Numerical solutions of fractional system, two-point BVPs using iterative reproducing kernel algorithm
We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product spaces are constructed in which the boundary conditions of...
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Institute of Mathematics, NAS of Ukraine
2018
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507387531624448 |
|---|---|
| author | Al-Smadi, M. Altawallbeh, Z. Ateiwi, A. M. Komashinskaya, I. V. Аль-Смаді, М. Алтавабех, З. Атеіві, А. М. Комашинська, І. В. |
| author_facet | Al-Smadi, M. Altawallbeh, Z. Ateiwi, A. M. Komashinskaya, I. V. Аль-Смаді, М. Алтавабех, З. Атеіві, А. М. Комашинська, І. В. |
| author_sort | Al-Smadi, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:19:33Z |
| description | We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution
of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product
spaces are constructed in which the boundary conditions of the systems are satisfied. The reproducing kernel functions
are constructed to get an accurate algorithm for the investigation of fractional systems. The developed procedure is based
on generating the orthonormal basis with an aim to formulate the solution throughout the evolution of the algorithm. The
analytic solution is represented in the form of a series in the reproducing kernel Hilbert space with readily computed
components. In this connection, some numerical examples are presented to show the good performance and applicability
of the developed algorithm. The numerical results indicate that the proposed algorithm is a powerful tool for the solution
of fractional models arising in different fields of sciences and engineering. |
| first_indexed | 2026-03-24T02:08:30Z |
| format | Article |
| fulltext |
UDC 517.5
Z. Altawallbeh (Tafila Techn. Univ., Jordan),
M. Al-Smadi (Ajloun College, Al-Balqa Appl. Univ., Jordan),
I. Komashynska (Univ. Jordan, Amman, Jordan),
A. Ateiwi (Al-Hussein Bin Talal Univ., Ma’an, Jordan)
NUMERICAL SOLUTIONS OF FRACTIONAL SYSTEM, TWO-POINT BVPS
USING ITERATIVE REPRODUCING KERNEL ALGORITHM
ЧИСЕЛЬНИЙ РОЗВ’ЯЗОК ДРОБОВИХ СИСТЕМ ДВОТОЧКОВИХ
ГРАНИЧНИХ ЗАДАЧ ЗА ДОПОМОГОЮ IТЕРАТИВНОГО
ВIДНОВЛЮЮЧОГО ЯДЕРНОГО АЛГОРИТМУ
We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution
of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product
spaces are constructed in which the boundary conditions of the systems are satisfied. The reproducing kernel functions
are constructed to get an accurate algorithm for the investigation of fractional systems. The developed procedure is based
on generating the orthonormal basis with an aim to formulate the solution throughout the evolution of the algorithm. The
analytic solution is represented in the form of a series in the reproducing kernel Hilbert space with readily computed
components. In this connection, some numerical examples are presented to show the good performance and applicability
of the developed algorithm. The numerical results indicate that the proposed algorithm is a powerful tool for the solution
of fractional models arising in different fields of sciences and engineering.
Запропоновано ефективний обчислювальний метод, а саме iтеративний вiдновлюючий ядерний алгоритм для на-
ближеного розв’язування систем дробового порядку для двоточкових часових граничних задач у сенсi Капуто.
Побудовано два розширенi гiльбертовi простори, в яких виконуються граничнi умови для систем. Також побудо-
вано вiдновлювальнi ядернi функцiї, щоб отримати точний алгоритм для вивчення дробових систем. Розроблена
процедура базується на генерацiї ортонормального базису з метою формулювання розв’язку для всiєї еволюцiї ал-
горитму. Аналiтичний розв’язок представлено у виглядi ряду у вiдновлювальному ядерному просторi Гiльберта з
компонентами, що легко обчислюються. У зв’язку з цим ми наводимо деякi чисельнi приклади, щоб продемонстру-
вати гарну роботу та застосовнiсть розробленого алгоритму. Чисельнi результати показують, що даний алгоритм є
потужним iнструментом для розв’язування дробових моделей, якi з’являютъся в рiзних областях науки i технiки.
1. Introduction. Recently, fractional differential equations received increasing attention as a superb
tool for modeling many problems in different fields of sciences and engineering. This concept is
not unique and there exist several definitions of fractional-order derivative including Grunwald –
Letnikov’s definition, Riemann – Liouville’s definition, Caputo’s definition, and Riesz’s definition.
This generalized calculus is an extension of the classical calculus theory of noninteger order [1 – 5].
On the other hand, the fractional derivatives supply a popularity implement for the definition of
memory and hereditary characteristics that involve the whole history of the function in a weighted
form. In this sense, the FDEs have a nonlocal property, which means that the next state of the system
depends not only upon the current state but also upon the history of all previous states. This is the
fundamental advantage of using FDEs compared with classical integer-order counterpart. Therefore,
there has been increasing interest in the subject of a fractional calculus which can give a more
realistic interpretation of natural phenomena. Moreover, several systems in interdisciplinary fields
can be described by FDEs including turbulence, signal processing, and quantum evolution. In spite
of this, most of nonlinear fractional systems do not have closed form solutions, so analytical and
numerical methods must be used.
c\bigcirc Z. ALTAWALLBEH, M. AL-SMADI, I. KOMASHYNSKA, A. ATEIWI, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5 599
600 Z. ALTAWALLBEH, M. AL-SMADI, I. KOMASHYNSKA, A. ATEIWI
The purpose of this study is to investigate and implement a computational iterative technique, the
reproducing kernel method (RKM), in finding approximate solutions for a certain class of fractional
system, two-point BVPs in Caputo sense. More specifically, we consider system of differential
equations of fractional-order in the following form:
D\alpha iui(t) = fi
\bigl(
t, ui(t), u
\prime
i(t)
\bigr)
, 0 < t < T, (1)
associated with two-point boundary conditions
ui(0) = ai, u(T ) = bi, i = 1, 2, . . . , N, (2)
where ai, bi \in \BbbR , 1 < \alpha i \leq 2, D\alpha i denotes the Caputo fractional derivative of order \alpha i, i =
= 1, 2, . . . , N, fi(t, ui, u
\prime
i) \in \scrW 1[0, T ], i = 1, 2, . . . , N, are sufficiently analytical given functions
such that BVPs (1) and (2) satisfies the existence and uniqueness of the solutions, and ui \in \scrW 3[0, T ]
are unknown functions to be determined.
The RKM was developed as an efficient numerical method for treating different kind of singular
differential equations [6, 7], integral equations [8], integrodifferential equations [9 – 14], and fuzzy
differential equations [15 – 17]. It is an alternative process for getting analytic Taylor series solution.
It has been successfully put into practiced to handle the approximate solution of periodic boundary-
value problems [18, 19], the approximate solution of MHD squeezing fluid flow [20], the solution
of difference equations [21], Duffing equations with integral boundary conditions [22], and parabolic
problems with nonclassical conditions [23]. While the numerical solvability for different categories
can be found in [24 – 26].
The present analysis extends the application of the RKM for obtaining approximate solutions
of FDE system in Caputo sense. The structure of the present article is as follows. In Section 2,
we utilized some necessary definitions and results from the fractional calculus theory. In Section
3, theoretical and analytical basis with representation of solutions are introduced in Hilbert space.
In Section 4, numerical examples are simulated to show the reasonableness of the theory and to
demonstrate the high performance of the proposed method. Finally, some conclusions are summarized
in the last section.
2. Background and preliminaries. The fractional calculus is a name for the theory of inte-
grals and derivatives of arbitrary-order that generalizes the notions of integer-order differentiation
and integration. Herein, we adopt the Caputo fractional derivative sense which is a modification
of Riemann – Liouville sense because the initial conditions that defined during the formulation of
the system are similar to those conventional conditions of integer-order. In this section, the main
descriptions and features of the fractional calculus theory are illustrated. For more details about the
mathematical properties of FDEs, we refer to [2 – 5].
Definition 1. A real function f(x), x > 0, is said to be in the space C\mu , \mu \in \BbbR , if there exists
a real number \rho > \mu such that f(x) = x\rho f1(x), where f1(x) is continuous in [0,\infty ), and it is said
to be in the space Cn
\mu if f (n)(x) \in C\mu , n \in \BbbN .
Definition 2. The Riemann – Liouville fractional integral operator of order \alpha \geq 0 of a function
f(x) \in C\mu , \mu \geq - 1, is defined as
J\alpha
s f(x) =
1
\Gamma (\alpha )
x\int
s
(x - \xi )\alpha - 1f(\xi ) d\xi , \alpha > 0, x > s \geq 0,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
NUMERICAL SOLUTIONS OF FRACTIONAL SYSTEM, TWO-POINT BVPS USING ITERATIVE . . . 601
J0
s f(x) = f(x),
where \Gamma is the well-known Gamma-function.
The operator J\alpha
s has the following properties: for f \in C\mu , \mu \geq - 1, \alpha , \beta > 0, x > s \geq 0,
c \in \BbbR , and \gamma > - 1, one can get J\alpha
s J
\beta
s f(x) = J\alpha +\beta
s f(x) = J\beta
s J\alpha
s f(x), J
\alpha
s c =
c
\Gamma (\alpha + 1)
(x - s)\alpha ,
and J\alpha
s (x - s)\gamma =
\Gamma (\gamma + 1)
\Gamma (\gamma + 1 + \alpha )
(x - s)(\alpha +\gamma ).
The Riemann – Liouville derivative has certain disadvantages when trying to model real-world
phenomena with fractional differential equations. Thus, we shall introduce a modified fractional
differential operator D\alpha
s proposed by Caputo in his work on the theory of viscoelasticity.
Definition 3. The Riemann – Liouville fractional derivative of order \alpha > 0 of f \in Cn
- 1, n \in \BbbN ,
is defined as
\widehat D\alpha
s f(x) =
\left\{
dn
dxn
Jn - \alpha
s f(x), n - 1 < \alpha < n, x > s \geq 0,
dn
dxn
f(x), \alpha = n.
Definition 4. The Caputo fractional derivative of order \alpha > 0 of f \in Cn
- 1, n \in \BbbN , is defined
as
D\alpha
s f(x) =
\left\{
Jn - \alpha
s f (n)(x), n - 1 < \alpha < n, x > s \geq 0,
dn
dxn
f(x), \alpha = n.
Remark 1. For n - 1 < \alpha \leq n, n \in \BbbN , x > s \geq 0, and f \in Cn
- 1, one can get
J\alpha
s D
\alpha
s f(x) = f(x) -
n - 1\sum
k=0
f (k)(s+)
(x - s)k
k !
,
D\alpha
s J
\alpha
s f(x) = f(x).
The operator D\alpha
s has the following properties: for f \in Cn
- 1, \alpha > 0, x > s \geq 0, c \in \BbbR , and
\gamma > - 1, one can get D\alpha
s c = 0, and D\alpha
s (x - s)\gamma =
\Gamma (\gamma + 1)
\Gamma (\gamma - \alpha + 1)
(x - s)(\gamma - \alpha ).
3. Theoretical and analytical basis of the method. In this section, we construct a representation
solution for fractional system associated to given boundary conditions, in which the solution provided
in terms of a rapidly convergent series in the reproducing kernel space with components that can be
elegantly computed.
Definition 5. Let \scrH be a Hilbert space of function \scrF : \Omega \rightarrow \scrH on a set \Omega . A function K :
\Omega \times \Omega \rightarrow \BbbR is a reproducing kernel of \scrH if the following conditions are satisfied: firstly, K(\cdot , \tau ) \in \scrH
for each \tau \in \Omega ; secondly,
\bigl\langle
\scrF (\cdot ),K(\cdot , \tau )
\bigr\rangle
= \scrF (\tau ) for each \scrF \in \scrH and each \tau \in \Omega .
Definition 6. The reproducing kernel Hilbert space \scrW 1[0, T ] is defined as \scrW 1[0, T ] = \{ u(t) is
one-variable absolutely continuous real-valued function on [0, T ] and u\prime (t) \in L2[0, T ]\} . The inner
product and the norm of \scrW 1[0, T ] are given, respectively, by
\bigl\langle
u1(t), u2(t)
\bigr\rangle
\scrW 1
= u1(0)u2(0) +
T\int
0
u\prime 1(\xi )u
\prime
2(\xi )d\xi ,
and \| u(t)\| 2\scrW 1
= \langle u(t), u(t)\rangle \scrW 1
, where u1, u2 \in \scrW 1[0, T ].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
602 Z. ALTAWALLBEH, M. AL-SMADI, I. KOMASHYNSKA, A. ATEIWI
Remark 2. The Hilbert space \scrW m[a, b], is called a reproducing kernel if for each fixed t in [a, b],
there exist a function Kt(s) \in \scrW m[a, b] such that \langle u(s),Kt(s)\rangle \scrW m = u(t) for any u(s) \in \scrW m[a, b]
and s \in [a, b].
Theorem 1. The Hilbert space \scrW 1[0, T ] is a complete reproducing kernel with the reproducing
kernel function
\scrR t(s) =
\left\{ 1 + s, s \leq t,
1 + t, s > t.
Anyhow, to solve system (1) and (2) by using the RKM, it is necessary to construct the repro-
ducing kernel space \scrW 3[0, T ] in which every function satisfies the homogenous boundary conditions
u(0) = 0 and u(T ) = 0.
Definition 7. The reproducing kernel Hilbert space \scrW 3[0, T ] is defined as \scrW 3[0, T ] = \{ u(t) :
u, u\prime , u\prime \prime are one-variable absolutely continuous real-valued functions on [0, T ] and u\prime \prime \prime (t) \in L2[0, T ],
u(0) = u(T ) = 0\} . The inner product and the norm of \scrW 3[0, T ] are given, respectively, by
\bigl\langle
u1(t), u2(t)
\bigr\rangle
\scrW 3
=
1\sum
i=0
u
(i)
1 (0)u
(i)
2 (0) + u1(T )u2(T ) +
T\int
0
u\prime \prime \prime 1 (\xi )u
\prime \prime \prime
2 (\xi ) d\xi , (3)
and \| u(t)\| 2\scrW 3
= \langle u(t), u(t)\rangle \scrW 3
, where u1, u2 \in \scrW 3[0, T ].
Lemma 1. The Hilbert space \scrW 3[0, T ] is a complete reproducing kernel with the reproducing
kernel function
Qt(s) =
\left\{
\sum 6
i=0
pi(t)s
i - 1, s \leq t,\sum 6
i=0
qi(t)s
i - 1, t > s,
where the unknown coefficients pi(t) and qi(t), i = 1, . . . , 6, can be uniquely obtained by utilizing
the following assumptions:
Qt(0) = 0, Qt(T ) = 0, Q
(3)
t (0) = 0,
Q
(i)
t (T ) = 0, i = 3, 4, Q\prime
t(0) +Q
(4)
t (0) = 0,
Q
(i)
t (s+) = Q
(i)
t (s - ), i = 0, 1, . . . , 4,
Q
(5)
t (s+) - Q
(5)
t (s - ) = - 1.
(4)
Consequently, by using the Mathematica for handling the above-mentioned generalized differen-
tial equations (4), the reproducing kernel function is given by
Qt(s) =
\left\{
s
120T 2
\Biggl[
- 6T 3t2s+ t2s( - 120 + t3 + s3) - 5Tt( - 24s+ t3s +
+t( - 24 + s3)) + T 2(10t3s - s4 + 5t( - 24 + s3)))
\Biggr]
, s \leq t,
- t
120T 2
\Biggl[
6T 3ts2 - ts2( - 120 + t3 + s3) + T 2(t4 + 120s -
- 5t3s - 10ts3) + 5Ts( - 24s+ t3s+ t( - 24 + s3))
\Biggr]
, s > t.
(5)
Herein, it is worth to mention that \{ un(t)\} \infty n=1 is a compact subset of the space C[0, T ], which
means that \{ un(t)\} \infty n=1 are equicontinuous functions. To see this, use the property of Qt(s) such
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
NUMERICAL SOLUTIONS OF FRACTIONAL SYSTEM, TWO-POINT BVPS USING ITERATIVE . . . 603
that
| un(t2) - un(t1)| =
\bigm| \bigm| \langle u(s), Qt2(s) - Qt1(s)\rangle \scrW 3
\bigm| \bigm| \leq
\leq \| u(s)\| \scrW 3
\| Qt2(s) - Qt1(s)\| \scrW 3
\leq \itM \| Qt2(s) - Qt1(s)\| \scrW 3
.
By “Mean-value theorem of differentials” and the symmetry of Qt(s), it follows that\bigm| \bigm| Qt2(s) - Qt1(s)
\bigm| \bigm| = | Qs (t2) - Qs(t1)| =
\bigm| \bigm| \bigm| \bigm| ddtQs(t)
\bigm| \bigm| \bigm| \bigm|
t=\tau
| t2 - t1| \leq \itN | t2 - t1| .
Thus, if \gamma \leq | t2 - t1| \leq
\epsilon
\itM \itN
, then | un(t2) - un(t1)| < \epsilon .
In order to illustrate the RKHS methodology of the proposed model, we consider the differential
operator L\scrW : \scrW 3[0, T ] \rightarrow \scrW 1[0, T ] such that L\scrW u(t) = D\alpha u(t). Then, BVPs (4) and (5) can be
equivalently converted into the form
L\scrW ui(t) = fi
\bigl(
t, ui(t), u
\prime
i(t)
\bigr)
,
ui(0) = 0, ui(T ) = 0, i = 1, 2, . . . , N,
(6)
where ui(t) \in \scrW 3[0, T ] and fi (\tau , ui, vi) \in \scrW 1[0, T ] as ui = ui(\tau ), vi = u\prime i(\tau ) \in \scrW 3[0, T ],
\tau \in [0, T ].
Let \varphi i(t) = Qti(t) and \psi i(t) = L\ast
\scrW \varphi i(t), where \{ ti\} \infty i=1 is countable dense subset of [0, T ], and
L\ast
\scrW is the adjoint operator of L\scrW . Thus, in terms of the properties of reproducing-kernel, it holds
\langle u(t), \psi i(t)\rangle \scrW 3
= \langle u(t), L\ast
\scrW \varphi i(t)\rangle \scrW 3
= \langle L\scrW u(t), \varphi i(t)\rangle \scrW 1
= L\scrW u(ti), i = 1, 2, . . . .
Lemma 2. The operator L\scrW : \scrW 3[0, T ] \rightarrow \scrW 1[0, T ] is a bounded linear operator.
Proof. It is so easy to see that L\scrW is a linear operator. Thus, it is enough to show that L\scrW is
bounded operator. From the space \scrW 1[0, T ], we have
\| L\scrW u(t)\| 2\scrW 1
= \| D\alpha u(t)\| 2\scrW 1
= \langle D\alpha u(t), D\alpha u(t)\rangle \scrW 1
= [D\alpha u(0)]2 +
T\int
0
\biggl[
d
d\xi
D\alpha u(\xi )
\biggr] 2
d\xi .
By reproducing property of Qt(s) and since D\alpha Qt(s) is uniformly bounded about t and s, we obtain
u(t) =
\bigl\langle
u(s), Qt(s)
\bigr\rangle
\scrW 3
, L\scrW u(t) =
\bigl\langle
u(s), D\alpha Qt(s)
\bigr\rangle
\scrW 3
and
d
dt
L\scrW u(t) =
\biggl\langle
u(s),
d
dt
D\alpha Qt(s)
\biggr\rangle
\scrW 3
.
By Schwarz inequality, we get
| L\scrW u(t)| =
\bigm| \bigm| \bigl\langle u(s), D\alpha Qt(s)
\bigr\rangle
\scrW 3
\bigm| \bigm| \leq \| u\| \scrW 3\| D\alpha Qt(s)\| \scrW 3 = \mu 1\| u\| \scrW 3
and \bigm| \bigm| \bigm| \bigm| ddtL\scrW u(t)
\bigm| \bigm| \bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm|
\biggl\langle
u(s),
d
dt
D\alpha Qt(s)
\biggr\rangle
\scrW 3
\bigm| \bigm| \bigm| \bigm| \bigm| \leq \| u\| \scrW 3
\bigm\| \bigm\| \bigm\| \bigm\| ddtD\alpha Qt(s)
\bigm\| \bigm\| \bigm\| \bigm\|
\scrW 3
= \mu 2\| u\| \scrW 3 ,
where \mu 1 and \mu 2 are positive constants.
Thus
\bigl[
D\alpha u(0)
\bigr] 2 \leq \mu 21\| u\| 2\scrW 3
and
\int T
0
\biggl[
d
d\xi
D\alpha u(\xi )
\biggr] 2
d\xi \leq T\mu 22\| u\| 2\scrW 3
. Hence, \| L\scrW u(t)\| \scrW 1
\leq
\leq \mu \| u(t)\| \scrW 3
, where \mu =
\sqrt{}
\mu 21 + T\mu 22.
Lemma 2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
604 Z. ALTAWALLBEH, M. AL-SMADI, I. KOMASHYNSKA, A. ATEIWI
Theorem 2. Let \{ ti\} \infty i=1 be a dense subset of the interval [0, T ], then the sequence
\bigl\{
\psi \upsilon
i (t)
\bigr\} \infty
i=1
,
\upsilon = 1, 2, . . . , N, is a complete function system of the space \scrW 3[0, T ] such that \psi \upsilon
i (t) =
= D\alpha
sQt(s)| s=ti
.
Proof. From reproducing property of Qt(s), we get
\psi \upsilon
i (t) = L\ast
\scrW \varphi
\upsilon
i (t) =
\bigl\langle
L\ast
\scrW \varphi
\upsilon
i (s), Qt(s)
\bigr\rangle
\scrW 3
=
=
\bigl\langle
\varphi \upsilon
i (s), L\scrW Qt(s)
\bigr\rangle
\scrW 1
= L\scrW Qt(ti) = D\alpha
sQt(s)
\bigm| \bigm|
s=ti
.
Since \{ ti\} \infty i=1 is dense in the interval [0, T ]. For each u\upsilon (t) in \scrW 3[0, T ], if
\bigl\langle
u\upsilon (t), \psi
\upsilon
i (t)
\bigr\rangle
\scrW 3
=
=
\bigl\langle
L\scrW u\upsilon (t), \varphi
\upsilon
i (t)
\bigr\rangle
\scrW 1
= L\scrW u\upsilon (ti) = 0, i = 1, 2, . . . , then from the density of \{ ti\} \infty i=1 and
continuity of u\upsilon (t), \upsilon = 1, 2, . . . , N, we have u\upsilon (t) = 0.
Theorem 2 is proved.
The reproducing kernel solution will be obtained by calculating a truncated series based on the
orthonormal functions
\bigl\{
\=\psi \upsilon
i (t)
\bigr\} \infty
i=1
of the space \scrW 3[0, T ], which is constructed from \{ \psi \upsilon
i (t)\}
\infty
i=1 by
using the Gram – Schmidt process such that
\=\psi \upsilon
i (t) =
i\sum
k=1
\mu \upsilon ik\psi
\upsilon
k (t), (7)
where \mu \upsilon ik are orthogonalizatio coefficients, \mu \upsilon ii > 0, i = 1, 2, . . . , n.
Theorem 3. If \{ ti\} \infty i=1 is dense on the interval [0, 1] and u\upsilon (t) \in \scrW 3[0, T ] is a unique solution
of Eq. (6), then the exact solution could be represented by
u\upsilon (t) =
\infty \sum
i=1
i\sum
k=1
\mu \upsilon ikf\upsilon
\bigl(
tk, u\upsilon (tk), u
\prime
\upsilon (tk)
\bigr)
\=\psi \upsilon
i (t). (8)
Proof. For each u\upsilon (t) \in \scrW 3[0, T ], the series
\sum \infty
i=1
\bigl\langle
u\upsilon (t), \=\psi
\upsilon
i (t)
\bigr\rangle
\scrW 3
\=\psi \upsilon
i (t) is convergent.
From the Fourier series expansion, u\upsilon (t) can be written as follows:
u\upsilon (t) =
\infty \sum
i=1
\bigl\langle
u\upsilon (t), \=\psi
\upsilon
i (t)
\bigr\rangle
\scrW 3
\=\psi \upsilon
i (t) =
\infty \sum
i=1
\Biggl\langle
v(t),
i\sum
k=1
\mu \upsilon ik\psi
\upsilon
k (t)
\Biggr\rangle
\scrW 3
\=\psi i(t) =
=
\infty \sum
i=1
i\sum
k=1
\mu \upsilon ik \langle u\upsilon (t), L\ast
\scrW \varphi
\upsilon
k(t)\rangle \scrW 3
\=\psi \upsilon
i (t) =
=
\infty \sum
i=1
i\sum
k=1
\mu \upsilon ik \langle L\scrW u\upsilon (t), \varphi
\upsilon
k(t)\rangle \scrW 1
\=\psi \upsilon
i (t) =
\infty \sum
i=1
i\sum
k=1
\mu \upsilon ikL\scrW u\upsilon (tk) \=\psi
\upsilon
i (t) =
=
\infty \sum
i=1
i\sum
k=1
\mu \upsilon ikf\upsilon
\bigl(
tk, u\upsilon (tk), u
\prime
\upsilon (tk)
\bigr)
\=\psi \upsilon
i (t).
Therefore, the form of Eq. (8) is the exact solution of Eq. (6).
Theorem 3 is proved.
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NUMERICAL SOLUTIONS OF FRACTIONAL SYSTEM, TWO-POINT BVPS USING ITERATIVE . . . 605
Since \scrW 3[0, T ] is a Hilbert space, the series
\sum \infty
i=1
\sum i
k=1
\mu \upsilon ik
\bigl\langle
L\scrW u\upsilon (t), \varphi
\upsilon
k(t)
\bigr\rangle
\scrW 1
\=\psi \upsilon
i (t) <\infty .
Hence, the truncated series
u\upsilon ,n(t) =
n\sum
i=1
i\sum
k=1
\mu \upsilon ikf\upsilon
\bigl(
tk, u\upsilon (tk), u
\prime
\upsilon (tk)
\bigr)
\=\psi \upsilon
i (t) (9)
is convergent in the sense of \| \cdot \| \scrW 3[0,T ] and the numerical solution of Eq. (6) can be directly
calculated by Eq. (9).
Corollary 1. The approximate solution u\upsilon ,n(t) and its derivative u(m)
\upsilon ,n (t), m = 1, 2, are con-
verging uniformly to the exact solution u\upsilon (t) and its derivative u(m)
\upsilon (t) as n\rightarrow \infty , respectively.
Proof. For any t \in [0, T ], it easy to see that
\bigm| \bigm| \bigm| u(i)\upsilon ,n(t) - u(i)\upsilon (t)
\bigm| \bigm| \bigm| = \bigm| \bigm| \bigm| \bigm| \Bigl\langle u\upsilon ,n(t) - u\upsilon (t), Q
(i)
t (s)
\Bigr\rangle
\scrW 3
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm\| \bigm\| \bigm\| Q(i)
t (s)
\bigm\| \bigm\| \bigm\|
\scrW 3
\| u\upsilon ,n(t) - u\upsilon (t)\| \scrW 3
\leq
\leq M\upsilon
i \| u\upsilon ,n(t) - u\upsilon (t)\| \scrW 3
, M\upsilon
i \in \BbbR , i = 0, 1, 2.
Hence, if \| u\upsilon ,n(t) - u\upsilon (t)\| \scrW 3
\rightarrow 0 as n \rightarrow \infty , then the approximate solution u(i)\upsilon ,n(t), i = 0, 1, 2,
are converge uniformly to the exact solution u\upsilon (t) and its derivative, respectively.
Remark 3. In order to apply the IRM for solving system (6) numerically, we have the following
two cases based on the structure of the function f\upsilon .
Case 1. If system (6) is linear, then the exact and approximate solutions can be obtained directly
from equations (8) and (9), respectively.
Case 2. If system (6) is nonlinear, then the exact and approximate solutions can be obtained by
using the following process: according to exact solution in equation (8), the representation of the
solution of system (6) can be denoted by
u\upsilon (t) =
\infty \sum
i=1
B\upsilon
i
\=\psi \upsilon
i (t),
where B\upsilon
i =
\sum i
k=1
\mu \upsilon ikf\upsilon
\bigl(
tk, u\upsilon ,k - 1(tk), u
\prime
\upsilon ,k - 1(tk)
\bigr)
. So, we will approximate the unknown B\upsilon
i
using the known \Lambda \upsilon
i as follows: set the initial data such that u\upsilon ,0(t1) = u\prime \upsilon ,0(t1) = 0, and define the
n-term approximation to u\upsilon (t) by
u\upsilon ,n(t) =
n\sum
i=1
\Lambda \upsilon
i
\=\psi \upsilon
i (t), (10)
where the coefficients \Lambda \upsilon
i of \=\psi \upsilon
i (t), i = 1, 2, . . . , n, are given by
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606 Z. ALTAWALLBEH, M. AL-SMADI, I. KOMASHYNSKA, A. ATEIWI
\Lambda \upsilon
1 = \mu \upsilon 11f\upsilon
\bigl(
t1, u\upsilon ,0(t1), u
\prime
\upsilon ,0(t1)
\bigr)
=\Rightarrow u\upsilon ,1(t) = \Lambda \upsilon
1
\=\psi \upsilon
1 (t),
\Lambda \upsilon
2 =
2\sum
k=1
\mu \upsilon 2kf\upsilon
\bigl(
tk, u\upsilon ,k - 1(tk), u
\prime
\upsilon ,k - 1(tk)
\bigr)
=\Rightarrow u\upsilon ,2(t) =
2\sum
i=1
\Lambda \upsilon
i
\=\psi \upsilon
i (t),
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
\Lambda \upsilon
n =
n\sum
k=1
\mu \upsilon nkf\upsilon
\bigl(
tk, u\upsilon ,k - 1(tk), u
\prime
\upsilon ,k - 1(tk)
\bigr)
=\Rightarrow u\upsilon ,n(t) =
n\sum
i=1
\Lambda \upsilon
i
\=\psi \upsilon
i (t).
(11)
In the iterative process of Eq. (10), we can guarantee that the numerical solution un satisfies the
constraints conditions of Eq. (6).
For the error behavior, if \varepsilon n = | un(t) - u(t)| , where un(t) is given in Eq. (10). Then, one can
write \| \varepsilon n\| 2\scrW 3
=
\bigm\| \bigm\| \bigm\| \sum \infty
i=n+1
\Lambda \upsilon
i
\=\psi \upsilon
i
\bigm\| \bigm\| \bigm\| 2
\scrW 3
=
\sum \infty
i=n+1
(\Lambda \upsilon
i )
2 and \| \varepsilon n - 1\| 2\scrW 3
=
\bigm\| \bigm\| \bigm\| \sum \infty
i=n
\Lambda \upsilon
i
\=\psi \upsilon
i
\bigm\| \bigm\| \bigm\| 2
\scrW 3
=
=
\sum \infty
i=n
(\Lambda \upsilon
i )
2 . Clearly, \{ \varepsilon n\} \infty n=1 is decreasing in the sense of \| \cdot \| \scrW 3
. Since
\sum \infty
i=1
\Lambda \upsilon
i
\=\psi \upsilon
i (t) is
convergent series, then \| \varepsilon n\| \scrW 3
\rightarrow 0 as n\rightarrow \infty .
4. Applications and numerical examples. In this section, numerical examples are studied to
demonstrate the performance, accuracy and applicability of the present method for both linear and
nonlinear problems. Results obtained are compared with the exact solution of each example and are
found to be in good agreement with each other. In the process of computation, all the symbolic and
numerical computations performed by using Mathematica software package.
Algorithm 1. To approximate the solution un(t) of u(t) for Eqs. (1) and (2), do the following
steps.
Step 1. Fixed t in [0, T ] and set s \in [0, T ];
if s \leq t, let Qs(t) =
6\sum
i=1
pi(t)s
i - 1;
else let Qs(t) =
6\sum
i=1
Qi(t)s
i - 1.
Step 2. Choose n collocation points and do the following subroutine:
set ti =
i - 1
N - 1
, i = 1, 2, . . . , N ;
set \psi \upsilon
i (t) = \scrD \alpha
sQs(t)| s=ti
.
Step 3. Obtain the orthogonalization coefficients \mu \upsilon i\rho as follows:
let c\nu ik =
\Bigl\langle
\psi \upsilon
i (t), \psi
\upsilon
k(t)
\Bigr\rangle
\scrW 3
, and do the following subroutine:
for i = 1, set \mu \upsilon 11 = \| \psi \upsilon
1\|
- 1
\scrW 3
;
for i = 2, . . . , N, set \mu \upsilon ii =
\Biggl(
\| \psi \upsilon
i \|
2
\scrW 3
-
i - 1\sum
k=1
(c\upsilon ik)
2
\Biggr) - 1/2
;
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NUMERICAL SOLUTIONS OF FRACTIONAL SYSTEM, TWO-POINT BVPS USING ITERATIVE . . . 607
else (for \rho < i), set \mu \upsilon i\rho = -
\left( i - 1\sum
k=\rho
c\upsilon ik\mu
\upsilon
k\rho
\right) \Biggl( \| \psi \upsilon
i \|
2
\scrW 3
-
i - 1\sum
k=1
(c\upsilon ik)
2
\Biggr) - 1/2
.
Step 4. For i = 1, 2, . . . , N, set
\psi
\upsilon
i (t) =
\sum i
k=1
\mu \upsilon ik\psi
\upsilon
k (t).
Step 5. Set t1 = 0, and choose an initial approximation u0(t1) = u(t1), u
\prime
0(t1) = u\prime (t1);
for i = 1, set \Lambda \upsilon
1 = \mu \upsilon 11f\upsilon
\bigl(
t1, u\upsilon ,0(t1), u
\prime
\upsilon ,0(t1)
\bigr)
and u\upsilon ,1(t) = \Lambda \upsilon
1\psi
\upsilon
1 ;
for i = 2, 3, . . . , n, set \Lambda \upsilon
i =
i\sum
k=1
\mu \upsilon nkf\upsilon
\bigl(
tk, u\upsilon ,k - 1(tk), u
\prime
\upsilon ,k - 1(tk)
\bigr)
;
set u\upsilon ,n(t) =
n\sum
i=1
\Lambda \upsilon
i
\=\psi \upsilon
i (t).
Outcome: the numerical solution u\upsilon ,n(t).
Stop.
By applying Algorithm 1 throughout the numerical computations, we present some tabulate data
and graphical results that discussed quantitatively at some selected grid points on [0, 1].
Example 1. Consider the following linear fractional system:
D\alpha 1u1(t) + u\prime 2(t) =
1
2
e
t
2 + (2 + t)et,
D\alpha 2u2(t) - u1(t) + u\prime 1(t) =
1
4
e
t
2 + et,
(12)
with two-point boundary conditions
u1(0) = 0, u1(1) = e,
u2(0) = 1, u2(1) =
\surd
e,
(13)
where 1 < \alpha i \leq 2, i = 1, 2, 0 \leq t \leq 1 and u1(t), u2(t) \in \scrW 3[0, 1].
The exact solutions at \alpha 1 = \alpha 2 = 2 are u1(t) = tet and u2(t) = e
t
2 . Using the proposed method,
taking ti =
i - 1
N - 1
, i = 1, 2, . . . , N. The numerical results at some selected grid points are given in
Tables 1 and 2.
Example 2. Consider the following nonlinear fractional system:
D\alpha 1u1(t) = u1(t)u
\prime
2(t) + 2et - e2t,
D\alpha 2u2(t) = \mathrm{l}\mathrm{n}(u2(t)) - 2u\prime 1(t) + 3et - 1,
D\alpha 3u3(t) = u22(t)u3(t) - u1(t) + e - t - 1,
(14)
with two-point boundary conditions
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
608 Z. ALTAWALLBEH, M. AL-SMADI, I. KOMASHYNSKA, A. ATEIWI
Table 1. Numerical results for solution u1(t) in Example 1
\alpha 1
t
2 1.9 1.75 1.5
0.1 0.110518 0.110523 0.110998 0.184325
0.2 0.244283 0.244292 0.249768 0.317781
0.3 0.404961 0.404975 0.417821 0.476476
0.4 0.596735 0.596753 0.600455 0.681002
0.5 0.824367 0.824392 0.932478 0.944067
0.6 1.093281 1.093310 1.110259 1.348214
0.7 1.409642 1.409670 1.293007 1.337854
0.8 1.780443 1.780490 1.978111 1.984125
0.9 2.213665 2.213712 2.423435 2.629512
1.0 2.718286 2.718361 2.880985 2.973447
Table 2. Numerical results for solution u2(t) in Example 1
\alpha 2
t
2 1.9 1.75 1.5
0.1 1.05127059 1.05128524 1.051230521 1.051415781
0.2 1.10516988 1.10517581 1.105068425 1.105237792
0.3 1.16183265 1.16185772 1.161668114 1.161423754
0.4 1.22140061 1.22143654 1.221170831 1.221227945
0.5 1.28402267 1.28410678 1.283725058 1.286784210
0.6 1.34985544 1.34987211 1.349486888 1.352745691
0.7 1.41906353 1.41912458 1.418620408 1.419824567
0.8 1.49181998 1.49190054 1.491298115 1.497587212
0.9 1.56830674 1.56843729 1.567701336 1.574500647
1.0 1.64871504 1.64887596 1.648020689 1.654788521
u1(0) = 0, u1(1) = e - 1,
u2(0) = 1, u2(1) = e,
u3(0) = 1, u3(1) =
1
e
,
(15)
where 1 < \alpha i \leq 2, 0 \leq t \leq 1 and ui(t) \in \scrW 3[0, 1], i = 1, 2, 3.
The exact solutions at \alpha 1 = \alpha 2 = \alpha 3 = 2 are u1(t) = et - 1, u2(t) = et and u3(t) = e - t. Using
the proposed method, taking ti =
i - 1
N - 1
, i = 1, 2, . . . , N. The numerical results at some selected
grid points are given in Table 3 and Fig. 1.
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NUMERICAL SOLUTIONS OF FRACTIONAL SYSTEM, TWO-POINT BVPS USING ITERATIVE . . . 609
Table 3. Numerical solutions and errors at \alpha 1 = \alpha 2 = \alpha 3 = 2, for Example 2
u1(t) u2(t) u3(t)
ti
Absolute Error Relative Error Absolute Error Relative Error Absolute Error Relative Error
0.16 9.9129\times 10 - 7 9.1507\times 10 - 7 1.0969\times 10 - 6 5.2652\times 10 - 7 2.2703\times 10 - 6 1.9349\times 10 - 6
0.32 2.0486\times 10 - 6 1.7457\times 10 - 6 2.2681\times 10 - 6 1.0435\times 10 - 6 4.7692\times 10 - 6 3.4672\times 10 - 6
0.48 3.1704\times 10 - 6 2.4939\times 10 - 6 3.5107\times 10 - 6 1.5457\times 10 - 6 7.4503\times 10 - 6 4.6269\times 10 - 6
0.64 4.3776\times 10 - 6 3.1788\times 10 - 6 4.8481\times 10 - 6 2.0394\times 10 - 6 1.0321\times 10 - 5 5.4861\times 10 - 6
0.80 5.6937\times 10 - 6 3.8166\times 10 - 6 6.3057\times 10 - 6 2.5305\times 10 - 6 1.3392\times 10 - 5 6.1058\times 10 - 6
0.96 7.1443\times 10 - 6 4.4207\times 10 - 6 7.9124\times 10 - 6 3.0245\times 10 - 6 1.6677\times 10 - 5 6.5366\times 10 - 6
(a) (b)
Fig. 1. Plots of system in Example 2 at \alpha i = 2, i = 1, 2, 3, n = 101: exact solutions (a) and approximate solutions (b).
The results of numerical analysis are approximate as much as is required within a logical error
ratio that will be stored in a fixed number of digits. It is clear from the tables that the numerical
solutions are in close agreement with the exact solutions for all examples, while the accuracy is in
advanced by using only few term of the RKM iterations. This is an indication of stability of the
presented method. In Fig. 1, the approximation values within a graphically plotted indicate that the
solution approach smoothly to the t-axis by satisfying their boundary conditions. Indeed, decreasing
the step-size increases the accuracy of the results while increasing the time required to simulate the
problem.
5. Concluding remarks. The main concern of this work has been to propose an efficient
numeric technique for the solutions of a class of time-fractional system in Caputo sense subjected
to appropriate boundary conditions. The goal has been achieved by introducing the IRKM to solve
this class of FDEs. A regularization procedure based on the reproducing kernel theory is utilized
to improve the regularity and localization of the method. The behavior of approximate solution for
different values of fractional-order \alpha is shown quantitatively as well as graphically. We can conclude
that the IRKM is powerful and promising technique in finding approximate solution for both linear
and nonlinear problems. In the proposed algorithm, the solution u(t) and the approximate solution
un(t) are represented in the form of series in \scrW 3[0, T ]. Moreover, the approximate solution and its
derivative converge uniformly to the exact solution and its derivative, respectively.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 5
610 Z. ALTAWALLBEH, M. AL-SMADI, I. KOMASHYNSKA, A. ATEIWI
References
1. Yang X. J. Advanced local fractional calculus and its applications. – New York: World Sci. Publ., 2012.
2. Podlubny I. Fractional differential equations. – New York: Acad. Press, 1999.
3. Miller K. S., Ross B. An introduction to the fractional calculus and fractional differential equations. – New York:
John Wiley and Sons, Inc., 1993.
4. Caputo M. Linear models of dissipation whose Q is almost frequency independent, Part II // J. Roy. Austral. Soc. –
1967. – 13. – P. 529 – 539.
5. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and applications of fractional differential equations // North-
Holland Math. Stud. – Amsterdam: Elsevier Sci. B.V., 2006. – 204.
6. Geng F. Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space
method // Appl. Math. and Comput. – 2009. – 215. – P. 2095 – 2102.
7. Geng F., Qian S. P. Reproducing kernel method for singularly perturbed turning point problems having twin boundary
layers // Appl. Math. Lett. – 2013. – 26. – P. 998 – 1004.
8. Gumah G., Moaddy K., AL-Smadi M., Hashim I. Solutions of uncertain Volterra integral equations by fitted reproducing
kernel Hilbert space method // J. Funct. Spaces. – 2016. – Article ID 2920463. – 11 p.
9. Komashynska I., Al-Smadi M. Iterative reproducing kernel method for solving second-order integrodifferential equati-
ons of Fredholm type // J. Appl. Math. – 2014. – 2014. – P. 1 – 11.
10. Abu Arqub O., Al-Smadi M., Momani S. Application of reproducing kernel method for solving nonlinear Fredholm –
Volterra integrodifferential equations // Abstr. and Appl. Anal. – 2012. – 2012. – P. 1 – 16.
11. Abu Arqub O., Al-Smadi M. Numerical algorithm for solving two-point, second-order periodic boundary value
problems for mixed integro-differential equations // Appl. Math. and Comput. – 2014. – 243. – P. 911 – 922.
12. Al-Smadi M., Abu Arqub O., Momani S. A computational method for two-point boundary value problems of fourth-
order mixed integrodifferential equations // Math. Probl. Eng. – 2013. – 2013. – P. 1 – 10.
13. Abu Arqub O., Al-Smadi M., Shawagfeh N. Solving Fredholm integro-differential equations using reproducing kernel
Hilbert space method // Appl. Math. and Comput. – 2013. – 219. – P. 8938 – 8948.
14. Altawallbeh Z., Al-Smadi M., Abu-Gdairi R. Approximate solution of second-order integrodifferential equation of
Volterra type in RKHS method // Int. J. Math. Anal. – 2013. – 7, №. 44. – P. 2145 – 2160.
15. Abu Arqub O., Al-Smadi M., Momani S., Hayat T. Numerical solutions of fuzzy differential equations using reproducing
kernel Hilbert space method // Soft Comput. – 2016. – 20, №. 8. – P. 3283–3302.
16. Abu Arqub O., Al-Smadi M., Momani S., Hayat T. Application of reproducing kernel algorithm for solving second-
order, two-point fuzzy boundary value problems // Soft Comput. – 2016. – P. 1 – 16.
17. Saadeh R., Al-Smadi M., Gumah G., Khalil H., Khan R. A. Numerical investigation for solving two-point fuzzy
boundary value problems by reproducing kernel approach // Appl. Math. and Inform. Sci. – 2016. – 10, №. 6. –
P. 2117 – 2129.
18. Al-Smadi M., Abu Arqub O., Shawagfeh N., Momani S. Numerical investigations for systems of second-order periodic
boundary value problems using reproducing kernel method // Appl. Math. and Comput. – 2016. – 291. – P. 137 – 148.
19. Al-Smadi M., Abu Arqub O., El-Ajou A. A numerical iterative method for solving systems of first-order periodic
boundary value problems // J. Appl. Math. – 2014. – 2014. – P. 1 – 10.
20. Inc M., Akgül A. Approximate solutions for MHD squeezing fluid flow by a novel method // Bound. Value Probl. –
2014. – 18. – P. 1 – 17.
21. Akgül A., Inc M., Karatas E. Reproducing kernel functions for difference equations // Discrete and Contin. Dyn. Syst.
Ser. S. – 2015. – 8, №. 6. – P. 1055 – 1064.
22. Geng F., Cui M. New method based on the HPM and RKHSM for solving forced Duffing equations with integral
boundary conditions // J. Comput. and Appl. Math. – 2009. – 233. – P. 165 – 172.
23. Zhou Y., Cui M., Lin Y. Numerical algorithm for parabolic problems with non-classical conditions // J. Comput. and
Appl. Math. – 2009. – 230. – P. 770 – 780.
24. Moaddy K., Al-Smadi M., Hashim I. A novel representation of the exact solution for differential algebraic equations
system using residual power-series method // Discrete Dyn. Nat. and Soc. – 2015. – Article ID 205207. – 12 p.
25. Al-Smadi M., Altawallbeh Z. Solution of system of Fredholm integro-differential equations by RKHS method // Int.
Contemp J. Math. Sci. – 2013. – 8, №. 11. – P. 531 – 540.
26. Inc M., Akgül A., Geng F. Reproducing kernel Hilbert space method for solving Bratu’s problem // Bull. Malays.
Math. Sci. Soc. – 2015. – 38. – P. 271 – 287.
Received 11.09.16
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|
| id | umjimathkievua-article-1580 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3a/a3ad39eaad3a7b1438119fe7b658433a.pdf |
| spelling | umjimathkievua-article-15802019-12-05T09:19:33Z Numerical solutions of fractional system, two-point BVPs using iterative reproducing kernel algorithm Чисельний розв’язок дробових систем двоточкових граничних задач за допомогою iтеративного вiдновлюючого ядерного алгоритму Al-Smadi, M. Altawallbeh, Z. Ateiwi, A. M. Komashinskaya, I. V. Аль-Смаді, М. Алтавабех, З. Атеіві, А. М. Комашинська, І. В. We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product spaces are constructed in which the boundary conditions of the systems are satisfied. The reproducing kernel functions are constructed to get an accurate algorithm for the investigation of fractional systems. The developed procedure is based on generating the orthonormal basis with an aim to formulate the solution throughout the evolution of the algorithm. The analytic solution is represented in the form of a series in the reproducing kernel Hilbert space with readily computed components. In this connection, some numerical examples are presented to show the good performance and applicability of the developed algorithm. The numerical results indicate that the proposed algorithm is a powerful tool for the solution of fractional models arising in different fields of sciences and engineering. Запропоновано ефективний обчислювальний метод, а саме iтеративний вiдновлюючий ядерний алгоритм для на- ближеного розв’язування систем дробового порядку для двоточкових часових граничних задач у сенсi Капуто. Побудовано два розширенi гiльбертовi простори, в яких виконуються граничнi умови для систем. Також побудо- вано вiдновлювальнi ядернi функцiї, щоб отримати точний алгоритм для вивчення дробових систем. Розроблена процедура базується на генерацiї ортонормального базису з метою формулювання розв’язку для всiєї еволюцiї алгоритму. Аналiтичний розв’язок представлено у виглядi ряду у вiдновлювальному ядерному просторi Гiльберта з компонентами, що легко обчислюються. У зв’язку з цим ми наводимо деякi чисельнi приклади, щоб продемонструвати гарну роботу та застосовнiсть розробленого алгоритму. Чисельнi результати показують, що даний алгоритм є потужним iнструментом для розв’язування дробових моделей, якi з’являютъся в рiзних областях науки i технiки. Institute of Mathematics, NAS of Ukraine 2018-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1580 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 5 (2018); 599-610 Український математичний журнал; Том 70 № 5 (2018); 599-610 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1580/562 Copyright (c) 2018 Al-Smadi M.; Altawallbeh Z.; Ateiwi A. M.; Komashinskaya I. V. |
| spellingShingle | Al-Smadi, M. Altawallbeh, Z. Ateiwi, A. M. Komashinskaya, I. V. Аль-Смаді, М. Алтавабех, З. Атеіві, А. М. Комашинська, І. В. Numerical solutions of fractional system, two-point BVPs using iterative reproducing kernel algorithm |
| title | Numerical solutions of fractional system,
two-point BVPs using iterative reproducing kernel algorithm |
| title_alt | Чисельний розв’язок дробових систем двоточкових
граничних задач за допомогою iтеративного
вiдновлюючого ядерного алгоритму |
| title_full | Numerical solutions of fractional system,
two-point BVPs using iterative reproducing kernel algorithm |
| title_fullStr | Numerical solutions of fractional system,
two-point BVPs using iterative reproducing kernel algorithm |
| title_full_unstemmed | Numerical solutions of fractional system,
two-point BVPs using iterative reproducing kernel algorithm |
| title_short | Numerical solutions of fractional system,
two-point BVPs using iterative reproducing kernel algorithm |
| title_sort | numerical solutions of fractional system,
two-point bvps using iterative reproducing kernel algorithm |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1580 |
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