Appl. Comput. Math., V.17, N.2, 2018, pp.151-160
A RELIABLE ALGORITHM FOR SOLVING LINEAR AND NONLINEAR ¨ SCHRODINGER EQUATIONS SHAHER MOMANI1,3 , OMAR ABU ARQUB2 , BANAN MAAYAH1 , FERAS YOUSEF1 , AHMED ALSAEDI3 Abstract. This paper is devoted to study the analytical series solutions for the Schr¨ odinger partial differential equations. By a general residual power series method, we construct the approximate analytical series solutions for linear and nonlinear Schr¨ odinger equations. The proposed technique is fully compatible with the complexity of this problem and obtained results are highly encouraging. These applications show that residual power series method is a simple, effective and powerful method for seeking analytical series solutions of partial differential equations. Keywords: Schr¨ odinger Equation, Multiple Power Series, Residual Power Series Method, PDEs. AMS Subject Classification: 32A05, 41A58.
1. Introduction Schr¨odinger partial differential equations (SPDEs) constitute a universal nonlinear model describing physical systems, with many applications in hydrodynamics, optics, nonlinear acoustics and quantum condensates. Particular interest is shown in nonlinear optics with further applications to optical communications. During recent years high bit rate information is transmitted in modern telecommunication photonic networks through dielectric waveguide structures, which commonly have a circular symmetry cross section i.e. optical fiber, achieving optical networks with capacities exceeding 1 Tb/s. The development of low loss optical fibers is one of the most remarkable progresses in photonic technology in order to save resources (signal generators) and to implement in a cost efficient way high bit rate optical links. Early optical fibers were extremely lossy exhibiting a 1000 dB/km attenuation loss depending of course on the wavelength used and defining optical windows. Nowadays fabrication progress resulted in today optical fibers of 0.2 dB/km loss offering high bit rate and long range transmission. Another problem in the lightwave technology was the use of appropriate devices required for optimum signal processing of the optical signal. In the early photonic devices light was transformed into an electrical signal and vice versa, resulting in increment of loss. According to today’s demands of lightwave technology, the signal processing in a photonic network has to be performed through an all-optical and reconfigurable mechanism. Furthermore, during the last two decades, numerous theoretical and experimental works show that periodic photonic structures offer remarkable potentiality in all-optical data processing applications. Also in optical soliton dynamics, the model of optical propagation is described by the nonlinear Schr¨odinger equation. The purpose of the present article is to extend the application of the residual power series (RPS) method to construct multiple PS solution of SPDEs. The main advantage of this method is the simplicity in computing the coefficients of terms of the series solution by using 1
Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan e-mail:
[email protected] 2 Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan e-mail:
[email protected] 3 Nonlinear Analysis and Applied Mathematics Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia Manuscript received 19 August 2015. 151
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the differential operators only and not as the other well-known analytic techniques that need the integration operators which is difficult in most cases. Moreover, the proposed method can be easily applied in the spaces of higher dimension solution and can be applied without any limitation on the nature of the equation and the type of classification. For convenience, the reader is kindly requested to go through [4, 5, 7 − 11] in order to know more details about the RPS method, including its properties, its construction, its motivation for use, its characteristics over conventional method, and its applications, Also, for similar methods the reader can be go through [1 − 3, 12 − 15, 17, 19] . More precisely, we provide approximate solutions for the following linear and nonlinear SPDEs [6, 16, 18 − 21]: iΥt (x, t) + Υxx (x, t) = 0, (1) iΥt (x, t) + Υxx (x, t) + γ |Υ (x, t)|2 Υ (x, t) = 0, subject to Υ (x, 0) = ζ (x) , (2) where x ∈ R, t ≥ 0, γ ∈ R, i2 = −1, and |.| is the modulus. This article is arranged as follows. In the next section, some definitions and theorems regarding the RPS method and multiple PS are given. The solution of the SPDEs by RPS method and the main theoretical results are presented in Section 3. Numerical experiments in which two certain types of the SPDEs are given to show the good performance and reliability of the proposed procedure. At the end of the article, some concluding remarks are given.
A series
∞ ∑
2. Excerpts of RPS method cm tm = c0 + c1 t + c2 t2 + . . . is called PS about t = 0. Suppose that f has a
m=0
PS representation at t = 0 of the form f (t) =
∞ ∑
cm tm , 0 ≤ t < R. If f (m) are continuous
m=0 (m)
on (0, R), m = 0, 1, 2, . . . , then cm are given as cm = f m!(t0 ) , m = 0, 1, 2, . . . , where R is the radius of convergence. Definition 1. A PS of the form ∞ ∑ fm (x)tm = f0 (x) + f1 (x) t + f2 (x) t2 + . . . , x ∈ I, t ≥ 0, (3) m=0
is called a multiple PS about t = 0, where t is a variable and fm are functions of x called the coefficients of the series. Theorem 1. Suppose that u (x, t) has a multiple PS representation at t = 0 of the form u (x, t) =
∞ ∑
fm (x)tm , x ∈ I, 0 ≤ t < R.
(4)
m=0 (m)
If ut
(x, t) are continuous on I × (0, R), m = 0, 1, 2, . . . , then the coefficients fm are given as (m)
(x, 0) , m = 0, 1, 2, . . . , (5) m! is the radius of convergence in which Rc is the radius of convergence of fm (x) =
where R = minc∈I Rc ∞ ∑ the PS fm (c)tm .
ut
m=0
Proof. Assume that u (x, t) is a function of two variables that can be represented by a multiple PS. Note that if t = 0 in (5), then all terms after the first are vanished. Thus, f0 (x) = u (x, 0) .
(6)
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Applying
∂ ∂t
153
one time on (4), we get ut (x, t) = f1 (x) + 2!f2 (x) t +
3! f3 (x) t2 + . . . , 2!
(7)
substitute t = 0 in (7), we get By applying
∂ ∂t
f1 (x) = ut (x, 0) . two times on (4), one obtains (2)
ut (x, t) = 2!c2 + 3!c3 t +
4! 2 c4 t + . . . . 2!
(8)
(9)
The substitution of t = 0 into (9) gives (2)
ut (x, 0) . 2! m-times on (4) and substituting t = 0, then f2 (x) =
Applying
∂ ∂t
(10)
(m)
fm (x) =
ut
(x, 0) , m = 0, 1, 2, . . . , m!
(11)
which are the same form of (5). 3. Solution of SPDEs by RPS method In this section, the procedure how to solve the SPDEs using the RPS method is presented in details. Firstly, we put the complex functions Υ (x, t) and ζ (x) as Υ (x, t) = u (x, t) + iv (x, t) , ζ (x) = f (x) + ig (x) , from (1), we can write [ ( ) ] i ut (x, t) + vxx (x, t) + γ u2 (x, t) + v 2 (x, t) v (x, t) [ ( ) ] − vt (x, t) − uxx (x, t) − γ u2 (x, t) + v 2 (x, t) u (x, t) = 0,
(12)
(13)
and (2) can be separated as u (x, 0) + iv (x, t0 ) = f (x) + ig (x) . Thus, the SPDEs can be converted into system of PDEs as ( ) ut (x, t) + vxx (x, t) + γ u2 (x, t) + v 2 (x, t) v (x, t) = 0, ( 2 ) vt (x, t) − uxx (x, t) − γ u (x, t) + v 2 (x, t) u (x, t) = 0,
(14)
(15)
subject to u (x, 0) = f (x) , v (x, 0) = g (x) . Suppose that the solutions takes form ∞ ∑ nα u (x, t) = fn (x) tn! , x ∈ I, 0 ≤ t < R1 , v (x, t) =
n=0 ∞ ∑
n=0
nα
(16)
(17)
gn (x) tn! , x ∈ I, 0 ≤ t < R2 ,
where R1 = minc∈I R1c is the radius of convergence in which R1c is the radius of convergence of ∞ ∑ the PS fn (c)tn and R2 = mind∈I R2d is the radius of convergence in which R2d is the radius n=0
of convergence of the fractional PS
∞ ∑ n=0
gn (d)tn .
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Next, uk (x, t) =
k ∑ n=0 k ∑
vk (x, t) =
n=0
n
fn (x) tn! , x ∈ I, 0 ≤ t, (18) n gn (x) tn! ,
x ∈ I, 0 ≤ t.
Obviously, u (x, t) and v (x, t) satisfy conditions of (2), so u (x, 0) = f0 (x) = f (x) and v (x, 0) = g0 (x) = g(x). Also, the initial approximation of u (x, t) is u0 (x, t) = f (x) and the initial approximation of v (x, t) is v0 (x, t)= g (x). As a result k ∑
uk (x, t) = f (x) + vk (x, t) = g (x) +
n
n=1 k ∑ n=1
fn (x) tn! , x ∈ I, 0 ≤ t, k = 1, 3, 4, . . . , (19)
n gn (x) tn! ,
x ∈ I, 0 ≤ t, k = 1, 3, 4, . . . .
Define the residual functions, Res1 and Res2 , for (15) as ( ) Res1 (x, t) = ut (x, t) + vxx (x, t) + γ u2 (x, t) + v 2 (x, t) v (x, t) , ( ) Res2 (x, t) = vt (x, t) − uxx (x, t) − γ u2 (x, t) + v 2 (x, t) u (x, t) ,
(20)
and the k-th truncated residual functions
( ) Res1k (x, t) = (uk )t (x, t) + (vk )xx (x, t) + γ u2k (x, t) + vk2 (x, t) vk (x, t) , ( ) Res2k (x, t) = (vk )t (x, t) − (uk )xx (x, t) − γ u2k (x, t) + vk2 (x, t) uk (x, t) .
(21)
As in [4, 5, 7 − 11], clearly Res1 (x, t) = 0 and lim Res1k (x, t) =Res1 (x, t), whilst Res2 (x, t) = 0 k→∞ (n−1) 1
and lim Res2k (x, t) =Res2 (x, t). Thus, Dt k→∞
(n−1)
Res (x, t) = 0 and Dt
n−1
n−1
Res2 (x, t) = 0, n = n−1
1 1 2 ∂ ∂ ∂ 1, 2, 3, . . . k. In the meantime, ∂t n−1 Res (x, 0) = ∂tn−1 Resn (x, 0) = 0 and ∂tn−1 Res (x, 0) = n−1 ∂ Res2n (x, 0) = 0 , n = 1, 2, 3, . . . k. ∂tn−1 To obtain the rules of fn (x) and gn (x), n = 1, 2, 3, . . . , k in (19), we apply the following: ∂ n−1 Res1n (x, 0) ∂tn−1 n−1 ∂ Res2n (x, 0) ∂tn−1
= 0, n = 1, 2, 3, . . . , k, = 0, n = 1, 2, 3, . . . , k.
(22)
In fact, to determine the form of f1 (x) and g1 (x), substitute the 1-st truncated series u1 (x, t) and v1 (x, t) into the 1-st truncated residual functions Res11 (x, t) and Res21 (x, t), to get ( ) Res11 (x, t) = (u1 )t (x, t) + (v1 )xx (x, t) + γ u21 (x, t) + v12 (x, t) v1 (x, t) , ( ) (23) Res21 (x, t) = (v1 )t (x, t) − (u1 )xx (x, t) − γ u21 (x, t) + v12 (x, t) u1 (x, t) . But since u1 (x, t) = f (x) + f1 (x) t and v1 (x, t) = g (x) + g1 (x) t, then (21) leads to ′′
Res11 (x, t) = f1 (x) + f 2 (x)g(x)γ + g 3 (x)γ + g (x) + 2tf (x)g(x)f1 (x)γ + tf 2 (x)g1 (x)γ ′′ +3tg 2 (x)g1 (x)γ + tg1 (x) + t2 g(x)f1 2 (x)γ + 2t2 f (x)f1 (x)g1 (x)γ +3t2 g(x)g12 (x)γ + t3 f12 (x)g1 (x)γ + t3 g13 (x)γ,
(24)
′′
Res21 (x, t) = g1 (x) − f 3 (x)γ − f (x)g 2 (x)γ − f (x) − 2tf (x)g(x)g1 (x)γ − tg 2 (x)f1 (x)γ ′′ −3tf 2 (x)f1 (x)γ − tf1 (x) − t2 f (x)g12 (x)γ − 2t2 g(x)f1 (x)g1 (x)γ −3t2 f (x)f12 (x)γ − t3 f1 (x)g12 (x)γ − t3 f13 (x)γ. The substituting of t = 0 through (24) and (25), yields ( ) ′′ f1 (x) = − f 2 (x)g(x)γ + g 3 (x)γ + g (x) , ′′
g1 (x) = f (x) g 2 (x) γ + f 3 (x) γ + f (x).
(25)
(26)
S. MOMANI et al.: A RELIABLE ALGORITHM FOR SOLVING LINEAR ...
Hence, the 1-st RPS solution can be expressed as ( ) ′′ u1 (x, t) = u0 (x, t) − f 2 (x)g(x)γ + g 3 (x)γ + g (x) t, ( ) ′′ v1 (x, t) = v0 (x, t) + f (x) g 2 (x) γ + f 3 (x) γ + δf (x) t.
155
(27)
Similarly, to find out the form of f2 (x) and g2 (x), substitute u2 (x, t) = f (x) + f1 (x) t + and v2 (x, t) = g (x) + g1 (x) t + 12 g2 (x) t2 into Res12 (x, t) and Res22 (x, t) to obtain
1 2 2 f2 (x) t
′′
Res12 (x, t) = f1 (x) + f 2 (x) g (x) γ + g 3 (x) γ + g (x) + tf2 (x) + tf2 (x) ′′ + tf 2 (x) g1 (x) γ + 3tg 2 (x) g1 (x) γ + tg1 (x) ′′ + 12 t2 g2 (x) + t2 g (x) f12 (x) γ + . . . ,
(28)
′′
Res12 (x, t) = g1 (x) − g 2 (x) f (x) γ − f 3 (x) γ − f (x) − tg2 (x) − 2tf (x) g (x) g1 (x) γ − tg 2 (x) f1 (x) γ − 3tf 2 (x) f1 (x) γ ′′ ′′ − tf1 (x) − 21 t2 f2 (x) − t2 f (x) g12 (x) γ − . . . . Now, applying
∂ ∂t
on both sides of (28) and (29) gives
1 ∂ ∂t Res2 (x, t) = f2 (x) + 2f ′′ + 3g 2 (x) g1 (x) γ + g1
(x) g (x) f1 (x) γ + f 2 (x) g1 (x) γ ′′ (x) + tg2 (x) + 2tg (x) f12 (x) γ + . . . ,
∂ 2 2 ∂t Res2 (x, t) = g2 (x) − 2f (x) g (x) g1 (x) γ − g (x) f1 (x) γ ′′ ′′ − 3f 2 (x) f1 (x) γ − f1 (x) − tf2 (x) − 2tf (x) g12 (x) γ − n−1
(29)
....
(30)
(31)
n−1
1 2 ∂ ∂ By ∂t n−1 Resn (x, 0) = 0 and ∂tn−1 Resn (x, 0) = 0 for n = 2 and solving the resultant system for f2 (x) and g2 (x), one get ( ) ′′ f2 (x) = − 2f (x) g (x) f1 (x) γ + f 2 (x) g1 (x) γ + 3g 2 (x) g1 (x) γ + g1 (x) , (32) ′′ g2 (x) = 2f (x) g (x) g1 (x) γ + g 2 (x) f1 (x) γ + 3f 2 (x) f1 (x) γ + f1 (x) .
Therefore, the 2-nd RPS solution is ) ( ′′ u2 (x, t) = u1 (x, t) − 2f (x) g (x) f1 (x) γ + f 2 (x) g1 (x) γ + 3g 2 (x) g1 (x) γ + 21 g1 (x) t2 , ) ( 1 ′′ 2 2 2 2 (x, t) = v1 (x, t) + 2f (x) g (x) g1 (x) γ + g (x) f1 (x) γ + 3f (x) f1 (x) γ + 2 f1 (x) t . (33) Applying the same procedure for n = 3, leads to ′′
f3 (x) = −f 2 (x)g2 (x)γ − 3g 2 (x)g2 (x)γ − g2 (x) − 2f (x)f2 (x)g(x)γ − 2γ(2f (x)f1 (x)g1 (x) − 2γg(x)(3g12 (x) + f12 (x)),
(34)
′′
g3 (x) = g 2 (x)f2 (x)γ + 3f 2 (x)f2 (x)γ + f2 (x) + 2g(x)g2 (x)f (x)γ + 2γ(2g(x)f1 (x)g1 (x) + 2γf (x)(3f12 (x) + g12 (x)). In fact, the 3-rd RPS solution will be ( ) ′′ u3 (x, t) = u2 (x, t) − 21 f 2 (x)g2 (x)γ + 3g 2 (x)g2 (x)γ + g2 (x) t2 ( ) − 21 2f (x)f2 (x)g(x)γ + γ(2f (x)f1 (x)g1 (x) + g(x)(3g12 (x) + f12 (x)))2 t2 , ( ) ′′ v3 (x, t) = v2 (x, t) + 12 g 2 (x)f2 (x)γ + 3f 2 (x)f2 (x)γ + f2 (x) t2 ( ) + 21 +2g(x)g2 (x)f (x)γ + γ(2g(x)f1 (x)g1 (x) + f (x)(3f12 (x) + g12 (x)))2 t2 . In order to finalize the solution, we should write ψn (x, t) = un (x, t) + ivn (x, t) , x ∈ R, t ≥ 0, i2 = −1,
(35)
(36)
(37)
(38)
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which is equivalent to the following n-th truncated series of ψ (x, t) ψn (x, t) =
n ∑ tk k=0
k!
ζk (x) , x ∈ R, t ≥ 0, i2 = −1.
(39)
4. Numerical experiments In this section, the procedure of using RPS method to construct approximate analytical solution to several SPDEs is presented. Example 1. Consider the linear SPDE: iψt (x, t) − ψxx (x, t) = 0, x ∈ R, t ≥ 0,
(40)
ψ (x, 0) = e3ix .
(41)
subject to The solution metholodogy is as folllwes, put ψ (x, t) = u (x, t) + iv (x, t), then ut (x, t) − vxx (x, t) = 0 and vt (x, t) + uxx (x, t) = 0, subject to u (x, 0) = cos (3x) and v (x, 0) = sin (3x). Taking f0 (x) = u (x, 0) = cos (3x) and g0 (x) = v (x, 0) = sin (3x); then u0 (x, t) = cos (3x) and v0 (x, t) = sin (3x). According to the previous results, the k-th truncated residual functions of (39) are as follow: Res1k (x, t) = (uk )t (x, t) − vk (x, t) , (42) Res2k (x, t) = (vk )t (x, t) + uk (x, t) . Now, the k-th truncated series of the multiple PS expansions of u (x, t) and v (x, t) about t = 0 is k ∑ n uk (x, t) = cos (3x) + fn (x) tn! , k = 1, 2, 3, . . . , n=1 (43) k ∑ n vk (x, t) = sin (3x) + gn (x) tn! , k = 1, 2, 3, . . . . n=1
To find f1 (x) and g1 (x) in (42), substitute u1 (x, t) and v1 (x, t) into Res11 (x, t) and Res21 (x, t), respectively, to get the following results: ′′
Res11 (x, t) = 9sin (3x) + f1 (x) − tg1 (x) , ′′ Res21 (x, t) = −9cos ( 3x) + g1 (x) + tf1 (x) .
(44)
Depending on the results of (43) in the case of n = 1, the substituting of t = 0 back yields f1 (x) = −9sin (3x), g1 (x) = 9cos (3x).
(45)
Hence, the 1-st order RPS solutions of (39) and (40) is u1 (x, t) = cos (3x) − 9sin (3x)t, v1 (x, t) = sin (3x) + 9cos (3x)t.
(46)
To find out f2 (x) and g2 (x), substitute u2 (x, t) and v2 (x, t) into Res12 (x, t) and Res22 (x, t) as ′′
′′
Res12 (x, t) = 9sin (3x) − 9sin (3x) + tf2 (x) − tg1 (x) − 21 t2 g2 (x) , ′′ ′′ Res22 (x, t) = −9cos (3x) + 9cos (3x) + tg2 (x) + tf1 (x) + 21 t2 f2 (x) . Applying
∂ ∂t
(47)
one time on the both sides of (46) gives ′′ ∂ 1 ∂t Res2 (x, t) = f2 (x) − g1 ′′ 2 ∂ ∂t Res2 (x, t) = g2 (x) + f1
′′
(x) − tg2 (x) , ′′ (x) + tf2 (x) .
(48)
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157
n−2
1 2 ∂ ∂ As ∂t n−2 Resn (x, 0) = 0 and ∂tn−2 Resn (x, 0) = 0 for n = 2 and solving the resulting algebraic equations with respect to f2 (x) and g2 (x), respectively, we obtain
f2 (x) = −(9)2 cos (3x), g2 (x) = −(9)2 sin (3x).
(49)
Therefore, the 2-nd order RPS solutions of (39) and (40) is 2
u2 (x, t) = cos (3x) − 9sin (3x)t − (9)2 cos (3x) 21 t , 2
v2 (x, t) = sin (3x) + 9cos (3x)t − (9)2 sin (3x) 21 t .
(50)
By applying the same procedure for n = 3 and n = 4, will to the following results: f3 (x) = (9)3 sin (3x), g3 (x) = −(9)3 cos (3x),
(51)
f4 (x) = (9)4 cos (3x), (52) g4 (x) = (9)4 sin (3x). If we collect the previous results for fn (x) and gn (x), n = 0, 1, 2, 3, 4, then the 4-th order RPS solution of (39) and (40) is 1 1 1 u4 (x, t) = cos (3x) − 9sin (3x)t − (9)2 cos (3x) t2 + (9)3 sin (3x) t3 + (9)4 cos (3x) t4 , (53) 2 3! 4! 1 1 t2α − (9)3 cos (3x) t3 + (9)4 sin (3x) t4 . 2 3! 4! Using Euler formula for complex numbers, the exact solution will be ( ) 2 ψ (x, t) = e3ix 1 + 9it + (9i)2 12 t + (9i)3 3!1 t3 + (9i)4 4!1 t4 + . . . v4 (x, t) = sin (3x) + 9cos (3x)t − (9)2 sin (3x)
= e3i(x+3t)
(54)
(55)
which are exactly the same as solutions obtained by homotopy analysis method [14], variational iteration method [26], Adomian decomposition method [24], and homotopy-perturbation method [24]. Example 2. Consider the following one-dimensional nonlinear time-fractional SE: iψt (x, t) + ψxx (x, t) + 2|ψ (x, t)|2 ψ (x, t) = 0, x ∈ R, t ≥ 0,
(56)
subject to ψ (x, 0) = eix . If one put ψ (x, t) = u (x, t) + iv (x, t), then (55) and (56) can be converted into ( ) ut (x, t) + vxx (x, t) + 2 u2 (x, t) + v 2 (x, t) v (x, t) = 0, ( ) vt (x, t) − uxx (x, t) − 2 u2 (x, t) + v 2 (x, t) u (x, t) = 0,
(57)
(58)
subject to u (x, 0) = cos (x), (59) v (x, 0) = sin (x). Using RPS method, taking f0 (x) = f (x) and g0 (x) = g (x), and starting with the initial approximations u0 (x, t) = cos (x) and v0 (x, t) = sin (x) with the k-th truncated residual functions of (21); the following forms for the unknown coefficients fn and gn , n = 0, 1, 2, 3, 4 are obtained: f0 (x, t) = cos (x), f1 (x, t) = −sin (x), f2 (x, t) = −cos (x), (60) f3 (x, t) = sin (x), f4 (x, t) = cos (x),
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g0 (x, t) = sin (x), g1 (x, t) = cos (x), g2 (x, t) = −sin (x), (61) g3 (x, t) = −cos (x), g4 (x, t) = sin (x). If we collect and extend the results of (59) and (60), then the 4-th order RPS solutions is t2 t3 t4 + sin (x) + cos (x) , 2! 3! 4! 3 2 t t4 t v4 (x, t) = sin (x) + cos (x)t − sin (x) − (5 − 2 (2!)) cos (x) + sin (x) . 2! 3! 4! Thus, the general pattern form is coinciding with the exact solution ( ) 2 3 4 ψ (x, t) = eix 1 + it + i2 t2! + i3 t3! + i4 t4! + . . . u4 (x, t) = cos (x) − sin (x)t − cos (x)
= ei(x+t)
(62) (63)
(64)
which are exactly the same as solutions obtained by homotopy analysis solution [6], variational iteration solution [18], Adomian decomposition solution [16], homotopy-perturbation solution [16], and He’s frequency solution [20]. 5. Conclusion This paper is concerned with the exact solutions of the SPDEs. By a general RPS method, we can obtain the exact solution in term of infinite convergence series. From the given applications, it shows the method is directly, simplicity and efficient and could be widespread use to many other PDEs with variable coefficients. On the other aspect as well, it has been observed that the method is easier to implement as compared to other techniques and can be used to solve nonlinear problems of a complex physical nature. 6. Acknowledgment The authors would like to acknowledge the University of Jordan for funding this research study. References [1] Abu Arqub, O. Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions, Computers & Mathematics with Applications, V.73, 2017, pp.1243-1261. [2] Abu Arqub, O. The reproducing kernel algorithm for handling differential algebraic systems of ordinary differential equations, Mathematical Methods in the Applied Sciences, V.39, 2016. pp.4549-4562. [3] Abu Arqub, O. Approximate solutions of DASs with nonclassical boundary conditions using novel reproducing kernel algorithm, Fundamenta Informaticae, V.146, 2016, pp.231-254. [4] Abu Arqub, O., El-Ajou, A., Al Zhour, Z., Momani, S. Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique, Entropy, V.16, 2014, pp.471-493. [5] Abu Arqub, O., El-Ajou, A., Momani, S. Construct and predicts solitary pattern solutions for nonlinear timefractional dispersive partial differential equations, Journal of Computational Physics, V.293, 2015, pp.385399. [6] Alomari, A.W., Noorani, M.S.M., Nazar, R. Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, V.14, 2009, pp.1196-1207. [7] El-Ajou, A., et al. A general form of the generalized Taylor’s formula with some applications, Applied Mathematics and Computation, V.256, 2015, pp.851-859. [8] El-Ajou, A., et al. New results on fractional power series: theories and applications, Entropy, V.15, 2013, pp.5305-5323.
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[9] El-Ajou, A., et al. Approximate analytical solution of the nonlinear fractional KdV-Burgers equation: A new iterative algorithm, Journal of Computational Physics, V.293, 2015, pp.81-95. [10] El-Ajou, A., et al. A novel expansion iterative method for solving linear partial differential equations of fractional order, Applied Mathematics and Computation, V.257, 2015, pp.119-133. [11] Komashynska, I., et al. An efficient analytical method for solving singular initial value problems of nonlinear systems, Applied Mathematics and Information Sciences, V.10, 2016, pp.647-656. [12] Laskin, N. Fractals and quantum mechanics, Chaos, V.10, 2000, pp.780-790. [13] Liu, J.G., Zeng, Z.F. Extended generalized hyperbolic-function method and new exact solutions of the generalized Hamiltonians and NNV equations by the symbolic computation, Fundamenta Informaticae, V.132, 2014, pp.501-517. [14] Magin, R.L., Ingo, C., Colon-Perez, L., Triplett, W., Mareci, T.H. Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy, Microporous and Mesoporous Materials, V.178, 2013, pp.39-43. [15] Momani, S., et. al., Analytical approximations for Fokker-Planck equations of fractional order in multistep schemes, Appl. Comput. Math., V.15, N.3, 2016, pp.319-330. [16] Sadighi, A., Ganji, D.D. Analytic treatment of linear and nonlinear Schr¨ odinger equations: a study with homotopy-perturbation and Adomian decomposition methods, Physics Letters A, V.372, 2008, pp.465-469. [17] Shawagfeh, N., et al. Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method, Journal of Computational Analysis and Applications, V.16, 2014, pp.750-762. [18] Wazwaz, A.M. A study on linear and nonlinear Schrodinger equations by the variational iteration method, Chaos, Solitons and Fractals, V.37, 2008, pp.1136-1142. [19] Zhang, L.H., Ahmad, B., Wang, G.T. Existence and approximation of positive solutions for nonlinear fractional integro-differential boundary value problems on an unbounded domain, Appl. Comput. Math., V.15, N.2, 2016, pp.149-158. [20] Zhang, Y.N., Xua, F, Deng, L.L. Exact solution for nonlinear Schr¨ odinger equation by He’s frequency formulation, Computers and Mathematics with Applications, V.58, 2009, pp.2449-2451. [21] Zeng, Z.F., Liu, J.G., Jiang, Y, Nie, B. Transformations and soliton solutions for a variable-coefficient nonlinear Schr¨ odinger equation in the dispersion decreasing fiber with symbolic computation, Fundamenta Informaticae, V.145, 2016, pp.207-219.
Shaher Momani, for a photograph and biography, see Appl. Comput. Math., V.15, N.3, 2016, p.330.
Omar Abu Arqub, for a photograph and biography, see Appl. Comput. Math., V.15, N.3, 2016, p.330.
The photo is not presented.
Banan Maayah - received her Ph.D. from the University of Jordan (Jordan) in 2014. Presently, she is an Associate Professor of applied mathematics at the University of Jordan. Her research interests focus on numerical analysis and fractional differential equations.
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Feras Yousef - is an Assistant Professor of Mathematics at the University of Jordan (Jordan). He received his Ph.D. in Applied Mathematics from New Mexico State University (USA). His research interests are in the areas of nonlinear partial differential equations, soft matter systems, calculus of variations, geometric function theory, structural bioinformatics.
Ahmed Al-Saedi - is a Professor of Applied Mathematics at the King Abdulaziz University (Saudi Arabia). His research interests focus on fluid dynamics, nonlinear dynamics, numerical analysis, and heat and mass transfer.
