Integral Equations and their Relationship to Differential Equations with Initial Conditions

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Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs GLM 6 3-3 Geerl Leers Mhemcs GLM Wese: hp://wwwscecereflecocom/geerl-leers--mhemcs/ Geerl Leers Mhemcs Scece Refleco Iegrl Equos d her Reloshp o Dfferel Equos wh Il odos Mosef Ndr Som Guech Deprme of Mhemcs Uvers of Msl 8 Alger mosefdr@hoofr guechs7@gmlcom Asrc Iegrl d dfferel equos hve fudmel mporce he fucol lss d he prcce prolems d m doms of scefc reserch However he resoluo of dfferel equos wh cos coeffces s es u he resoluo of hese equos wh vrle coeffces s prccll dffcul or mpossle more pr of he cses Ths wor prese lcl mehod whch rsform dfferel equos wh l codos o Volerr equos of secod d effce mehods for pprome umercl soluo of hese equos he lss of he esece of her soluos d he covergece of he Ideg erms/kewords Dfferel equo Volerr egrl equo Euler mehod fe dffereces mehod rpezodl mehod SUBJET LASSIFIATION MS[]: 45D5 45E5 45L5 45L d 65R Iroduco We Two m pes of egrl equos wll pper hs pper: her mes occur he le elow Suppose h f : R d : R re couous fucos d h re coss Volerr o-homogeeous of secod d Volerr homogeeous of secod d f d d where d he fuco s clled he erelof he egrl equo 3 P g e

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Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs 5 P g e B egro he equo 3 we ge B dzd z Afer he replceme lemm we ge B A d The A B d If we replce he vlues of ẏ d he frs equo of he prolem we wll ge equo of he form A B B f d d he F d where d A B B f F Thus we hve o-homogeeous Volerr egrl equo of he secod d The prevous emple dces fudmel reloshp ewee Volerr egrl equos d ordr ler dfferel equo of he secod-order Acull he soluo of dfferel equo of he pe d d f d d 4 wh couous coeffces ogeher wh he l codos e reduced o he soluo of cer Volerr egrl equo of he secod d F d where! d ]! [ f F

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Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs where re he scsss of he pro pos of he egro ervl [ ] or erpolo odes Le us choose cos egro sep h h For he equo d cosder he dscree se of pos cqures he form d f d f j f d f 6 j j j j Applg he Trpezodl formul 5 o he egrl 6 we rrve he ssem of equos where he odes j h j f h j j j j d f h j j j j f j f j d j j j see for emple [ 3 ] d j re pprome vlues of he uow fuco orollr 4 See for emple 4 Le e couous erel wh m d The for ll f he egrl equo of secod d d f hs uque soluo d he successve ppromo d f uforml coverge o he ec soluo for ll Proof See for emple 4 5 Numercl Soluo of secod-order dfferel equo wh l codos There re m umercl mehods for he soluo of dfferel equos of he secod-order h c e used o o pprome soluos of dfferel equos Such ppromos re ecessr whe o ec soluo c e foud I our sud we wll use wo fmous d smple mehods Euler d fe dffereces If we ppl he Euler formul or fe dffereces F-D mehod he prolem of l codos we wll rrve ler ssem of equos see for emple [ 3 ] 7 P g e

Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs 6 Numercl Emples I hs seco we prese few umercl emples d ll hese emples gve were ru wh MATLAB R Frs Le ssf he dfferel equo wh he l vlues ep ep ep Ths correspods o he followg Volerr egrl equo 7 ep d ep 8 For whch he ec soluo s ep Tle omprso he resuls solue for he egrl equo 8 ec Trpezodl mehod e+ e+ e+ 9484e- 9468e- 547e-4 8873e- 8845e- 837e-4 3 748e- 7443e- 38658e-4 4 673e- 66985e- 46559e-4 5 6653e- 66e- 546e-4 6 5488e- 5486e- 5564e-4 7 49659e- 496e- 5758e-4 8 44933e- 44876e- 576e-4 9 4657e- 46e- 5558e-4 36788e- 36735e- 599e-4 Tle omprso he resuls solue for dfferel equo 7 Euler F-D e+ e+ e+ e+ 9e- 48374e-3 9e- 48374e-3 8e- 7738e-3 89e- 96344e-3 3 7388e- 89356e-3 7647e- 4344e- 4 666e- 876e-3 654e- 894e- 5 599e- 7356e-3 5839e- 3336e- 6 5438e- 49957e-3 56e- 7548e- 7 4946e- 9854e-3 4655e- 3533e- 8 4583e- 55e-3 446e- 357e- 9 484e- 574e-3 3678e- 3875e- 3774e- 9574e-3 359e- 4965e- 8 P g e

Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs Le ssf he dfferel equo wh he l vlues cos cos cos s 4 4 4 Ths correspods o he followg Volerr egrl equo cos s cos d 4 4 For whch he ec soluo s cos Tle 3 omprso of resuls solue for he egrl equo ec Trpezodl mehod e+ e+ e+ 995e- 995e- 669e-6 987e- 989e- 4536e-5 3 95534e- 95539e- 53968e-5 4 96e- 95e- 9886e-5 5 87758e- 8777e- 396e-4 6 8534e- 8553e- 8969e-4 7 76484e- 7658e- 453e-4 8 6967e- 697e- 99e-4 9 66e- 694e- 33396e-4 543e- 5467e- 36665e-4 9 T 4 omprso des resuls solue for dfferel equo 9 Euler F-D e+ e+ e+ e+ e+ 49958e-3 e+ 49958e-3 99e- 99334e-3 995e- 83e- 3 975e- 474e- 9779e- 5456e- 4 944e- 934e- 9446e- 44e- 5 95e- 396e- 9588e- 896e- 6 85398e- 8649e- 869e- 36578e- 7 79868e- 33839e- 86e- 46757e- 8 73658e- 39874e- 7569e- 59385e- 9 66883e- 47e- 69666e- 7545e- 5967e- 5645e- 63463e- 9435e- 9 P g e

Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs Le ssf he dfferel equo wh he l vlues 4 3 4 8 Ths correspods o he followg Volerr egrl equo 3 5 3 6 d 4 3 6 For whch he ec soluo s Tle 5 omprso of resuls solue for he egrl equo ec Trpezodl mehod e+ e+ e+ e+ e+ 878e-5 4e+ 4e+ 67e-4 3 9e+ 9e+ 446e-4 4 6e+ 63e+ 33e-4 5 5e+ 54e+ 45e-4 6 36e+ 365e+ 5378e-4 7 49e+ 496e+ 6359e-4 8 64e+ 648e+ 7668e-4 9 8e+ 89e+ 8996e-4 3e+ 3e+ 538e-3 Tle 6 omprso des resuls solue for dfferel equo Euler F-D e+ e+ e+ e+ e+ e- e+ e- e+ e- e+ 9995e- 3 6e+ 35e- 6e+ 9968e- 4 e+ 43e- e+ 39894e- 5 999e+ 5e- 3e+ 49733e- 6 997e+ 656e- 36e+ 59435e- 7 495e+ 754e- 4e+ 6894e- 8 559e+ 84e- 568e+ 7876e- 9 783e+ 9747e- 79e+ 8759e- 897e+ 8e- 945e+ 95497e- 3 P g e

Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs 7 ocluso we c coclude ll dfferel equo wh l codos correspods o Volerr egrl equo of he secod d d he umercl soluo of egrl equo for emple Trpezodl mehod s he eer h he umercl soluo of correspode dfferel equo for emple Euler or fe dffereces mehod Refereces [] P ollsdfferel d egrl equo Oford uvers of Press New Yor 6 [] H HochsdIegrl equos Polechc se of Brool New Yor 973 [3] S GuechRelo des équos egrles e dffereelles Thess of Mser uvers of Msl Alger [4] M Ndr ours sur les équos égrles Uvers of Msl Alger 8 3 P g e

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