(Subscript[f, 0])=1;IN=20;For[j=1,j≤IN,j++,fj=1+(∫_0^x)((x-η)*((∂_η)(∂_η)(Subscript[f, j-1])+h((∂_η)(∂_η)(Subscript[f, j-1])+((2/η)*(∂_η)(Subscript[f, j-1]))+1)))\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]];

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=0.5;h=-0.3;m

NumberForm[0.9583333333333144`,8]

e=1-((x^2)/3!);

x=0.5;e

NumberForm[0.9583333333333334`,8]

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=1;h=-0.3;m

NumberForm[0.8333333333332575`,8]

x=1;e

N[(5/6)]

NumberForm[0.8333333333333334`,8]

0.83329,0.83337

Clear[m];m=ReplaceAll[(Subscript[f, 20]),η->x];Clear[x];x=1;

Clear[h];Plot[m,{h,-0.68,0.01},PlotRange->{{-0.68,0.01},{0.6,1}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=1.5;h=-0.3;m

NumberForm[0.6249999999996589`,8]

x=1.5;e

NumberForm[0.625`,8]

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=2;h=-0.3;m

NumberForm[0.33333333333303017`,8]

x=2;e

N[(1/3)]

NumberForm[0.3333333333333333`,8]

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=4;h=-0.3;m

NumberForm[-1.6666666666678793`,8]

x=4;e

N[-(5/3)]

NumberForm[-1.6666666666666667`,8]

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=6;h=-0.3;m

NumberForm[-5.000000000005457`,8]

NumberForm[-5.000000000005457`,16]

x=6;e

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=8;h=-0.3;m

NumberForm[-9.666666666671517`,8]

x=8;e

N[-(29/3)]

NumberForm[-9.666666666666666`,8]

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=10;h=-0.3;m

NumberForm[-15.666666666671517`,8]

x=10;e

N[-(47/3)]

NumberForm[-15.666666666666666`,8]

m=ReplaceAll[(Subscript[f, 15]),η->x];e=1-((x^2)/3!);R=Abs[e-m];h=-1;R

R=((16384 ((Abs[x])^2))/3);Plot[R,{x,0,2},PlotRange->{{0,2},{100,22000}},Axes->False,Frame->True,FrameLabel->{x,"Abs error by standard VPM"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

Clear[R];R=1.1368683772161603`*^-13 ((Abs[x])^2);Plot[R,{x,0,2},PlotRange->{{0,2},{0,40.*(10^-14)}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"Abs error by VPM at h = -0.3"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

(Subscript[f, 0])=1;IN=30;For[j=1,j≤IN,j++,fj=1+(∫_0^x)((x-η)*((∂_η)(∂_η)(Subscript[f, j-1])+h((∂_η)(∂_η)(Subscript[f, j-1])+((2/η)*(∂_η)(Subscript[f, j-1]))+(Subscript[f, j-1]))))\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]];

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=0.1;h=-0.4;m

NumberForm[0.9983341664116462`,8]

e=Sin[x]/x;

x=0.1;e

NumberForm[0.9983341664682815`,8]

m=ReplaceAll[(Subscript[f, 15]),η->x];Clear[x];x=0.5;h=-0.3;m

NumberForm[0.958851050316157`,8]

x=0.5;e

NumberForm[0.958851077208406`,8]

Clear[x];m=ReplaceAll[(Subscript[f, 15]),η->x];x=1;m;

NumberForm[0.8414709905496444`,16]

NumberForm[0.8414709277254097`,8]

x=1;e

N[Sin[1]]

NumberForm[0.8414709848078965`,8]

Plot[m,{h,-0.7,0.02},PlotRange->{{-0.7,0.02},{0.6,1}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

m=ReplaceAll[(Subscript[f, 20]),η->x];Clear[x];x=1;

m=ReplaceAll[(Subscript[f, 34]),η->x];Clear[x];x=5;h=-0.3;m

NumberForm[-0.19178489565018314`,8]

x=5;e

N[((Sin[5])/5)]

NumberForm[-0.1917848549326277`,7]

m=ReplaceAll[(Subscript[f, 29]),η->x];Clear[x];x=10;h=-0.348;m

NumberForm[-0.05439122818327746`,3]

x=10;e

N[((Sin[10])/10)]

NumberForm[-0.05440211108893698`,3]

Clear[h];m=ReplaceAll[(Subscript[f, 30]),η->x];h=-0.35;e=Sin[x]/x;Plot[{e,m},{x,0,18},PlotRange->{{0,18},{-0.5 ,1}},PlotRangeClipping->True,Axes->False,Frame->True,PlotStyle->{Dashed,Tiny},FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

Show[%2,AxesLabel->{HoldForm[x],HoldForm["y(x)"]},PlotLabel->None,LabelStyle->{12,GrayLevel[0]}]

Claear[R];R=Abs[1-0.166666641831398` (x^2)+0.008333333302289248` (x^4)-0.00019841271569021046` (x^6)+2.7557290422919323`*^-6 (x^8)-2.505146046161144`*^-8 (x^10)+1.6054826565414793`*^-10 (x^12)-7.634461639260959`*^-13 (x^14)+2.7907107824659238`*^-15 (x^16)-8.017494977971195`*^-18 (x^18)+1.829187041589004`*^-20 (x^20)-3.318600435812201`*^-23 (x^22)+4.77077685341358`*^-26 (x^24)-5.409768833621945`*^-29 (x^26)+4.822514048180152`*^-32 (x^28)-3.3745490246816274`*^-35 (x^30)+1.853476975706377`*^-38 (x^32)-7.996743026378863`*^-42 (x^34)+2.712469235301814`*^-45 (x^36)-7.23608451054005`*^-49 (x^38)+1.5172659147534922`*^-52 (x^40)-2.4953676013418724`*^-56 (x^42)+3.205919195624315`*^-60 (x^44)-3.195863291401625`*^-64 (x^46)+2.4464247173038027`*^-68 (x^48)-1.4158583814828796`*^-72 (x^50)+6.051646881123454`*^-77 (x^52)-1.8420081733321607`*^-81 (x^54)+3.7588162224823155`*^-86 (x^56)-4.5887335144173646`*^-91 (x^58)+2.522705076362767`*^-96 (x^60)-((Sin[x])/x)];

Plot[R,{x,0,14},PlotRange->{{0,14},{0,0.0011}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"Abs error by VPM at h = -0.35"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

Clear[h];m=ReplaceAll[(Subscript[f, 30]),η->x];e=Sin[x]/x;R=Abs[m-e];h=-0.35;R

Clear[h];m=ReplaceAll[(Subscript[f, 29]),η->x];e=Sin[x]/x;R=Abs[m-e];h=-0.348;

(Subscript[f, 10])

(Subscript[f, 10])=1+((341 (η^2))/2)+((1680877 (η^4))/157464)+((27308121511 (η^6))/123018750000)+((125123557527529 (η^8))/54032656230000000)+((162028673942911 (η^10))/11254230397620000000)+((4126989819822181 (η^12))/69047811800651376000000)+((1371404635775357 (η^14))/7967977991266278585600000)+((15292427593 (η^16))/42453429190895923200000)+((2420773 (η^18))/4902713700766801920000)+((η^20)/2432902008176640000);

Clear[h];m=ReplaceAll[(Subscript[f, 30]),η->x];e=Sin[x]/x;R=Abs[m-e];h=-1;R

R=Abs[1+((357913941 (x^2))/2)+((6140942219040120301117 (x^4))/549043018919064)+((9531996848116803048453190405427978011111 (x^6))/40906892636348605155944824218750000)+((3479833512088269094013682717248894180074873426276316465729929 (x^8))/1433647227779163556001640141127226660728454589843750000000)+((15795124764278195988440282515392194388565547279590219981902595881032911 (x^10))/1041182542334416603246536627602552701642018823015213012695312500000000)+((271643702979424371193898047877371202093313689795669612584298509468953115827803511243485077381 (x^12))/(4297488258888731056592924462262330444240409721875915750795444439827195298767089843750000000000))+((1773068906556577914230218946003169904441467219676331543373871932525905261028964009677479989978836513934106811761357 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(x^58))/(12064943478346374937804919240218671715636511775178320674327894840808021285119610322944000000000000000))+((x^60)/(8320987112741390144276341183223364380754172606361245952449277696409600000000000000))-((Sin[x])/x)];

Plot[R,{x,0,14},PlotRange->{{0,14},{20000,7*(10^12)}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"Abs error by standard VPM"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

a=1;(Subscript[f, 0])=m;IN=1;For[j=1,j≤IN,j++,b=((Subscript[f, j-1])^2);c=Expand[b];fj=a+(∫_0^x)((x-η)*((∂_η)(∂_η)(Subscript[f, j-1])+h((∂_η)(∂_η)(Subscript[f, j-1])+((2/η)*(∂_η)(Subscript[f, j-1]))+c)))\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]];

a=1;z=ReplaceAll[m,x->η];b=(z^2);c=Expand[b]
d=(∂_η)(∂_η)z
e=(∂_η)z
g=d+(h*(d+(((2*e)/η))+c))
i=(g*x)-(g*η)
s=Expand[i]
v=(∫_0^x)s\[DifferentialD]η;q=1+v

x=1;

Clear[h];
Plot[q,{h,-0.67,0.02},PlotRange->{{-0.67,0.02},{0.7,1}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

Clear[a];a=10;Clear[h];h=-0.4;m

f=;Plot[f,{x,0,4.3},PlotRange->{{0,4.3},{0,2.2}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

ff=;Plot[ff,{x,0,2.1},PlotRange->{{0,2.1},{0,8.2}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

fff=;Plot[fff,{x,0,2},PlotRange->{{0,2},{0,10.2}},Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

ffff=;Plot[ffff,{x,0,1.4},PlotRange->{{0,1.4},{0,20.4}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

(Subscript[f, 0])=1;IN=7;For[j=1,j≤IN,j++,a=((Subscript[f, j-1])^3);
b=((2/η)*(∂_η)(Subscript[f, j-1]));
c=(∂_η)(∂_η)(Subscript[f, j-1]);
d=c(1+h);e=h(a+b);q=(x-η)*(d+e);z=Expand[q];
fj=1+(∫_0^x)z\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]]

NumberForm[0.9983357861749972`,7]

0.9983358-0.99833720

Clear[x];x=1;Plot[(Subscript[f, 7]),{h,-0.65,-0.02},PlotRange->{{-0.65,-0.02},{0.7,1}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

NumberForm[0.8550574700898644`,16]

NumberForm[0.8550574700898644`,7]

(Subscript[f, 0])=1;IN=5;For[j=1,j≤IN,j++,A=Expand[(Subscript[f, j-1])];aa=(A^2);aaa=Expand[aa];bb=aaa*aaa;
a=Expand[bb];b=((2/η)*(∂_η)A);
c=(∂_η)(∂_η)A;
d=c(1+h);e=h(a+b);q=(x-η)*(d+e);z=Expand[q];
fj=1+(∫_0^x)z\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]]

Clear[x];x=0.1;h=-0.3;m

NumberForm[0.998336545152426`,16]

NumberForm[0.998336545152426`,7]

Clear[x];x=0.2;h=-0.39;m

NumberForm[0.9933861746743978`,16]

NumberForm[0.9933861746743978`,7]

Clear[x];x=0.5;h=-0.39;m

NumberForm[0.9603074811769037`,16]

NumberForm[0.9603074811769037`,6]

NumberForm[0.9603074811769037`,5]

Clear[x];x=1;h=-0.36;m

NumberForm[0.8607669347703885`,4]

Clear[x];x=1;Plot[m,{h,-0.8,0.2},PlotRange->{{-0.8,0.2},{0,2}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

(Subscript[f, 0])=1;IN=2;For[j=1,j≤IN,j++,a=((Subscript[f, j-1])^5);
b=((2/η)*(∂_η)(Subscript[f, j-1]));
c=(∂_η)(∂_η)(Subscript[f, j-1]);
d=c(1+h);e=h(a+b);q=(x-η)*(d+e);z=Expand[q];
fj=1+(∫_0^x)z\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]]

(Subscript[f, 1])

(Subscript[f, 2])

a=(((1+((x^2)/3)))^(-1/2));d=Expand[a]

e=(((1+(((x^2))/3)))^((-1)/2));Clear[x];x=0.1;e

m=ReplaceAll[(Subscript[f, 5]),η->x];Clear[x];x=0.1;h=-0.3;m

e=(((1+(((x^2))/3)))^((-1)/2));Clear[x];x=0.5;e

m=ReplaceAll[(Subscript[f, 5]),η->x];Clear[x];x=0.5;h=-0.39;m

e=(((1+(((x^2))/3)))^((-1)/2));Clear[x];x=1;e

N[((Sqrt[3])/2)]

Clear[x];m=ReplaceAll[(Subscript[f, 5]),η->x];x=1;Plot[m,{h,-0.8,0.2},PlotRange->{{-0.8,0.2},{0,2}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

m=ReplaceAll[(Subscript[f, 5]),η->x];

Clear[h];h=-0.1;m

Clear[h];h=-0.35;m

Clear[h];h=-0.4;m=ReplaceAll[(Subscript[f, 5]),η->x];c=(((1+(((x^2))/3)))^((-1)/2));R=Abs[m-c];Plot[R,{x,0,1},PlotRange->{{0,1},{0,0.000055}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"Abs error by VPM with h = - 0.4"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

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Plot[{g,l},{x,0,1},PlotRange->{{0,1},{0.862,1.01}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

R=Abs[1048576 (x^2)+(x^3)+((228509902438400 (x^4))/3486784401)+((209715 (x^5))/2097152)+((90801676953129760622968832 (x^6))/66505134601593017578125)+((60935741601185 (x^7))/17062072720293888)+((102682419842443114173348166306563063808 (x^8))/7219857701073252065157634735107421875)+((253552655302726175 (x^9))/3833759992447475122176)+((11839664399762422231933394603119750267361792 (x^10))/133196228621915006662077652912319183349609375)+((1831833730151205746497799783 (x^11))/2437438960041984000000000000000000)+((21588465656186658037194849603278659498933081681446258688 (x^12))/58288864038319933383170066569304025825859844991302490234375)+((12211668063783695990193193 (x^13))/2112447098703052800000000000000000)+((6643966811974986762813662081702586003664511341935452217893737542016 (x^14))/(6027409209249143896743587672278202521401182516677395126465402374267578125))+((3381554082464703559972438432495785671 (x^15))/105304211644730436535592366899200000000000000000)+((4938484050040992348124850007237631215123783905462854251893696 (x^16))/2007129273809238726854341549210190649817243595297167874280853271484375)+((1477658606396493045810390451115939303 (x^17))/10954312403728656966343779931914240000000000000000)+((67867448820956617870214219078395589912899451186860327365604442744325593 (x^18))/(15887926884773971249208455605692182122500306266131078242086719267517958862304687500))+((165972739898020973815626954817283370247 (x^19))/374243833166783219154688162551199432704000000000000000)+((272440458602721001549408226009890026972593408259085363040636048689676643817771 (x^20))/(45921380067125796607802227112556161021623100954344795917337992352924134537130173413085937500))+((1124526204272968422342612496208633 (x^21))/959599572222521074755610673208203673600000000000000)+((393210346492104620676702715181030548001401397929179496808798463940480027 (x^22))/(58326750898945204868209544637683462439149767029413093171184240505420826593163671875000000))+((698478878297563862016989184220107321634877 (x^23))/274871405310743392196030398526941853943358449254400000000000000)+((6360437254301939778553693186884079136276540863117327760689263645229184422859 (x^24))/(996433411859911160327342554710959205314792007230469181537623064347336144653464825590468750000000))+((27342338764966363407986781447108745279 (x^25))/5949597517548558272641350617466273894877888512000000000000000)+((123693076524710488001107959058003648464786565588851837028806184678528501673 (x^26))/(24193277210403536361813978358060908425514107073428096260220201562461726291564589843750000000000000))+((11208193380164872961483018076335015974543667 (x^27))/1597797157554153029397329520386348253446931577172498841600000000000000)+((10480427256847893666075520855562832635773521417706970008308036687151 (x^28))/(2988328517470423544658412213847972840885286191964805088418468587971941111500390625000000000000))+((6553738902348436798248260596732888127 (x^29))/720063987087341691569892091549403253301647486216044544000000000000)+((5125985057639536961158074272244539300486367449221311650701524031523 (x^30))/(2464661934294760824282680044184882032822176695326737782055213942006859921539161250000000000000000))+((18674805907314077434446889769227 (x^31))/1858306992982650053169052184318306764264991015043072000000000000)+((4600818948797329763676812390696585891316539501771897116371661 (x^32))/(4293320100536298207171173559934687991686106596707437392832534645685930575030100000000000000000))+((1166920514783200434235205999 (x^33))/124470588750276926638220135841309924555687519584256000000000000)+((166323584350250341213427371371702315046640059463351 (x^34))/(347803174880451650820897087469904419447549479286454175586765881679958016000000000000000))+((14813804963775095035632282949 (x^35))/2042630339064979759213853531702888449925593718417522688000000000000)+((48398322511255276222248307177702930991 (x^36))/(267060022125353559667321259105487411360080224840413552096468172800000000000000))+((2657921108939705071 (x^37))/592171142938663306034456372435461578791772678940262400000000)+((50478556706600783122514860073 (x^38))/(911296570869850193433335105677195385031293531875212441157632000000000000))+((1568274265798307 (x^39))/751979384470439546328756069754152846947765540880384000000000)+((25876414060339 (x^40))/2128603238849523330007550243666717738653334686924800000000000)+((11167027 (x^41))/17305554019436738569015790357287398641011104153600000000)+((x^42)/702503058876439949271571303122255784968192000000000)+((x^43)/10069210510562305939559188678085666251210752000000000)];Plot[R,{x,0,1},PlotRange->{{0,1},{0,0.000055}},Axes->False,Frame->True,FrameLabel->{x,"Abs error by standard VPM"},FrameStyle->{Directive[Thick,Black,16],Directive[Thick,Black,16]}]

(Subscript[f, 0])=0;IN=20;For[j=1,j≤IN,j++,fj=(∫_0^x)((x-η)*((∂_η)(∂_η)(Subscript[f, j-1])+h((∂_η)(∂_η)(Subscript[f, j-1])+((2/η)*(∂_η)(Subscript[f, j-1]))+(Subscript[f, j-1])-6-(12*η)-(η^2)-(η^3))))\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]];

R=Abs[-2.3283064365386963`*^-10 (x^2)+3.637978807091713`*^-12 (x^3)-2.9103830456733704`*^-11 (x^4)-1.0982148523908108`*^-9 (x^5)+4.795310815097764`*^-10 (x^6)+2.413898414488358`*^-9 (x^7)-2.7982238748336385`*^-10 (x^8)-5.894081889223912`*^-10 (x^9)+3.405263571121253`*^-11 (x^10)+4.1774137100880626`*^-11 (x^11)-1.4818265429809852`*^-12 (x^12)-1.2216384418547565`*^-12 (x^13)+2.951102037623176`*^-14 (x^14)+1.771671965019947`*^-14 (x^15)-3.097675280600553`*^-16 (x^16)-1.425817738489917`*^-16 (x^17)+1.876154165625612`*^-18 (x^18)+6.863182747196759`*^-19 (x^19)-6.975039265479061`*^-21 (x^20)-2.0822689469497865`*^-21 (x^21)+1.6627955450983752`*^-23 (x^22)+4.1342503437861774`*^-24 (x^23)-2.621903059455595`*^-26 (x^24)-5.5174368721204324`*^-27 (x^25)+2.7936171363513035`*^-29 (x^26)+5.041066498536167`*^-30 (x^27)-2.03828568599924`*^-32 (x^28)-3.188179061961498`*^-33 (x^29)+1.0242014411125759`*^-35 (x^30)+1.4012606432471062`*^-36 (x^31)-3.5355769081989285`*^-39 (x^32)-4.263764692460021`*^-40 (x^33)+8.2798534760586`*^-43 (x^34)+8.859858999052445`*^-44 (x^35)-1.280016020216826`*^-46 (x^36)-1.222303622301248`*^-47 (x^37)+1.2392356413777078`*^-50 (x^38)+1.0613260513353075`*^-51 (x^39)-6.748859922790293`*^-55 (x^40)-5.206708464606624`*^-56 (x^41)+1.5651342921332256`*^-59 (x^42)+1.0919541573022505`*^-60 (x^43)];Plot[R,{x,0,3},PlotRange->{{0,3},{0,9.*(10^-7)}},Axes->False,Frame->True,FrameLabel->{x,"Abs error by VPM at h = -0.4 "},FrameStyle->{Directive[Thick,Black,16],Directive[Thick,Black,16]}]

z=Clear[x];m=ReplaceAll[(Subscript[f, 20]),η->x];x=1;Plot[m,{h,-0.65,-0.02},PlotRange->{{-0.65,-0.02},{2.005,1.941}},Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Thick,Bold,Black,16],Directive[Thick,Black,Bold,16]}]

a=(x^2)+(x^3);Plot[a,{x,0.01,20},PlotRange->{{0.01,20},{0.1,8000.1}},Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Thick,Black,Bold,16],Directive[Thick,Black,Bold,16]}]

m=ReplaceAll[(Subscript[f, 20]),η->x];h=-0.4;m

m=;Plot[m,{x,0,20},PlotRange->{{0,20},{0,8000.1}},Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Thick,Bold,Black,16],Directive[Thick,Black,Bold,16]}]

Clear[x];x=3;a=ReplaceAll[(Subscript[f, 20]),η->x];Plot[a,{h,-0.65,-0.02}]

Clear[x];x=2;Clear[h];h=-0.34;b=ReplaceAll[(Subscript[f, 30]),η->x];b

a=(x^2)+(x^3);b=1.0000000002328306` (x^2)+0.999999999996362` (x^3)+2.9103830456733704`*^-11 (x^4)+1.0982148523908108`*^-9 (x^5)-4.795310815097764`*^-10 (x^6)-2.413898414488358`*^-9 (x^7)+2.7982238748336385`*^-10 (x^8)+5.894081889223912`*^-10 (x^9)-3.405263571121253`*^-11 (x^10)-4.1774137100880626`*^-11 (x^11)+1.4818265429809852`*^-12 (x^12)+1.2216384418547565`*^-12 (x^13)-2.951102037623176`*^-14 (x^14)-1.771671965019947`*^-14 (x^15)+3.097675280600553`*^-16 (x^16)+1.425817738489917`*^-16 (x^17)-1.876154165625612`*^-18 (x^18)-6.863182747196759`*^-19 (x^19)+6.975039265479061`*^-21 (x^20)+2.0822689469497865`*^-21 (x^21)-1.6627955450983752`*^-23 (x^22)-4.1342503437861774`*^-24 (x^23)+2.621903059455595`*^-26 (x^24)+5.5174368721204324`*^-27 (x^25)-2.7936171363513035`*^-29 (x^26)-5.041066498536167`*^-30 (x^27)+2.03828568599924`*^-32 (x^28)+3.188179061961498`*^-33 (x^29)-1.0242014411125759`*^-35 (x^30)-1.4012606432471062`*^-36 (x^31)+3.5355769081989285`*^-39 (x^32)+4.263764692460021`*^-40 (x^33)-8.2798534760586`*^-43 (x^34)-8.859858999052445`*^-44 (x^35)+1.280016020216826`*^-46 (x^36)+1.222303622301248`*^-47 (x^37)-1.2392356413777078`*^-50 (x^38)-1.0613260513353075`*^-51 (x^39)+6.748859922790293`*^-55 (x^40)+5.206708464606624`*^-56 (x^41)-1.5651342921332256`*^-59 (x^42)-1.0919541573022505`*^-60 (x^43);R=Abs[a-b];R

(Subscript[f, 0])=0;IN=2;For[j=1,j≤IN,j++,fj=(∫_0^x)((x-η)*((∂_η)(∂_η)(Subscript[f, j-1])+h((∂_η)(∂_η)(Subscript[f, j-1])+((2/η)*(∂_η)(Subscript[f, j-1]))+1+(Subscript[f, j-1])+((((Subscript[f, j-1])^2))/(2!))+((((Subscript[f, j-1])^3))/(3!))+((((Subscript[f, j-1])^4))/(4!)))))\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]];

f5=ReplaceAll[(Subscript[f, 5]),x->η];
a=Expand[f5]
b=a*a;
c=Expand[b]
d=c*a;
e=Expand[d]
f=e*a;
g=Expand[f]
i=1+a+(c/(2!))+(e/(3!))+(g/(4!));
k=(∂_η)(∂_η)a;
l=(2/η)*((∂_η)a);
m=(x-η)*(k+h(k+l+i));
v=Expand[m]
(Subscript[f, 6])=(∫_0^x)v\[DifferentialD]η;

(Subscript[f, 1])

(Subscript[f, 2])

t=h (η^2)+((3 (h^2) (η^2))/2)+(((h^2) (η^4))/24)+(((h^3) (η^6))/240)+(((h^4) (η^8))/2688)+(((h^5) (η^10))/34560);h=-0.39;t

1/120

N[(1/120)]

h=-0.35;t

Clear[x];x=0.1;h=-0.37;(Subscript[f, 6])

NumberForm[-0.0016658368283409132`,10]

NumberForm[-0.0016658368283409132`,8]

Clear[x];x=0.2;h=-0.37;(Subscript[f, 6])

NumberForm[-0.006653448633229459`,8]

Clear[x];x=0.5;h=-0.4;(Subscript[f, 6])

NumberForm[-0.04115260520706706`,16]

0.0411539568−0.0411526052

Clear[x];x=1;Plot[(Subscript[f, 6]),{h,-0.7,0.02},PlotRange->{{-0.7,0.02},{-0.2,0}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Tiny,Plain,Black,16],Directive[Tiny,Plain,Black,16]}]

Clear[x];x=1;h=-0.4;(Subscript[f, 6])

NumberForm[-0.15882808142904348`,16]

NumberForm[-0.15882808142904348`,8]

Clear[x];x=1.5;h=-0.394;(Subscript[f, 6])

NumberForm[-0.33801484072214794`,10]

Clear[x];x=2;h=-0.35;(Subscript[f, 6])

NumberForm[-0.5599574275798109`,10]

Clear[x];x=2.5;h=-0.28;(Subscript[f, 6])

NumberForm[-0.810499714450844`,10]

(Subscript[f, 0])=1;IN=27;For[j=1,j≤IN,j++,fj=1+(∫_0^x)((x-η)*((∂_η)(∂_η)(Subscript[f, j-1])+h((∂_η)(∂_η)(Subscript[f, j-1])+((2/η)*(∂_η)(Subscript[f, j-1]))-(2*(2*(η^2)+3)*(Subscript[f, j-1])))))\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]];m=ReplaceAll[(Subscript[f, 27]),η->x];

m=1+1.0000000298023224` (x^2)+0.5000001192092896` (x^4)+0.166666679084301` (x^6)+0.04166665859520435` (x^8)+0.008333326637512073` (x^10)+0.0013888812100049108` (x^12)+0.0001984073460334912` (x^14)+0.00002479905197105836` (x^16)+2.754842446606176`*^-6 (x^18)+2.753296604041111`*^-7 (x^20)+2.4998225253702877`*^-8 (x^22)+2.0777751966877256`*^-9 (x^24)+1.5904989052683248`*^-10 (x^26)+1.1264264653952812`*^-11 (x^28)+7.405806095707329`*^-13 (x^30)+4.5304906255918324`*^-14 (x^32)+2.5828254725921443`*^-15 (x^34)+1.3733550998329666`*^-16 (x^36)+6.814944873154029`*^-18 (x^38)+3.156204766013333`*^-19 (x^40)+1.364708888189327`*^-20 (x^42)+5.507825068397556`*^-22 (x^44)+2.07599499861641`*^-23 (x^46)+7.304362805529603`*^-25 (x^48)+2.4014885153336945`*^-26 (x^50)+7.372003641854419`*^-28 (x^52)+2.1162846797524605`*^-29 (x^54)+5.673789222986904`*^-31 (x^56)+1.4237948886470211`*^-32 (x^58)+3.337107414680714`*^-34 (x^60)+7.32725645877425`*^-36 (x^62)+1.5023278382960685`*^-37 (x^64)+2.8874956501876813`*^-39 (x^66)+5.178977734167096`*^-41 (x^68)+8.7099859334195`*^-43 (x^70)+1.3653438922575066`*^-44 (x^72)+2.00613452907686`*^-46 (x^74)+2.742628582134059`*^-48 (x^76)+3.5096595272007965`*^-50 (x^78)+4.1692567822925665`*^-52 (x^80)+4.621666109637998`*^-54 (x^82)+4.743467669624632`*^-56 (x^84)+4.515442010636202`*^-58 (x^86)+3.9702419792619605`*^-60 (x^88)+3.200518907923527`*^-62 (x^90)+2.3810743736668238`*^-64 (x^92)+1.588221353307427`*^-66 (x^94)+9.816932860286082`*^-69 (x^96)+5.197888555251174`*^-71 (x^98)+2.5965892273924914`*^-73 (x^100)+1.001630080243913`*^-75 (x^102)+3.8689502283517496`*^-78 (x^104)+8.556892785566624`*^-81 (x^106)+2.3640812781693997`*^-83 (x^108);

Clear[x];x=1;Plot[m,{h,-0.69,-0.01},PlotRange->{{-0.69,-0.01},{2.77,2.27}},PlotRangeClipping->True,Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Thick,Black,Bold,16],Directive[Thick,Black,Bold,16]}]

a=(E^( (x^2)));Series[a,{x,0,10}]

1/120

N[(1/120)]

N[(1/24)]

N[(1/6)]

N[E]

NumberForm[2.718281828459045`,16]

a=(E^( (x^2)));x=1;a

N[E]

NumberForm[2.718281828459045`,16]

λ=(∑_(i=1)^k)(((η^(i-1))*(ξ^(k-i))*(((-1))^(i-1)))/((i-1)!*(k-i)!));k=2;λ

Simplify[-((((-η+ξ))^2)/(η-ξ))]

a=(E^( (x^2)));Plot[a,{x,0,2},PlotRange->{{0,2},{0,30}},Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Thick,Black,Bold,16],Directive[Thick,Black,Bold,16]}]

Clear[h];h=-0.4;Plot[m,{x,0,2},PlotRange->{{0,2},{0,30}},Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Thick,Black,Bold,16],Directive[Thick,Black,Bold,16]}]

Clear[h];h=-0.4;a=(E^( (x^2)));R=Abs[a-m];Plot[R,{x,0,1.1},PlotRange->{{0,1.1},{.03*(10^-8),1.38*(10^-7)}},Axes->False,Frame->True,FrameLabel->{x,"Abs error by VPM with 
h = - 0.4"},FrameStyle->{Directive[Thick,Black,16],Directive[Thick,Black,16]}]

Clear[h];h=-0.399;m;a=(E^( (x^2)));b=a-m

a=(E^( (x^2)));b=;m=a-b;

Clear[x];x=0.01;m

Clear[x];x=0.02;m

Clear[x];x=0.05;m

Clear[x];x=0.1;m

Clear[x];x=0.2;m

Clear[x];x=0.5;m

Clear[x];x=1;b

NumberForm[2.569936263796535`*^-9,16]

Clear[h];h=-0405;a=(E^( (x^2)));b=a-m;

Clear[x];x=0.7;b

Clear[x];x=0.8;b

Clear[x];x=0.9;m

(Subscript[f, 0])=0;IN=20;For[j=1,j≤IN,j++,fj=(∫_0^x)((x-η)*((∂_η)(∂_η)(Subscript[f, j-1])+h((∂_η)(∂_η)(Subscript[f, j-1])+((8/η)*(∂_η)(Subscript[f, j-1]))+(η*(Subscript[f, j-1]))-(44*(η^2))+(30*η)-(η^5)+(η^4))))\[DifferentialD]η;(Subscript[f, j])=ReplaceAll[fj,x->η]];

Clear[x];x=1;m=ReplaceAll[(Subscript[f, 20]),η->x];Plot[m,{h,-0.41,0.06},PlotRange->{{-0.41,0.06},{-0.19,0.7}},Axes->False,Frame->True,FrameLabel->{h,"y(1)"},FrameStyle->{Directive[Thick,Black,Bold,16],Directive[Thick,Black,Bold,16]}]

m=ReplaceAll[(Subscript[f, 20]),η->x];

e=(x^4)-(x^3);m=-0.9999999999417923` (x^3)+1.000000000005457` (x^4)-5.8548721426632255`*^-9 (x^6)+6.191351076267892`*^-8 (x^7)+9.268166678566558`*^-9 (x^9)-2.3787315228673833`*^-8 (x^10)-6.780679004392276`*^-10 (x^12)+1.005264672719608`*^-9 (x^13)+1.1691548194804526`*^-11 (x^15)-1.2245713992290396`*^-11 (x^16)-7.562351801961108`*^-14 (x^18)+6.125606300277903`*^-14 (x^19)+2.295509523030224`*^-16 (x^21)-1.5142524708766304`*^-16 (x^22)-3.7290425220237523`*^-19 (x^24)+2.0717049925110878`*^-19 (x^25)+3.5311561158838966`*^-22 (x^27)-1.691706587685578`*^-22 (x^28)-2.0673487106460846`*^-25 (x^30)+8.692284040764373`*^-26 (x^31)+7.802629818535006`*^-29 (x^33)-2.9186551588425576`*^-29 (x^34)-1.955789064775094`*^-32 (x^36)+6.579633161418378`*^-33 (x^37)+3.323205989683167`*^-36 (x^39)-1.0144496948139708`*^-36 (x^40)-3.876495843140441`*^-40 (x^42)+1.0817336648374708`*^-40 (x^43)+3.120532953548872`*^-44 (x^45)-8.010146065143172`*^-45 (x^46)-1.7285910257716376`*^-48 (x^48)+4.1036716946217645`*^-49 (x^49)+6.504755474034255`*^-53 (x^51)-1.4348645285105138`*^-53 (x^52)-1.6176370149208974`*^-57 (x^54)+3.329217977971766`*^-58 (x^55)+2.5215616339037936`*^-62 (x^57)-4.859466349348207`*^-63 (x^58)-2.2126586865441037`*^-67 (x^60)+4.005871797084719`*^-68 (x^61)+8.272913504143455`*^-73 (x^63)-1.4111288835338992`*^-73 (x^64);Plot[{e,m},{x,0.1,11},PlotRange->{{0.1,11},{10 ,9000}},Axes->False,Frame->True,FrameLabel->{x,"y(x)"},FrameStyle->{Directive[Thick,Black,Bold,16],Directive[Thick,Black,Bold,16]}]

h=-0.2;m

λ=(∑_(i=1)^k)(((η^(i-1))*(ξ^(k-i))*(((-1))^(i-1)))/((i-1)!*(k-i)!));k=3;λ

Simplify[-((((-η+ξ))^3)/(2 (η-ξ)))]