctio by operatioal atri of derivative D ; ad approiate phrases icludig powers of ukow fuctio by Berstei polyoial atri his yields a equatio based o the ukow U which is the approiatio u o eaie the accuracy of the forula we preset soe eaples We will show the differece betwee the accurate ad approiatio solutios i these eaples upo kowig Berstei Polyoials ad heir Properties Defiitio of Berstei Polyoials Basis are de- he Berstei basis polyoials of degree fied by: i i Bi i i By usig bioial epasio of show that i () oe ca Copyright 3 SciRes IJMNA

2 B BASIRA M A SHAHDADI i i i k i i i k k ik With the aid of Equatio (3) the Berstei vector B B ca be writte i the for A (3) () (5) Fuctio Approiatio A fuctio f square itegrable i () ay be epressed i ters of Berstei basis [3] I practice oly the first ters Berstei polyoials are cosidered Hece if we write the i i (7) i f u B U U u u u (8) U Q f (9) Q is said dual atri of ad is give i [3] We ca also approiate the fuctio k s L by double Fourier epasio as follows ij i j j i k s k B B s K s k k k k k k K k k k ad k ij eleets are give by () k i j Bi k s Bj s Bi Bi Bi s Bi s () for i j Due of (9) we obtai K Q k s s Q () 3 Operatioal Matri of Itegratio I perforig arithetic ad other operatios o Berstei basis we frequetly ecouter the itegratio of the vector defied i Equatio () by t d t P (3) is the operatioal atri for itegratio ad is give i [3] P Product Operatioal Matri It is always ecessary to evaluate the product of ad which is called the product atri of Berstei polyoials basis Let () with the vector he ultiplyig the atri U which is defied i Equatio (8) we obtai U U ˆ (5) ˆ is a atri ad is called the coefficiet atri he details of obtaiig this atri are give i [3] U 3 Operatioal Matri of Derivative he differetiatio of vector i Equatio () ca be epressed as: D (6) D is the operatioal atri of derivatives for Berstei polyoials Fro (5) we have A ad the i i i A i i i (6) Copyright 3 SciRes IJMNA

3 B BASIRA M A SHAHDADI 3 Defiig A atri V ad vector (7) as V (8) Equatio (7) ay the be restated as AV (9) We ow epad vector i ters of get B A A B A 3 () A So AVB () We herefore we have the operatioal atri of derivative as D AVB () If we approiate u U the for ( is the order of derivatives) we get u U U D (3) Operatioal Matrices for Nuerical Solutio of NVFIDE o obtai the approiate solutio of the () uder the iitial coditios the followig atri ethod is used Fuctio u is approiated by usig a fiite uber of ters i (7) as j j u u B U j he approiate fuctios by Berstei polyoials ca be fraed by: k s K s k s K s () k s ad k s (5) K ad K are defied with () For uerical ipleetatio of the ethod eplaied i this sec tio we eed to evaluate us ad us ad are positive itegers as follows u s U u s U (6) Fro () ad (6) we have us U XU U U (7) ˆ UUU the vector U U Uˆ is a -vector; the for us 3 we get U U UU U 3 u s U U ˆ 3 (8) herefore with this ethod we ca approiate us ad us arbitrary ad Suppose that this ethod holds for us U We obtai it for as follows 3 3 u s U U U U U Uˆ U (9) We have a siilar relatio for So the copoets of u ad u ca be coputed i ters of copoets of ukow vector U I this case for the Volterra ad Fredhol part of () we have ad d K su s K suds KUˆ sds KUˆ P k s u s s l d l K QU k s u s s ds K s U s ds K s s dsu l l (3) (3) Copyright 3 SciRes IJMNA

4 B BASIRA M A SHAHDADI Also to approiate the differetial part of Equatio () akig use of Equatio (3) we have u q u r u U D q U D r U (3) After substitutig the approiate Equatios ()-(3) i () we get KU P K QU f U D q U D r U ˆ l Ad also for the iitial coditios we have U t U D t (33) (3) he iitial coditios Equatio (3) give two liear equatios Sice the uber of ukows for a vector U i (33) is the we collocate Equatio (33) i Newto-Cotes poits as p p p (35) For collocatig Equatio (33) we have used the Newto-Cotes poits because of their siplicity ad their good utility i our ipleetatio as regards the speed ad accuracy of aswers However we ca use other poits like the Gauss poits Cleshaw-Curtis poits Lobatto poits etc Here we preset the fial syste: ( ) ˆ pk QU f p U D p q p U D p r p U p p KU P p p U t U D t After solvig oliear syste (36) we get (36) U ; the we have the approiate solutio of Equatio () 5 Nuerical Eaples o illustrate the effectiveess of the proposed ethod i the preset paper several eaples are preseted i this sectio Eaple Cosider the oliear Fredhol itegrodifferetial equatio [5] u y uy dy u (37) u he eact solutio is u he results of proposed ethod for this eaple are ehibited i able with choices of For this coplicate NFIDE with a sall uber of Berstei basis fuctios we get acceptable results By applyig the ethod i Sectio for we have 6 A Q able Approiate ad eact solutios for Eaple Preset ethod with = Preset ethod with = 3 Preset ethod with = Preset ethod with = 5 Eact solutio Copyright 3 SciRes IJMNA

5 B BASIRA M A SHAHDADI 5 D P K We obtai the approiate solutios of the proble for 3 ad 6 respectively u 3 u3 3 u u Eaple Cosider the oliear Fredhol itegrodifferetial equatio as follows [6]: 5 u y d u y y 3 (38) u u he eact solutio is u akig we copare the obtaied solutio with eact solutio ad ethod [6] i able By applyig the ethod i Sectio we have for 5 3 A Q D P K Q We obtai the approiate solutios of the proble u for Eaple 3 Cosider the oliear Volterra itegrodifferetial equatio as follows: 3 5 u u d u y y u u (39) the eact solutio is u able 3 with choices of illustrates the uerical results for this eaple able Approiate ad eact solutios for Eaple Preset ethod with k = Preset ethod with k = Preset ethod with = Eact solutio Copyright 3 SciRes IJMNA

6 6 B BASIRA M A SHAHDADI By applyig the ethod i Sectio we have for A D K Q P Eaple Cosider the oliear Volterra-Fredhol itegro-differetial equatio u u u y uy dy y uy dy f () u u 7 f he eact solutio is u he results of proposed ethod o this eaple are ehibited i able with choices of For this coplicated NVFIDE we get acceptable results with a sall uber of Berstei basis fuctios We obtai the approiate solutios of the proble for 3 5 ad 6 respectively u u u u able 3 Approiate ad eact solutios for Eaple Preset ethod with = Preset ethod with = Preset ethod with = Preset ethod with = Eact solutio able Approiate ad eact solutios for Eaple Preset ethod with = Preset ethod with = Preset ethod with = Preset ethod with = Eact solutio Copyright 3 SciRes IJMNA

7 B BASIRA M A SHAHDADI 7 I this regard we have reported i figures ad a table the values of the eact solutio u the approiate solutio u ad the absolute error fuctio e u u at the selected poits of the give iterval We have zooed out Figures ad to a great etet because the eact ad approiate solutios are very close Eaple 5 Fially cosider the oliear Volterra- Fredhol itegro-differetial equatio [7] u u uy dy y y uy dy f () u u f he eact solutio is u he results of the proposed ethod for this eaple are ehibited i able 5 with choices of For this coplicated NVFIDE we get acceptable results with a sall uber of Berstei basis fuctios We obtai the approiate solutios of the proble for 3 ad 6 respectively u u u u I this regard we have reported i figures ad a table the values of the eact solutio u the approiate solutio u ad the absolute error fuctio u u at the selected poits of the give iterval Figure Copariso of the eact solutio ad the approiate solutios for Eaple 5 Figure Copariso of the absolute error fuctios for Eaple 5 able 5 Approiate ad eact solutios for Eaple 5 Preset ethod with = Preset ethod with = 3 Preset ethod with = Preset ethod with = 6 Eact solutio Copyright 3 SciRes IJMNA

8 8 B BASIRA M A SHAHDADI Figure 3 Copariso of the eact solutio ad the approiate solutios for Eaple 5 Figure Copariso of the absolute error fuctios for Eaple 5 We have zooed out Figures 3 ad to a great etet because the eact ad approiate solutios are very close While i [7] the adopted ethod yields o solutio for 3 takes tie to give the approiate aswer for 5 ad is proper oly for our ethod is sigificat ot oly because it yeilds solutios for ay but also because the approiate solutios it gives are very close to eact oes 6 Coclusio I this paper we have proposed a uerical solutio to solve oliear Volterra-Fredhol itegro-differetial equatios with iitial coditio by Berstei polyoials operatioal atrices ad derived operatioal atri We use forula for uerical eaples ad it is obvious that the uerical solutio coicides with the eact solutio eve with a few Berstei polyoials used i the approiatio Fially errors show that the approiatio becoes ore accurate whe is icreased herefore for better results it is recoeded to use a larger REFERENCES [] M A Abdou O Asyptotic Methods for Fredhol- Volterra Itegral Equatio of the Secod Kid i Cotact Probles Joural of Coputatioal ad Applied Matheatics Vol 5 No 3 pp 3-6 [] F Bloo Asyptotic Bouds for Solutios to a Syste of Daped Itegro-Differetial Equatios of Electroagetic heory Joural of Matheatical Aalysis ad Applicatios Vol pp 5-5 [3] M A Jaswo ad G Sy Itegral Equatio Methods i Potetial heory ad Elastostatics Acadeic Press Lodo 977 [] S Jiag ad V Rokhli Secod Kid Itegral Equatios for the Classical Potetial heory o Ope Surface II Joural of Coputatioal Physics Vol 95 No 3 pp -6 [5] P Schiavae ad C Costada ad A Mioduchowski Itegral Methods i Sciece ad Egieerig Birkhauser Bosto [6] B I Setai O a Itegral Equatio for Aially- Syetric Probles i the Case of a Elastic Body Cotaiig a Iclusio Joural of Applied Matheatics ad Mechaics Vol 55 No 3 99 pp [7] N N Voitovich ad O O Reshyak Solutios of Noliear Itegral Equatio of Sythesis of the Liear Atea Arrays BSUAE Joural of Applied Electroagetis Vol No 999 pp 3-5 [8] L M Delves ad J Walsh Nuerical Solutio of Itegral Equatio Oford Uiversity Press Lodo 97 [9] G Micula ad P Pavel Differetial ad Itegral Equatios through Practical Probles ad Eercises Kluwer 99 [] R K Miller Noliear Volterra Itegral Equatios Melo Park 967 [] E H Doha A H Bhrawy ad M A Saker Itegrals of Berestei Polyoials: A Applicatio for the Solutio of High Eve-Order Differetial Equatios Applied Matheatics Letters Vol No pp [] R Farouki ad V Raja Algoriths for Polyoials i Berstei For Coputer Aided Geoetric Desig Vol 5 No 988 pp -6 [3] K Malekejad E Hasheizadeh ad B Basirat Coputatioal Method Based o Berestei Operatioal Matrices for Noliear Volterra Fredhol-Haerstei Itegral Equatios Couicatios i Noliear Sciece ad Nuerical Siulatio Vol 7 No pp 5-6 doi:6/jcss3 [] S A Yousefi ad M Behroozifar Operatioal Matrices of Berstei Polyoials ad heir Applicatios Iteratioal Joural of Systes Sciece Vol No 6 pp doi:8/ [5] Z Avazzadeh M Heydari ad G B Loghai Nuerical Solutio of Fredhol Itegral Equatios of the Secod Kid by Usig Itegral Mea Value heore Applied Matheatical Modellig Vol 35 No 5 pp [6] F Mirzaee he RHFs for Solutio of Noliear Fredhol Itegro-Differetial Equatios Applied Mathea- Copyright 3 SciRes IJMNA

9 B BASIRA M A SHAHDADI 9 tical Modellig Vol 5 No 7 pp [7] W Wag A Algorith for Solvig the High-Order Noliear Volterra-Fredhol Itegro-Differetial Equa- tio with Mechaizatio Applied Matheatics ad Coputatio Vol 7 No 6 pp -3 Copyright 3 SciRes IJMNA