# Solving Fractional Partial Differential Equations with Corrected Fourier Series Method

Authors: Nor Hafizah Zainal; Adem Kılıçman
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.


---

## Body

## 1. Introduction

In recent years, differential equations of fractional orders have been appearing more and more frequently in various research and applications in the fluid mechanics, viscoelasticity, biology, physics, and engineering; see [1, 2]. There are some methods usually used in solving the fractional partial differential equations such as Laplace and Fourier transform, variational iteration method, and differential transform methods. In this study, we want to use the corrected Fourier series method in solving the problems.In [3], corrected Fourier series method has been used in solving classical PDEs problems. The corrected Fourier series is a combination of the uniformly convergent Fourier series and the correction functions and consists of algebraic polynomials and Heaviside step function.


## 2. Basic Definitions

The Riemann-Liouville fractional integral is the most popular definition that we always find in the study of fractional calculus.Definition 1.
The Riemann-Liouville fractional integral operator of orderα
>
0 of a function f
(
x
) is defined as
(1)
J
α
f
(
x
)
=
1
Γ
(
α
)
∫
0
x
(
x
-
s
)
α
-
1
f
(
s
)
d
s
.Jumarie’s modified Riemann-Liouville derivative of orderα is defined by the following definition.Definition 2.
Letf
:
R
→
R, x
→
f
(
x
) denote a continuous function. Its fractional derivative of order α is defined as follows:forα
<
0,
(2)
D
x
α
=
1
Γ
(
-
α
)
∫
0
x
(
x
-
s
)
-
α
-
1
(
f
(
s
)
-
f
(
0
)
)
d
s
,
forα
>
0,
(3)
D
x
α
=
1
Γ
(
1
-
α
)
d
d
x
∫
0
x
(
x
-
s
)
-
α
(
f
(
s
)
-
f
(
0
)
)
d
s
,
hhhhhhhhhhhhhhhhhhhhhhh
where
0
<
α
<
1
,
0
D
x
α
f
(
x
)
=
1
Γ
(
n
-
α
)
d
n
d
x
n
∫
0
x
(
x
-
s
)
n
-
α
-
1
[
f
(
s
)
-
f
(
0
)
]
d
s
,where n
-
1
<
α
<
n with n
∈
N.Definition 3 (see [4–7]).
Fractional derivative of compounded function is defined as(4)
d
α
f
≅
Γ
(
1
+
α
)
d
f
,
0
<
α
<
1
.Definition 4 (see [4–7]).
The integral with respect to(
d
t
)
α is defined as the solution of the fractional differential equation
(5)
d
x
=
f
(
x
)
(
d
t
)
α
,
t
≥
0
,
x
(
0
)
=
0
,
0
<
α
<
1
.Lemma 5 (see [4–7]).
Letf
(
x
) denote a continuous function; then the solution of y
(
x
), y
(
0
)
=
0, (5) is defined as
(6)
y
=
∫
0
x
f
(
ξ
)
(
d
ξ
)
α
=
α
∫
0
x
(
x
-
ξ
)
α
-
1
f
(
ξ
)
d
ξ
,
hhhhhhhhhhhhhhhhhhhhhhhhh
0
<
α
<
1
.


## 3. Corrected Fourier Series

The CFS is described in the form of(7)
u
(
x
,
t
)
=
∑
|
n
|
<
∞
∑
|
m
|
<
∞
A
n
m
e
i
α
n
x
e
i
β
m
t
+
∑
|
m
|
<
∞
(
a
1
m
x
+
a
2
m
x
2
2
!
+
a
3
m
x
3
3
!
)
e
i
β
m
t
+
∑
|
n
|
<
∞
(
b
1
n
t
+
b
2
n
t
2
2
!
+
b
3
n
t
3
3
!
)
e
i
α
n
x
+
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
x
l
l
!
t
l
0
l
0
!
,
where α
n
=
2
n
π
/
x
0 and β
m
=
2
m
π
/
t
0.Due to the periodicity of eithere
i
α
n
x or e
i
β
m
t, we can cancel out the first three terms on the right-hand side of (7) because they are identically zero. Based on the endpoints values of u
(
x
,
t
) and its partial derivative, we obtain the following linear equations:
(8)
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
x
l
-
j
(
l
-
j
)
!
t
l
0
-
j
0
(
l
0
-
j
0
)
!
H
(
l
-
j
)
H
(
l
0
-
j
0
)
|
0
x
0
|
0
t
0
=
∂
j
+
j
0
u
(
x
,
t
)
∂
x
j
∂
t
j
0
|
0
x
0
|
0
t
0
,
where j
=
0,1
,
2 and j
0
=
0,1
,
2.Next, we want to determine the coefficientsa
1
m, a
2
m, and a
3
m. With respect to x, the endpoints effect of u
(
x
,
t
) and its partial derivatives yields
(9)
∂
2
u
∂
x
2
|
0
x
0
=
∑
|
m
|
≤
M
a
3
m
x
0
e
i
β
m
t
+
∑
l
0
=
1
3
d
3
l
0
x
0
t
l
0
l
0
!
,
∂
u
∂
x
|
0
x
0
=
∑
|
m
|
≤
M
(
a
2
m
x
0
+
a
3
m
x
0
2
2
!
)
e
i
β
m
t
+
∑
l
=
2
3
∑
l
0
=
1
3
d
l
l
0
x
0
l
-
1
(
l
-
1
)
!
t
l
0
l
0
!
,
u
|
0
x
0
=
∑
|
m
|
≤
M
(
a
1
m
x
0
+
a
2
m
x
0
2
2
!
+
a
3
m
x
0
3
3
!
)
e
i
β
m
t
+
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
x
0
l
l
!
t
l
0
l
0
!
.
Again, due to the periodicity of e
i
α
n
x, the first and third terms of (7) and its partial derivatives are identically zero. Then, by applying the Fourier projection on the basis function e
i
β
m
t, we solved for a
1
m
,
a
2
m, and a
3
m:
(10)
a
3
m
=
1
x
0
(
F
2
〈
∂
2
u
∂
x
2
|
0
x
0
〉
m
-
∑
l
0
=
1
3
d
3
l
0
x
0
F
2
〈
t
l
0
l
0
!
〉
m
)
,
a
2
m
=
1
x
0
×
(
F
2
〈
∂
u
∂
x
|
0
x
0
〉
m
-
∑
l
=
2
3
∑
l
0
=
1
3
d
l
l
0
x
0
l
-
1
(
l
-
1
)
!
F
2
〈
t
l
0
l
0
!
〉
m
-
a
3
m
x
0
2
2
!
∑
l
=
2
3
∑
l
0
=
1
3
d
l
l
0
x
0
l
-
1
(
l
-
1
)
!
F
2
〈
t
l
0
l
0
!
〉
m
)
,
a
1
m
=
1
x
0
×
(
F
2
〈
u
|
0
x
0
〉
m
-
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
x
0
l
l
!
F
2
〈
t
l
0
l
0
!
〉
m
-
a
2
m
x
0
2
2
!
-
a
3
m
x
0
3
3
!
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
x
0
l
l
!
F
2
〈
t
l
0
l
0
!
〉
m
)
.
Similar in the case with respect to t, we have
(11)
b
3
n
=
1
t
0
(
F
1
〈
∂
2
u
∂
t
2
|
0
t
0
〉
n
-
∑
l
=
1
3
d
l
3
F
1
〈
x
l
l
!
〉
n
t
0
)
,
b
2
n
=
1
t
0
×
(
F
1
〈
∂
u
∂
t
|
0
t
0
〉
n
-
∑
l
=
1
3
∑
l
0
=
2
3
d
l
l
0
F
1
〈
x
l
l
!
〉
n
×
t
0
l
0
-
1
(
l
0
-
1
)
!
-
b
3
n
t
0
2
2
!
∑
l
=
1
3
∑
l
0
=
2
3
d
l
l
0
F
1
〈
x
l
l
!
〉
n
)
,
b
1
n
=
1
t
0
×
(
F
1
〈
u
|
0
t
0
〉
n
-
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
F
1
〈
x
l
l
!
〉
n
t
0
l
0
l
0
!
-
b
2
n
t
0
2
2
!
-
b
3
n
t
0
3
3
!
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
F
1
〈
x
l
l
!
〉
n
t
0
l
0
l
0
!
)
.


## 4. Fractional Corrected Fourier Series

In this paper, we consider the following general form of the linear time-fractional equation:(12)
∂
α
u
∂
t
α
+
a
0
(
x
)
u
(
x
,
t
)
+
a
1
(
x
)
∂
u
∂
x
+
a
2
(
x
)
∂
2
u
∂
x
2
=
f
(
x
,
t
)
and subject to the initial conditions
(13)
u
(
x
,
0
)
=
f
(
x
)
,
as
0
<
α
≤
1
,
x
∈
[
0
,
x
0
]
,
t
∈
[
0
,
t
0
]
,
u
(
x
,
0
)
=
f
(
x
)
,
∂
u
(
x
,
0
)
∂
t
=
g
(
x
)
,
as
1
<
α
≤
2
,
x
∈
[
0
,
x
0
]
,
t
∈
[
0
,
t
0
]
.
For the case of α
=
1, the fractional equation reduces to the classical linear PDE and is similar to the case of α
=
2.Definition 6.
Form to be the smallest integer that exceeds α, the modified Riemann-Liouville time-fractional derivative operator of order α
>
0 is defined as
(14)
D
*
t
α
u
(
x
,
t
)
=
∂
α
u
∂
t
α
=
{
1
Γ
(
m
-
α
)
∂
m
∂
t
m
×
∫
0
t
(
t
-
τ
)
m
-
α
-
1
(
u
(
x
,
τ
)
-
u
(
x
,
0
)
)
d
τ
,
for
m
-
1
<
α
<
m
,
∂
m
u
(
x
,
t
)
∂
t
m
,
for
α
=
m
∈
N
,
where m
-
1
<
α
<
m with m
∈
N.To obtain the nine unknownsd
l
l
0
(
l
,
l
0
=
1,2
,
3) in (7), we solve the following linear equations:
(15)
∑
l
=
1
3
∑
l
0
=
1
3
d
l
l
0
λ
l
0
x
l
-
j
(
l
-
j
)
!
t
1
+
l
0
-
α
H
(
l
-
j
)
H
(
l
0
-
j
0
)
|
0
x
0
|
0
t
0
=
∂
j
+
(
j
0
+
α
)
u
(
x
,
t
)
∂
x
j
∂
t
j
0
+
α
|
0
x
0
|
0
t
0
,
where j
=
0,1
,
2, j
0
=
0,1
,
2, and λ
l
0
=
(
(
-
1
)
2
l
0
)
(
Γ
(
α
-
(
l
0
+
2
)
)
·
(
(
l
0
+
2
)
-
α
)
/
Γ
(
1
-
α
)
Γ
(
α
-
1
)
).Then, we arrange the order of equations asj
=
0, j
0
=
0,1
,
2; j
=
1, j
0
=
0,1
,
2; and j
=
2, j
0
=
0,1
,
2; we can have the nine unknowns d
11, d
12, d
13, d
21, d
22, d
23, d
31, d
32, and d
33 in a vector form. Then, we solve this coefficient in the matrix form. By solving the matrix, we can determine the coefficients d
l
l
0
(
l
=
l
0
=
1,2
,
3
).


## 5. Numerical Results

Problem 1 (see [2]).
We consider the linear time-fractional wave equation(16)
∂
α
u
(
x
,
t
)
∂
t
α
=
1
2
x
2
∂
2
u
(
x
,
t
)
∂
x
2
,
1
<
α
≤
2
,
t
>
0
u
(
x
,
0
)
=
x
,
∂
u
(
x
,
0
)
∂
t
=
x
2
.
The exact solution for the caseα
=
2 is given by u
(
x
,
t
)
=
x
+
x
2
sinh
⁡
t (see Table 1).
For comparison between fractional corrected Fourier series (FCFS) and variational iteration method (VIM) where we taket
=
0.2. (See Tables 2 and 3.)
To be cleared here, forα
=
2, we solve it by using original corrected Fourier series.Table 1
t
x
α
=
1.25
α
=
1.5
α
=
1.75
α
=
2
CFS
Exact
0.2
0.25
0.248566
0.351241
0.295434
0.248567
0.262584
0.50
0.542423
0.602263
0.548652
0.542423
0.550334
0.75
0.832454
0.852586
0.801804
0.832454
0.863252
1.00
1.115475
1.102422
1.054999
1.115475
1.201334Table 2
α
α
=
1.25
α
=
1.5
α
=
1.75
x
FCFS
VIM
FCFS
VIM
FCFS
VIM
0.25
0.248566
0.263175
0.351241
0.262840
0.295434
0.262670
0.50
0.542423
0.552700
0.602263
0.551362
0.548652
0.550680
0.75
0.832454
0.868575
0.852586
0.865566
0.801804
0.864029
1.00
1.115475
1.210800
1.102422
1.205450
1.054999
1.202718Table 3
x
α
=
2
VIM
CFS
0.25
0.262584
0.245396
0.50
0.550334
0.516298
0.75
0.863252
0.787390
1.00
1.201334
1.058374Problem 2 (see [2]).
Consider(17)
∂
α
u
∂
t
=
∂
2
u
∂
x
2
,
0
<
t
<
1
,
0
<
x
<
1
,
0
<
α
<
1
,
where u
(
x
,
0
)
=
sin
(
x
). For t
=
0.2, see Tables 4 and 5.Table 4
α
0.25
0.5
0.75
x
FCFS
VIM
FCFS
VIM
FCFS
VIM
0.25
0.184853
0.155391
0.180014
0.155803
0.180899
0.179254
0.50
0.290771
0.211629
0.280927
0.301919
0.283769
0.347363
0.75
0.382190
0.300891
0.369521
0.429263
0.374601
0.493874
1.00
0.521740
0.371445
0.506676
0.529918
0.512320
0.609679Table 5
t
x
α
=
1
VIM
CFS
0.2
0.25
0.202557
0.245396
0.50
0.392520
0.516298
0.75
0.558079
0.787390
1.00
0.688938
1.058374Problem 3 (see [8]).
Let us first consider the following time-fractional differential equation as follows:(18)
∂
α
u
(
x
,
t
)
∂
t
α
=
1
2
x
2
∂
2
u
(
x
,
t
)
∂
x
2
,
0
<
α
≤
1
,
t
>
0
u
(
x
,
0
)
=
x
2
,
u
(
0
,
t
)
=
0
,
u
(
1
,
t
)
=
e
t
.
The exact solution for the caseα
=
1 is given by u
(
x
,
t
)
=
x
2
e
t (see Table 6).
For comparison between corrected Fourier series and other methods see Tables7 and 8.Table 6
t
x
α
=
0.25
α
=
0.5
α
=
7.5
α
=
1
FCFS
exact
1
0.25
0.077299
0.140985
0.136142
0.173193
0.169892
0.50
0.699350
0.697982
0.597600
0.690081
0.679570
0.75
1.350993
1.305838
1.256308
1.618244
1.529034
1.00
2.157733
2.196163
2.187799
2.692081
2.718282Table 7
t
x
α
=
0.25
α
=
0.5
α
=
0.75
FCFS
Homotopy
FCFS
Homotopy
FCFS
Homotopy
1
0.25
0.077299
0.131454
0.140985
0.133024
0.136142
0.130504
0.50
0.699350
0.525816
0.697982
0.532095
0.597600
0.522016
0.75
1.350993
1.183085
1.305838
1.197213
1.256308
1.174540
1.00
2.157733
2.103263
2.196163
2.128379
2.187799
2.088065Table 8
t
x
α
=
1
FCFS
Homotopy
CFS
1
0.25
0.173193
0.166667
0.170034
0.50
0.690081
0.666667
0.612833
0.75
1.618244
1.500000
1.592353
1.00
2.692081
2.666667
2.682901


## 6. Conclusion

In this paper, with the presence of the modified Riemann-Liouville derivative, the corrected Fourier series has been proposed to solve the fractional partial differential problems. The solutions of problems are shown for different values in the given Tables1–8. It is shown that there is smaller error in between CFS method and other methods.



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*Source: 958931-2014-05-26.xml*