Solving ODEs with Matlab Instructors Manual - L.F. Shampine.pdf
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SolvingODEswithMatlab:
Instructor’sManual
L.F.ShampineandI.Gladwell
MathematicsDepartment
SouthernMethodistUniversity
Dallas,TX75275
S.Thompson
DepartmentofMathematics&Statistics
RadfordUniversity
Radford,VA24142
c
2002,L.F.Shampine,I.Gladwell&S.Thompson
2
Contents
1GettingStarted 5
1.1Introduction.........................................
5
1.2Existence,Uniqueness,andWell-Posedness .......................
5
1.3StandardForm.......................................
7
1.4ControloftheError....................................
10
1.5QualitativeProperties...................................
11
2InitialValueProblems 13
2.1Introduction.........................................
13
2.2NumericalMethodsforIVPs ...............................
13
2.2.1One–StepMethods.................................
13
LocalErrorEstimation..............................
13
Runge–KuttaMethods...............................
13
ExplicitRunge–KuttaFormulas..........................
13
ContinuousExtensions...............................
14
2.2.2MethodswithMemory...............................
15
AdamsMethods..................................
15
BDFmethods....................................
15
ErrorEstimationandChangeofOrder......................
15
ContinuousExtensions...............................
15
2.3SolvingIVPsinMatlab .................................
20
2.3.1EventLocation...................................
21
2.3.2ODEsInvolvingaMassMatrix..........................
22
2.3.3LargeSystemsandtheMethodofLines.....................
22
2.3.4Singularities.....................................
23
3BoundaryValueProblems 25
3.1Introduction.........................................
25
3.2BoundaryValueProblems.................................
25
3.3BoundaryConditions....................................
25
3.3.1BoundaryConditionsatSingularPoints.....................
25
3.3.2BoundaryConditionsatInfinity.........................
26
3.4NumericalMethodsforBVPs...............................
28
3.5SolvingBVPsinMatlab .................................
29
4DelayDi‹erentialEquations 33
4.1Introduction.........................................
33
4.2DelayDi‹erentialEquations................................
33
4.3NumericalMethodsforDDEs...............................
34
4.4SolvingDDEsinMatlab .................................
34
4.5OtherKindsofDDEsandSoftware............................
36
3
4
CONTENTS
Chapter1
GettingStarted
1.1Introduction
1.2Existence,Uniqueness,andWell-Posedness
SolutionforExercise1.1.
Itiseasilyverifiedthatbothsolutionsreturnedby
dsolve
aresolutions
oftheIVP.Thisfactdoesnotconflictwiththebasicexistenceanduniquenessresultbecausethat
resultisforIVPswritteninthestandard(explicit)form
y
0
=
f
(
t;y
)
; y
(
t
0
)=
y
0
Infact,whenwewritethegivenIVPinthisform,weobtaintwoIVPs,
y
0
=
f
1
(
t;y
)=
p
1
y
2
; y
(0)=0
and
y
0
=
f
2
(
t;y
)=
p
1
y
2
; y
(0)=0
Itiseasilyverifiedthatbothfunctions,
f
1
and
f
2
,satisfyaLipschitzconditiononaregioncontaining
theinitialcondition,hencebothIVPshaveauniquesolution.Forexample,
@f
1
@y
=
y
p
1
y
2
0
:
5
0
:
75
for
0
:
5
y
0
:
5andforall
t
.InthiswaywefindthatthegivenIVPhasexactlytwosolutions.
SolutionforExercise1.2.
Bydefinition,
f
(
t;y
)satisfiesaLipschitzconditionwithconstant
L
in
aregionif
jf
(
t;u
)
f
(
t;v
)
jLjuvj
forall(
t;u
)
;
(
t;v
)intheregion.Ifthisfunction
f
(
t;y
)satisfiesaLipschitzconditionon
jtj
1
;jyj
1,then
p
p
p
j
juj
j
0
jj
=
jujLjuj
=
Lju
0
j
Thisimpliesthat
1
p
juj
L
5
p
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