boundary value problems_2

7/24/2019 Boundary Value Problems_2

1/27

BOUNDARY VALUE

PROBLEMS

BY

NOOR HIDAYATMATH FMIPA UB


2/27

Introduction

BVP for higher order ordinr! di"erenti# e$ution%re fre$uent#! encountered in &iction%'

The%e re$uire the deter(intion of function of

%ing#e inde&endent )ri*#e %ti%f!ing gi)endi"erenti# e$ution nd %u*+ect to %&eci,ed)#ue% t the *oundrie% of the %o#ution do(in'


3/27

Introduction

Four i(&ortnt -ind% of .#iner/ B01Drich#et 1

Neu(nn 1

Ro*in 1 2

Periodic 1

Second nd fourth order BVP re (o%t co((on inengineering &iction%'

Four i(&ortnt -ind% of .#iner/ B01Drich#et 1

Neu(nn 1

Ro*in 1 2

Periodic 1

Second nd fourth order BVP re (o%t co((on inengineering &iction%'


4/27

IVP re$uire deter(intion of the

function %u*+ect to %&eci,ed )#ue .%/t one of the do(in .t!&ic##! /'

BVP in)o#)ing %econd order ODE% ndfunction )#ue% re %&eci,ed t the

t3o end of the %o#ution do(in.t!&ic##! nd /'

Mn! &ro*#e(% in engineering nd%cience cn *e for(u#ted % BVP%'

E4(&e%1 %ted! %tte conduction hettrn%fere in thin heted 3ire5 e#ectric&otenti# in%ide thin conductor5de6ection of thin e#%tic thred under#od nd (n! other

IVP re$uire deter(intion of the

function %u*+ect to %&eci,ed )#ue .%/t one of the do(in .t!&ic##! /'

BVP in)o#)ing %econd order ODE% ndfunction )#ue% re %&eci,ed t the

t3o end of the %o#ution do(in.t!&ic##! nd /'

Mn! &ro*#e(% in engineering nd%cience cn *e for(u#ted % BVP%'

E4(&e%1 %ted! %tte conduction hettrn%fere in thin heted 3ire5 e#ectric&otenti# in%ide thin conductor5de6ection of thin e#%tic thred under#od nd (n! other


5/27

So#ution

ODE1 7ener# %o#ution1

B01 2 Uni$ue %o#ution%1

B01 No %o#ution e4i%t%'

ODE1 7ener# %o#ution1

B01 2 Uni$ue %o#ution%1

B01 No %o#ution e4i%t%'


6/27

Higher Order E$ution% nd

S!%te(% of E$ution%

IVP1

5

IVP1

5


7/27

I01

Introducing the ne3 )ri*#e%1

I01

Introducing the ne3 )ri*#e%1


8/27

E-ui)#en Si!te( 1E-ui)#en Si!te( 1


9/27

A#gorith(

8e De)e#o& #gorith(% for %o#)ing %!%te( of.#iner or non#iner/ ode of the *oundr!)#ue t!&e'

Such e$ution ri%e in de%cri*ingdi%tri*uted5 %ted! %tte (ode# in 9:D%&ti#

DE re trn%for(ed into %!%te( of .#iner ndnon#iner/ #ge*ric e$ution% through di%cretit;ion &roce%%


10/27

Three Method%

Finite di"erence (ethod% u%ing thedi"erence &&ro4i(tion

Shooting (ethod% *%ed on(ethod% for initi# )#ue &ro*#e(%

Method of 3eighted re%idu#% u%ingnotion% of function# &&ro4i(tion


11/27

Finite Di"erence Method%

Derivativeat

Diference approximation Truncation error

Diference approximation Truncation error


12/27

Finite Di"erence Method% for LinerPro*#e(%

DE

B0

Di%criti;tion

Introduce (e%h in *! di)iding the inter)#into e$u# %u*inter)#% of %i;e '

Thi% &roduce% t3o *oundr! (e%h &oint nd5 nd N interior (e%h &oint

DE

B0

Di%criti;tion

Introduce (e%h in *! di)iding the inter)#into e$u# %u*inter)#% of %i;e '

Thi% &roduce% t3o *oundr! (e%h &oint nd5 nd N interior (e%h &oint


13/27

V#ue% < re -no3n5 *ut the )#ue% (u%t*e deter(ined'

Fro( the T!#or e4&n%ion% for < thefo##o3ing centered di"erence for(u# i%

o*tined

V#ue% < re -no3n5 *ut the )#ue% (u%t*e deter(ined'

Fro( the T!#or e4&n%ion% for < thefo##o3ing centered di"erence for(u# i%

o*tined


14/27

5

If the higher order ter(% re di%crdfro( the *o)e for(u#e nd the&&ro4i(tion re u%ed in origin#DE5 the %econd order ccurte ,nite

di"erence &&ro4i(tion to the BVP*eco(e% %!%te( of %i(u#tneou%#iner #ge*ric e$ution%

5

If the higher order ter(% re di%crdfro( the *o)e for(u#e nd the&&ro4i(tion re u%ed in origin#DE5 the %econd order ccurte ,nite

di"erence &&ro4i(tion to the BVP*eco(e% %!%te( of %i(u#tneou%#iner #ge*ric e$ution%


15/27

In (tri4 nottion the %!%te(*eco(e%

3here i% tridigon# (tri4

In (tri4 nottion the %!%te(*eco(e%

3here i% tridigon# (tri4


16/27


17/27

Liner Finite Di"erence MethodA#gorith(

7i)en function end &oint 5 B0 5nu(*er of %u*inter)# '

Set

So#)e tridigon# %!%te(


18/27

Theore( of Uni$uene%%

If < re continuou% nd onthe &ro*#e( h% uni$ue%o#ution &ro)ided 3here 'Further5 if i% continuou% onthe truncution error i%

If < re continuou% nd onthe &ro*#e( h% uni$ue%o#ution &ro)ided 3here 'Further5 if i% continuou% onthe truncution error i%


19/27

DIRI0HLET PROBLEMS

DE

B0

A%%u(e A%%u(e1 %o#ution% e4i%t

0on%truction%che(e

Method%1 FDE S!%te( of Liner E$ution

DE

B0

A%%u(e A%%u(e1 %o#ution% e4i%t

0on%truction%che(e

Method%1 FDE S!%te( of Liner E$ution


20/27

E4(e

DE

B0

Di%-riti%%i

Bgi inter)# (en+di %u*inter)# dengn u+ung:u+ungn! 5 di(n

DE

B0

Di%-riti%%i

Bgi inter)# (en+di %u*inter)# dengn u+ung:u+ungn! 5 di(n


21/27

E4(e

A&ro-%i(%i

7un-n &ro-%i(%i *edhingg5 %ehingg did&t

Su*%titu%i

A&ro-%i(%i

7un-n &ro-%i(%i *edhingg5 %ehingg did&t

Su*%titu%i


22/27


23/27

2

2


24/27


25/27

Se-rng co* %e#e%i-n untu-


26/27

Tug%

Pi#ih %edi-itn! %tu contoh BVP dri %econdorder ODE 3ith con%tnt coe>cient'

Pi#ih %edi-itn! %tu contoh BVP dri %econdorder ODE 3ith )ri*#e coe>cient'

But#h #gorit( ,nite di"erence (ethod%

But#h &rogr( untu- (etode ter%e*ut'

?e#%-n tentng Finite Di"erence Method% for

Non#iner &ro*#e(%1 #gorith(5 contoh5 &rogr( Bnding-n h%i# &enggunn #terntif

&ro-%i(%i dn %ert (etode &en!e#e%in


27/27

TERIMA KASIHSEMOGA BERMANFAAT

boundary value problems_2

Documents