axioms
A icle
On he Nume ical Solu ion o O dina y, In e al and
Fuzzy Di e en ial Equa ions by Use o F-T ans o m
Da ide Radi 1,2, Lae e So ini 3and Luciano S e anini 3,*
1Depa men o Economics and Managemen , Uni e si y o Pisa, Via C. Ridol i, 10, 56124 Pisa (PI), I aly;
[email p o ec ed] o [email p o ec ed]
2Depa men o Finance, Facul y o Economics, VŠB—Technical Uni e si y o Os a a, Sokolská . 33,
70121 Os a a, Czech Republic
3DESP, Depa men o Economics, Socie y, Poli ics, Uni e si y o U bino Ca lo Bo, Via A. Sa i 42,
61029 U bino, I aly; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 24 No embe 2019; Accep ed: 28 Janua y 2020; Published: 5 Feb ua y 2020


Abs ac :
An in e es ing p ope y o he in e se F- ans o m
ˆ
o a con inuous unc ion
on
a gi en in e al
[a
,
b]
says ha he in eg als o
ˆ
and
on
[a
,
b]
coincide. Fu he mo e, he same
p ope y can be es ablished o he es ic ions o he unc ions o all subin e als
[a
,
pk]
o he uzzy
pa i ion o
[a
,
b]
used o de ine he F- ans o m. Based on his ac , we p opose a new me hod
o he nume ical solu ion o o dina y di e en ial equa ions (ini ial- alue o dina y di e en ial
equa ion (ODE)) ob ained by app oxima ing he de i a i e
·
x( )
ia F- ans o m, hen compu ing
(an app oxima ion o ) he solu ion
x( )
by exac in eg a ion. Fo an ODE, a global second-o de
app oxima ion is ob ained. A simila cons uc ion is hen applied o in e al- alued and (le el-wise)
uzzy di e en ial equa ions in he se ing o gene alized di e en iabili y (gH-de i a i e). P ope ies
o he new me hod a e analyzed and a compu a ional sec ion illus a es he pe o mance o he
ob ained p ocedu es, in compa ison wi h well-known e icien algo i hms.
Keywo ds:
F- ans o m; ini ial- alue ODE; nume ical ODE sol e ; in e al di e en ial equa ions;
gH-De i a i e; uzzy di e en ial equa ions
1. In oduc ion
The uzzy ans o m (F- ans o m) o a con inuous unc ion
:[a
,
b]−→ R
was in oduced by
Pe ilie a in [
1
,
2
]. This special uzzy me hod is pa icula ly appealing and use ul o handle many
eal-wo ld p oblems and an ex ensi e esea ch ac i i y has bo h analyzed i s p ope ies and i s ields
o applica ions; o he li e a u e ela ed o his pape we e e o, e.g.,
[3–11]
and he e e ences he ein.
In ecen esea ch, a en ion has been paid o he nume ically app oxima ed solu ions o o dina y
di e en ial equa ions (ODEs)
·
x( ) = F(
,
x)
o a ious ypes. In pa icula , i is shown ha by using he
in e se F- ans o m, i is possible o ob ain good app oxima ions o he solu ion
x( )
. The me hods ha
use he F- ans o m a e (compu a ionally) supe io wi h espec o o he ones such as he second-o de
Runge–Ku a algo i hm o basic mul i-s ep algo i hms (see [
12
–
16
]). In he inal sec ion o his pape ,
we will p esen some commen s and a p elimina y compa a i e alua ion o he p oposed me hods.
In his pape , we p opose a nume ical me hod whe e he in e se F- ans o m is used o
app oxima e he de i a i e
·
x( )
; he solu ion
x( )
is hen ob ained by exac in eg a ion o he
app oxima ed de i a i e: his is allowed by an in e es ing p ope y which says ha he in eg al
o he in e se F- ans o m o
on
[a
,
b]
coincides (exac ly) wi h he in eg al o he unc ion
i sel ;
his idea was p esen ed in a p elimina y o m in [17].
Axioms 2020,9, 15; doi:10.3390/axioms9010015 www.mdpi.com/jou nal/axioms
Axioms 2020,9, 15 2 o 37
Fo an o dina y di e en ial equa ion, including he case o a sys em, a global second-o de
app oxima ion is ob ained. P ope ies o he new me hod a e analyzed and a compu a ional
sec ion illus a es he pe o mance o he ob ained p ocedu es, in compa ison wi h well-known
e icien algo i hms.
A simila cons uc ion is hen applied o in e al- alued (IDE) and (le el-wise) uzzy di e en ial
equa ions (FDE) in he se ing o gene alized di e en iabili y (gH-de i a i e, see [
18
–
20
]). IDEs and
FDEs a e designed o model unce ain y and i s p opaga ion in a dynamical se ing and i is well
known ha he uzzy case can be exp essed in e ms o a amily o IDEs by adop ing he le el-wise
ep esen a ion o uzzy numbe s and uzzy- alued unc ions. Fo a mode n in oduc ion o FDEs
unde Hukuha a and gene alized de i a i e, we e e o chap e 9 o Bede’s book [
21
]. The in e es ed
eade is also e e ed o he ecen book [22], in pa icula o chap e 4, and he e e ences he ein.
This pape is o ganized in o i e sec ions. In Sec ion 2we ecall some o he basic de ini ions and
p ope ies o he F- ans o m as con ained in [
1
,
2
,
23
]. Then, in Sec ion 3, we desc ibe ou app oach
o he nume ical solu ion o o dina y (and sys ems o ) di e en ial equa ions wi h ini ial condi ions,
usually e e ed as Cauchy p oblem. The F- ans o m is used o app oxima e he de i a i e o he
solu ion o be ounded and he unknown unc ion is de e mined by poin -wise by exac in eg a ion o
he de i a i e; his sec ion also con ains he main p ope ies o he me hod and a p oo o i s (global)
con e gence. Nume ical examples and a compu a ional compa ison wi h o he well-pe o ming
algo i hms is p esen ed in Sec ion 4. Sec ion 5conside s he case o in e al di e en ial equa ions
and ex ends he use o F- ans o m o he nume ical solu ion o IDEs and FDEs unde (le el-wise)
gene alized Hukuha a di e en iabili y; in pa icula , he swi ching phenomenon is analyzed and ule o
manage he swi ching is p oposed and implemen ed in he p oposed p ocedu e. Se e al compu a ional
examples o in e al and uzzy di e en ial equa ion a e p esen ed and discussed in Sec ion 6. Sec ion 7
p esen s some conclusions and p esen some ideas o u he wo k.
2. P elimina ies
We b ie ly ecall he basic de ini ions and p ope ies o he F- ans o m se ing. Fo all he de ails,
we e e o he pape s [1,2,23].
A uzzy pa i ion
(P
,
A)
o he compac in e al
[a
,
b]
is de ined by a ini e decomposi ion
P=
{a=p1<p2<
...
<pn=b}
o
[a
,
b]
wi h
n
poin s and by a amily
A={A1
,
A2
,...,
An}
o
n
con inuous basic unc ions
Ak:[a
,
b]−→ [
0,1
]
wi h he ollowing p ope ies ( he decomposi ion
P
is
no equi ed o be uni o m):
1. Ak( ) =
0 o
/∈]pk−1
,
pk+1[
(
k=
2,...,
n−
1),
A1( ) =
0 o
/∈[p1
,
p2[
,
An( ) =
0 o
/∈
]pn−1,pn];
2. Ak(pk) = 1 o all k=1, 2, ..., nand A1( ) + A2( ) + ... +An( ) = 1 o all ∈[a,b];
3.
o
k=
2,...,
n−
1,
Ak
is inc easing on
[pk−1
,
pk]
and dec easing on
[pk
,
pk+1]
,
A1
is dec easing on
[p1,p2],Anis inc easing o [pn−1,pn].
Le us de ine he ollowing in eg als





I−
1=0 , I+
1=Rp2
p1A1( )d
I−
k=Rpk
pk−1Ak( )d ,I+
k=Rpk+1
pkAk( )d
I−
n=Rpn
pn−1A1( )d ,I+
n=0
(1)
and
Ik=I−
k+I+
k o k=1, ..., n. (2)
Axioms 2020,9, 15 3 o 37
The di ec F- ans o m o
wi h espec o
(P
,
A)
is he
n
- uple o eal numbe s
F(P,A)=
(F1,F2,..., Fn)de ined as
F1=1
I1
p2
R
p1
( )Ak( )d ,Fn=1
In
pn
R
p1
( )Ak( )d , (3)
Fk=1
Ik
pk+1
R
pk−1
( )Ak( )d ,k=2, ..., n−1
and, ob ained om he di ec uzzy ans o m
F(P,A)
, he in e se F- ans o m (iF- ans o m) o
is he
unc ion (P,A):[a,b]−→ Rgi en by
(P,A)( ) = n
∑
k=1FkAk( ) o ∈[a,b]. (4)
The ollowing p ope ies (see [1]) a e he undamen als o he F- ans o m se ing.
P oposi ion 1. ( om [1]) I :[a,b]−→ Ris a con inuous unc ion hen;
1.
o any posi i e eal
ε
, he e exis s a uzzy pa i ion
(Pε
,
Aε)
such ha he co esponding iF- ans o m
(Pε,Aε):[a,b]−→ Rsa is y
 ( )− (Pε,Aε)( )<ε o all ∈[a,b].
2. o all k =1, ..., n,
Fk= (P,A)(pk)(5)
(pk) = Fk+O(h2)as h −→ 0 (6)
whe e h =max{pk+1−pk;k=1, ..., n−1}.
3.
he di ec and in e se F- ans o ms a e linea , i.e., o any
λ∈R
and o any wo con inuous unc ions
,
g:[a
,
b]−→ R
, wi h di ec F- ans o ms
F(P,A)
and
G(P,A)
wi h espec o he same pa i ion
(P,A); hen
3.1. he di ec F- ans o ms o λ and +g a e, espec i ely, λF(P,A)and F(P,A)+G(P,A),
3.2. he in e se F- ans o m o λ and +g a e, espec i ely,λ (P,A)and (P,A)+g(P,A).
In [10], he ollowing p ope y has been es ablished:
P oposi ion 2.
Le
,
g:[a
,
b]−→ R
be con inuous unc ions and le
(P
,
A)
be a uzzy pa i ion o
[a
,
b]
. Then
(i) he iF- ans o m sa is ies
b
Ra
(P,A)( )d =
b
Ra
( )d (7)
(ii) i ang g ha e he same di ec F- ans o m componen s F(P,A)=G(P,A), hen
Zb
a ( )d =Zb
ag( )d .
I is in e es ing o obse e ha a uzzy pa i ion has a “nes ing” p ope y (i s p oo is immedia e):
Axioms 2020,9, 15 4 o 37
P oposi ion 3.
Le
(P
,
A)
be a uzzy pa i ion o in e al
[a
,
b]
wi h
P={a=p1<p2<... <pn=b}
and
conside any subin e al [a,pk]wi h k =2, ..., n; le (Pk,Ak)be he uzzy pa i ion o [a,pk]de ined by
Pk={a,p2,..., pk}(8)
Ak={A1,..., e
Ak}
whe e he las basic unc ion
e
Ak
is gi en by he es ic ion o he basic unc ion
Ak
o he subin e al
[pk−1
,
pk]
.
The uzzy pa i ion
(Pk
,
Ak)
,
k=
1,2,...,
n
is called he
k
- h nes ed pa i ion associa ed wi h
(P
,
A)
and clea ly
(Pn,An) = (P,A).
A consequences o he p ope ies abo e is he ollowing p oposi ion:
P oposi ion 4.
Le
:[a
,
b]−→ R
be a con inuous unc ion and le
F(P,A)= (F1
,
F2
,...,
Fn)T
be i s
F- ans o m wi h espec o
(P
,
A)
. Then, o any
k=
2,...,
n−
1, he F- ans o m o he es ic ion o
o he
subin e al
[a
,
pk]
(wi h espec o he
k
- h nes ed pa i ion
(Pk
,
Ak)
), is gi en by
F(k)= (F1
,...,
Fk−1
,
e
Fk)T
whe e only he las componen e
Fkis changed wi h espec o he componen s in F(P,A)and is gi en by
e
Fk=
pk
Ra
( )Ak( )d
pk
Ra
Ak( )d
.(9)
We ha e, o all k =2, ..., n−1( he cen al summa ion is assumed o be ze o i k =2),
(Pk,Ak)( ) = F1A1( ) +
k−1
∑
j=2
FjAj( ) + e
Fke
Ak( )
and pk
Ra
( )d =F1
pk
Ra
A1( )d +
k−1
∑
j=2
Fj
pj+1
Ra
Aj( )d +e
Fk
pk
Ra
Ak( )d . (10)
Wi h he no a ion in (1), equali y (10) is w i en ( o k>1) as
pk
Ra
( )d =F1I+
1+
k−1
∑
j=2
FjIj+e
FkI−
k(11)
and we ha e:
P oposi ion 5.
Le
:[a
,
b]−→ R
be a con inuous unc ion, le
(P
,
A)
be a uni o m uzzy pa i ion o
[a
,
b]
wi h
h=b−a
n−1
and le
F(P,A)= (F1
,
F2
,...,
Fn)T
be he F- ans o m o
; le also
F(k)= (F1
,...,
Fk−1
,
e
Fk)T
be as
in (9) o any k =2, ..., n−1. Then
I−
ke
Fk=h
2 (pk) + O(h3)as h −→ 0
IkFk=h (pk) + O(h3)as h −→ 0.
P oo . Conside 2 ≤k≤n−1. We ha e I−
ke
Fk=
pk
R
pk−1
( )Ak( )d and, by he apezoidal in eg a ion,
I−
ke
Fk=h
2[ (pk−1)Ak(pk−1) + (pk)Ak(pk)] + O(h3);
Axioms 2020,9, 15 5 o 37
on he o he hand,
Ak(pk) =
1 and
Ak(pk−1) =
0, so
I−
ke
Fk=h
2 (pk) + O(h3)
. Fo he second equali y
(conside ha Ak(pk+1) = 0) we ha e
IkFk=I−
ke
Fk+h
2 (pk) + O(h3)
=h
2 (pk) + h
2 (pk) + O(h3) = h (pk) + O(h3).
3. Nume ical Solu ion o Ini ial-Value ODE by F-T ans o m
Le us conside he ollowing ini ial- alue o dina y di e en ial equa ion (ODE):
(·
x( )= ( ,x( )) , ∈[ 0, 1]
x( 0)=x0(12)
We assume he usual equi emen s on unc ion
(
,
x)
ha ensu e exis ence and unici y o he
solu ion x( ), ∈[ 0, 1].
We a e in e es ed o ind an app oxima ion o he inal alue
x( 1)
o he solu ion
x( )
. Le
(P
,
A)
be a ixed (bu a bi a y) uni o m uzzy pa i ion o
[ 0, 1]
wi h
p1= 0
,
pk=pk−1+h
,
k=
2,...,
n
and
h= 1− 0
n−1
; le
A={A1,.., An}
be he basic unc ions. Le
·
x(P,A)
deno e he iF- ans o m o
·
x( )
wi h
(exac ) di ec F- ans o m componen s (Fk)k=1,...,n.
In he es o he pape , we will make use o he ollowing unc ions and no a ion: o any basic
unc ion Ak∈A,k=1, 2,..., n, we will deno e by Bk he in eg al unc ion de ined by
Bk( )=Z
0
Ak(s)ds.
The componen s o he di ec F- ans o m o
·
x( )
will be deno ed by
F1
,
F2
,...,
Fn
and he in e se
F- ans o m o ·
x(P,A)o is he unc ion
·
x(P,A)( ) = n
∑
k=1FkAk( ) o ∈[a,b]. (13)
The ollowing p oposi ion ollows immedia ely om (10).
P oposi ion 6.
I he exac alues
(Fk)k=1,...,n
o he di ec F- ans o m componen s o he unc ion
−→
( ,x( ))
on
[ 0, 1]
wi h espec o he uzzy pa i ion
(P
,
A)
a e known, hen he inal alue
x( 1)
o he
solu ion o (12) is exac ly gi en by
x( 1)=x0+
n
∑
k=1
FkIk.
Fu he mo e, a he in e media e poin s
pk
,
k=
2,...,
n−
1o he decomposi ion
P
, we ha e ha he
solu ion x (pk)is exac ly gi en by
x(pk)=x0+
k−1
∑
j=1
FjIj+e
FkI−
k(14)
P oo . By de ini ion, we ha e
·
x(P,A)( )=
n
∑
k=1
FkAk( )

Axioms 2020,9, 15 6 o 37
and, om p ope y (ii) in P oposi ion 1, also
Z 1
0
·
x(P,A)( )d =Z 1
0
·
x( )d ;
we hen ob ain
x( 1)=x0+Z 1
0
·
x(P,A)(s)ds =x0+
n
∑
k=1
FkBk( 1).
By P oposi ion 2 applied o he de i a i e unc ion
·
x( )= ( ,x( ))
, wi h
(P
,
A)
on
[ 0, 1]
,
and using (1)–(2), we ha e Bj(pk)=Ijand Bk(pk)=I−
kand he conclusion ollows.
Rema k 1. Conside ha in gene al we ha e x( )6=x0+R
0·
x(P,A)(s)ds, o /∈ {p1,..., pn}.
F om he p ope ies o F- ans o m as in P oposi ion 1 (poin 3.) we inally ha e he ollowing
P oposi ion 7.
I he exac alues
(Fk)k=1,...,n
o he di ec F- ans o m componen s o he unc ion
−→
( ,x( ))
on
[ 0, 1]
wi h espec o a uzzy pa i ion
(P
,
A)
a e known, hen he solu ion
x( )
o (12), o all
k=1, ..., n−1, sa is ies
x( )=x(P,A,x0)(pk) + Z
pk
(s,x(s)) ds, o all ∈[pk,pk+1[,
whe e
x(P,A,x0)( ) = x0+Z
0
·
x(P,A)(s)ds (15)
P oo . Apply he iden i y x( )=x0+Rpk
0 (s,x(s)) ds +R
pk (s,x(s)) ds and (14).
3.1. An F-T ans o m Algo i hm o ODE
In iew o P oposi ions 6and 7, we hen need a way o compu e o o app oxima e he di ec
F- ans o m componen s (Fk)k=1,...,no ·
x( ).
By de ini ion, we ha e ha (he e p0=p1and pn+1=pn), o k=1, 2,..., n,
IkFk=Zpk+1
pk−1
·
x( )Ak( )d =Zpk+1
pk−1
( ,x( ))Ak( )d
I−
ke
Fk=Zpk
pk−1
·
x( )Ak( )d =Zpk
pk−1
( ,x( ))Ak( )d
and, om P oposi ion 5,
IkFk=h (pk,x(pk)) + O(h3)
I−
ke
Fk=h
2 (pk,x(pk)) + O(h3).
As a inal s ep, subs i u e x(pk) = x0+
k−1
∑
j=1
FjIj+e
FkI−
kand ob ain, o h−→ 0,
IkFk=h pk,x0+
k−1
∑
j=1
FjIj+e
FkI−
k!+O(h3)(16)
I−
ke
Fk=h
2 pk,x0+
k−1
∑
j=1
FjIj+e
FkI−
k!+O(h3). (17)
Axioms 2020,9, 15 7 o 37
App oxima ed alues
Gk
and
e
Gk
o
Fk
and
e
Fk
, espec i ely, can be compu ed by sol ing he
ollowing equa ions (we will assume ha he sums below will be ze o i k=1)
IkGk=h pk,x0+
k−1
∑
j=1
GjIj+e
GkI−
k!(18)
I−
ke
Gk=h
2 pk,x0+
k−1
∑
j=1
GjIj+e
GkI−
k!. (19)
To sol e Equa ions (18) and (19) le us w i e hem o di e en alues o
k=
1,...,
n
. The i s
componen
G1
can be de e mined om he ini ial condi ion
·
x( 0)= ( 0
,
x( 0)) = (p1
,
x0)
,
ob ained om I1F1=h (p1,x(p1)) + O(h3)by igno ing he e m O(h3):
G1=h ( 0,x0)
I1. (20)
When compu ing G2and e
G2, he alue o G1is known, and Equa ions (18) and (19) become
I−
2e
G2=h
2 p2,x0+G1I1+e
G2I−
2(21)
I2G2=h p2,x0+G1I1+e
G2I−
2. (22)
F om he i s equa ion we de e mine
e
G2
and, by subs i u ing in o he second, we compu e
G2=2I−
2e
G2/I2.
Fo a gene al
k
, he alues
G1
,...,
Gk−1
a e known and we need o sol e Equa ion (19) only o
e
Gk
I−
ke
Gk=h
2 pk,x0+
k−1
∑
j=1
GjIj+e
GkI−
k!; (23)
hen we se Gk=2I−
ke
Gk/Ik.
Summa izing, we de e mine he alues
e
Gk
and
Gk
i e a i ely om (23) s a ing wi h (20).
Each Equa ion (23) has he o m o a ixed-poin p oblem o e
Gk
e
Gk=h
2I−
k
pk,x0+
k−1
∑
j=1
GjIj+e
GkI−
k!
and can be sol ed by any ze o inde ou ine o , conside ing ha he sys em is in (nonlinea ) iangula
o m, by any i e a i e p ocedu e.
We ha e he ollowing app oxima ion p ope y o he solu ion o (12).
Theo em 1. Le Gk= (G1, ..., Gk−1,e
Gk)be solu ions o (20)–(23) and de ine
xGk(pk)=x0+
k−1
∑
j=1
GjIj+e
GkI−
k o k =1, 2, ..., n.
Then, o he solu ion x (pk)o (12) a he poin s pk, k =1, 2,..., n, we ha e
x(pk)=xGk(pk)+O(kh3)as h −→ 0,
x( 1) = xGn( 1)+ ( 1− 0)O(h2)as h −→ 0.
Axioms 2020,9, 15 8 o 37
P oo . F om (14), (24) and (25) we ha e
|x(pk)−xGk(pk)| ≤
k−1
∑
j=1|Fj−Gj|Ij+|e
Fk−e
Gk|I−
k
=
k
∑
j=1
O(h3) = kO(h3)as h−→ 0.
F om
h= 1− 0
n−1
we ge
n= ( 1− 0)O(1
h)
and
nO(h3)=( 1− 0)O(1
h)O(h3)=( 1− 0)O(h2)
as
h−→ 0 and he conclusion ollows.
The las heo em allows de elopmen o he ollowing algo i hm o he nume ical solu ion o
ODE based on F- ans o m.
Algo i hm ODE-FT
: Find an app oxima ed inal alue
x(P,A)( 1)
o he ODE
·
x( )= ( ,x( ))
,
∈[ 0, 1]wi h ini ial condi ion x( 0)=x0.
S ep 1.
Choose a uni o m uzzy pa i ion
(P
,
A)
o
[ 0, 1]
wi h
n
poin s
p1= 0
,
pk=pk−1+h
,
k=2, ..., nand h= 1− 0
n−1; le I−
k,I+
kand Ikas in (1)-(2).
S ep 2. Fo k=1, 2, ..., n, compu e he solu ions G1,..., Gk−1,e
Gko Equa ion (23) and de ine
x(P,A)(pk)=x0+
k−1
∑
j=1
GjIj+e
GkI−
k o k=1, 2,..., n.
S ep 3.
The inal alue
x(P,A)( 1)
(co esponding o
pn= 1
) is he desi ed app oxima ion o
x( 1)
wi h x( 1)−x(P,A)( 1)= ( 1− 0)O(h2)as h−→ 0.
Theo em 1ensu es ha he algo i hm ODE-FT is (globally) con e gen . Clea ly, he con e gence
o he algo i hm ODE-FT assumes ha he solu ions o Equa ion (23) a e sol ed wi h high p ecision,
independen on he numbe
n
o poin s
pk
o he uzzy pa i ion; in p ac ice, i he exac solu ions
G1
,...,
Gk−1
,
e
Gk
a e app oxima ed and subs i u ed in he algo i hm by quan i ies
G∗
1
,...,
G∗
k−1
,
e
G∗
k
, i is
equi ed ha hey a e such ha
Gj−G∗
j< olj
o small posi i e ole ances
olj<ε
and ixed small
ε>
0; in his case, aking in o accoun ha
Ij≤
2
h
,
I−
j≤h
and consequen ly
k−1
∑
j=1|Gj−G∗
j|Ij+|e
Gk−
e
G∗
k|I−
k≤(2k−1)εh=2k−1
n−1( 1− 0)ε, he p ecision o ODE-FT is such ha
|x(pk)−xG∗
k(pk)| ≤
k−1
∑
j=1|Fj−Gj|Ij+
k−1
∑
j=1|Gj−G∗
j|Ij+|e
Fk−e
Gk|I−
k+|e
Gk−e
G∗
k|I−
k
=
k
∑
j=1
O(h3) +
k−1
∑
j=1|Gj−G∗
j|Ij+|e
Gk−e
G∗
k|I−
kas h−→ 0
≤kO(h3) + 2k−1
n−1( 1− 0)εas h−→ 0
and, o
k=n
,
|x(pn)−xG∗
n(pn)| ≤ ( 1− 0)O(h2+
2
ε)as h−→
0
and n−→ ∞
. In he compu a ions
epo ed in his pape , we ha e sol ed he ixed-poin p oblems ei he exac ly (in he case o linea
di e en ial equa ions) o , in he nonlinea cases, wi h a ole ance
ol ≤
10
−12
in he absolu e di e ence
be ween wo successi e i e a es o he used equa ion sol e .
Axioms 2020,9, 15 9 o 37
3.2. Ex ension o Sys ems o ODEs
The ex ension o he desc ibed p ocedu e o sol e sys ems o o dina y di e en ial equa ions wi h
ini ial condi ions in he o m





.
xi( ) = i( ,x1( ),..., xn( )),i=1, 2,..., d
xi( 0) = xi,0,i=1, 2,..., d
∈[ 0, 1]
We can ind an app oxima ion o he inal alue o each unc ion
xi( 1)
,
i=
1,2,...,
d
, in e ms
o a ixed uzzy pa i ion
(P
,
A)
o
[ 0, 1]
, e.g.,
p1= 0
,
pk=pk−1+h
,
k=
2,...,
n
and
h= 1− 0
n−1
and
basic unc ions
A={A1,.., An}
. Le
·
xi,(P,A)
deno e he iF- ans o m o
·
xi( )
wi h di ec F- ans o m
componen s
(Fi,k)k=1,...,n
, hen, he inal alue
xi( 1)
can be ob ained, in e ms o he componen s
(Fi,k)k=1,...,nand he ini ial condi ion xi,0:
xi,(P,A)( ) = x0+Z
0
·
xi,(P,A)(s)ds =x0+
n
∑
k=1
Fi,kBk( )
whe e Bk( )=R
0Ak(s)ds,k=1, ..., n.
The di ec F- ans o m componen s o he unc ions
−→ i( ,x1( ),..., xd( ))
on
[ 0, 1]
wi h
espec o a uzzy pa i ion
(P
,
A)
a e gi en, in his case, by
d
simul aneous sys ems o equali ies (he e
p0=p1and pn+1=pn), o k=1, 2, ..., n,
IkFi,k=Zpk+1
pk−1
·
xi( )Ak( )d =Zpk+1
pk−1
i( ,x1( ),..., xd( ))Ak( )d
I−
ke
Fi,k=Zpk
pk−1
·
xi( )Ak( )d =Zpk
pk−1
i( ,x1( ),..., xd( ))Ak( )d
and, o all
i=
1,...,
d
and
k=
1,...,
n
, we ob ain, by subs i u ing
xi(pk) = xi,0 +
k−1
∑
j=1
Fi,jIj+e
Fi,kI−
k
,
o h−→ 0,
IkFi,k=h pk,x1,0 +
k−1
∑
j=1
F1,jIj+e
F1,kI−
k,..., xd,0 +
k−1
∑
j=1
Fd,jIj+e
Fd,kI−
k!+O(h3)(24)
I−
ke
Fi,k=h
2 pk,x1,0 +
k−1
∑
j=1
F1,jIj+e
F1,kI−
k,..., xd,0 +
k−1
∑
j=1
Fd,jIj+e
Fd,kI−
k!+O(h3). (25)
App oxima ed alues
Gi,k
and
e
Gi,k
o
Fi,k
and
e
Fi,k
, espec i ely, a e compu ed by sol ing he
sys ems o dequa ions
IkGi,k=h i pk,x1,0 +
k−1
∑
j=1
G1,jIj+e
G1,kI−
k,..., xd,0 +
k−1
∑
j=1
Gd,jIj+e
Gd,kI−
k!(26)
I−
ke
Gi,k=h
2 i pk,x1,0 +
k−1
∑
j=1
G1,jIj+e
G1,kI−
k,..., xd,0 +
k−1
∑
j=1
Gd,jIj+e
Gd,kI−
k!. (27)
The i s componen s Gi,1, om he ini ial condi ion ·
xi( 0)= i(p1,x1,0,..., xd,0), a e
Gi,1 =h i( 0,x1,0, ..., xd,0)
I1(28)
Axioms 2020,9, 15 16 o 37
P oblem No2: (Van de Pol equa ions) Solu ion in e al is ∈[0,300];









.
x1=x2
.
x2=µ(1−x2
1)x2−x1+asin(ω )
x1(0) = 2
x2(0) = 0.
The pa ame e s a e µ=50, a=3, ω=π
5.
This sys em is conside ed o be a ha d ODE and equi es e y small s ep size o de e mine
poin s whe e he solu ion changes suddenly (see Figu e 3). To cap u e his phenomenon, we use M
= 30,001. The solu ion by ode45 has op imal s ep size
hode45 =
2.196
×
10
−5
and inal solu ion ec o
x(1) = 1.700147566 ×100,x(2) = −1.819270275 ×10−2.
Wi h MFT = 30,001, we choose N = 501 o ha e hs ep = 2.0
×
10
−5
and ODE-FT inds he inal
alue
x(
1
) =
1.700150689
×
10
0
,
x(
2
) = −
1.819263248
×
10
−2
. The compa ison gi es A eDi FTRK =
1.398228278980211 ×10−4and MaxDi FTRK = 0.397462807473403.
Figu e 3.
P oblem No2 (Van de Pol): The wo componen s
xj( )
,
j=
1,2 a e displayed in he o de
om op o bo om; FT solu ion is ed-colo ed, RK solu ion is blue-colo ed. Rema k he eigh jumps o
he solu ion om posi i e o nega i e alues o ice e sa.
P oblem No3: (Rössle ’s equa ions) Solu ion in e al is ∈[0, 20];





.
x1=−x1−x3x1(0) = 1
.
x2=x1+αx2x2(0) = 1
.
x3=β+x3(x1−γ)x3(0) = 1
The pa ame e s a e α=0.2, β=0.2, γ=5.0.
The hi d equa ion is nonlinea , bu his p oblem is conside ed o be no nume ically ha d.
The solu ion by ode45 has op imal s ep size
hode45 =
0.00144 and inal solu ion ec o
x(
1
) =
−3.722813228 ×10−1,x(2) = 7.646204266 ×100,x(3) = 3.722813233 ×10−1.

Axioms 2020,9, 15 17 o 37
Wi h MFT = 101, we choose N = 41 o ha e hs ep = 0.005 and ODE-FT inds he inal alue
x(
1
) = −
3.722813228
×
10
−1
,
x(
2
) =
7.646166172
×
10
0
,
x(
3
) =
3.722813233
×
10
−1
. The compa ison
(see Figu e 4) gi es
A eDi FTRK = 1.0 ×10−5∗(0.025127112474346,0.947056407667963,0.015362751349660),
MaxDi FTRK = 1.0 ×10−4∗(0.030880110357123, 0.380942339370804,0.039961137723310).
Figu e 4.
P oblem No3 (Rössle sys em):The h ee componen s
xj( )
,
j=
1,2, 3 a e displayed in he
o de om op o bo om; FT solu ion is ed-colo ed, RK solu ion is blue-colo ed.
P oblem No4: (Lo enz’s sys em) Solu ion in e al is ∈[0,100];





.
x1=a(x2−x1)x1(0) = 10
.
x2=x1(b−x3)−x1x2(0) = 20
.
x3=x1x2−cx3x3(0) = 10
The pa ame e s a e a=10, b=28.0, c=8
3.
I is well known ha his nonlinea sys em is ha d o sol e nume ically as i exhibi s chao ic
ajec o ies. Rou ine ode45 sol es he Lo enz sys em by op imal s ep size
hode45 =
9.37
×
10
−7
and
compu es he inal alue
x(
1
) = −
2.480991301578959
×
10
0
,
x(
2
) =
4.324784533758371
×
10
−1
,
x(
3
) =
2.470283374011438
×
10
1
. Taking M = 30,001 he s ep size hs ep = 1.0
×
10
−6
is ob ained wi h N = 1001;
he ound inal alue using ODE-FT is
x(
1
) = −
2.453228212514948
×
10
0
,
x(
2
) =
3.798179329138090
×
10
−1
,
x(
3
) =
2.457087763696305
×
10
1
wi h A eDi FTRK = (0.004740929429043, 0.006537238822312,
0.008150973471168) and MaxDi FTRK = (0.149405722393993, 0.285676182819206, 0.328764820957794).
See Figu e 5 o he ajec o ies o componen s
xj( )
,
j=
1,2,3 and Figu e 6 o he 3D ep esen a ions
o he solu ions.
Axioms 2020,9, 15 18 o 37
Figu e 5.
P oblem No4 (Lo enz sys em): The h ee componen s
xj( )
,
j=
1,2, 3 a e displayed in he
o de om op o bo om; FT solu ion is ed-colo ed, RK solu ion is blue-colo ed. Rema k ha his
sys em, wi h he gi en alues o pa ame e s, exhibi s a s ong sensi i i y o changes in ini ial condi ions
and he nume ical solu ions a e a amous example o chao ic ajec o y.
Figu e 6.
P oblem No4 (Lo enz sys em): 3D ep esen a ions o he solu ions ob ained by ODE-FT (
le
)
and ode45 ( igh ); hey appea o be coinciden o g aphical p ecision.
Axioms 2020,9, 15 19 o 37
We ha e also sol ed his sys em by a un o ODE-FT wi h MFT = 30,001, N = 101 and by unning
he wo MATLAB ou ines ode113 and ode15i; he esul ing solu ions a e isualized in Figu e 7and we
see ha all he ou ines end o gene a e ajec o ies wi h e y di e en beha io o la ge alues o
ime .
Figu e 7.
P oblem No4 (Lo enz sys em): The h ee componen s
xj( )
,
j=
1,2, 3 a e ob ained also
by ou ines ode15i (cyan colo ) and ode113 (g een colo ); FT solu ion is ed-colo ed, RK solu ion is
blue-colo ed. Rema k ha di e ences be ween he solu ions o his sys em a e essen ially due o e y
small di e ences in he solu ions ound by he di e en me hods.
P oblem No5
: (Pe iodic sys em wi h pe iod
T=
8, see [
24
], Sec ion 6.8): Solu ion in e al is
∈[0,8];















.
x1=x3x1(0) = 1−k
.
x2=x4x2(0) = 0
.
x3=−α2x1
(x2
1+x2
2)3
2x3(0) = 0
.
x4=−α2x2
(x2
1+x2
2)3
2x4(0) = α1+k
1−k
The pa ame e s a e
k=
0.25,
α=π
4q1+k
1−k
; we ema k ha om pe iodici y, he exac solu ion has
xi(8) = xi(0),i=1,...,4.
The ini ial condi ion is x(1) = 7.50 ×10−1,x(2) = 0.0, x(3) = 0.0, x(4) = 1.013944668993403.
Wi h MFT = 801, N = 201 and he s ep size hs ep = 5.0
×
10
−5
, he inal solu ion ound by ODE-FT
is
x(
1
) =
7.500000000000394
×
10
−1
,
x(
2
) = −
1.890855
×
10
−8
,
x(
3
) =
1.923939
×
10
−8
,
x(
4
) =
1.013944668993378, and he one compu ed by ode45 is
x(
1
) =
7.499999999999980
×
10
−1
,
x(
2
) =
−
5.566185
×
10
−14
,
x(
3
) =
5.123376
×
10
−14
,
x(
4
) =
1.013944668993403; compa a i ely (see Figu e 8),
we ge
A eDi FTRK = 1.0 ×10−8∗(0.4273611,0.3695634,0.3303905,0.3234898),
MaxDi FTRK = 1.0 ×10−7∗(0.1278607, 0.18908499,0.19239342,0.13127243).
Axioms 2020,9, 15 20 o 37
Figu e 8.
P oblem No5 (Pe iodic sys em): The ou componen s
xj( )
,
j=
1,2,3,4 a e displayed in he
o de om op o bo om; FT solu ion is ed-colo ed, RK solu ion is blue-colo ed and he wo coincide
a g aphical p ecision.
5. In e al (Fuzzy) Di e en ial Equa ions and F-T ans o m
In his sec ion, we conside in e al and uzzy di e en ial equa ions in he se ing o gene alized
Hukuha a di e en iabili y as desc ibed in [
18
,
19
,
25
]. A p elimina y e sion o his sec ion has been
p esen ed as a con e ence pape in [17].
Compac in e als o eal numbe s will be deno ed by he usual endpoin s no a ion
A= [a−
,
a+]
,
B= [b−
,
b+]
o by he midpoin no a ion
A= (ba
;
ea)
,
B= (bb
;
eb)
whe e
ba=1
2(a−+a+)
is he midpoin
and
a=1
2(a+−a−)
is he adius (hal -leng h). The se o all compac eal in e als will be deno ed
by KC.
The use o midpoin ep esen a ion o in e als has been ecen ly adop ed o s udy se e al opics
in he analysis o in e al- alued unc ions, he single a iable case (see [
26
,
27
] and he e e ences
he ein), o which he in e es ed eade is e e ed o a comple e desc ip ion o in e al- alued
gene alized Hukuha a de i a i e (gH-de i a i e o sho ).
Gi en wo in e als
A
,
B∈ KC
, he gH-di e ence is he in e al
C∈ KC
(i always exis s and is
unique) such ha
AgH B=C⇐⇒ ((i) A =B+C
o (ii) B =A−C. (33)
Using midpoin no a ion, we ha e
AgH B= (ba−bb
;
|ea−eb|)
and (i) is e i ied i
ea≥eb
and
C= (ba−bb
;
ea−eb)
, (ii) is e i ied when
ea≤eb
and
C= (ba−bb
;
eb−ea)
. I
ea=eb
hen clea ly
AgH B=
(ba−bb;0) = {ba−bb}is a single on ( eal numbe ).
Please no e ha i
AigH Bi
a e gH-di e ences o he same ype o all
i=
1,...,
n
, i.e., all sa is y
ei he (i) o (ii) abo e, hen (see [25])
n
∑
i=1AigH Bi=n
∑
i=1AigH n
∑
i=1Bi. (34)
Axioms 2020,9, 15 21 o 37
An in e al- alued unc ion
F:[a
,
b]→ KC
will be deno ed by
F( )=[ −( )
,
+( )]
o ,
in equi alen midpoin no a ion, by F( ) = ( b
( );e
( )), o ∈[a,b].
We will deno e by
RF
he se o uzzy numbe s, i.e. no mal, uzzy con ex, uppe semi-con inuous
and compac ly suppo ed uzzy se s de ined o e he eal line
R
. The
α
-le el se o
u
(o simply
i s
α
-cu ) is de ined by
[u]α={x|x∈R
,
u(x)≥α}
and, o
α=
0, i is he closu e o he suppo
[u]0=cl{x|x∈R,u(x)>0}.
A well-known esul (see [
21
]) allows us o ep esen a uzzy numbe as a pai
u=(u−,u+)
o
unc ions u−,u+:[0,1]−→ R, de ining he endpoin s o he α-cu s as [u]α= [u−
α,u+
α]=(b
uα;e
uα)
We e e o unc ions
u−
(.)
and
u+
(.)
as he lowe and uppe b anches o
u
, espec i ely; he wo
unc ions b
u(.),e
u(.)a e he (le el-wise) midpoin and adius unc ions.
Gi en uzzy numbe s
u
,
∈RF
, he le el-wise gene alized Hukuha a di e ence (LgH-di e ence
o sho ) is he amily o in e als
uLgH =[u]αgH [ ]α|α∈[0, 1].
A uzzy- alued unc ion
F:[a
,
b]→RF
will ha e
α
-cu s deno ed by
[F( )]α= [ −
α( )
,
+
α( )]
o ,
equi alen ly, by
(b
α( )
;
e
α( ))
, o
∈[a
,
b]
. Rema k ha o each
α∈[
0,1
]
, he unc ions
[F]α:[a
,
b]→
KC
de ined by
[F]α( ) = [F( )]α
a e p ope ly in e al- alued unc ions and hei amily (some imes
called hei bunch) {[F]α|α∈[0, 1]}gi es a unique and equi alen ep esen a ion o F.
The ollowing de ini ion o gene alized de i a i e can be applied bo h o in e al- alued unc ion
o o he le el-cu s o a uzzy- alued unc ion.
De ini ion 1 ([18]).Le 0∈]a,b[and h be such ha 0+h∈]a,b[.
- I
F:[a
,
b]→ KC
is an in e al- alued unc ion, i s gH-de i a i e a
0
is de ined o be he limi , i i
exis s,
F0
gH( 0) = lim
h→0F( 0+h)gH F( 0)
h.(35)
- I
F:]a
,
b[→RF
is a uzzy- alued unc ion, i s le el-wise gH-de i a i e (LgH-de i a i e o sho ) a
0
is
de ined o be he amily o he gH-de i a i es o [F]α, i hey exis o all α∈[0,1], i.e.,
F0
LgH( 0) = n([F]α)0gH( 0)|α∈[0,1]owhe e (36)
([F]α)0gH( 0) = lim
h→0
1
h[F( 0+h)]αgH [F( 0)]α.(37)
Fo an in e al- alued unc ion F:[a,b]→ KC,F( )=[ −( ), +( )], when −( )and +( )a e
bo h di e en iable, we can dis inguish wo cases, co esponding o (i) and (ii) o (33) (see [18])
De ini ion 2. Le F :[a,b]−→ KCand 0∈]a,b[wi h α( )and α( )bo h di e en iable a 0. We say ha
- F is (i)-gH-di e en iable a 0i
(i.) F0
gH( 0) = h −0( 0), +0( 0)i(38)
- F is (ii)-gH-di e en iable a 0i
(ii.) F0
gH( 0) = [ +0( 0), −0( 0)].(39)
I
F:]a
,
b[→RF
is a uzzy- alued unc ion, we de ine analogously (i)-LgH and (ii)-LgH di e en iabili y
o F, wi h he addi ional equi emen ha (38) (o (39), espec i ely) a e alid o all [F]α.

Axioms 2020,9, 15 22 o 37
As in [
18
], we say ha a poin
0∈]a
,
b[
is an
l
-c i ical poin o
F
i i is a c i ical poin o he leng h
unc ion
len([F( )]) = +( )− −( )
. A poin
0∈]a
,
b[
is a swi ching poin o he gH-di e en iabili y
o F, i in any neighbo hood Vo 0 he e exis poin s 1< 0< 2such ha
ype-I swi ch poin ): a
1
(38) holds while (39) does no hold and a
2
(39) holds and (38) does no
hold, o
ype-II swi ch poin ): a
1
(39) holds while (38) does no hold and a
2
(38) holds and (39) does
no hold.
Analogous de ini ions can be gi en le el-wise o a uzzy- alued unc ion.
5.1. Nume ical In e al ODE by F-T ans o m
An in e al di e en ial equa ion (IDE) wi h ini ial condi ion can be w i en in he o m
(·
xgH ( )=F( ,x( )) , ∈[ 0, 1]
x( 0)=x0(40)
whe e
F( ,x( )) = [F−( ,x( ))
,
F+( ,x( ))]
and
x( )= [x−( )
,
x+( )]
a e in e als o all
∈[ 0, 1]
and x0= [x−
0,x+
0]is an in e al ini ial alue.
The gH-de i a i e o x( )is deno ed by
·
xgH ( )= [ ·
x−
gH ( ),·
x+
gH ( )].
We will app oxima e
·
xgH ( )
by he iF- ans o ms o he wo unc ions
·
x−
gH ( )
and
·
x+
gH ( )
on he
same uzzy pa i ion (P,A), i.e.,
(·
x−
gH)(P,A)( )=
n
∑
j=1
F−
jAj( )(41)
(·
x+
gH)(P,A)( )=
n
∑
j=1
F+
jAj( )(42)
whe e o
j=
1,2,...,
n
, using he same no a ion as in Sec ion 2wi h
[a
,
b]=[ 0
,
1]
,
F−
j
and
F+
j
a e he
di ec F- ans o ms o (·
x−
gH)and (·
x+
gH).
F om he mono onici y p ope ies o F- ans o m (see [
4
,
10
] ), we know ha
F−
j≤F+
j
(because
·
x−
gH ( )≤·
x+
gH ( )
o all
) and we can de ine he in e als
Fj= [F−
j
,
F+
j]
. Consequen ly, in e ms
o in e al a i hme ic ope a ions, we also ha e he ollowing app oxima ion o he (in e al- alued)
in e se F- ans o m (·
xgH)(P,A)( )o ·
xgH:
(·
xgH)(P,A)( )=
n
∑
j=1
FjAj( ).
On he o he hand, he ollowing unc ion is well de ined:
H( ) =
R 0 n
∑
j=1
FjAj(s)ds!=
n
∑
j=1
FjBj( ), ∈[ 0, 1](43)
and
H( 0) =
0 (he e 0 s ands o he in e al
[
0,0
]
). The in e al- alued unc ion
H( )
will play
a cen al ole in ou me hod o sol e he IDE (40).
Axioms 2020,9, 15 23 o 37
Fo simplici y, de ine he ollowing in e al- alued unc ion
ϕ( ) =
n
∑
j=1
FjAj( ), ∈[ 0, 1]; (44)
Clea ly,
ϕ
is a con inuous in e al- alued unc ion and, om Theo em 43(i) in [
18
], he in eg al
unc ion H( ) =
R 0
ϕ(s)ds is gH-di e en iable wi h
H0
gH( ) = ϕ( ),H( 0) = 0.
The ollowing p ope y is p o ed in [17].
P oposi ion 8.
Conside he unc ion
H( )
in (43) and de ine he wo in e al- alued unc ions
Φ( )
and
Ψ( )
on [ 0, 1]by
Φ( ) = H( )gH (−x0)(45)
Ψ( ) = H( ) + x0.(46)
Then
(1) Ψ( )is gH-di e en iable wi h Ψ0gH( ) = ∑n
j=1FjAj( )and Ψ( 0) = x0.
(2) I all he gH-di e ences
H( )gH (−x0)
a e o he same ype o all
, hen
Φ( )
is gH-di e en iable wi h
Φ0gH( ) = ∑n
j=1FjAj( )and Φ( 0) = x0.
Consequen ly, i unc ion
ϕ:[ 0, 1]−→ KC
is gene a ed by a ixed uzzy pa i ion
(P,A)
as in (44)
and H( ) = ∑n
j=1FjBj( ), hen we ha e he ollowing wo cases:
(1) I he in e als Fj,j=1, ..., na e such ha
ϕ( )=F( ,H( ) + x0) o ∈[ 0, 1]
hen Ψ( )de ined on [ 0, 1]as in (46) is a solu ion o (40).
(2) I he in e als Fj,j=1, ..., na e such ha
ϕ( )=  ,H( )gH (−x0) o ∈[ 0, 1]
hen
Φ( )
de ined on
[ 0, 1]
as in (45) is a solu ion o (40), p o ided ha all he gH-di e ences
H( )gH (−x0)a e o he same ype o all ∈[ 0, 1].
Rema k 2. F om he p ope ies o gH-di e ence, we ha e ha
Ψ( )gH x0=H( )
Φ( )gH x0= (H( )gH (−x0)) gH x0.
I is in e es ing o obse e ha unc ion
Ψ( )
is (i)-gH-di e en iable a all poin s, while unc ion
Φ( )
is (i)-gH-di e en iable i he di e ences
H( )gH (−x0)
a e o ype (i), i.e., i
H( ) = Φ( )−x0
,
and is (ii)-gH-di e en iable i he di e ences
H( )gH (−x0)
a e o ype (ii), i.e., i
x0=Φ( )−H( )
.
Consequen ly, solu ion
Ψ( )= [Ψ−( ),Ψ+( )]=(b
Ψ( );e
Ψ( ))
has always inc easing leng h, bu he same is no ue o solu ion
Φ( )= [Φ−( ),Φ+( )]=(b
Φ( );e
Φ( ))
Axioms 2020,9, 15 24 o 37
in he case whe e Φ( )is (ii)-gH-di e en iable.
Finally, le
G−
j
and
G+
j
be he
O(h3)
app oxima ions, espec i ely, o
F−
j
and
F+
j
simila o (18)–(19);
we can see ha he in e al- alued iF- ans o m app oxima ion
(·
xgH)(P,A)( )=∑n
j=1GjAj( )
o
·
xgH
is use ul in de e mining condi ions o a swi ching poin .
Indeed, i
pk−1<pk<pk+1
a e h ee adjacen poin s o
P
(2
≤k≤n−
1) and we suppose
ha no swi ching poin exis s in e nally o he wo subin e als
[pk−1
,
pk]
and
[pk
,
pk+1]
(i.e., possibly,
he swi ching is exac ly a pk) hen he solu ion x( )mus sa is y
x(pk)gH x(pk−1) = HL
kand x(pk+1)gH x(pk) = HR
k(47)
whe e HL
k=
pk
R
pk−1
·
xgH ( )d and HR
k=
pk+1
R
pk
·
xgH ( )d . (48)
On he o he hand, we ha e HL
k=Fk−1I+
k−1+FkI−
kand HR
k=FkI+
k+Fk+1I−
k+1.
Suppose now ha pkis a swi ching poin o he gH-di e en iabili y o x( ). We ha e wo cases:
(I): (i)- o-(ii) swi ch:
x( )
is (i)-gH-di e en iable on
]pk−1
,
pk[
and is (ii)-gH-di e en iable on
]pk
,
pk+1[
, i.e.,
x(pk) = x(pk−1) + HL
k
and
x(pk) = x(pk+1)−HR
k
so ha
x(pk+1) = (x(pk−1) +
HL
k)gH (−HR
k), whe e he di e ence is o ype (i);
(II): (ii)- o-(i) swi ch:
x( )
is (ii)-gH-di e en iable on
]pk−1
,
pk[
and is (i)-gH-di e en iable
on
]pk
,
pk+1[
, i.e.,
x(pk−1) = x(pk)−HL
k
and
x(pk+1) = x(pk) + HR
k
so ha
x(pk+1) =
x(pk−1)gH (−HL
k)+HR
k, whe e he di e ence is o ype (i).
Ins ead, i
pk
is no a swi ching poin and
x(pk+1) = x(pk−1) + HL
k+HR
k
we ha e a ype (i)
solu ion on
]pk−1
,
pk+1[
; o , i
x(pk−1) = x(pk+1)−HL
k+HR
k
we ha e a ype (ii) solu ion on
]pk−1
,
pk+1[
.
In e ms o midpoin no a ion o in e als, we can summa ize he discussion abo e as ollows:
Types o swi ching poin s:
Le
(P,A)
be a uzzy pa i ion o
[ 0, 1]
and
pk−1<pk<pk+1
(2
≤k≤
n−
1); le
x( ) = (b
x( )
;
e
x( ))
be a solu ion o (40) and
HL
k= ( b
HL
k
,
e
HL
k)
and
HR
k= ( b
HR
k
,
e
HR
k)
be gi en as
in (48). Then, he midpoin alues sa is y
b
x(pk) = b
x(pk−1) + b
HL
k
b
x(pk+1) = b
x(pk−1) + b
HL
k+b
HR
k.
Assuming ha only pkis e en ually a swi ching poin , we ha e he ollowing ou cases
(a) i pkis a (i)- o-(ii) swi ch, hen
e
x(pk) = e
x(pk−1) + e
HL
k
e
x(pk+1) = e
x(pk−1) + e
HL
k−e
HR
k≥0
(b) i pkis a (ii)- o-(i) swi ch, hen
e
x(pk) = e
x(pk−1)−e
HL
k≥0
e
x(pk+1) = e
x(pk−1)−e
HL
k+e
HR
k≥0
(c) i x( )is (i)-gH- di e en iable on [pk−1,pk+1], hen
e
x(pk) = e
x(pk−1) + e
HL
k
e
x(pk+1) = e
x(pk−1) + e
HL
k+e
HR
k
Axioms 2020,9, 15 25 o 37
(d) i x( )is (ii)-gH- di e en iable on [pk−1,pk+1], hen
e
x(pk) = e
x(pk−1)−e
HL
k≥0
e
x(pk+1) = e
x(pk−1)−(e
HL
k+e
HR
k)≥0.
The e a e no gene al ules o loca e a swi ching poin . Deno e by
x( ) = [x−( )
,
x+( )] =
(b
x( )
;
e
x( ))
a solu ion o he IDE (40); i
x( )
is (i)-gH-di e en iable, i s leng h
e
x( )
will no dec ease,
while
e
x( )
will no inc ease i
x( )
is (ii)-gH-di e en iable. Some au ho s ha e no iced ha possibly,
he sequence o swi ching poin s can be p e-de ined a p io i, by posi ioning hem in he ime domain
o he in e al di e en ial equa ion; his is ue, a leas in p inciple, p o ided ha he ound solu ion
is gua an eed o ha e exac ly hem and no o he swi ching poin s, bu such pu ely exogenous p oposal
is no ully con incing.
An endogenous way may be p e e ed, simila o con ol s a egies, o connec he ype o
gH-di e en iabili y o he e olu ion o he ajec o y. Fo example, i seems easonable o loca e
he swi ching poin s depending on how he solu ion is e ol ing, by ixing a lowe
l( )
and an uppe
h eshold u( ), say 0 ≤L≤l( )≤u( )≤Uand equi ing ha l( )≤e
x( )≤u( ) o all ; hen,
(a) a (i)- o-(ii) swi ch is decided a =pki he ollowing condi ion is eached
(e
x(pk) = e
x(pk−1) + e
HL
k≤u(pk)
e
x(pk+1) = e
x(pk−1) + e
HL
k+e
HR
k>u(pk+1)
(b) a (ii)- o-(i) swi ch is decided a =pki he ollowing condi ion is eached
(e
x(pk) = e
x(pk−1)−e
HL
k≥l(pk)
e
x(pk+1) = e
x(pk−1)−e
HL
k−e
HR
k<l(pk+1).
In some sense, he ule abo e will con ol he inc easing and dec easing o “unce ain y” in
an endogenous way, wi hou any e e ence o he in e al ini ial condi ion
x0
o o he in e al- alued
unc ion F( ,x).
Fu he mo e, we a e essen ially ee o decide he ype o di e en iabili y a he ini ial poin ; i he
ini ial leng h
e
x0
is such ha
L<e
x0<U
, we can s a ei he wi h a ype c) o a ype d) and a unique
solu ion is hen ound by applying he decided swi ching ule.
A di e en pu ely endogenous ule can be ob ained by ollowing he inc ease o dec ease o
e
x( )
and ying o in e cep poin s
∗
whe e he unc ion
e
x( )
has a local maximum ( o a (i)- o-(ii) swi ch)
o a local minimum ( o a (ii)- o-(i) swi ch). A necessa y condi ion can be ob ained acco ding o he
ollowing simple esul :
Endogenous c i e ia o a swi ching poin :
Assume ha
x( ) = [x−( )
,
x+( )] = (b
x( )
;
e
x( ))
is
such ha
x−( )
and
x+( )
a e di e en iable so ha i s gH-de i a i e
·
xgH( )
can be exp essed in e ms o
he de i a i es
(x−)0( )
and
(x+)0( )
. Le
(P
,
A)
be a uzzy pa i ion o
[ 0
,
1]
and le
(·
xgH)(P,A)( )=
∑n
k=1FkAk( )
be he iF- ans o m o
·
xgH( )
wi h in e al- alued componen s
Fk= (b
Fk
;
e
Fk)
. I
pk
is a local
minimum o maximum o e
x( ), hen e
Fk=0+O(h2).
P oo .
F om he p ope ies o F- ans o m, we ha e
F−
k= ( ·
xgH)−(pk) + O(h2)
,
F+
k= ( ·
xgH)+(pk) +
O(h2)
and, in pa icula ,
e
Fk=g
·
xgH(pk) + O(h2)
. On he o he hand, om he di e en iabili y o
x−( )
and
x+( )
, we ha e 0
=d
d e
x( ))
o
=pk
, i.e.,
(x+)0(pk)−(x−)0(pk) =
0. I ollows ha
g
·
xgH(pk) = (x+)0(pk)−(x−)0(pk)
2=0, i.e., e
Fk=g
·
xgH(pk) + O(h2) = 0+O(h2).
We hen sugges he ollowing (pu ely endogenous) swi ching ule.
Axioms 2020,9, 15 32 o 37
Table 3.
(In e al- alued p oblem IDE2-a): Fo i e alues o
α∈{0,0.2,0.4,0.6,0.8}
, he h ee
swi ching poin s
w
co esponding o Me h=1 a e gi en in he i s column; he second column con ains
he in e al- alued solu ion
X( w) = [x−
α( w)
,
x+
α( w)]
. I
α=
1, he solu ion is single- alued and no
swi ching poin exis s.
α= 0.00 Ini ial alue = [−1.0, 1.0]
w = 1.7954×10−1X( w) = [−1.0090673275×10−1, 1.0090673275×10−1]
w = 2.5918×100X( w) = [−5.0989773431×10−1, 5.0989773431×10−1]
w = 4.3297×100X( w) = [−8.9894395242×10−1, 8.9894395242×10−1]
Final alue: [−8.35571783×10−1, 8.35571783×10−1]
α= 0.20 Ini ial alue: [−8.00×10−1, 8.00×10−1]
w = 2.7944×10−1X( w) = [−8.1796697089×10−1, 8.1796697089×10−1]
w = 2.5810×100X( w) = [−4.6779933026×10−1, 4.6779933026×10−1]
w = 4.3542×100X( w) = [−7.9775755945×10−1, 7.9775755945×10−1]
Final alue: [−7.52396593×10−1, 7.52396593×10−1]
α= 0.40 Ini ial alue: [−6.00×10−1, 6.00×10−1]
w = 3.7935×10−1X( w) = [−6.2502008124×10−1, 6.2502008124×10−1]
w = 2.5579×100X( w) = [−4.0293011585×10−1, 4.0293011585×10−1]
w = 4.3844×100X( w) = [−6.7059314969×10−1, 6.7059314969×10−1]
Final alue: [−6.41450486×10−1, 6.41450486×10−1]
α= 0.60 Ini ial alue: [−4.00×10−1, 4.00×10−1]
w = 4.8538×10−1X( w) = [−4.2690473012×10−1, 4.2690473012×10−1]
w = 2.5174×100X( w) = [−3.0871431900×10−1, 3.0871431900×10−1]
w = 4.4226×100X( w) = [−5.0673707018×10−1, 5.0673707018×10−1]
Final alue: [−4.91260393×10−1, 4.91260393×10−1]
α= 0.80 Ini ial alue: [−2.00×10−1, 2.00×10−1]
w = 6.0271×10−1X( w) = [−2.2003436269×10−1, 2.2003436269×10−1]
w = 2.4500×100X( w) = [−1.7731361249×10−1, 1.7731361249×10−1]
w = 4.4716×100X( w) = [−2.9073091262×10−1, 2.9073091262×10−1]
Final alue: [−2.85325600×10−1, 2.85325600×10−1]
α= 1.00 Ini ial alue: [ 0.00, 0.00]
Final alue: [0.00000000×100, 0.00000000×100]
Table 4.
(In e al- alued p oblem IDE2-b): Fo i e alues o
α∈{0,0.2,0.4,0.6,0.8}
, he swi ching
poin s
w
co esponding o Me h=2 a e gi en in he i s column; he second column con ains he
in e al- alued solu ion
X( w) = [x−
α( w)
,
x+
α( w)]
. The numbe o swi ching poin s changes wi h
α
and i α=1, he solu ion is single- alued and no swi ching poin exis s.
α= 0.00 Ini ial alue = [−1.0, 1.0]
w = 1.9321×10−1X( w) = [−9.8998479170×10−1, 9.8998479170×10−1]
Final alue: [−3.50656260×100, 3.50656260×100]
α= 0.20 Ini ial alue: [−8.00×10−1, 8.00×10−1]
w = 3.0866×10−1X( w) = [−7.7947575655×10−1, 7.7947575655×10−1]
Final alue: [−1.87166877×100, 1.87166877×100]
α= 0.40 Ini ial alue: [−6.00×10−1, 6.00×10−1]
w = 4.2600×10−1X( w) = [−5.7091483288×10−1, 5.7091483288×10−1]
w = 3.0348×100X( w) = [−9.5345951680×10−1, 9.5345951680×10−1]
w = 4.2887×100X( w) = [−8.4078653821×10−1, 8.4078653821×10−1]
Final alue: [−8.90550965×10−1, 8.90550965×10−1]
α= 0.60 Ini ial alue: [−4.00×10−1, 4.00×10−1]
w = 5.4899×10−1X( w) = [−3.6865624377×10−1, 3.6865624377×10−1]
w = 2.7506×100X( w) = [−5.2242232939×10−1, 5.2242232939×10−1]
w = 4.5230×100X( w) = [−3.3741623443×10−1, 3.3741623443×10−1]
Final alue: [−3.44451472×10−1, 3.44451472×10−1]
α= 0.80 Ini ial alue: [−2.00×10−1, 2.00×10−1]
w = 6.8094×10−1X( w) = [−1.7696972113×10−1, 1.7696972113×10−1]
w = 2.5277×100X( w) = [−2.1957335579×10−1, 2.1957335579×10−1]
w = 4.6445×100X( w) = [−8.1718701678e−02, 8.1718701678e−02]
Final alue: [−8.21485320e−02, 8.21485320e−02]
α= 1.00 Ini ial alue: [0.00, 0.00]
Final alue: [0.00000000×100, 0.00000000×100]

Axioms 2020,9, 15 33 o 37
P oblem FDE1: Solu ion in e al is ∈[0, 1
2];
(.
xgH( ) = −1
2x( ) + 2sin(3 )
x(0) = x0
whe e [x0]α= [−1+α,1 −α],α∈[0,1].
The s ep size in his case is h= 4.000 ×10−5.
The solu ion ound wi h bo h Me h = 1 and Me h = 2 a e uzzy- alued wi h leng hs o he
α
-cu s all inc easing (Me h = 1, Figu e 13) o all dec easing (Me h = 2, Figu e 14) and he e a e no
swi ching poin s.
The
α
-cu s o he ini ial condi ion
x0
and he inal uzzy solu ion o Me h = 1 and Me h = 2 a e
inse ed in Table 5.
Table 5.
(Fuzzy- alued p oblem FDE1): Fo ele en alues o
α∈ni−1
10 |i=1,...,10o
, he able con ains
he in e al le el-wise ini ial condi ion (column 2), he inal in e al alue wi h Me h = 1 (column 3)
and he inal in e al alue wi h Me h = 2.
αIni ial Condi ion Final solu ion (Me h = 1) Final solu ion (Me h = 2)
0.0 [−1.0000×100, 1.0000×100] [−7.9066×100, 6.8715×100] [−6.5292×10−1,−3.8225×10−1]
0.1 [−9.0000×10−1, 9.0000×10−1] [−7.1677×100, 6.1326×100] [−6.3939×10−1,−3.9579×10−1]
0.2 [−8.0000×10−1, 8.0000×10−1] [−6.4288×100, 5.3937×100] [−6.2586×10−1,−4.0932×10−1]
0.3 [−7.0000×10−1, 7.0000×10−1] [−5.6899×100, 4.6548×100] [−6.1232×10−1,−4.2285×10−1]
0.4 [−6.0000×10−1, 6.0000×10−1] [−4.9510×100, 3.9158×100] [−5.9879×10−1,−4.3639×10−1]
0.5 [−5.0000×10−1, 5.0000×10−1] [−4.2121×100, 3.1769×100] [−5.8526×10−1,−4.4992×10−1]
0.6 [−4.0000×10−1, 4.0000×10−1] [−3.4732×100, 2.4380×100] [−5.7172×10−1,−4.6345×10−1]
0.7 [−3.0000×10−1, 3.0000×10−1] [−2.7343×100, 1.6991×100] [−5.5819×10−1,−4.7699×10−1]
0.8 [−2.0000×10−1, 2.0000×10−1] [−1.9954×100, 9.6022×10−1] [−5.4465×10−1,−4.9052×10−1]
0.9 [−1.0000×10−1, 1.0000×10−1] [−1.2565×100, 2.2132×10−1] [−5.3112×10−1,−5.0405×10−1]
1.0 [0.0000×100, 0.0000×100] [−5.1759×10−1,−5.1759×10−1] [−5.1759×10−1,−5.1759×10−1]
Figu e 13.
P oblem FDE1, Me h = 1: Fuzzy- alued gH-di e en iable solu ion (
le
) and i s gH-
de i a i e ( igh ). The e a e no swi ching poin s.
Figu e 14.
P oblem FDE1, Me h = 2: Fuzzy- alued gH-di e en iable solu ion (
le
) and i s
gH-de i a i e ( igh ). Fo all α-cu s, he e a e no swi ching poin s.
Axioms 2020,9, 15 34 o 37
P oblem FDE2: Solu ion in e al is ∈[0, 4π];
(.
xgH( ) = sin( )x( )
x(0) = x0
wi h [x0]α= [−1+α,1 −α],α∈[0,1]. In his case, he s ep size is h= 1.257 ×10−4.
The uzzy solu ion is pe iodic wi h pe iod
T=
2
π
. Fo bo h Me h = 1 and Me h = 2, he e a e
h ee in e nal swi ching poin s a
w ∈Sw ={π, 2π,3π}
, whe e he leng h o
.
xgH
is ze o and he
leng h o he solu ion
x( )
is maximal (a
w =π
,3
π
) o minimal (a
w =
2
π
.) A poin s
=
2
π
and
=4π he solu ion coincides wi h he ini ial condi ion (see Figu es 15 and 16).
Figu e 15.
P oblem FDE2, Me h = 1: gH-di e en iable solu ion (
le
) and i s gH-de i a i e (
igh
).
The e a e h ee swi ching poin s, in he same posi ion o all α-cu s.
Figu e 16.
P oblem FDE2, Me h = 2: gH-di e en iable solu ion (
le
) and i s gH-de i a i e (
igh
).
The e a e h ee swi ching poin s, in he same posi ion o all α-cu s.
7. Concluding Commen s and Fu he Wo k
In his pape , we see ha he F- ans o m app oxima ion se ing allows good nume ical p ocedu es
o sol e o dina y di e en ial equa ions (ODEs) and o app oach he nume ical solu ion o in e al and
uzzy di e en ial equa ions. The compu a ional compa ison o he p oposed ODE-FT me hod wi h
o he well-known and well beha ing nume ical ou ines a ailable in MATLAB, such as ode45, ode15i
o ode113, posi ions F- ans o m among he mos p omising ma hema ical ools o he app oxima ion
o unc ions.
One o ou conclusions is hen ha de eloping nume ical p ocedu es based on F- ans o m
is a p omising a ea o esea ch, an icipa ed by some successes in ecen esea ch such as [
13
–
16
];
a comple e compa ison o (and be ween) he a ious F- ans o m-based p oposed me hods and ou
app oach was no a scope o ou s udy, whe e we ha e chosen s anda d well-pe o ming ou ines
as benchma ks and we ha e e alua ed algo i hm ODE-FT wi h he same and su icien ly small s ep
size
h
on ypical (including ha d) ODEs. Two o he examples in Sec ion 4a e also p esen ed in [
14
],
Axioms 2020,9, 15 35 o 37
whe e he quan i y MSE (mean squa ed e o o app oxima e solu ion and he exac one) is compu ed.
Ex4 is Example 1 in [
14
] and Ex5 is Example 3 in [
14
]. The bes MSE quan i ies ob ained by [
14
] and by
ODE-FT a e epo ed in Table 6:
Table 6.
Fo he wo ODE p oblems in example Ex4 and Ex5, using di e en s ep-sizes, he able
con ains he compu ed MSE o he wo solu ion- a iables
x1( )
and
x2( )
, ob ained by ODE-FT and
Scheme II in [14].
Example/(Algo i hm, S ep Size) MSE(x1)MSE(x2)
Ex4/(Scheme II in [14], h = 0.01) 2.241 ×10−23.775 ×10−4
Ex4/(ODE-FT, h = 0.01) 1.865 ×10−62.027 ×10−6
Ex5/(Scheme II in [14], h = 0.1) 1.721 ×10−54.102 ×10−5
Ex5/(ODE-FT, h = 0.1) 1.242 ×10−61.685 ×10−5
Ex5/(ODE-FT, h = 0.01) 1.264 ×10−10 2.693 ×10−9
I seems ha in gene al, he di e en algo i hms beha e simila ly, a leas o he chosen s ep
size
h=
0.1 and
h=
0.01 (conside ha such
h
is a big one and alues a ound
h=
0.00001 o less
a e mo e adequa e o a compa ison, as sugges ed, e.g., by he alues used in ou ine
ode45
ha
chooses
h
dynamically). Possibly, mo e e icien and elabo a ed implemen a ions o he p oposed
algo i hms will equi e some addi ional analysis o F- ans o m p ope ies o allow a iable-o de
and/o s ep-size con ol.
As a ool o nume ical solu ion o in e al (IDEs) and uzzy (FDEs) di e en ial equa ions in e ms
o gH-de i a i e, he F- ans o m allows an immedia e app oach o handle he swi ching phenomenon,
a s ill open p oblem in his a ea. This app oach is ob ained by he applica ion o Equa ion (43) and
p oposi ion 8, which is possible because he in e al- alued unc ion
H( )
o e s an app oxima ion o
he in e al solu ion
x( )
a all poin s
∈[ 0
,
1]
and no only a he disc e ized poin s
pk
, as usual in
he (explici ) single- o mul i-s ep ODE sol e s.
Fu he esea ch can be planned in he design and expe imen a ion o e icien nume ical
p ocedu es o sol e eal-wo ld applica ions. In his di ec ion, a possible imp o emen in he
app oxima ion can be ob ained by highe -o de
Fd
- ans o m (see [
11
] o ecen esul s o i s
p ope ies), by in oducing local polynomials o , mo e gene ally, local pa ame ic unc ions
Fk( ;θ)
in place o cons an di ec componen s
Fk
(coe icien s o he polynomials o pa ame e s
θ∈Rd
a e
hen es ima ed by leas squa es). In hese cases, he in e se F- ans o m o a unc ion
( )
on
[a
,
b]
has
he o m
d
(P,A)( ) = n
∑
k=1Fk(
;
θ(k))Ak( )
, wi h es ima ed pa ame e s
θ(k)
o he
k
- h di ec componen .
I is wo h o ema k ha he same in eg al p ope y used in his pape o he s anda d F- ans o m
(P,A)( )is also alid o d
(P,A)( ), i.e.,
b
Ra
( )d =
b
Ra
d
(P,A)( )d =n
∑
k=1
b
Ra
Fk( ;θ(k))Ak( )d . (49)
I should be in e es ing o see i highe -o de F- ans o m app oxima ions will be able o gene a e
high o de s
O(hq)
,
q>
2 o con e gence, and o ob ain possibly inc easing o de s
q
by inc easing
d
( wo nume ic schemes o o de q=2 based on he F2- ans o m a e ob ained in [16]).
Simila esul s can be ob ained by conside ing he disc e e F- ans o m
(P,A)( j)
on a da a se o
poin s
S=( j, j),j=1, 2,..., m
; he in eg als a e subs i u ed by summa ions and we ha e (see [
10
])
m
∑
j=1 (P,A)( j) = m
∑
j=1 j. (50)
Finally, i is wo h men ioning he possibili y o applying he ideas p esen ed in his pape o
he nume ical solu ion o o he kinds o di e en ial and in eg al equa ions, such as delay di e en ial
Axioms 2020,9, 15 36 o 37
equa ions (e.g., [
28
]), di e en ial equa ions on ime scales, ac ional di e en ial equa ions ([
29
])
and implici di e en ial algeb aic equa ions (DAE, see, e.g., [
30
,
31
]); a gene al DAE wi h addi ional
cons ain s, on an in e al [ 0, 1]is exp essed in he o m









F ,x( ),·
x( )=0
G( ,x( ))=0
H( ,x( ))≤0
x( 0)=x0.
(51)
In his cases, he disc e iza ion o
[a
,
b]
by a uzzy pa i ion
(P
,
A)
and he subs i u ion o
·
x( )
and
x( )
wi h he unc ions
·
x(P,A)( )
and
x(P,A,x0)( )
a poin s
pk∈P
will ans o m he DAE in o
a s anda d easibili y p oblem, consis ing o inding easible solu ions o he ans o med sys em a
poin s pk,k=1, ..., n.
Au ho Con ibu ions:
All au ho s con ibu ed equally o he inal e sion o his pape . All au ho s ha e ead
and ag eed o he published e sion o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
Acknowledgmen s:
Da ide Radi g a e ully acknowledges he suppo o he Czech Science Founda ion (GACR)
unde p ojec [20-16701S] and he VŠB-TU Os a a unde he SGS p ojec SP2020/11.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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c
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