# The Forward Euler method for first order differential equations



The Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value.



In [5]:

import math



### The forward Euler method

$\frac{dy}{dx}=f(x,y) \quad y=y_0 \ when \ x=x_0 \quad x_0<x<X$

The forward Euler method calculates the values $y_1, y_2, y_3,... ,y_{N−1}, y_{N}$ using the formula $y_i - y_{i−1} = \frac{f(x_{i-1},\ y_{i-1})}{h} \quad i = 1, 2, 3, . . . , N$

This may be written as the explicit formula $y_i = y_{i-1} + hf(x_{i-1},\ y_{i-1}) \quad i = 1, 2, 3,... , N$

Methods where an explicit expression for $y_i$ may be written down are known as explicit methods.



In [3]:

def forward_euler(f, x0, y0, X, N):
h = (X - x0) / N
ys = [y0]

for i in range(1, N + 1):
ys.append(ys[-1] + h * f((i - 1) * h, ys[-1]))

return ys



### Example(s)



In [6]:

print(forward_euler(lambda x, y: y + pow(math.e, x), 0, 1, 1, 4))




[1, 1.5, 2.1960063541719355, 3.1571882603899515, 4.475735329640608]




In [8]:

print(forward_euler(lambda x, y: y + pow(math.e, x), 0, 1, 1, 40)[-1])




5.320403930032723




In [27]:

e_approx = forward_euler(lambda x, y: pow(math.e, x), 0, 1, 1, 1000)

for it, e in enumerate(e_approx):
print("Iteration:", it, math.e, e, abs(e - math.e))




Iteration: 0 2.718281828459045 1 1.718281828459045
Iteration: 1 2.718281828459045 1.001 1.7172818284590452
Iteration: 2 2.718281828459045 1.0020010005001665 1.7162808279588786
Iteration: 3 2.718281828459045 1.0030030025015004 1.7152788259575447
Iteration: 4 2.718281828459045 1.0040060070060037 1.7142758214530414
Iteration: 5 2.718281828459045 1.005010015016681 1.7132718134423641
Iteration: 6 2.718281828459045 1.0060150275375404 1.7122668009215047
Iteration: 7 2.718281828459045 1.0070210455735944 1.7112607828854507
Iteration: 8 2.718281828459045 1.0080280701308613 1.7102537583281838
Iteration: 9 2.718281828459045 1.0090361022163656 1.7092457262426795
Iteration: 10 2.718281828459045 1.0100451428381394 1.7082366856209057
Iteration: 11 2.718281828459045 1.0110551930052236 1.7072266354538215
Iteration: 12 2.718281828459045 1.0120662537276683 1.7062155747313767
Iteration: 13 2.718281828459045 1.0130783260165344 1.7052035024425107
Iteration: 14 2.718281828459045 1.014091410883894 1.704190417575151
Iteration: 15 2.718281828459045 1.0151055093428325 1.7031763191162126
Iteration: 16 2.718281828459045 1.0161206224074482 1.702161206051597
Iteration: 17 2.718281828459045 1.0171367510928542 1.701145077366191
Iteration: 18 2.718281828459045 1.0181538964151795 1.7001279320438656
Iteration: 19 2.718281828459045 1.0191720593915694 1.6991097690674757
Iteration: 20 2.718281828459045 1.0201912410401868 1.6980905874188583
Iteration: 21 2.718281828459045 1.0212114423802134 1.6970703860788316
Iteration: 22 2.718281828459045 1.022232664431851 1.6960491640271942
Iteration: 23 2.718281828459045 1.0232549082163214 1.6950269202427237
Iteration: 24 2.718281828459045 1.0242781747558687 1.6940036537031764
Iteration: 25 2.718281828459045 1.0253024650737592 1.6929793633852859
Iteration: 26 2.718281828459045 1.0263277801942836 1.6919540482647615
Iteration: 27 2.718281828459045 1.027354121142757 1.690927707316288
Iteration: 28 2.718281828459045 1.0283814889455205 1.6899003395135246
Iteration: 29 2.718281828459045 1.029409884629942 1.688871943829103
Iteration: 30 2.718281828459045 1.030439309224417 1.687842519234628
Iteration: 31 2.718281828459045 1.0314697637583705 1.6868120647006746
Iteration: 32 2.718281828459045 1.032501249262257 1.6857805791967881
Iteration: 33 2.718281828459045 1.0335337667675621 1.684748061691483
Iteration: 34 2.718281828459045 1.0345673173068035 1.6837145111522416
Iteration: 35 2.718281828459045 1.0356019019135316 1.6826799265455135
Iteration: 36 2.718281828459045 1.0366375216223314 1.6816443068367137
Iteration: 37 2.718281828459045 1.0376741774688223 1.6806076509902228
Iteration: 38 2.718281828459045 1.0387118704896605 1.6795699579693846
Iteration: 39 2.718281828459045 1.039750601722539 1.678531226736506
Iteration: 40 2.718281828459045 1.0407903722061893 1.6774914562528558
Iteration: 41 2.718281828459045 1.0418311829803817 1.6764506454786634
Iteration: 42 2.718281828459045 1.0428730350859272 1.6754087933731179
Iteration: 43 2.718281828459045 1.043915929564678 1.674365898894367
Iteration: 44 2.718281828459045 1.0449598674595286 1.6733219609995165
Iteration: 45 2.718281828459045 1.046004849814417 1.672276978644628
Iteration: 46 2.718281828459045 1.0470508776743257 1.6712309507847194
Iteration: 47 2.718281828459045 1.0480979520852827 1.6701838763737624
Iteration: 48 2.718281828459045 1.0491460740943623 1.6691357543646828
Iteration: 49 2.718281828459045 1.0501952447496867 1.6680865837093584
Iteration: 50 2.718281828459045 1.0512454651004268 1.6670363633586183
Iteration: 51 2.718281828459045 1.0522967361968028 1.6659850922622423
Iteration: 52 2.718281828459045 1.053349059090086 1.664932769368959
Iteration: 53 2.718281828459045 1.0544024348325993 1.6638793936264458
Iteration: 54 2.718281828459045 1.0554568644777187 1.6628249639813264
Iteration: 55 2.718281828459045 1.0565123490798738 1.6617694793791713
Iteration: 56 2.718281828459045 1.0575688896945492 1.6607129387644959
Iteration: 57 2.718281828459045 1.0586264873782858 1.6596553410807593
Iteration: 58 2.718281828459045 1.0596851431886813 1.6585966852703637
Iteration: 59 2.718281828459045 1.0607448581843917 1.6575369702746534
Iteration: 60 2.718281828459045 1.0618056334251318 1.6564761950339133
Iteration: 61 2.718281828459045 1.0628674699716771 1.655414358487368
Iteration: 62 2.718281828459045 1.0639303688858643 1.6543514595731807
Iteration: 63 2.718281828459045 1.0649943312305923 1.6532874972284528
Iteration: 64 2.718281828459045 1.0660593580698237 1.6522224703892214
Iteration: 65 2.718281828459045 1.0671254504685852 1.65115637799046
Iteration: 66 2.718281828459045 1.0681926094929695 1.6500892189660756
Iteration: 67 2.718281828459045 1.0692608362101355 1.6490209922489096
Iteration: 68 2.718281828459045 1.07033013168831 1.647951696770735
Iteration: 69 2.718281828459045 1.0714004969967887 1.6468813314622563
Iteration: 70 2.718281828459045 1.0724719332059371 1.645809895253108
Iteration: 71 2.718281828459045 1.0735444413871913 1.6447373870718538
Iteration: 72 2.718281828459045 1.0746180226130597 1.6436638058459854
Iteration: 73 2.718281828459045 1.0756926779571236 1.6425891505019214
Iteration: 74 2.718281828459045 1.0767684084940383 1.6415134199650068
Iteration: 75 2.718281828459045 1.0778452152995346 1.6404366131595105
Iteration: 76 2.718281828459045 1.0789230994504193 1.6393587290086258
Iteration: 77 2.718281828459045 1.0800020620245765 1.6382797664344686
Iteration: 78 2.718281828459045 1.081082104100969 1.637199724358076
Iteration: 79 2.718281828459045 1.082163226759639 1.636118601699406
Iteration: 80 2.718281828459045 1.0832454310817095 1.6350363973773356
Iteration: 81 2.718281828459045 1.0843287181493844 1.6339531103096607
Iteration: 82 2.718281828459045 1.085413089045951 1.632868739413094
Iteration: 83 2.718281828459045 1.0864985448557805 1.6317832836032646
Iteration: 84 2.718281828459045 1.0875850866643286 1.6306967417947165
Iteration: 85 2.718281828459045 1.0886727155581375 1.6296091129009076
Iteration: 86 2.718281828459045 1.089761432624836 1.6285203958342092
Iteration: 87 2.718281828459045 1.090851238953141 1.627430589505904
Iteration: 88 2.718281828459045 1.0919421356328594 1.6263396928261857
Iteration: 89 2.718281828459045 1.0930341237548875 1.6252477047041576
Iteration: 90 2.718281828459045 1.0941272044112138 1.6241546240478313
Iteration: 91 2.718281828459045 1.095221378694919 1.6230604497641261
Iteration: 92 2.718281828459045 1.0963166477001773 1.6219651807588678
Iteration: 93 2.718281828459045 1.097413012522258 1.620868815936787
Iteration: 94 2.718281828459045 1.0985104742575262 1.6197713542015189
Iteration: 95 2.718281828459045 1.0996090340034435 1.6186727944556016
Iteration: 96 2.718281828459045 1.1007086928585696 1.6175731356004754
Iteration: 97 2.718281828459045 1.1018094519225636 1.6164723765364815
Iteration: 98 2.718281828459045 1.1029113122961847 1.6153705161628604
Iteration: 99 2.718281828459045 1.1040142750812931 1.614267553377752
Iteration: 100 2.718281828459045 1.105118341380852 1.613163487078193
Iteration: 101 2.718281828459045 1.1062235122989277 1.6120583161601174
Iteration: 102 2.718281828459045 1.107329788940691 1.610952039518354
Iteration: 103 2.718281828459045 1.108437172412419 1.6098446560466262
Iteration: 104 2.718281828459045 1.1095456638214949 1.6087361646375502
Iteration: 105 2.718281828459045 1.1106552642764105 1.6076265641826346
Iteration: 106 2.718281828459045 1.111765974886766 1.606515853572279
Iteration: 107 2.718281828459045 1.1128777967632726 1.6054040316957725
Iteration: 108 2.718281828459045 1.113990731017752 1.6042910974412932
Iteration: 109 2.718281828459045 1.1151047787631383 1.6031770496959068
Iteration: 110 2.718281828459045 1.1162199411134797 1.6020618873455654
Iteration: 111 2.718281828459045 1.1173362191839384 1.6009456092751067
Iteration: 112 2.718281828459045 1.1184536140907928 1.5998282143682523
Iteration: 113 2.718281828459045 1.1195721269514378 1.5987097015076073
Iteration: 114 2.718281828459045 1.1206917588843863 1.5975900695746588
Iteration: 115 2.718281828459045 1.1218125110092705 1.5964693174497746
Iteration: 116 2.718281828459045 1.1229343844468425 1.5953474440122026
Iteration: 117 2.718281828459045 1.1240573803189757 1.5942244481400694
Iteration: 118 2.718281828459045 1.1251814997486662 1.5931003287103789
Iteration: 119 2.718281828459045 1.1263067438600336 1.5919750845990115
Iteration: 120 2.718281828459045 1.127433113778322 1.590848714680723
Iteration: 121 2.718281828459045 1.1285606106299013 1.5897212178291438
Iteration: 122 2.718281828459045 1.1296892355422687 1.5885925929167763
Iteration: 123 2.718281828459045 1.130818989644049 1.587462838814996
Iteration: 124 2.718281828459045 1.1319498740649965 1.5863319543940486
Iteration: 125 2.718281828459045 1.1330818899359956 1.5851999385230495
Iteration: 126 2.718281828459045 1.1342150383890623 1.5840667900699827
Iteration: 127 2.718281828459045 1.1353493205573453 1.5829325079016998
Iteration: 128 2.718281828459045 1.1364847375751268 1.5817970908839183
Iteration: 129 2.718281828459045 1.1376212905778238 1.5806605378812213
Iteration: 130 2.718281828459045 1.1387589807019896 1.5795228477570555
Iteration: 131 2.718281828459045 1.1398978090853142 1.5783840193737309
Iteration: 132 2.718281828459045 1.1410377768666262 1.5772440515924189
Iteration: 133 2.718281828459045 1.1421788851858934 1.5761029432731517
Iteration: 134 2.718281828459045 1.1433211351842243 1.5749606932748208
Iteration: 135 2.718281828459045 1.144464528003869 1.573817300455176
Iteration: 136 2.718281828459045 1.1456090647882202 1.5726727636708249
Iteration: 137 2.718281828459045 1.146754746681815 1.57152708177723
Iteration: 138 2.718281828459045 1.1479015748303354 1.5703802536287097
Iteration: 139 2.718281828459045 1.1490495503806095 1.5692322780784356
Iteration: 140 2.718281828459045 1.1501986744806132 1.568083153978432
Iteration: 141 2.718281828459045 1.1513489482794703 1.5669328801795748
Iteration: 142 2.718281828459045 1.152500372927455 1.56578145553159
Iteration: 143 2.718281828459045 1.153652949575992 1.5646288788830531
Iteration: 144 2.718281828459045 1.154806679377658 1.5634751490813872
Iteration: 145 2.718281828459045 1.1559615634861828 1.5623202649728622
Iteration: 146 2.718281828459045 1.157117603056451 1.5611642254025941
Iteration: 147 2.718281828459045 1.1582747992445017 1.5600070292145434
Iteration: 148 2.718281828459045 1.1594331532075315 1.5588486752515136
Iteration: 149 2.718281828459045 1.1605926661038946 1.5576891623551505
Iteration: 150 2.718281828459045 1.1617533390931036 1.5565284893659415
Iteration: 151 2.718281828459045 1.1629151733358318 1.5553666551232133
Iteration: 152 2.718281828459045 1.1640781699939136 1.5542036584651315
Iteration: 153 2.718281828459045 1.1652423302303456 1.5530394982286995
Iteration: 154 2.718281828459045 1.1664076552092884 1.5518741732497567
Iteration: 155 2.718281828459045 1.1675741460960667 1.5507076823629784
Iteration: 156 2.718281828459045 1.1687418040571718 1.5495400244018733
Iteration: 157 2.718281828459045 1.1699106302602615 1.5483711981987835
Iteration: 158 2.718281828459045 1.1710806258741624 1.5472012025848827
Iteration: 159 2.718281828459045 1.17225179206887 1.546030036390175
Iteration: 160 2.718281828459045 1.1734241300155508 1.5448576984434943
Iteration: 161 2.718281828459045 1.1745976408865426 1.5436841875725025
Iteration: 162 2.718281828459045 1.1757723258553565 1.5425095026036886
Iteration: 163 2.718281828459045 1.1769481860966775 1.5413336423623676
Iteration: 164 2.718281828459045 1.1781252227863659 1.5401566056726792
Iteration: 165 2.718281828459045 1.1793034371014586 1.5389783913575865
Iteration: 166 2.718281828459045 1.18048283022017 1.5377989982388751
Iteration: 167 2.718281828459045 1.1816634033218933 1.5366184251371517
Iteration: 168 2.718281828459045 1.1828451575872017 1.5354366708718434
Iteration: 169 2.718281828459045 1.1840280941978494 1.5342537342611957
Iteration: 170 2.718281828459045 1.1852122143367734 1.5330696141222717
Iteration: 171 2.718281828459045 1.1863975191880938 1.5318843092709513
Iteration: 172 2.718281828459045 1.1875840099371155 1.5306978185219295
Iteration: 173 2.718281828459045 1.1887716877703294 1.5295101406887157
Iteration: 174 2.718281828459045 1.1899605538754134 1.5283212745836317
Iteration: 175 2.718281828459045 1.1911506094412339 1.5271312190178112
Iteration: 176 2.718281828459045 1.1923418556578462 1.5259399728011989
Iteration: 177 2.718281828459045 1.193534293716497 1.524747534742548
Iteration: 178 2.718281828459045 1.194727924809624 1.523553903649421
Iteration: 179 2.718281828459045 1.195922750130859 1.5223590783281862
Iteration: 180 2.718281828459045 1.1971187708750268 1.5211630575840183
Iteration: 181 2.718281828459045 1.1983159882381487 1.5199658402208964
Iteration: 182 2.718281828459045 1.1995144034174419 1.5187674250416032
Iteration: 183 2.718281828459045 1.2007140176113218 1.5175678108477233
Iteration: 184 2.718281828459045 1.2019148320194026 1.5163669964396425
Iteration: 185 2.718281828459045 1.203116847842499 1.5151649806165461
Iteration: 186 2.718281828459045 1.2043200662826266 1.5139617621764185
Iteration: 187 2.718281828459045 1.205524488543004 1.512757339916041
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Iteration: 190 2.718281828459045 1.2091449902958866 1.5091368381631585
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Iteration: 195 2.718281828459045 1.2152033489390655 1.5030784795199796
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Iteration: 216 2.718281828459045 1.2409818479030332 1.4772999805560119
Iteration: 217 2.718281828459045 1.2422229502820339 1.4760588781770112
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Iteration: 219 2.718281828459045 1.2447088814517908 1.4735729470072543
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Iteration: 230 2.718281828459045 1.258470731474511 1.4598110969845342
Iteration: 231 2.718281828459045 1.2597293314844404 1.4585524969746047
Iteration: 232 2.718281828459045 1.2609891907238897 1.4572926377351554
Iteration: 233 2.718281828459045 1.262250310452718 1.456031518006327
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