G.2089Page1GlobalResearchjournalofNaturalScience&Technology(GRJNST)Volume:04-Issue3(2026),2089ISSNP:2790-7643ISSNE:2790-7651www.grjnst.nethttps://doi.org/10.53762/grjnst.04.03.10EfficientExplicitImprovedSchemeforNumericalSolutionofCauchyProblemsReceived:29March2026.Accepted:21April2026.Published:18May2026MuhammadDaudKandhro(CorrespondingAuthor)InstituteofMathematicsandComputerScience,UniversityofSindh,JamshoroEmail:daud.kandhro@gmail.comORCHIDID:https://orcid.org/0009-0001-7160-2047KhushbuRajputDepartmentofBasicEngineeering,SindhAgricultureUniversity,TandojamEmail:rajputkhushi971@gmail.comORCHIDID:https://orcid.org/0000-0003-4850-6038InstituteofMathematicsandComputerScience,ZohaibAliUniversityofSindh,JamshoroEmail:zohaib.ali@usindh.edu.pkORCHIDID:https://orcid.org/0009-0005-8077-2004GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10Copyright©2026GRJNST.ThisarticleispublishedunderanOpenAccessmodel.ItismadeavailabletothepublicunderthetermsoftheCreativeCommonsAttribution4.0International(CCBY4.0)license,whichpermitsunrestricteduseanddistributionG.2089Abstract:Page2Thisstudyworkisfocusontodesignexplicitsplendidconvergingmethodtoestimatenumericalresultofinitial-value-problems(IVP’s)fordifferentialequationinnatureordinary.Thestabilitycriteriaoftheimprovedschemeareinvestigated,andcorrespondingstabilityregionisdepicted.Erroranalysiscarriedoutalsoaffirmsaccuracyhavingthirdorder.Error(LTE)isderivedbyutilizingTaylor’sseriesexpansiontoskiphigherordertermaftermatchingcorrespondingcoefficients.Addingapartialderivativeinsidefunctionevaluationraisesconvergenceandreducesbotherrors(maximumandlast).Basedonresultsforimprovedschemetoillustratetheaccuracyandefficiency.Thecomparisonsoftheimprovedandexistingschemeshavingsameorderoflocalaccuracyarediscussed.Lookingatnumericalresults,theimprovedschemegivesthebetterresultinthecomparisonofexistingfewsameordermethod.Stabilityisprovedtoexaminebehaviorofmethodanditsregionisvisualized.Consistencyisinvestigatedthatisshowinghowerrorshrinktozero.thenumericalresultisvalidatedandillustratedgraphicallyviautilizingMATLAB2023asoftware.Keywords:NumericalMethods,Thirdorder,LocalTruncationError,Convergence1.INTRODUCTIONOrdinarydifferentialequations(ODEs)playacrucialroleinmodellingproblemsarisinginscienceandengineering.Inparticular,Cauchyproblems(orinitialvalueproblems-IVPs)areencounteredinpopulationdynamics,fluiddynamicsandcontroltheory.Theseproblemsmaynotalwayshaveanalyticalsolutions,andthereisaneedforefficientandaccuratenumericaltechniques.Manynumericalmethodshavebeenproposedtofindapproximatesolutionswithdifferentlevelsofaccuracy.Butthereisaneedforfastandaccurateschemes.Inthiswork,ourgoalistoaddressthisissuebydesigninganewnumericalintegrationtechnique.Thegoalistoimprovetheaccuracyofthemethodwithoutcompromisingitssimplicity.Wehavedevelopedanefficientexplicitthird-orderimprovedmethodforsolvingCauchyproblemsforordinarydifferentialequations.Themodificationofthenewschemeisbasedontheadjustmentofclassicalnumericalschemesandintroducingcorrectionterms.Theyaredonetoimprovetheorderofconvergenceofthescheme,GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10G.2089Page3whilekeepingitefficient.Thisschemeisexplicit,easytoimplementandefficient.Itdoesnotrequirethesolutionofnonlinearequations,asisrequiredbyimplicitschemes.Theschemeisbasedonthemethod,whichhasthird-orderaccuracy.Hence,itcanbeusedtosolvemanyproblems.TheproposedschemeisderivedusingaTaylorseriesexpansion,astandardapproachinnumericalanalysistodevelophigher-ordermethods.Theseriesexpansionoftheexactsolutionisusedtocomparethecoefficientswiththenumericalsolution,thusrevealingthelocaltruncationerror.Termsofhigherorderarediscardedtoobtainthedesiredaccuracy.Thepresenceofpartialderivativetermsintheevaluationofthefunctionincreasestheaccuracy.Thisfacilitatesthetracingofsolutioncurveineachstep.Thisleadstobetterconvergenceoftheproposedscheme.Theschemeistheoreticallyproventobethird-orderaccurate.Adetailederroranalysisfortheschemeisconducted.Thelocaltruncationerror(LTE)iscomputedbyTaylorseriesexpansion.Theanalysisconfirmstheorderoftheleadingerrortermasfourandthustheconvergenceoftheschemeasthirdorder.BesidesLTE,theglobalerrorisalsoinvestigatedbynumericalexperiments.Theseresultsindicatethemaximumerroraswellasthefinalerrorissignificantlyreduced.Hence,thenewschemeissuperiortoconventionalapproaches.Thisshowstheadvantageofaddingmorecorrectionprocedures.Anotherimportantaspectofthemethodisthestability.Weinvestigatethestabilityofthemethodtoassessitsperformanceunderdifferentstepsizes.Weuseasuitabletestequationtogetthestabilityfunction.Usingthis,wecomputeanddrawthestabilityregion.Theplotgivesanideaofhowthemethodwillperforminvariouscases.Weobservethatthenewschemehasagoodstabilityregion.Asaresult,theschemecanbeappliedtosolveawidevarietyofproblemswithoutstabilityproblems.Theoreticalanalysisalsoindicatesthatthenewschemeisconsistent.Anumericalschemeissaidtobeconsistentifthelocaltruncationerror(LTE)approacheszeroasthestepsizeshrinkstozero.TheLTEformuladerivedisindicativeofthis.Thisguaranteesthenumericalsolutionwillconvergetotheanalyticsolutionasthestepsizegoestozero.This,togetherwithstability,ensuresconvergenceasperthetheoryofnumericalanalysis.Hence,theschememeetsthenecessaryconditionsforagoodnumericalscheme.Thisenhancestheproposedmethod'sreliability.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10G.2089Page4Toassesstheeffectivenessofthenewapproach,anumberofexamplesarepresented.Thesearechosentodemonstratetheaccuracyandefficiencyofthescheme.Thesearecomparedwiththesolutionsofexistingthird-orderschemes.Resultsarepresentedintablesandfigurestoclearlyshowthedifferences.Theenhancedmethodhasbetteraccuracywithsmallererrormeasures.Thisprovesittobebetterthantraditionalschemesofthesamelevel.Theresultsdemonstratetheefficiencyandaccuracyoftheproposedscheme.AllnumericalsimulationsandgraphicalvalidationsareperformedinMATLAB2023a.Theproposedmethodiseasytoapplyasitisexplicit.Plotsoftheexactandapproximatesolutionsvisuallyverifytheaccuracyofthescheme.Theresultsdemonstratethatthenewschemeisclosetotheexactsolution.Thisconfirmsthetheoreticalresultsobtainedinthepaper.Inconclusion,theproposedmethodisaccurate,stableandefficient.Thismethodcanthusbeseenasanimportantadditiontothenumericalsolutionofordinarydifferentialequations.WeinitiatebyconsideringgeneralformofODE’s2.DERIVATIONItisassumed,eqn.(1)onaspecificintegrationintervalconfessesauniquesolution.respectively.Integareeqn.(1),itExactandnumericalresultissignifiedbyadoptsnewformandIndiscretemannerIfWhereandaretheslopes,determinedbyGRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.101,,00yxyyxfdxdynndxyxfyyhxxnnnn00,1iisinnwhyy1123322111wwwhyynn21,ww3wnnyxfw,1ynnfwhwhyhxfw112212122,G.2089TaylorSeries(TS)expensionforisandSlopesandintoattainingsystem.areexpendedthruusingTaylor’sSeriesthensurrogateinpowerofhupto.FinallycoefficientsofareequatingwithPage5,forAbovesysyemisnonolinearwhichhasequationasand10variablequantities.Nowweinvestigatethissytemforsolution.Soithasmultiplesolutionbecauseoffreevariables.OneoftheaccurateamndreliablysolutionwehaveNewproposedmethodhasderivennowfurtherinvestigationismandatorytocheckaccuracy,convergenceandcomputeerror.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10ynnfwhwwhyhxfw131223213133,nnyxY,3241332412418153324124161212131612121,523234232hOfffffffffffffffffffffffhffffffffhfffhhfxyyxGxxyyxyyyxyyyyyyxyxxyyxyyxxyxxxyxyyyxyxxyxnn2w3w1w2w3w233h612161431216121216113231323232313221232213313212323331332122323313212323222332232323213211122131215431255453151,3232,32,wwwhyyfwhhwhwyhxfwhfhwyhxfwyxfwnnynnynnnnG.2089Page63.LOCALTRUNCATIONERRORTheenhancedextendedmethodhavinglocaltruncationerrorisexpressedaswhereisgainedbyTSandWhereseriesisutilizedtoexpandthesearoundexpandproposedacceleratedexplicitMethodpresentineqn(2)uptoisusedasanapproximatesolution.Taylor.Nowweandsimilartermscollectin,weget4.CONSISTENCYANALYSISDefinition4.1NMtogetherIVPwithanincrementfunctionbeconsistent,ifisacceptedtoFromproposedmethodProceedsonbothsideHenceprovedtheconsistentwithatleastthirdorderaccuracyforimprovedmethod.5.LINEARSTABILITYANALYSISGRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.101nT11nnyhxGThxG1ny62161241815)72165(241964849721)(721)648492161(5422321hOhffffffffffffffffffffffffTxxxyxxyyxyyyyxxyxxyxyxyxyyyyynhyxnn;,321543121;,wwwhyxnn0limhnnynnynnnnhhnnhyxffwhhwhwyhxfhfhwyhxfyxfwwwhyx,5453151,32532,324,3lim121543lim121;,lim12211032100nnnnhyxfhyx,;,lim0G.2089Page7StabilityisshownbytakingDahlquist’stestproblemwhichisinformwehaveattainedpolynomialfunction,wecalleditstabilitypolynomialbyemplying(5)onthistestproblem.Infigure1unshadedregiondosplayslinearstabilityregion.Substitutingaboveallvaluesin(5),thestabilityfunctionisoriginatedwhere6.NUMERICALINVESTIGATIONToinvestigatethebehaviourofmethodseithertheyhavebetterconvergesornot,weexaminedandtestedfewproblemsforprovingbetterresult.Wehavechoosenthreemethods,twomethodaretakenfromopenliterauteandoneisproposed.Beforetakingmethod,onethingiskeepinmindallmethodshavesameorder.SoherewiththehelpofMATLAB2023a,errorsuchabsolutelastandmaximumisevaluatedbyusingthesemethods,graphofeachproblemisdisplayedseparately.Ingraphthreecurvylinesaredrawn.Numericalllandgraphicallyresultgivetheclearevidenceonwhichproposedmethodisbetterinallaspects.RK3MandRK3HMbothlessconvergesthandevelopedone.GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10Cqyyxyqdxdy,0;033223222136321;321;qhqhhqyqwqhhqyqwyqwnnn2312122zzzzzRhqzG.2089Page8Table1.ShowsNumericallyIllustrationofProblem1Problem1.,Step-size/method0.10.050.0250.0125RK3HMRK3MProposed2.2406e-0042.1363e-0040e05.1698e-0054.9489e-0050e01.2497e-0051.1988e-0050e03.0773e-0062.9553e-0060e02.1008e-0051.5944e-0050e02.5084e-0061.8970e-0060e03.0722e-0072.3198e-0070e03.8027e-0082.8697e-0080e05.8246e-0062.2251e-0060e05.7261e-0076.9545e-0080e06.2151e-0083.1638e-0090e07.1587e-0091.1110e-0090e0GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.1010,3yyxydxdyxexExact22422G.2089Page9Table2.ShowsNumericallyIllustrationofProblem1Problem2,Step-size/method0.10.05RK3HMRK3MProposed1.5241e-0031.5241e-0030e03.8551e-0043.8551e-2.2290e-0052.2290e-0050e02.8590e-0062.8590e-2.6463e-0062.0957e-0060e04.1078e-0074.1078e-007GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.1010,2yyxdxdy36931xExactG.2089Page100.0250.01250040e09.7000e-0059.7000e-0050e02.4332e-0052.4332e-0050e00060e03.6139e-0073.6139e-0070e04.5408e-0084.5408e-0080e00e06.1315e-0086.1315e-0080e08.3075e-0098.3075e-0090e0GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10G.2089Page11Table3.ShowsNumericallyIllustrationofProblem3Problem3.Step-size/method0.10.050.0250.0125RK3HMRK3MProposed2.2846e-0031.9600e-0030e06.5681e-0045.5494e-0040e01.8632e-0041.5560e-0040e05.2173e-0054.3185e-0050e01.4261e-0051.3062e-0050e01.8175e-0061.6664e-0060e02.2894e-0072.0996e-0070e02.8709e-0082.6336e-0080e05.1507e-0063.8481e-0070e06.2727e-0073.3522e-0080e07.7070e-0089.3205e-0090e09.5465e-0091.4863e-0090e0GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.101)0(,sinyyxdxdyxExactcos23G.2089Page12Table4.ShowsNumericallyIllustrationofProblem4Step-size/method0.10.05Problem4.,RK3HMRK3MProposed3.1171e-0033.1171e-0030e08.2411e-0048.2411e-0040e06.3568e-0056.3568e-0050e08.4079e-0068.4079e-0060e01.4165e-0051.3359e-0050e01.6414e-0061.2474e-0060e0GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.1010,2yyxdxdyexExact33G.2089Page130.0250.01252.1194e-0042.1194e-0040e05.3744e-0055.3744e-0050e01.0805e-0061.0805e-0060e01.3694e-0071.3694e-0070e01.9677e-0071.2857e-0070e02.4068e-0081.4347e-0080e07.RESULTSANDDISCUSSIONSTheproposedmethodisconstructedtofindmoresuitableandaccuratesolutionofIVP’sforODE’s.Proposedmethodhasaccuracyorderthreesowecompareitwithorderthirdmethods.ATdifferentstepsizesuchas0.1,0.05,0.025and0.0125erroriscomputedincludingabsolutelastandmaximumandCPUtimeisobservedinseconds.Fromthedataandgraphslistedaboveclearlyobservedthattheextendedproposedschemeproducedasmallererrorthenexistingmethod,havingthesameorderofaccuracyandCPUtime.TheresultshaveachievednumericallybyproposedmethodwhichcomesquicklyclosertoexactpointofthesolutionincomparisonofRK3MandGRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10G.2089Page14RK3HM.Conclusively,theamendedproposedschemeismosteffectivetosolveCauchydifferentialequationsandconvergesfastertotherequiredsolution.7.CONCLUSIONThisstudyworkhadachievedtheseobjectyive,firstlyitfocusontodesignexplicitsplendidconvergingmethodtoestimatenumericalresultofIVP’sfordifferentialequationinnatureordinary.Secondly,Thestabilitycriteriaoftheimprovedschemeareinvestigated,andcorrespondingstabilityregionisdepicted.Thirdly,Erroranalysiscarriedoutalsoaffirmsaccuracyhavingthirdorder.Error(LTE)isderivedbyutilizingTaylor’sseriesexpensiontoskiphigherordertermaftermatchingcorrespondingcoefficients.Addingapartialderivativeinsidefunctionevalutionraisesconvergenceandreducesbotherrors(maximumandlast).Basedonresultsforimprovedschemetoillustratetheaccuracyandefficiency.Thecomparisonsoftheimprovedandexistingschemeshavingsameorderoflocalaccuracyarediscussed.Lookingatnumericalresults,theimprovedschemegivesthebetterresultinthecomparisonofexistingfewsameordermethod.Stabilityisprovedtoexaminebehaviorofmethodanditsregionisvisualized.Finally,Consistencyisinvestigatedthatisshowinghowerrorshrinktozero.thenumericalresultisvalidatedandillustreatedgraphicallyviautilizingMATLAB2023asoftwareREFERENCES[1]AbdulMajidWazwak‘’AModifiedThirdOrderRungeKuttaMethod”AlQudsuniversitycollegeofscienceandtechnology(1990).[2]AshiriboSenaponWusu,MosesAdebowaleAkanbiandSolomonAdebolaOkunuga‘’AThree-StageMultiderivativeExplicitRunge-KuttaMethod’’,AmericanJournalofComputationalMathematics,2013,3,121-126[3]Butcher,JohnCharles.Numericalmethodsforordinarydifferentialequations.JohnWiley&Sons,2016.[4]Burden,RichardL.,andJ.DouglasFaires."Numericalanalysis.2001."Brooks/Cole,USA(2001).GRJNST,Volume:04-Issue3(2026)/ISSNP:2790-7643ArticleID:2089https://doi.org/10.53762/grjnst.04.03.10G.2089Page15[5]DennisG.Zill,WarrenS.Wright"DifferentialEquationswithBoundary-ValueProblems."EighthEdition.[6]ErwinKreyszig,HerbertKreyszig,Edward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