Numerical Methods for Partial Differential Equations

Received: 28 June 2017

Revised: 21 August 2017

Accepted: 26 August 2017

DOI: 10.1002/num.22209

RESEARCH ARTICLE

Numerical algorithm for solving time-fractional partial integrodifferential equations subject to initial and Dirichlet boundary conditions Omar Abu Arqub1 1 Department

of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan

2 Department

of Applied Science, Ajloun College, Al-Balqa Applied University, Ajloun 26816, Jordan Correspondence Omar Abu Arqub, Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan. Email: [email protected]

Mohammed Al-Smadi2 Recently, many new applications in engineering and science are governed by a series of time-fractional partial integrodifferential equations, which will lead to new challenges for numerical simulation. In this article, we propose and analyze an efficient iterative algorithm for the numerical solutions of such equations subject to initial and Dirichlet boundary conditions. The algorithm provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n-term of exact solutions, numerical solutions of linear and nonlinear time-fractional equations of nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such integrodifferential equations. KEYWORDS

fractional calculus theory, reproducing kernel algorithm, partial integrodifferential equation

Numer Methods Methods Partial Partial Differential Differential Eq. Eq. 2017;1–21. Numer 2018;34:1577–1597. wileyonlinelibrary.com/journal/num wileyonlinelibrary.com/journal/num

© 2017 Wiley Periodicals, 1 © 2017 Wiley Periodicals, Inc.Inc. 1577

2 1578

1

ABU ARQUB AND and AL-SMADI

I NT R O DU C T I O N

The subject of the fractional calculus theory has gained considerable popularity and importance during the past almost two decades, mainly due to their attractive applications in diverse and widespread fields of physics and engineering, such as viscoelasticity, electromagnetic theory, rheology, diffusive transport, electrical networks, and fluid flow [1–5]. Fractional-order models are often more adequate than classical integer-order descriptions, because fractional-order derivatives and integrals embed the description of the memory and hereditary properties of different substances. Consequently, the field of time-fractional partial integrodifferential equations (PIDEs) has attracted interest of researchers in several important phenomenons in chemistry, hydrology, fluid mechanic, physics, gas dynamics, and signal processing (see, for instance, [6–17] and the references therein). Developing analytical and numerical methods for the solutions of time-fractional PIDEs is a very important task. Indeed, it is difficult to obtain exact solutions form in general for most cases. Therefore, attempts have been made to propose analytical methods that approximate the exact solutions of such equations. The main goal of this article is to construct an efficient iterative reproducing kernel algorithm (RKA) to solve a class of time-fractional PIDEs with various order types of the fractional derivatives. More precisely, we consider the following general form of the time-fractional PIDE: ∂tαα u (x, t) + κ1 ∂x u (x, t) + κ2 ∂x22 u (x, t) +

t

0

k (x, t, s) ∂x22 u (x, s) ds = f (x, t, u (x, t)) ,

(1)

subject to the following initial and boundary conditions:

u (x, 0) = ω (x) , u (0, t) = υ1 (t) ,

(2)

u (1, t) = υ2 (t) .

To covering this research precisely, the following assumptions for Equations (1) and (2) will be assumed: • 0 ≤ x, t, s ≤ 1, 0 < α ≤ 1, and κ1 , κ2 ∈ R,

• k (x, t, s) is continuous arbitrary kernel function over the cube [0, 1]3 , • f is continuos real-valued function in the given domain, • u is an unknown function to be determined,

• ω (x), υ1 (t), and υ2 (t) are continuos real-valued functions in the given domain,

• ∂x = ∂/∂x, ∂x22 = ∂ 2 /∂x 2 , and so on,

• ∂tαα denote the Caputo time-fractional derivative of order α. In this aspect, ∂tαα = ∂ α /∂t α and ∂tαα u (x, t) =

1 � (1 − α)

0

t

(t − τ )−α ∂τ u (x, τ ) dτ ,

0 < τ < t,

0 < α < 1.

(3)

A brief survey of the RK theory and the RKA can be summarized shortly as follows: this theory was used first at 1907 as a solver for harmonic and biharmonic Dirichlet problems [18]. In 1950, it was established by knitting it with the RK functions [19]. This theory, which is proxy in the RKA, has been coming out well used to diverse application in applied mathematics and engineering modeling [20–23]. Newly, large-scale researches has been applied the RKA for the solutions of several integral and differential operators alongside with their theories [24–45].

ABU ARQUB and AND AL-SMADI ABU

3 1579

Although the RKA is a relatively new, it has several key features, points of views. Some of its advantages are; first, it can produce good globally smooth numerical solutions, and with ability to solve many fractional PIDE with complex constraint conditions, which are difficult to solve; secondly, the numerical solutions and their derivatives are converge uniformly to the exact solutions and their derivatives, respectively; third, the algorithm is mesh-free, needs no time discretization, and easily implemented. While, the RKA results in two main drawbacks: first, the complexity in obtaining variable coefficients in the corresponding RK functions, especially, for nonhomogeneous constrains conditions; secondly, the RKA normally suffers from computational burden when applied on sequential machines. This means that the time required for solving certain problem will be relatively large. After this brief introduction, the organization of the article is as follows: • For clarity of presentation, in Section 2, several Hilbert spaces and several RK functions are listed, • To formulate the solution, in Section 3, two extended Hilbert spaces and two extended RK functions are fitted, • In Section 4 problem formulation and computational algorithm are presented,

• In Section 5, essential theoretical results are presented to fixtures the stable numerical solution, • To capture the behavior of the numerical solution, error estimations and error bounds are derived in Section 6, • Numerical outcomes are discussed to demonstrate the accuracy and the applicability of the presented algorithm as utilized in Section 7, • Finally, in Section 8 concluding remarks are listed.

2 R E Q U I RE M E N T S F R O M T H E RE PRODUCING KE RNE L TH E O RY The main idea in this section is to present several inner product spaces that satisfying the constraints initial and boundary conditions of the given time-fractional PIDEs. To do so, some basic requirements are presented. Let K be a Hilbert space of real-valued functions defined on a non-empty set �. A function � : � × � → R is a reproducing kernel of K if it satisfies the following: • for each t ∈ �, � (·, t) ∈ K,

• for each θ ∈ K and each t ∈ �, �θ (·) , � (·, t)�K = θ (t).

A Hilbert space which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS). this section, we denote �z�2 = �z (∗) , z (∗)� , where z ∈ , ∗ ∈ [0, 1], and 1 Through 2 1 ∈ W2 , W2 , W2 , W23 . Definition 1 [24] Suppose that z� is in L 2 [0, 1]. The space W21 [0, 1] is defined as W21 [0, 1] = {z = z (t) : z is absolutely continuous function on [0, 1]}. While, its metric system structure lie in �z1 (t) , z2 (t)�W 1 = z1 (0) z2 (0) + 2

1 0

z1� (t) z2� (t) dt.

(4)

4 1580


Theorem 1 [24] The Hilbert space W21 [0, 1] is a complete RK with structure kernel function Rs{1} (t) = 1 + min (s, t) .

(5)

In the simiar fashion, if [0, 1] is the domain space in the x direction, then the space � �21 [0, 1] can be defined as �z1 (x) , z2 (x)�W� 1 = z1 (0) z2 (0) + 1 z1� (x) z2� (x) dx and W 0 2 � Ry{1} (x) = 1 + min (x, y) . Definition 2 [25] Suppose that z�� is in L 2 [0, 1]. The space W22 [0, 1] is defined as W22 [0, 1] = {z = z (t) : z, z� are absolutely continuous functions on [0, 1] and z (0) = 0}. While, its metric system structure lie in �z1 (x) , z2 (x)�W 2 = 2

1 � i=0

z1(i) (0) z2(i) (0) +

�

1 0

z1�� (t) z2�� (t) dt.

(6)

Theorem 2 [25] The Hilbert space W22 [0, 1] is a complete RK with structure kernel function ⎧ ⎨st + 1 st 2 − 1 t 3 , t ≤ s, 2 6 Rs{2} (t) = ⎩st + 1 s2 t − 1 s3 , t > s. 2 6

(7)

Definition 3 [26] Suppose that z�� is in L 2 [0, 1]. The space W23 [0, 1] is defined as W23 [0, 1] = {z = z (x) : z, z� , z�� are absolutely continuous functions on [0, 1] and z (0) = z (1) = 0}. While, its metric system structure lie in �z1 (x) , z2 (x)�W 3 = 2

1 � i=0

z1(i) (0) z2(i) (0) + z1 (1) z2 (1) +

�

1 0

z1�� (x) z2�� (x) dx.

(8)

Theorem 3 [26] The Hilbert space W23 [0, 1] is a complete RK with structure kernel function

Ry{3} (x) =

3

⎧ � � � � �� 2 3 3 3 3 3 3 3 ⎪ ⎨ y x y 126−x −y + y y −10x −5x −24+y +5x y x −24 +x y −24 , ⎪ ⎩

�

�

� � �

�

120 �

x xy2 126−x 3 −y3 + x x 3 −10y3 −5y −24+x 3 120

� � �� +5y x y3 −24 +y x 3 −24

,

x ≤ y, x > y.

(9)

F I T T I N G A P P R O P R I A T E I N N E R PRO DUCT SPACE S

Here, two extended inner product spaces and two extended RK functions are fitted to formulate the solutions � spaces. Henceforth, we denote the following markers: � = [0, 1] ⊗ [0, 1], � in the� �mentioned i+j ∂xi t j = ∂ i /∂x i ∂ j /∂t j , whenever i, j = 1, 2 and �u�2 = �u (∗, ◦) , u (∗, ◦)� , where u ∈ , ∗, ◦ ∈ �, ∈ {H, W }. Next, we combined the results in the previous section to create new RKHSs that satisfying the initial and boundary conditions of Equation (2).


5 1581

Definition 4 Suppose that ∂x33 ∂t33 u is in L 2 (�). The Hilbert space W (�) can be defined as W (�) = {u = u (x, t) : ∂x22 ∂t22 u is complete continuous function in � and u (x, 0) = u (0, t) = u (1, t) = 0}. While, its metric system structure lie in �u1 (x, t) , u2 (x, t)�W =

1 j=0

+

j

j

�∂t j u1 (x, 0) , ∂t j u2 (x, 0) �W 3 2

1 1 0

j=0

1

+

0

j ∂t22 ∂xj u1

1

0

j (0, t) ∂t22 ∂xj u2

(0, t) +

∂t22 u1

(1, t) ∂t22 u2

(1, t) dt

∂x33 ∂t22 u1 (x, t) ∂x33 ∂t22 u2 (x, t) dxdt.

(10)

Theorem 4 The Hilbert space W (�) is a complete RK with structure kernel function R(y,s) (x, t) = Ry{3} (x) Rs{2} (t) ,

(11)

such that for any u (x, t) ∈ W (�), we have �u (x, t) , R(y,s) (x, t)�W = u (y, s) and R(y,s) (x, t) = R(x,t) (y, s), where Ry{3} (x) and Rs{2} (t) are the RK functions of the spaces W23 [0, 1] and W22 [0, 1], respectively. Proof By using properties of the inner product spaces W23 [0, 1] and W22 [0, 1] with respect to the differentials dx and dt, respectively, one can find �u (x, t) , Ry{3} (x) Rs{2} (t)�W =

1 j=0

+ + =

=

2

0

1 0

j=0

+

j=0 1

0

j j ∂t22 ∂xj u (0, t) ∂t22 ∂xj Ry{3}

(0) Rs{2}

(t) +

∂t22 u (1, t) ∂t22 Ry{3}

(1) Rs{2}

(t) dt

∂x33 ∂t22 u (x, t) ∂x33 ∂t22 Ry{3} (x) Rs{2} (t) dxdt

j

j

�∂t j u (x, 0) , Ry{3} (x) ∂t j Rs{2} (0)�W 3 2

1 1 0

1 0

1

j=0

0

1

j ∂t22 ∂xj u (0, t) ∂t22 Rs{2}

j (t) ∂xj Ry{3}

(0) +

∂t22 u (1, t) Ry{3}

(1) ∂t22 Rs{2}

(t) dt

∂x33 ∂t22 u (x, t) ∂x33 Ry{3} (x) ∂t22 Rs{2} (t) dxdt

j

j

∂t j �u (x, 0) , Ry{3} (x)�W 3 ∂t j Rs{2} (0) 2

j=0

+

j

1 1

1

+

j

�∂t j u (x, 0) , ∂t j Ry{3} (x) Rs{2} (0)�W 3

0

1

∂t22 Rs{2}

(t) ∂t22

1 j=0

j j ∂xj u (0, t) ∂xj Ry{3}

(0) +

u (1, t) Ry{3}

(1) +

1 0

∂x33 u (x, t) ∂x33 Ry{3}

(x) dx dt

6 1582


= = =

1 j=0

1 j=0

j

j

∂t j u (y, 0) ∂t j Rs{2} (0) + j

j

∂t j u (y, 0) ∂t j Rs{2} (0) +

�u (y, t) , Rs{2}

1

∂t22 Rs{2} (t) ∂t22 �u (x, t) , Ry{3} (x)�W 3 dt 2

0 1

∂t22 Rs{2} (t) ∂t22 u (y, t) dt

0

(t)�W 2 = u (y, s) .

(12)

2

Thus, �u (x, t) , R(y,s) (x, t)�W = u (y, s). While on the other hand, R(y,s) (x, t) = �R(y,s) (ξ , ζ ) , R(x,t) (ξ , ζ )�W = �R(x,t) (ξ , ζ ) , R(y,s) (ξ , ζ )�W = R(x,t) (y, s). Definition 5 4 Suppose that ∂x ∂t u is in L 2 (�). The Hilbert space H (�) can be defined as H (�) = {u = u (x, t) : u is complete continuous function in �}. While, its metric system structure lie in �u1 (x, t) , u2 (x, t)�H = �u1 (x, 0) , u2 (x, 0)�W 1 2 1 1 1 + ∂t u1 (0, t) ∂t u2 (0, t) dt + ∂xt2 u1 (x, t) ∂xt2 u2 (x, t) dxdt. 0

0

0

(13)

Theorem 5 The Hilbert space H (�) is a complete RK with structure kernel function Ry{1} (x) Rs{1} (t) , r(y,s) (x, t) =

(14)

such that for any u (x, t) ∈ H (�), we have �u (x, t) , r(y,s) (x, t)�H = u (y, s) and 21 [0, 1] Ry{1} (x) and Rs{1} (t) are the RK functions of spaces W r(y,s) (x, t) = r(x,t) (y, s), where 1 and W2 [0, 1], respectively. Proof

Similar to the proof of Theorem 4.

4 P R O B L E M F O R M U L A T I O N AND CO M PUT AT IONAL ALGO RI T H M To set the stage for solving the time-fractional PIDEs of Equations (1) and (2), it’s necessary first to formulate it in a manner not only reflecting the situation being modeled, but so as to be amenable to computational technique. Through the remainder sections, we will use the following markers: P = P (x, t, u (x, t)), Pk = P (xk , tk , u (xk , tk )), and Pnk = P (xk , tk , un (xk , tk )) whenever k = 1, 2, 3, . . . , ∞. To apply the RKA, we must homogenized the nonhomogeneous constraints conditions by suitable transformations. This normalizing is needed to put the mentioned conditions into the space W (�) to make an agreement feasible between the constructed spaces and the corresponding RK functions which are on �, for the convenience, we still denote the solution of the new equation by u (x, t). So, let ∂tαα u (x, t) + κ1 ∂x u (x, t) + κ2 ∂x22 u (x, t) +

t

0

k (x, t, s) ∂x22 u (x, s) ds = P (x, t, u (x, t)) ,

(15)


u (x, 0) = 0, u (0, t) = 0,

u (1, t) = 0.

(16)


7 1583

For the conduct of proceedings, we define the fractional differential linear operator � : W (�) → H (�) such that

�u (x, t) := ∂tαα u (x, t) + κ1 ∂x u (x, t) + κ2 ∂x22 u (x, t) +

0

t

k (x, t, s) ∂x22 u (x, s) ds.

(17)

Thus, the time-fractional PIDEs to be solved is governed by the following equivalent functional equation: �u (x, t) = P (x, t, u (x, t)) .

(18)

To build an orthogonal function systems of the space W (�), we choose a countable dense subset ∗ ∗ {(xi , ti )}∞ i=1 in �, define ϕi (x, t) = r(xi ,ti ) (x, t) and ψi (x, t) = � ϕi (x, t), where � : H (�) → W (�) is the adjoint operator of � and is uniquely determined. ∞ The normalized orthonormal function systems ψ i (x, t) i=1 of W (�) is usually constructed from the process of the Gram-Schmidt orthogonalization of {ψi (x, t)}∞ i=1 as ψ i (x, t) =

i

μik ψk (x, t).

(19)

k=1

To apply the RKA, we divide the finite domain � into a p × q mesh point with the space step size �x = p1 in the x direction of [0, 1] and the time step size �t = q1 in the t direction of [0, 1], respectively, in which p and q are positive integers. Anyhow the grid points (xl , tm ) in the space-time domain � are defined simultaneously as (xl , tm ) = (l�x, m�t) ,

l = 0, 1, 2, · · · , p,

m = 0, 1, 2, · · · , q.

(20)

The numerical solvability of the time-fractional PIDEs presented in this work can be done by applying the following computational algorithm. The steps in the algorithm are explained in more detail in the next sections. Algorithm 1 To approximate the solution un (x, t) of u (x, t) for Equations (18) and (16) on �, do the following main steps. Step 1: Choose n = pq collocation points in the domain �; Step 2: Set ψi (xi , ti ) = �(y,s) R (x, t) |(y,s)=(xi ,ti ) ; Step 3: Obtain the orthogonalization coefficients μik ; Step 4: Set ψ i (xi , ti ) = il=1 μik ψi (xi , ti ) , i = 1, 2, . . . , n; Step 5: Choose an initial approximation u0 (x1 , t1 ); Step 6: Set i = 1; Step 7: Set Bi = il=1 μik Pkk−1 ; Step 8: Set ui (xi , ti ) = ik=1 nj=1 Bk ψ k (xk , tk ); Step 9: If i < η, then set i = i + 1 and go to step 7, else stop.

5

F I X T U RE S O F T H E S T A B L E NUM E RICAL SO L UT IO N

∞ In this section, we will show that {ψi (x, t)}∞ is i=1 is a complete function system, while R(xi ,ti ) (x, t) i=1 a linearly independent in the space W (�). After that, an iterative formulas of obtaining the numerical

8 1584


solutions are utilized. Through the remainder sections, we suppose that the sequence {(xi , ti )}∞ i=1 is dense on �. At first, depending on the the Schwarz inequality it is easy to see that � : W (�) → H (�) is a bounded linear operator, that is ��u (x, t) �2W 1 ≤ M�u�2W with M > 0. For more details see the proof 2 of Lemma 1in [27]. Lemma 1 The sequence {ψi (x, t)}∞ i=1 is a complete function system in W (�) with ψi (x, t) = �(y,s) R (x, t) |(y,s)=(xi ,ti ) .

(21)

Proof Here, �(y,s) indicates that the operator � applies to the function of (y, s). Indeed ψi (x, t) = �∗ ϕi (x, t) = ��∗ ϕi (y, s) , R(x,t) (y, s)�W = �ϕi (y, s) , �(y,s) R(x,t) (y, s)�H = �(y,s) R(x,t) (y, s) |(y,s)=(xi ,ti )

= �(y,s) R(y,s) (x, t) |(y,s)=(xi ,ti ) ∈ W (�) .

(22)

Now, for each fixed u ∈ W (�), let �u (x, t) , ψi (x, t)�W = 0, i = 1, 2, . . . . Then, �u (x, t) , ψi (x, t)�W = �u (x, t) , �∗ ϕi (x, t)�W = ��u (x, t) , ϕi (t)�H = �u (xi , ti ) = 0. −1 While, {(xi , ti )}∞ i=1 is dense on �, we must have �u (x, t) = 0 from the existence of � , it follows that u = 0. Theorem 6 The sequence {R(xi ,ti ) (x, t)}∞ is a linearly independent in W (�). i=1

Proof It is adequate to show that {R(xi ,ti ) (x, t)}m is a linearly independent for each i=1 m ≥ 1. In fact, if {ci }mi=1 satisfies mi=1 ci R(xi ,ti ) (x, t) = 0, taking hk (x, t) ∈ W (�) such that hk (xl , tl ) = δl,k for each l = 1, 2, · · · , m. Then

0 = hk (x, t), = =

m i=1

m i=1

m i=1

ci R(xi ,ti ) (x, t)

W

ci �hk (x, t) , R(xi ,ti ) (x, t)�

W

ci hk (xi , ti ) = ck .

(23)

Thus ck = 0 for k = 1, 2, · · · , m.

In the next theorem, μik are the orthogonalization coefficients of the orthonormal systems ψ ij (t) and these basis can be obtained directly from the Gram-Schmidt orthogonalization process of Equation (19). Theorem 7. Suppose that Ai = 18 and 16, then

i

k=1

μik Pk . If u ∈ W (�) is the solution of Equations

u (x, t) =

∞ i=1

Ai ψ i (x, t) .

(24)


9 1585

∞ Proof Since, �u (x, t) , ϕi (x, t)�W = u (xi , ti ) for ∞each u ∈ W (�), while, i=1 Ai ψ i (x, t) is the Fourier series expansion about ψ i (x, t) i=1 , then it is a convergent in the sense of � · �W . Thus, u (x, t) = = = = = In other words,

∞

i=1

i ∞ i=1 k=1

∞ i=1

�u (x, t) , ψ i (x, t)�W ψ i (x, t)

�u (x, t) ,

i ∞ i=1 k=1

i ∞

i k=1

μik ψk (x, t) �W ψ i (x, t)

μik �u (x, t) , �∗ ϕk (x, t)�W ψ i (x, t) μik �u (xk , tk ) ψ i (x, t)

i=1 k=1

∞

Ai ψ i (x, t) .

(25)

i=1

Ai ψ i (x, t) is the exact solution of Equations (18) and (16).

For numerical computations, put (x1 , t1 ) = (0, 0), then from the boundary conditions of Equation (16), the value of u (x1 , t1 ) is known. Set u0 (x1 , t1 ) = u (x1 , t1 ) and define the n -term numerical solution of u (x, t) using the truncating version as un (x, t) =

n

Ai ψ i (x, t) .

(26)

i=1

In order that W (�) is a Hilbert space, then the series ∞ i=1 Ai ψ i (x, t) < ∞. Thus, we can guarantee that the numerical solution un (x, t) satisfies the constraints conditions of Equation (16).

6

C O NVE R G E N C E A N A L Y S I S A ND E RROR E ST IM AT IO NS

In this part, the theoretical basis of the proposed algorithm under some hypotheses for the obtained approximation is studied in terms of convergence and error estimates in the space W (�). Because �un �W is bounded, {(xi , ti )}∞ i=1 is dense on �, and the solution of Equations (18) and (16) is unique. Then the n-term numerical solution un (x, t) must converges to the exact solution u (x, t) as n → ∞. j

Theorem 8 The partial derivatives of the numerical solution ∂xi i ∂t j un (x, t) are conj verging uniformly to the partial derivatives of the exact solution ∂xi i ∂t j u (x, t), whenever i = 0, 1, 2, j = 0, 1 as n → ∞. Proof Since W (�) is a Hilbert space, from Equations (24) and (26), it is follows that, �u − un �W → 0 as n → ∞. Again, since i j j j ∂xi ∂t j u (x, t) − ∂xi i ∂t j un (x, t) = �u (y, s) − un (y, s) , ∂xi i ∂t j �R(x,t) (y, s)�W

10 1586

ABU ARQUB AND and AL-SMADI j

≤ �u − un �W �∂xi i ∂t j �R(x,t) (y, s) �W ≤ Mi,j �u − un �W .

(27)

j j Thus, ∂xi i ∂t j u (x, t) − ∂xi i ∂t j un (x, t) → 0 as n → ∞.

For errors estimations, the following assumptions have been made in defining the proof of requested 1 1 with xi = (i − 1) hx and ht = n−1 with tj = (j − 1) ht , and n is a positive two next theorems: hx = n−1 integer, where i = 1, 2, . . . , n and j = 1, 2, . . . , n. In the case of two variables functions type, we may use a norm that represents the maximum absolute value at each point, that is �u (x, t) �∞ = max |u (x, t)|. (x,t)∈�

Theorem 9 Let u (x, t) and un (x, t) be the exact and the numerical solutions of Equations (18) and (16), respectively. If u (x, t) − un (x, t) is the nature error at (x, t) ∈ � in the space W (�), then there is a positive constant C such that �u (x, t) − un (x, t) �∞ ≤ C hx2 + hx ht .

(28)

Proof In each subdomain [xi , xi+1 ] × tj , tj+1 ⊂ �, one can write

∂x u (x, t) − ∂x un (x, t) = ∂x u (x, t) − ∂x u xi , tj + ∂x un xi , tj − ∂x un (x, t) + ∂x u xi , tj − ∂x un xi , tj . (29) Expand ∂x u (x, t) about the point xi , tj using Taylor series of function of two variables, we get ∂x u (x, t) = ∂x u xi , tj + (hx ∂x + ht ∂t ) ∂x u xi + ζ hx , tj + ζ ht + · · · , ζ ∈ [0, 1] .

(30)

By means of the continuation of ∂xt u and ∂x22 u on �, one can show that �∂x u (x, t) − ∂x un xi , tj �∞ = O (hx + ht ) .

(31)

On the other aspect as well, we know that ∂x un xi , tj − ∂x un (x, t) ≤

x t 2 |∂xs un (x, s)| ds. ∂y2 un y, tj dy + xi

tj

(32)

In the means of ∞ th norm, it follows that

�∂x un xi , tj − ∂x un (x, t) �∞ = O (hx + ht ) .

(33)

Again, given any ε > 0, using Theorem 8, there is an n sufficiently large such that �∂x u xi , tj − ∂x un xi , tj �∞ < ε.

(34)

As ε is arbitrary and by combining Equations (29)–(34) for the chosen value of n, we must have �∂x u (x, t) − ∂x un (x, t) �∞ = O (hx + ht ) .

(35)


11 1587

Using the integral property for differentiable functions, we have u (x, t) − un (x, t) = u (xi , t) − un (xi , t) +

xi

x

∂y u (y, t) − ∂y un (y, t) dy.

(36)

Using Equations (35) and (36) and applying Theorem 8, we can straightforwardly see that �u (x, t) − un (x, t) �∞ ≤ C hx2 + hx ht ,

where C > 0.

(37)

Theorem 10 Let u (x, t) and un (x, t) be the exact and the numerical solutions of Equations (18) and (16), respectively. If u (x, t) − un (x, t) is nature error at (x, t) ∈ � in the space W (�) with ∂x33 ∂t u (x, t) , ∂x22 ∂t22 u (x, t) ∈ (�) such that �∂x33 ∂t u (x, t) �∞ and �∂x22 ∂t22 u (x, t) �∞ are bounded, then there is a positive constant D such that �u (x, t) − un (x, t) �∞ ≤ D ht hx3 + ht2 hx2 .

(38)

Proof In each subdomain [xi , xi+1 ] × tj , tj+1 ⊂ �, one can write

∂x22 ∂t u (x, t) − ∂x22 ∂t un (x, t) = ∂x22 ∂t u (x, t) − ∂x22 ∂t u xi , tj + ∂x22 ∂t un xi , tj − ∂x22 ∂t un (x, t) + ∂x22 ∂t u xi , tj − ∂x22 ∂t un xi , tj . (39) Expand ∂x22 ∂t u (x, t) about the point xi , tj using Taylor series of function of two variables, we get

∂x22 ∂t u (x, t) = ∂x22 ∂t u xi , tj + (hx ∂x + ht ∂t ) ∂x22 ∂t u xi + ζ hx , tj + ζ ht + · · · , ζ ∈ [0, 1] .

(40)

By means of the continuation of ∂x33 ∂t u and ∂x22 ∂t22 u on �, one can show that �∂x22 ∂t u (x, t) − ∂x22 ∂t un xi , tj �∞ = O (hx + ht ) .

(41)

On the other aspect as well, one can write 2 ∂ 2 ∂t un xi , tj − ∂ 22 ∂t un (x, t) ≤ x

x

xi

x

3 ∂ 3 ∂t un y, tj dy + x

tj

In the means of ∞ th norm, it follows that

t

2 2 ∂ 2 ∂ 2 un (x, s) ds. x

�∂x22 ∂t un xi , tj − ∂x22 ∂t un (x, t) �∞ = O (hx + ht ) .

s

(42)

(43)

Again, given any ε > 0, using Theorem 8, there is an n sufficiently large such that �∂x22 ∂t u xi , tj − ∂x22 ∂t un xi , tj �∞ < ε.

(44)

12 1588


As ε is arbitrary and by combining Equations (39)–(44) for the chosen value of n, we must have �∂x22 ∂t u (x, t) − ∂x22 ∂t un (x, t) �∞ = O (hx + ht ) .

(45)

Using the integral property for differentiable functions, we have x

∂x ∂t u (x, t) − ∂x ∂t un (x, t) = ∂x ∂t u (xi , t) − ∂x ∂t un (xi , t) + ∂t u (x, t) − ∂t un (x, t) = ∂t u (xi , t) − ∂t un (xi , t) + u (x, t) − un (x, t) = u (x, ti ) − un (x, ti ) +

xi x

xi

t

ti

∂y22 ∂t u (y, t) − ∂y22 ∂t un (y, t) dy. (46)

∂y ∂t u (y, t) − ∂y ∂t un (y, t) dy.

(∂s u (x, s) − ∂s un (x, s)) ds.

(47) (48)

Using Equations (45) and (48) and applying Theorem 8, we can straightforwardly see that

where D > 0.

�u (x, t) − un (x, t) �∞ ≤ D ht hx3 + ht2 hx2 ,

(49)

Theorem 11 Let u (x, t) and un (x, t) be the exact and the numerical solutions of Equations (18) and (16), respectively. If u (x, t) − un (x, t) is the nature error at (x, t) ∈ � in the space W (�), then {u (x, t) − un (x, t)}∞ n=1 is decreasing in the sense of the norm of W (�) and �u (x, t) − un (x, t) �W → 0 as n → ∞. Proof

Using prevoius results described in this article, one can write �u (x, t) −

un (x, t) �2W

=

∞

i=n+1

i k=1

μik Pk

2

.

(50)

Consequently {u (x, t) − un (x, t)}∞ n=1 is decreasing in the sense of the norm of � · �W . ∞ In addition, since i=1 Ai ψ i (x, t) is convergent. Thus, �u (x, t) − un (x, t) �2W → 0 as n → ∞.

7

N UME R I C A L O U T C O M E S

In numerical analysis problems there are some basic unknowns. If they are found, the behavior of the entire structure can be predicted. The basic unknowns or the field variables which are encountered are displacements in the applied mathematics and engineering problems. This section presents the numerical solutions for three different time-fractional PIDEs using the RKA. The results reveal that the algorithm is highly accurate, rapidly converge, and convenient to handle various physical problems in fractional calculus. Next, throughout the numerical computations; some graphical results, tabulate data, and numerical comparison are presented and discussed quantitatively at some selected grid points on � to illustrate the numerical solutions for the following time-fractional PIDEs. In the process of computation, all the symbolic and numerical computations are performed by using Mathematics 9 software package.


Example 1

13 1589

Consider the following linear time-fractional PIDE:

∂tαα u (x, t) + u (x, t) − ∂x u (x, t) + ∂x22 u (x, t) +

x

0

ex+s sα ∂x22 u (x, s) ds = g (x, t) ,

(51)


u (x, 0) = 1 − x 2 , 1 u (0, t) = 1 + �(1+α) tα ,

u (1, t) =

1 tα , �(1+α)

(52)

1 where 0 ≤ x, t ≤ 1 and 1 < α ≤ 2. Here, the exact solution is u (x, t) = 1 − x 2 + �(1+α) tα .

Example 2 ∂tαα u (x, t)

Consider the following nonlinear time-fractional PIDE: 3

2

+ u (x, t) + u (x, t) +

∂x22 u (x, t)

−

0

x

(x + t) ex sα ∂x22 u (x, s) ds = g (x, t) ,

(53)


u (x, 0) = 0, u (0, t) = 0,

(54)

u (1, t) = 0,

where 0 ≤ x, t ≤ 1 and 1 < α ≤ 2. Here, the exact solution is u (x, t) = x2 (1 − x)3 t 3α+1 . Example 3

Consider the following nonlinear time-fractional PIDE:

∂tαα u (x, t) + eu(x,t) + cos u (x, t) + ∂x u (x, t) − 2∂x22 u (x, t) −

0

x

(t + s) ∂x22 u (x, s) ds = g (x, t) , (55)

subject to the following initial and boundary conditions: u (x, 0) = 0, u (0, t) = t 2α ,

u (1, t) = cos (7) t 2α ,

where 0 ≤ x, t ≤ 1 and 1 < α ≤ 2. Here, the exact solution is u (x, t) = t 2α cos (7x). Remark 1 In the previous examples, the function g (x, t) is chosen, so that, the given exact solution is satisfied in the both LHS and RHS. The RKA presented in this article is used to numerically solve the given time-fractional PIDE. The input data to the algorithm is divided into two parts; the RKA related parameters and the time-fractional PIDE related parameters. Anyhow, the input data to the algorithm are as follows:

(56)

14 1590


Parameter or concept � = [0, 1] ⊗ [0, 1] p = 20 q = 20 �x = 1/20 �t = 1/20 (xl , tm ) = (l�x, m�t) α ∈ (0, 1] ω (x) υ1 (t) υ2 (t) R(y,s) (x, t) = Ry{3} (x) Rs{2} (t) Ry{1} (x) Rs{1} (t) r(y,s) (x, t) = � μik

Description Dirct product between the x direction domain and the t direction domain The number of mishpoint in the x direction domain of [0, 1] The number of mishpoint in the t direction domain of [0, 1] The space step size in the x direction domain of [0, 1] The space step size in the x direction domain of [0, 1] The two-dimensional partition of the domain � in un (xl , tm ) The order of the fractional derivative in the Caputo sense The function form of initial condition at t = 0 The function form of boundary condition at x = 0 The function form of boundary condition at x = 1 The structure of the RK function of the Hilbert space W (�) The structure of the RK function of the Hilbert space H (�) The orthogonalization coefficients of the orthonormal systems ψ ij (t)

For more detail and clarification, through this section, we will do the following main analyzes: • The absolute errors of approximating u (x, t) are tabulated for Examples 1, 2, and 3.

• The absolute errors of approximating ∂t u (x, t), ∂x u (x, t), and ∂x22 u (x, t) are tabulated for Example 1.

• The approximate solutions are sketched for the Examples 1 and 2. • The absolute value of εn (x, t) are plotted for Example 1.

• The effect of number of nodes on the values of �u (x, t) �∞ are tabulated for Example 3. With a view to demonstrate the agreement between the exact and the RKA approximate solutions, Tables 1, 2, and 3 show the absolute error of approximate solution of Examples 1, 2, and 3, respectively, obtained at various (x, t) in � when α ∈ {0.25, 0.5, 0.75, 1}. From the table, it can be seen that the error estimate show that the accuracy of the numerical solutions is closely related to the fill time tm = m�t, m = 0, 1, . . . , q, where �t = q1 and fill distance xl = l�x, l = 0, 1, . . . , p, where �x = p1 . So, more accurate numerical solutions can be obtained using more mesh points. In the RKA, it is possible to pick any point in the domain �, and as well the n -term approximate partial derivatives ⎧ n � ⎪ ⎪ ⎪ Ai ψ i (x, t) , ⎪ ⎨∂t un (x, t) = i=1

n � ⎪ ⎪ ω ⎪ ⎪ t) = ∂ u Ai ∂xωω ψ i (x, t) , ω = 1, 2, (x, ⎩ xω n i=1

are applicable. In Table 4, numerical results of approximating these derivatives for Example 2 at various values of (x, t) ∈ � have been tabulated. In fact, these approximate partial

(57)


T A B LE 1 x 0.2

t 0.25 0.5 0.75 1

0.4

0.25 0.5 0.75 1

0.6

0.25 0.5 0.75 1

0.8

0.25 0.5 0.75 1

15 1591

Absolute errors of approximating the solution in Example 1 over the domain � α = 0.25

8.4857818172 × 10

−7

7.3853535349 × 10

−7

9.7840259738 × 10

−7

4.0140404038 × 10

−7

1.2142348482 × 10

−6

α = 0.5

4.7531272718 × 10

−7

5.1742572056 × 10

−7

9.2799151474 × 10

−7

α = 0.75

9.6848484787 × 10

−8

3.2767676765 × 10

−7

9.9121212127 × 10

−7

3.6527030312 × 10

−7

3.8233005800 × 10

−7

α=1

3.0438449317 × 10−8 1.2548974615 × 10−7

9.0535757558 × 10−7

8.6412748604 × 10−7

6.3541125542 × 10−7

2.9128773570 × 10−7

−7

−7

−8

3.9917575689 × 10−8

2.5321212127 × 10

2.2239393949 × 10 3.9648484851 × 10

−7

1.0102121206 × 10

−6

6.0612728170 × 10

−7

9.0795460039 × 10

−7

3.0560611513 × 10

−7

7.5509242424 × 10

−7

6.1121212536 × 10−7

5.2690700462 × 10−7

−7

−8

1.3697636359 × 10

1.4969090412 × 10

−7

1.6148049429 × 10

−6

5.4449515079 × 10

−7

8.0210606057 × 10

−7

9.4954546448 × 10−7 8.0994110169 × 10

−8

6.5453151626 × 10−7

9.1315347950 × 10

8.7386320666 × 10−7 7.0962207573 × 10

−8

5.9898788152 × 10−7

7.7837878746 × 10

3.0240587643 × 10−7 8.7920875518 × 10

−8

4.7655843716 × 10

−7

8.3397726502 × 10

−7

6.1439393573 × 10−7 3.2819214013 × 10

−8

6.2744873075 × 10

−8

7.3241048258 × 10

−7

5.8592308783 × 10−7

5.3826831437 × 10−7

1.1863939342 × 10−7 5.1636886338 × 10−7 6.1172809161 × 10−7

5.2690706204 × 10−8 3.3950908751 × 10−7 3.7646923801 × 10−7

7.4340908984 × 10−7 1.5219224658 × 10−8 4.5450309355 × 10−8

1.4564386774 × 10−7

2.6913415740 × 10−7

derivatives are converge uniformly to exact partial derivatives and completely agree well with Theorem 8. The geometric behaviors of the memory and hereditary properties of the RKA solutions and their level characteristics are investigated next. Anyhow, the comparisons of between the computational values of the RKA solutions for different values of α ∈ {0.25, 0.5, 0.75, 1} for Example 2 have been depicted on the domain � as shown in Figure 1, while, Figure 2 show the comparisons of for Example 3 on the same domain and at the same fractional level. T A B LE 2 x 0.2

t 0.25 0.5 0.75 1

0.4

0.25 0.5 0.75 1

0.6

0.25 0.5 0.75 1

0.8

0.25 0.5 0.75 1

Absolute errors of approximating the solution in Example 2 over the domain � α = 0.25

7.6703559656 × 10

−8

5.1472977725 × 10

−7

1.0426235519 × 10

−7

5.6622695799 × 10

−7

4.3005412840 × 10

−8

3.1837986369 × 10

−7

6.0150773051 × 10−7 4.7740537084 × 10

−7

9.6887553428 × 10−7 2.9326522899 × 10

−7

9.9757877351 × 10

−7

3.9242161187 × 10

−7

5.1872185301 × 10

−7

9.9167175582 × 10

−7

2.4735624127 × 10−7 4.9334339947 × 10

−7

α = 0.5

5.8511620438 × 10

−8

3.6535738150 × 10

−7

2.0213895542 × 10

−8

2.9547221225 × 10

−7

5.2602206423 × 10−7 2.1385227655 × 10

−7

5.0875177320 × 10

−7

4.8366651351 × 10

−8

7.8709207676 × 10

−7

7.5835358704 × 10−7 5.4696741114 × 10

−7

9.9661977390 × 10

−7

3.4228651236 × 10

−7

3.5555245614 × 10

−7

2.1631449445 × 10−7 4.7804845941 × 10

−7

α = 0.75

5.6472355681 × 10

−9

2.7797158446 × 10

−7

1.2402194604 × 10

−8

1.2629963742 × 10

−8

4.1199552925 × 10−7 2.0722229477 × 10

−7

5.0746201752 × 10

−7

3.7101537853 × 10

−8

6.7223310030 × 10

−7

7.1298183402 × 10−7 5.6446738128 × 10

−7

9.9617658300 × 10

−7

1.5725874252 × 10

−7

2.2995938642 × 10

−7

1.8364442922 × 10−7

2.9479942500 × 10

−7

α=1

5.0212167131 × 10−9 1.2517309477 × 10−8 8.5871290745 × 10−8 3.2519767235 × 10−7 1.1027361277 × 10−8 2.0395818146 × 10−8 5.5751695949 × 10−7 6.9103446289 × 10−7 3.7435447187 × 10−8 4.1328852261 × 10−7

4.2651149834 × 10−7 9.9273787557 × 10−7 6.7006823201 × 10−8 1.4403573889 × 10−7 1.6758132181 × 10−7

1.6715949092 × 10−7

16 1592


T A B LE 3 x

Absolute errors of approximating the solution in Example 3 over the domain α = 0.25

t

0.2

0.25 0.5 0.75 1

0.4

0.25 0.5 0.75 1

0.6

0.25 0.5 0.75 1

0.8

0.25 0.5 0.75 1

T A B LE 4

9.3010091651 × 10

−7

3.3381141273 × 10

−7

3.3379932063 × 10

−7

9.3008926916 × 10

−7

3.2760805743 × 10

−6

9.2797295262 × 10

−7

2.0758316852 × 10

−6

9.2797935644 × 10−7 −6

3.2761687906 × 10

3.5561494238 × 10−7 2.9427119793 × 10

−6

3.5563246170 × 10

−7

3.0569597571 × 10

−6

8.7615774299 × 10

−7

8.7616745377 × 10−7 3.0571377901 × 10

−6

α = 0.5

8.8823193534 × 10

−7

2.8203745606 × 10

−7

8.8822377431 × 10

−7

3.4045676722 × 10

−7

9.8641871604 × 10

−7

3.1805767042 × 10

−7

2.8206209435 × 10

−7

9.8643767621 × 10−7

3.4047059860 × 10

−7

8.9520450426 × 10−7 3.1808178025 × 10

−7

8.9518484157 × 10

−7

2.8172781510 × 10

−7

8.5087741276 × 10−7 2.8173926056 × 10 8.5086105818 × 10

−7 −7

α = 0.75

9.4766061134 × 10

−7

5.6943964206 × 10

−7

9.4764983569 × 10

−7

5.4764155218 × 10

−7

9.1858052889 × 10

−7

2.0291319114 × 10

−7

7.8954905758 × 10

−7

6.3489082214 × 10

−7

9.7722694226 × 10

−7

6.6016174460 × 10

−7

9.1858919591 × 10−7 5.4764956770 × 10

−7

7.8954223407 × 10−7

1.6791057044 × 10

−7

α=1

8.4029904435 × 10−7

5.4042223674 × 10−7

5.4040694270 × 10−7 8.4029259351 × 10−7 7.7100246166 × 10−7 4.6615522266 × 10−7 4.6614203043 × 10−7

7.7099654280 × 10−7 1.4955493064 × 10−7 6.5768851043 × 10−7

6.5769776032 × 10−7 1.4956945083 × 10−7

9.7721849679 × 10−7

8.1791128844 × 10−7

−7

1.5375115564 × 10−7

6.3490711004 × 10

1.5373430045 × 10−7 6.9100368287 × 10−7

Absolute errors of approximating the partial derivatives in Example 2 over the domain ∂x22 u (x, t)

x

t

∂t u (x, t)

0

0.25

9.9691734867 × 10−9

9.3168265829 × 10−8

8.0228267891 × 10−7

2.4024599443 × 10

−7

2.2601637976 × 10

−7

2.7622878738 × 10−7

6.0876898522 × 10

−9

4.9485273657 × 10

−8

7.6150954037 × 10

−7

3.3291666757 × 10

−8

3.0579166665 × 10

−7

0.5 0.75 1 0.2

0.25 0.5 0.75 1

0.4

0.25 0.5 0.75 1

0.6

0.25 0.5 0.75 1

0.8

0.25 0.5 0.75 1

1

0.25 0.5 0.75 1

9.6149141537 × 10

∂x u (x, t) −8

2.6386070084 × 10−7 2.5097949163 × 10

−8

5.8257547118 × 10

−8

2.8368880578 × 10

−9

4.8644344242 × 10

−7

2.4078119445 × 10

−7

6.7269374829 × 10−7 6.5644349823 × 10

−7

1.0765366287 × 10−7

7.8957547034 × 10−7

7.7122397601 × 10

−7

1.3516666675 × 10

−7

3.3259834131 × 10

−7

5.4516666648 × 10

−7

9.0111111165 × 10

−8

2.2224666663 × 10

−7

2.7258333324 × 10

−7

5.0224666670 × 10

−7

8.1706911616 × 10

−7

5.1905022079 × 10

−8

4.2136199487 × 10

−9

4.4304042051 × 10

−7

9.3307427777 × 10

−8

6.0665463873 × 10

−7

3.7883161924 × 10

−8

8.5180454081 × 10

−8

2.6633333405 × 10−8 1.7473809523 × 10

−7

1.5145880085 × 10−7

4.7518584490 × 10

−8

8.1809207567 × 10−9 2.0135633620 × 10

−9

5.4886666767 × 10−8 3.4816666661 × 10

−7

2.7621878117 × 10−7 8.0227263218 × 10−7 8.3338586253 × 10−7

7.6616750282 × 10−7

2.3340625070 × 10−7 7.6616546331 × 10−7

1.8073113650 × 10−7 8.1402279672 × 10−7 8.1403424534 × 10−7 1.8074868358 × 10−7 9.9583757551 × 10−7 5.6942908471 × 10−7 6.6014955128 × 10−7

9.9585511504 × 10−7

5.0124569613 × 10−7

6.9933408020 × 10−7

−8

9.0619201294 × 10−7

1.8254686296 × 10

9.5185762228 × 10−7 3.9851823333 × 10−7

3.4163896677 × 10−8

4.7135733292 × 10−7

−8

6.2156993148 × 10−7

9.7760445997 × 10

6.0904613105 × 10−7 4.6999968306 × 10−7


17 1593

F I GU RE 1 Comparisons of between the computational values of the RKA solutions at various values of (x, t) ∈ � when α ∈ {0.25, 0.5, 0.75, 1} for Example 2: black α = 1; blue: α = 0.75; green: α = 0.5; red: α = 0.25 [Color figure can be viewed at wileyonlinelibrary.com]

It is clear from the Figures 1 and 2 that each of the graphs are nearly coinciding and similar in their behaviors with good agreement with the RKA solutions when the ordinary derivatives are considered. While, one can note that the RKA solutions are continuously depends on the time-fractional derivatives assumed. As a fast scan, the process described by the RKA is slightly more skewed to the down and drag to the right than that modeled by the standard equations (when ordinary derivatives are considered). In the meantime, from the graphs, it can be seen that, the RKA solutions are stable and convergent. The graph planning of the absolute error function εn (x, t) = |u (x, t) − un (x, t)| and its level characteristics is discussed geometrically at the middle space x direction. To specific, the n-term of εn (x, t) is computed as follows: 20 1 α εn (0.5, t) = 0.75 + Ai ψ i (0.5, t) . t − � (1 + α) i=1

Figure 3 gives the relevant data of the RKA results at various value of (0.5, t) ∈ � when α ∈ {0.25, 0.5, 0.75, 1} for Example 1. Regarding the convergence speed, it is obvious that the difference between the exact and the RKA nodal values decreases initially till a maximum level fractional derivatives values are reached.

(58)

18 1594


F I G URE 2 Comparisons of between the computational values of the RKA solutions at various values of (x, t) ∈ � when α ∈ {0.25, 0.5, 0.75, 1} for Example 3: black α = 1; blue: α = 0.75; green: α = 0.5; red: α = 0.25 [Color figure can be viewed at wileyonlinelibrary.com]

The last target is to show the effect of number of nodes on behavior of �u (x, t) �∞ in order to check numerically the order of convergence of the RKA. Anyhow, Table 5 gives the relevant data for Example 3 at n ∈ {25, 100, 225, 400, 625}. It is observed that 1 1 the reduction in the step size, n = pq = �x , of � results in a reduction in the error �t and correspondingly an improvement in the accuracy of the obtained solution. This goes in agreement with the known fact that the error is monotone decreasing where more accurate solutions are achieved using a reduction in the step size, while, the cost to be paid while going in this direction is the rapid increase in the number of iterations required for convergence.

8

CO NCL U D I N G R E M A R K S

The RKA based on several effects such as: RK functions, constraints conditions functions type, orthogonal function systems, and orthonormal generation basis. The intrinsic worth of the proposed algorithm lies in its ability to handle efficiently and reliably the major challenges associated with the timefractional PIDEs in terms of highly nonlinearity, nonhomogeneity, fractional level characteristics, and the nature of BCs appear. The solutions were represented in the form of series in the extened inner product spaces which proved higher accuracy in numerical computations. It is observed that the gained solutions bifurcate and produce similar patterns when α ∈ (0, 1] and the patterns coincides when α is


19 1595

F I GU RE 3 The absolute value of the error function εn (0.5, t) of the RKA solutions at various values of t ∈ [0, 1] when α ∈ {0.25, 0.5, 0.75, 1} for Example 1: black α = 1; blue: α = 0.75; green: α = 0.5; red: α = 0.25 [Color figure can be viewed at wileyonlinelibrary.com]

T A B LE 5

The effect of number of nodes on the values of �u (x, t) �∞ in Example 3 over the domain �

�x

�t

1/5

1/5

1/10

1/10

1/15 1/20 1/25

1/15 1/20 1/25

α = 0.25

α = 0.5

α = 0.75

α=1

3.93468 × 10−3

6.14499 × 10−4

4.49426 × 10−4

9.40120 × 10−4

−5

2.65936 × 10

−6

1.15969 × 10

−6

6.67421 × 10−6

7.35228 × 10

−8

6.07875 × 10

−8

6.96755 × 10−4

8.08924 × 10 3.27617 × 10 9.73485 × 10

−6 −7

4.86355 × 10−5 9.86438 × 10

−7

7.65257 × 10−5 9.77227 × 10

−7

9.14145 × 10−5 8.40299 × 10−7 3.72427 × 10−8

close to 1. The comparative studies based on the absolute natural error function sense shows that the RKA approximate values are much acceptable in terms of accuracy and stability. As our findings suggest, this series solution methodology can be applied to much more complicated nonlinear fractional PIDEs.

ACKNOW LEDGMEN T The authors would like to express their gratitude to the unknown referees for carefully reading the paper and their helpful comments.

ORCI D Omar Abu Arqub

http://orcid.org/0000-0001-9526-6095

20 1596


R E FERENCES [1] F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London, UK, 2010. [2] G. M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford University Press, Oxford, UK, 2005. [3] I. Podlubny, Fractional differential equations, Academic Press, San Diego, CA, USA, 1999. [4] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives theory and applications, Gordon and Breach, New York, 1993. [5] A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, Netherlands, 2006. [6] S. Arshed, B-spline solution of fractional integro partial differential equation with a weakly singular kernel, Numer. Methods Partial Differ. Equ. vol. 33 (2017) pp. 1565–1581. [7] Y. Rostami, K. Maleknejad, Numerical solution of partial integro-differential equations by using projection method, Mediterr. J. Math. vol. 14 (2017) p. 113. [8] L. Huang, X. F. Li, Y. Zhao, X.Y. Duan, Approximate solution of fractional integro-differential equations by Taylor expansion method, Comput. Math. Appl. vol. 62 (2011) pp. 1127–1134. [9] D. S. Mohammed, Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomial, Math. Probl. Eng. vol. 2014 (2014) p. 5. [10] S. Momani, R. Qaralleh, An efficient method for solving systems of fractional integro-differential equations, Comput. Math. Appl. vol. 52 (2006) pp. 459–470. [11] E. Tohidi, M. M. Ezadkhah, S. Shateyi, Numerical solution of nonlinear fractional Volterra integro-differential equations via Bernoulli polynomials, Abstr. Appl. Anal. vol. 2014 (2014) p. 7. [12] Y. Wang, L. Zhu, Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method, Adv. Differ. Equ. vol. 2017 (2017) p. 27. [13] O. Abu Arqub, A. El-Ajou, S. Momani, Constructing and predicting solitary pattern solutions for nonlinear time-fractional dispersive partial differential equations, J. Comput. Phys. vol. 293 (2015) pp. 385–399. [14] A. El-Ajou, O. Abu Arqub, S. Momani, D. Baleanu, A. Alsaedi, A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl. Math. Comput. vol. 257 (2015) pp. 119–133. [15] A. El-Ajou, O. Abu Arqub, S. Momani, Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm, J. Comput. Phys. vol. 293 (2015) pp. 81–95. [16] S. S. Ray, New exact solutions of nonlinear fractional acoustic wave equations in ultrasound, Comput. Math. Appl. vol. 71 (2016) pp. 859–868. [17] M. D. Ortigueira, J. A. T. Machado, Fractional signal processing and applications, Signal Process. vol. 83 (2003) pp. 2285–2286. [18] S. Zaremba, L’equation biharminique et une class remarquable defonctionsfoundamentals harmoniques, Bull. Int. Acad. Sci. Cracov. vol. 39 (1907) pp. 147–196. [19] N. Aronszajn, Theory of reproducing kernels, Trans. Am. Math. Soc. vol. 68 (1950) pp. 337–404. [20] M. Cui, Y. Lin, Nonlinear numerical analysis in the reproducing kernel space, Nova Science, New York, NY, USA, 2009. [21] A. Berlinet, C. T. Agnan, Reproducing Kernel Hilbert space in probability and statistics, Kluwer Academic Publishers, Boston, Mass, USA, 2004. [22] A. Daniel, Reproducing kernel spaces and applications, Springer, Basel, Switzerland, 2003. [23] H. L. Weinert, Reproducing Kernel Hilbert spaces: Applications in statistical signal processing, Hutchinson Ross, 1982. [24] Y. Lin, M. Cui, L. Yang, Representation of the exact solution for a kind of nonlinear partial differential equations, Appl. Math. Lett. vol. 19 (2006) pp. 808–813. [25] Y. Zhoua, M. Cui, Y. Lin, Numerical algorithm for parabolic problems with non-classical conditions, J. Comput. Appl. Math. vol. 230 (2009) pp. 770–780. [26] L. H. Yang, Y. Lin, Reproducing kernel methods for solving linear initial-boundary-value problems, Electron. J. Differ. Equ. vol. 2008 (2008) pp. 1–11.


21 1597

[27] O. Abu Arqub, Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of timefractional partial differential equations subject to initial and Neumann boundary conditions, Comput. Math. Appl. vo. 73 (2017) pp. 1243–1261. [28] O. Abu Arqub, H. Rashaideh, The RKHS method for numerical treatment for integrodifferential algebraic systems of temporal two-point BVPs, Neural Comput. Appl. (2017) pp. 1–12. https://doi.org/10.1007/s00521-017-2845-7. [29] O. Abu Arqub, The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations, Math. Methods Appl. Sci. vol. 39 (2016) pp. 4549–4562. [30] O. Abu Arqub, M. Al-Smadi, N. Shawagfeh, Solving Fredholm integro-differential equations using reproducing Kernel Hilbert space method, Appl. Math. Comput. vol. 219 (2013) pp. 8938–8948. [31] O. Abu Arqub, M. Al-Smadi, Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations, Appl. Math. Comput. vol. 243 (2014) pp. 911–922. [32] S. Momani, O. Abu Arqub, T. Hayat, H. Al-Sulami, A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Voltera type, Appl. Math. Comput. vol. 240 (2014) pp. 229–239. [33] O. Abu Arqub, M. Al-Smadi, S. Momani, T. Hayat, Numerical solutions of fuzzy differential equations using reproducing Kernel Hilbert space method, Soft Comput. vol. 20 (2016) pp. 3283–3302. [34] O. Abu Arqub, M. Al-Smadi, S. Momani, T. Hayat, Application of reproducing kernel algorithm for solving secondorder, two-point fuzzy boundary value problems, Soft Comput. (2016) pp. 1–16. https://doi.org/10.1007/s00500016-2262-3. [35] O. Abu Arqub, Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm-Volterra integrodifferential equations, Neural Comput. Appl. vol. 28 (2017) pp. 1591–1610. https://doi.org/10.1007/s00521-015-2110-x. [36] O. Abu Arqub, Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm, Fundam. Inform. vol. 146 (2016) pp. 231–254. [37] O. Abu Arqub, B. Maayah, Solutions of Bagley-Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm, Neural Comput. Appl. (2016) pp. 1–15. https://doi.org/10.1007/s00521-016-2484-4. [38] O. Abu Arqub, Numerical solutions for the Robin time-fractional partial differential equations of heat and fluid flows based on the reproducing kernel algorithm, Int. J. Numer. Methods Heat Fluid Flow (2017). https://doi.org/10.1108/HFF-07-2016-0278. [39] O. Abu Arqub, N. Shawagfeh, Application of reproducing kernel algorithm for solving Dirichlet time-fractional diffusion-Gordon types equations in porous media, J. Porous Media (2017). In Press. [40] F. Z. Geng, S. P. Qian, Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers, Appl. Math. Lett. vol. 26 (2013) pp. 998–1004. [41] W. Jiang, Z. Chen, A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation, Numer. Methods Partial Differ. Equ. vol. 30 (2014) pp. 289–300. [42] F. Z. Geng, S. P. Qian, S. Li, A numerical method for singularly perturbed turning point problems with an interior layer, J. Comput. Appl. Math. vol. 255 (2014) pp. 97–105. [43] F. Z. Geng, M. Cui, A reproducing kernel method for solving nonlocal fractional boundary value problems, Appl. Math. Lett. vol. 25 (2012) pp. 818–823. [44] W. Jiang, Z. Chen, Solving a system of linear Volterra integral equations using the new reproducing kernel method, Appl.00 Math. Comput. vol. 219 (2013) pp. 10225–10230. [45] F. Z. Geng, S. P. Qian, Modified reproducing kernel method for singularly perturbed boundary value problems with a delay, Appl. Math. Model. vol. 39 (2015) pp. 5592–5597.

How to cite cite this thisarticle: article:Abu Abu Arqub Al-Smadi Numerical algorithm for solving timeArqub O,O, Al-Smadi M.M. Numerical algorithm for solving timefracfractional partial integrodifferential equations subject initialand andDirichlet Dirichletboundary boundary conditions. conditions. tional partial integrodifferential equations subject to toinitial https://doi.org/10.1002/num.22209 Numer. Methods Partial PartialDifferential DifferentialEq. Eq.2018;34:1577–1597. 2017;00:1–21. https://doi.org/10.1002/num.22209