journal membrane 29-03-16-2

8/18/2019 Journal Membrane 29-03-16-2

1/37

Automated Matlab Solver for Nonlinear Singular BVPs in Membrane

N. Ch. N. Suresh1*, Syamal K. Sen2, J. Vasundhara Devi3, and Ravi P. Agaral!

1*, 3 "VP#Pr$%. V. &a'shmi'an(ham )ns(i(u(e %$r Advaned S(udies and De+ar(men( $%

a(hema(is, "VP C$llege $% -ngineering A/, Visa'ha+a(nam#03!

suresh.ne''an(igv+e.a.in, 4vdevi.liasgv+e.a.in

2 "VP#Pr$%. V. &a'shmi'an(ham )ns(i(u(e %$r Advaned S(udies and als$ De+ar(men( $%

C$m+u(er Siene 5-ngineering, "VP C$llege $% -ngineering A/, Visa'ha+a(nam#03!

s'sen.liasgv+e.a.in

! De+ar(men( $% a(hema(is, 6e7as A5 8niversi(y#Kingsville, 9 8niversi(y :lvd., SC192, Kingsville, 6e7as 93;3#22

Ravi.Agaral(amu'.edu

Abstract A %as( $nverging i(era(ive a(la


2/37

A( (he edge, i.e., a( 1= x , e have (he $ndi(i$n

1/ 3/ y λ = >

here λ is a $ns(an( hih re+resen(s radial s(ress a( (he

here v is (he +$iss$nFs ra(i$.

)% e n$ in(r$due (he hange $% varia


3/37

6he $m+u(a(i$nal +r$edures ($ s$lve (he +r$


4/37

!/ vi@. +r$


5/37

)% ei(her θ 1 $r θ 2 are rela(ively aay %r$m (he $riginal value $% y a( 7Ea $riginal ini(ial

$ndi(i$n/, (hen e en$un(er an in(egra(i$n ($lerane +r$


6/37

n)$- /input!+no. of intervals+#-

feval!3pf14,14v4a4b4n#

else if r))$

disp!+Problem $ data+#

,1)input!+, value )+#-

lmda)input!+lmda value )+#- a).$$$- /!+starting value of 0+#-

b).2- /!+final value of 0+#-

n)$- / !+no. of intervals+#-

feval!3p$f14,14lmda4a4b4n#

else

disp!+invalid number assigned to r+#

end-

end-

end-

Problem P1 C$nsider (he Pr$!160!


7/37

title!+Gonvergence of solns E!0# bet%een lo%er and upper solns+#-

legend!+"o%er soln+4+pper soln+4 +True soln+4 +Iterated solns+#-

/SJGHN KI"J

/subprogram to solve the ode iterativelEfunction'C()p1f!lmda4a4b4h4lb4ub#

0span)a8h8b-

s9)si9e!0span#- l0)s9!14 #- p)-

/ l0 denotes the no of values of D0F at %hich %e are calculating the solution

flag)1- !!ub6lb#7!fub6flb###-

E)'ne%E4(-

'0r4E()ode12s!+Edash1+40span4E#-

si9e0r)si9e!0r#-s9r)si9e0r!141#-

if !s9r))l0#OO


8/37


9/37


10/37

.!332 .; .;!;

.!;;0 .; .;10

.!II .; .;91.0331 .; .;9!;

.0;;! .; .;99

.0II9 .; .;;;9.;331 .; .;;2!

.;;;! .; .;09I

.;II9 .; .;032

.933 .; .;!2

.9;;3 .; .;!3

.9II; .; .;39;.32I .; .;31I

.;;2 .; .;2;

.II; .; .;1II.I32I .; .;130

.I;;2 .; .;;I

.III0 .; .;1&$er s$ln a( 7E 8++er s$ln a( 7E

.; .9!2

%r E #2.30!e#

a


11/37

6he %$ll$ing gra+h ig.1/ %$r 'E.3 and lmdaE.; gives (he l$er, (he u++er, (he (rue,

and (he i(era(ed s$lu(i$ns.

Kig. 1 ,).$ and ).W for Problem P1 !integration

tolerance Problem not encounterd#

Example 2

>> membrn_bvp(1)Problem 1 datak value =0.7lmda value =0.6

6he s$lu(i$n


12/37

.1 .9I1; .9;I .991I .992

.33! .9I1! .9; .9919 .991

.;;9 .9II .9;1 .9911 .9911

… … …… … .I32I .;0!I .;2I .;203.;20!

.I;;2 .;!30 .;0 .;13 .;131

.III0 .;310 .0I02 .0III .;

Kig. ,).@ and ).W for Problem P1 !integration

tolerance problem not encounterd#

N$(ie (ha( e have n$( en$un(ered any

in(egra(i$n ($lerane +r$ membrn_bvp(1)Problem 1 datak value =0.8lmda value =0.5

6he s$lu(i$n


13/37

reducing the ste si!e below the smallest "alue allowed #1.776$%7e-01%& at time t.' (n ode1%s at )19 (n t1*2 at 9 (n 1*1 at 17 (n membrn+b" at 9

Warning: Failure at t=9.0$0)1)e-001. Unable to meet integrationtolerances withoutreducing the ste si!e below the smallest "alue allowed #1.776$%7e-01%& at time t.' (n ode1%s at )19 (n t1*2 at 9 (n 1*1 at 17 (n membrn+b" at 9Warning: Failure at t=9.62721%e-001. Unable to meet integrationtolerances withoutreducing the ste si!e below the smallest "alue allowed #1.776$%7e-01%& at time t.' (n ode1%s at )19

(n t1*2 at 9 (n 1*1 at 17 (n membrn+b" at 9

6he a


14/37

Kig. $ ,).= and ).2 for Problem P1!obviated

integration tolerance problem automaticallE#

Kig. ,)1.2 and ).W2 for Problem P1

!overcome integration tolerance problem#:

1!


15/37

Kig. 2 ,)$ and ).$@ for Problem P1 !integration

tolerance problem obviated#

>ere hen 'E3 and HE.39 e $me

ar$ss (he in(egra(i$n ($lerane +r$


16/37

Kig. @ ,)1 and )1.2 for Problem P1

!solutionfound onlE in t%o iterations#

Kig. = ,)1 and ) for Problem P1 !final solution

almost coincides %ith upper solution#

Kig. ,)1 and ).2 for Problem P1 !obviated

integration tolerance problem automaticallE#

)% e (a'e 'E2 and vary H, similar +a((erns are $


17/37

Kig. 11 ,) and )1.@2 for Problem P1 !lb and ub

solutions are close and true solution comes verE

close to ub solution#

Kig. 11 ,) and ). for Problem P1 !obviated

integration tolerance problem to get the solution#

Problem P C$nsider (he +r$


18/37

fprintf!+:n Gol 18 04 Gols 4 $4 ..4 last but one8 successive iterated sols E!0#4 Gollast8 true sol E!0# :n+#

C1)'0+ C(4

/fprintf!+/@.f /@.f /@.f /@.f /@.f /@.f /@.f :n+4 0+4 C#

s)si9e!C#- Cf ) s!84 #- ts ) C!84 Cf#-

plot!0+4 lb4 +r5+4 0+4 ub4 +b0+4 0+4 ts4 +,o+4 0+4 C#- 0label!+0+#- Elabel!+Solution E!0# for HJ ?E 5 ,7E 5 $E70)+#-

title!+Gonvergence of solns E!0# bet%een lo%er and upper solns+#-

legend!+"o%er soln+4+pper soln+4 +True soln+4 +Iterated solns+#-

/SJGHN KI"J

/subprogram pf

function'C()pf!v4 a4b4h4 lb4 ub#

0span)a8h8b-

s9)si9e!0span#- l0)s9!14 #- p)-

flag)1- Elb!l041#5Elb!l04#-

fub)!16v#>Eub!l041#5Eub!l04#-

ne%E)lb6!flb>!!ub6lb#7!fub6flb###-

E)'ne%E4(-

'0r4E()ode12s!+Edash1+40span4E#-

1


19/37


20/37

if !s9r))l0#OOE!l041#5E!l04#-)soln!84#-

C!84E!#70(-

Numerical results 6he %$reg$ing +r$grams +r$vide (he s$lu(i$n $% +r$ere (he value $% ($ in+u( +arame(ers ' and , are as'ed %$r in (he a(la<

+r$gram and (hen ill


21/37

6he s$lu(i$n


22/37

Q1 E

C$lumns 1 (hr$ugh 0 C$lumns ; (hr$ugh

.1 .99; .9!! .902

.9023 .33! .990 .9!39 .9!II

.9021

.;;9 .9999 .9!3 .9!I3.9012

.1 .99;; .9!2 .9!9I

.901

.133! .9902 .9!; .9!;0.9!0

.1;;9 .993! .93I .9!!0

.9!;9 .2 .9913 .93; .9!22

.9!!!

.2333 .9; .93!3 .93I0.9!1

.2;;; .9;;2 .9310 .93;0

.93I

.2III .9;32 .922 .9333.930

.3332 .90I; .92!0 .92I

.931 .3;;0 .9009 .92! .9203

.9299

.3III .901! .910I .92.9231

.!332 .9!;9 .91I .910

.911 .!;;0 .9!19 .900 .912

.9129

.!II .93;2 .;II0 .9!2

.9; .0331 .93! .;I32 .;I9I

.90

.0;;! .92!1 .;;3 .;I11.;I39

.0II9 .919! .;9I .;3

.;;0 .;331 .913 .;91 .;90I

.;99

.;;;! .92; .;;20 .;;9;

.;90 .;II9 .;I!0 .;03! .;0;

.;;1;

.933 .;0 .;!3 .;!I1.;021

.9;;3 .;9;; .;33! .;3I

.;!2

.9II; .;;; .;22! .;21.;312

.32I .;0;! .;19 .;1;0

.;1I .;;2 .;!03 .0I2 .;!2

.;9;

.II; .;33; .0! .0I1.0I!0

.I32I .;211 .090 .09;I

.0;

.I;;2 .;9 .0003 .0;2.0;0

.III0 .0I39 .03I .0!;

.00 .901! .9010 .9010

.9011 .9012 .9013

.900 .90; .90; .9!I2 .9!I3 .9!I3

.9!99 .9!9 .9!9I

.9!09 .9!0 .9!0 .9!30 .9!3; .9!3;

.9! .9!I .9!I

.939 .939I .939I

.93!; .93!9 .93! .931 .9312 .9312

.92;; .92;9 .92;

.9221 .9222 .9223 .9191 .9192 .9193

.911; .9119 .911

.90; .909 .90 .;II3 .;II! .;II0

.;I20 .;I2; .;I29

22


23/37

.;02 .;03 .;0!

.;99! .;99; .;99;

.;;I1 .;;I2 .;;I3 .;;2 .;;3 .;;!

.;09 .;0 .;0I

.;!; .;!9 .;! .;2I9 .;2II .;2II

.;12 .;1! .;1!

.;; .;;1 .;;2

.0I2 .0I3 .0I31 .09I .09I1 .09I1

.0;! .0;!2 .0;!2

.0!1 .0!3 .0!!

Kig. 1 ,) .@2 and X) .1 for Problem P

e $


24/37

Kig. 1$ ,).@2 and X). for Problem P

!integration tolerance problem not encounterd#

Kig. 1 ,).@2 and X).2 for Problem P !no. of

iterations re;uired to find the true solution is

small#

Kig. 12 ,).@2 and X)Y.W2 for Problem P !no. of

iterations to find the true solution is large#

Kig. 1W ,).@2 and X)6.= for Problem P

!integration tolerance problem is obviated#

Kig. 1@ ,)1.$2 and X).2 for problem P

!integration tolerance problem not encounterd#Kig. 1= ,)1.$2 and X).= for Problem P !the no.

of iterations are small to find true solution#

2!


25/37

Kig. 1 ,)1.$2 and X)Y.@ for Problem P !some of

the iterated solutions crosses ub solution curve but

bounce bac, into lb6ub solutions# Kig. ,)1.$2 and X)Y.= for Problem P

!obviated integration tolerance problem#

Kig. 1 ,) and X).2 for problem P

!integration tolerance problem not encounterd#

Kig. ,) and X).=2 for problem P

!the no. of iterations are small to find true solution#

Kig. $ ,) and X)6.2 for problem P!some of

the iterated solutions crosses ub solution curve but

bounce bac, into lb6ub solutions#

Kig. ,) and X)Y. for problem P

!obviated integration tolerance problem#

20


26/37

Problem P$ B!16!17t!


27/37

/step denotes the length of the interval bet%een lo%er and upper bounds of the

/ solution E!0#. step needs to be appropriatelE divided into subintervals

/%henever the tolerance problem crops up. e have ta,en the

/number of subintervals as 2 and this has %or,ed %ell for most of the problems

/considered. hen this number fails to obviate the tolerance problem bE

/indicating an error message %e increase suitablE this no. of subintervals.

'tlb4Elb()ode12s!+Edash$+4 tspan4 'lb4(#-

'tub4Eub()ode12s!+Edash$+4 tspan4 'ub4(#-

/ the follo%ing part of the program determines %hich of the bounds !lo%er7upper#

/ causes the integration tolerance problem and accordinglE modifE the interval.

/ If the problem occurs at the lo%er bound then %e increase the lo%er bound bE

/ an amount step71 else %e decrease the upper bound bE step71.

ntlb)si9e!tlb+#- mtlb)ntlb!14#-

ntub)si9e!tub+#- mtub)ntub!14#-

if !mtlb))lt mtub))lt#OO!mtlb)lt mtub)lt#

flb)Elb!lt41#6lmda- fub)Eub!lt41#6lmda-ne%t)lb6!flb>!ub6lb#7!fub6flb##-

t)'ne%t4(-

'tr4E()ode12s!+Edash$+4tspan4t#-

/'tr4E()ode$tb!+Edash$+4tspan4t#-

soln)'tr4 E(-

si9etr)si9e!tr#-s9r)si9etr!141#-

n)1

if !s9r))lt#OO!nQ1#

flag)-

else

p)p5$-

lb)lb6!step7p#4

ub)ub6!step7p#4

end-

n)n51

else

p)p5@4

if mtlb)lt

lb)lb5!step7p#-

else ub)ub6!step7p#-

end

/ Rere !17p#th part of the interval !lb4 ub# is deleted from consideration.

/ p value is being modified !incremented bE @# each time the integration

/ tolerance problem is encounterd.

/ e can change this depending on the conte0t and sensitiveness of the solution.

/ e have not come across the condition !m0lb)l0 m0ub)l0# in our

/ e0tensive numerical e0periments4 it is possible to encounter this case though.

end

29


28/37

else

disp!+no solution found+#

brea,-

end

end-

!ub6lb#7!fub6flb##-

t)'ne%t4(-

'tr4E()ode12s!+Edash$+4tspan4t#-

soln)'tr4 E(-

fr)E!lt41#6lmda-)soln!84#-

C!84!17E!1#?#5E!#7t(-

Numerical results for Problem P$

2


29/37

Example 5

>> membrn_bvp(3)Problem 3 datak value =0.5lmda value =0.6

6he s$lu(i$n


30/37

1.22 .92; .;I0 .;I0I

1.9219 .;I1 .;I12 .;I13

1.;313 .;I33 .;;3 .;;0 1.0!I .;2 .;11 .;13

1.!9;2 .;29 .;900 .;909

1.!I2 .;99 .;;I9 .;;II 1.3! .;9I .;;30 .;;39

1.2I1I .;;!0 .;0;I .;091

1.2!3 .;099 .;0 .;02

1.1I29 .;00 .;!29 .;!2I

1.1!0 .;!3 .;30 .;302

1.19; .;30 .;2; .;29 1.;I! .;2;9 .;13 .;10

1.33 .;19 .;I3 .;I0

1.0 .;; .0II .;

Kig. 2 ,) .2 and lmda).W !integration

tolerance problem not encounterd#

e $


31/37

Kig. = ,).2 and ). for problem P$

!integration tolerance problem obviated#

Kig. ,).2 and ).12 for problem P$

!integration tolerance problem that is encounterd

at different stages is obviated automaticallE#

)( is im+$r(an( ($ n$(e (ha( i% 'E.0 and HE.10 (hen e en$un(er in(egra(i$n($lerane +r$


32/37

Kig. $ ,)1.2 and ).2 for problem P$ !the true

solution almost coincides %ith upper solution#

Kig. $$ ,)1.2 and ).2 for problem P$ !obviated

integration tolerance problem to find solution#

Kig. $ ,) and ). for problem P$ !obviated

integration tolerance problem to find solution#

Kig. $2 ,) and ). for problem P$ !true solution

almost coincides %ith upper solution#

32


33/37

. GHNG"SIHNS

Need for lower and upper solutions )n (he suessive linear in(er+$la(i$n +r$edure ($s$lve (he n$nlinear singular :VPs e have made use $% ma(hema(ially derived u++er

and l$er s$lu(i$n


34/37

"h# four signifi!ant digits in the independent variable x N$ measuring devie an

+r$due an auray m$re (han .0T. $r any engineering a++lia(i$n, %$ur signi%ian(digi( in (he %inal resul( ($


35/37

*se of an appropriate method with effi!ient !omputer implementation 6here are manyme(h$ds suh as a sh$$(ing me(h$d and (he ! (h $rder Runge#Ku((a "ill#Runge#Ku((a

me(h$d ($ minimi@e r$unding err$rs/ in li(era(ure ($ s$lve BD-s ass$ia(ed i(h

)VPs?:VPs.hile m$re (han $ne me(h$d may


36/37

+e!imal digit versus signifi!ant digit a!!ura!# )n (his +a+er e have (al'ed a


37/37

I. -.V. Krishnamur(hy and S.K. Sen, Pro4rammin4 in *!56!", A%%ilia(ed -as(#

es( Press, Ne Delhi 23/.

1. S.K. Sen and . Chanda 1I92/G A general sheme %$r s$lving $rdinarydi%%eren(ial e=ua(i$ns under ($#+$in(

journal membrane 29-03-16-2

Documents