lectures in numerical analysis：在数值分析的讲座

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1、alwahip2011LecturesinNumericalAnalysisMethodsBushraH.AliwiDepartmentofMathematicsBABYLONUNIVERSITYbushra_aliwi@yahoo.comCourseforyear20113alwahip2011SolutionofOrdinaryDifferentialEquations:Someequationshasananalyticalsolution,whileotherhasnotsuch;Therearetwotypesofconditionsfortheordinarydifferentia

2、lequationswhichare;·InitialValueProblem:Conditionsarespecifiedatonlyonevalueoftheindependentvariable·Analyticallycansolve;withform;whereand·Bothinitialconditionshavebeenspecifiedat.·Anordinarydifferentialequationoforderrequiredconditionstobespecified.·BoundaryValueProblem:Conditionsarespecifiedattwo

3、valuesofindependentvariable,suchform;where,Orwhere,,,andMethodsofSolution:1.DirectmethodusingTaylarSeries.2.Onestepmethod(selfstartingmethod):-solutioniscarriedfromto.3.Multistepmethod;requiredinformationfor.OneStepMethod(SelfStartingMethod):-I.EulerMethod;Considerthe1storderinitialvalueproblem;,rep

4、lacingbyforwarddifference;wherethen;,orcanwrittenas;wecanuseonlytwotermsfromTaylerSeriesII.ModifiedEulerMethod;Theaccuracyofmethod(I)canbeimplementedifbetterapproximationisusedforthederivative;·Ineachstepwefirstcalculate(byEulerMethod);·Andthenfindnewvalueas;3alwahip2011,orformedas;Thismethodisalsok

5、nownas"Predictor_CorrectionMethod",becauseineachstepwefirstcalculate(byEulerMethod)andthencorrectit.SameexpressioncanobtainedbyusingTaylerSeriesif2ndderivativeisapproximatedbyforwarddifferences;whereso;I.Runge_KuttaMethod;The4thorderformulagivenas;Where;EulerandModifiedEulerareinreality1stand2ndorde

6、rRunge_KuttaMethod.ThemethodrequiresfourevaluationofftogetmoreaccurateresultsandthestepsizeshouldbesufficientlysmallExample:Solvethedifferentialequation;,,by;(1)EulerMethod(2)ModifiedEulerMethod(3)Ruge_KuttaMethod,alsofindonusingSolution:(1)EulerMethod;(2)ModifiedEulerMethod;(3)Ruge_KuttaMethod;3alw

7、ahip2011Sothevalueofthroughthelawcalculatedas;Thenthetableofcalculatingforinclosedintervalthroughthisthreemethodsas;byEulerbyModifiedEulerbyRunge_KuttaExactvaluefor011110.11.21.2211.22211.22210.21.442