Solving Fractional Partial Differential Equations with Corrected Fourier Series Method

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.

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 order 𝛼>0 of a function 𝑓(𝑥) is defined as 𝐽^𝛼𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠. (1)

Jumarie’s modified Riemann-Liouville derivative of order 𝛼 is defined by the following definition.

Definition 2. Let 𝑓∶ℜ→ℜ, 𝑥→𝑓(𝑥) denote a continuous function. Its fractional derivative of order 𝛼 is defined as follows: for 𝛼<0, 𝐷^𝛼_𝑥=1Γ(−𝛼)^𝑥₀(𝑥−𝑠)^−𝛼−1𝑓(𝑠)−𝑓(0)𝑑𝑠, (2) for 𝛼>0, 𝐷^𝛼_𝑥=1Γ()𝑑1−𝛼𝑑𝑥^𝑥₀(𝑥−𝑠)^−𝛼𝑓(𝑠)−𝑓(0)𝑑𝑠,where0<𝛼<1,₀𝐷^𝛼_𝑥𝑓(𝑥)=1Γ()𝑑𝑛−𝛼𝑑𝑥^𝑛𝑛𝑥₀(𝑥−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)−𝑓(0)𝑑𝑠, (3)where 𝑛−1<𝛼<𝑛 with 𝑛∈𝑁.

Definition 3 (see [4–7]). Fractional derivative of compounded function is defined as 𝑑^𝛼𝑓≅Γ(1+𝛼)𝑑𝑓,0<𝛼<1. (4)

Definition 4 (see [4–7]). The integral with respect to (𝑑𝑡)^𝛼 is defined as the solution of the fractional differential equation 𝑑𝑥=𝑓(𝑥)(𝑑𝑡)^𝛼,𝑡≥0,𝑥(0)=0,0<𝛼<1. (5)

Lemma 5 (see [4–7]). Let 𝑓(𝑥) denote a continuous function; then the solution of 𝑦(𝑥), 𝑦(0)=0, (5) is defined as 𝑦=^𝑥₀𝑓(𝜉)(𝑑𝜉)^𝛼=𝛼^𝑥₀(𝑥−𝜉)^𝛼−1𝑓(𝜉)𝑑𝜉,0<𝛼<1. (6)

3. Corrected Fourier Series

The CFS is described in the form of 𝑢(𝑥,𝑡)=_{|𝑛|<∞|𝑚|<∞}𝐴_𝑛𝑚𝑒^{𝑖𝛼𝑛𝑥}𝑒^{𝑖𝛽𝑚𝑡}+_|𝑚|<∞𝑎_1𝑚𝑥+𝑎_2𝑚𝑥²2!+𝑎_3𝑚𝑥³𝑒3!^{𝑖𝛽𝑚𝑡}+_|𝑛|<∞𝑏_1𝑛𝑡+𝑏_2𝑛𝑡²2!+𝑏_3𝑛𝑡³𝑒3!^{𝑖𝛼𝑛𝑥}+³³_{𝑙=1𝑙0=1}𝑑_𝑙𝑙0𝑥^𝑙𝑡𝑙!𝑙^𝑙0₀!, (7) where 𝛼_𝑛=2𝑛𝜋/𝑥₀ and 𝛽_𝑚=2𝑚𝜋/𝑡₀.

Due to the periodicity of either 𝑒^{𝑖𝛼𝑛𝑥} or 𝑒^{𝑖𝛽𝑚𝑡}, 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: ³³_{𝑙=1𝑙0=1}𝑑_𝑙𝑙0𝑥^𝑙−𝑗!𝑡𝑙−𝑗𝑙₀−𝑗^{𝑙0−𝑗0}₀!𝐻𝑙−𝑗𝐻𝑙₀−𝑗₀^𝑥0₀^𝑡0₀=𝜕^𝑗+𝑗0𝑢(𝑥,𝑡)𝜕𝑥^𝑗𝜕𝑡^𝑗0𝑥0₀^𝑡0₀, (8) where 𝑗=0,1,2 and 𝑗₀=0,1,2.

Next, we want to determine the coefficients 𝑎_1𝑚, 𝑎_2𝑚, and 𝑎_3𝑚. With respect to 𝑥, the endpoints effect of 𝑢(𝑥,𝑡) and its partial derivatives yields 𝜕²𝑢𝜕𝑥^2𝑥0₀=_{|𝑚|≤𝑀}𝑎_3𝑚𝑥₀𝑒^{𝑖𝛽𝑚𝑡}+³_𝑙0=1𝑑_3𝑙0𝑥₀𝑡^𝑙0𝑙₀!,𝜕𝑢𝜕𝑥^𝑥0₀=_{|𝑚|≤𝑀}𝑎_2𝑚𝑥₀+𝑎_3𝑚𝑥²₀𝑒2!^{𝑖𝛽𝑚𝑡}+³³_{𝑙=2𝑙0=1}𝑑_𝑙𝑙0𝑥₀^𝑙−1()!𝑡𝑙−1𝑙^𝑙0₀!,𝑢|^𝑥0₀=_{|𝑚|≤𝑀}𝑎_1𝑚𝑥₀+𝑎_2𝑚𝑥²₀2!+𝑎_3𝑚𝑥³₀𝑒3!^{𝑖𝛽𝑚𝑡}+³³_{𝑙=1𝑙0=1}𝑑_𝑙𝑙0𝑥^𝑙₀𝑡𝑙!𝑙^𝑙0₀!. (9) Again, due to the periodicity of 𝑒^{𝑖𝛼𝑛𝑥}, 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^{𝑖𝛽𝑚𝑡}, we solved for 𝑎_1𝑚,𝑎_2𝑚, and 𝑎_3𝑚: 𝑎_3𝑚=1𝑥₀𝐹₂𝜕²𝑢𝜕𝑥^2𝑥0_0𝑚−³_𝑙0=1𝑑_3𝑙0𝑥₀𝐹₂𝑡^𝑙0𝑙₀!_𝑚,𝑎_2𝑚=1𝑥₀×𝐹₂𝜕𝑢𝜕𝑥^𝑥0_0𝑚−³³_{𝑙=2𝑙0=1}𝑑_𝑙𝑙0𝑥₀^𝑙−1()!𝐹𝑙−1²𝑡^𝑙0𝑙₀!_𝑚−𝑎_3𝑚𝑥²₀,𝑎2!_1𝑚=1𝑥₀×𝐹₂𝑢|^𝑥0_0𝑚−³³_{𝑙=1𝑙0=1}𝑑_𝑙𝑙0𝑥^𝑙₀𝐹𝑙!²𝑡^𝑙0𝑙₀!_𝑚−𝑎_2𝑚𝑥²₀2!−𝑎_3𝑚𝑥³₀.3! (10) Similar in the case with respect to 𝑡, we have 𝑏_3𝑛=1𝑡₀𝐹₁𝜕²𝑢𝜕𝑡^2𝑡0_0𝑛−³_𝑙=1𝑑_𝑙3𝐹₁𝑥^𝑙𝑙!_𝑛𝑡₀,𝑏_2𝑛=1𝑡₀×𝐹₁𝜕𝑢𝜕𝑡^𝑡0_0𝑛−³³_{𝑙=1𝑙0=2}𝑑_𝑙𝑙0𝐹₁𝑥^𝑙𝑙!_𝑛×𝑡^𝑙0₀⁻¹𝑙₀−1!−𝑏_3𝑛𝑡²₀,𝑏2!_1𝑛=1𝑡₀×𝐹₁𝑢|^𝑡0_0𝑛−³³_{𝑙=1𝑙0=1}𝑑_𝑙𝑙0𝐹₁𝑥^𝑙𝑙!_𝑛𝑡^𝑙0₀𝑙₀!−𝑏_2𝑛𝑡²₀2!−𝑏_3𝑛𝑡³₀.3! (11)

4. Fractional Corrected Fourier Series

In this paper, we consider the following general form of the linear time-fractional equation: 𝜕^𝛼𝑢𝜕𝑡^𝛼+𝑎₀(𝑥)𝑢(𝑥,𝑡)+𝑎₁(𝑥)𝜕𝑢𝜕𝑥+𝑎₂(𝑥)𝜕²𝑢𝜕𝑥²=𝑓(𝑥,𝑡) (12) and subject to the initial conditions 𝑢(𝑥,0)=𝑓(𝑥),as0<𝛼≤1,𝑥∈0,𝑥₀,𝑡∈0,𝑡₀,𝑢()(𝑥),()𝑥,0=𝑓𝜕𝑢𝑥,0(𝑥),𝜕𝑡=𝑔as1<𝛼≤2,𝑥∈0,𝑥₀,𝑡∈0,𝑡₀. (13) 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. For 𝑚 to be the smallest integer that exceeds 𝛼, the modified Riemann-Liouville time-fractional derivative operator of order 𝛼>0 is defined as 𝐷^𝛼_∗𝑡𝑢()=𝜕𝑥,𝑡^𝛼𝑢𝜕𝑡^𝛼=1Γ()𝜕𝑚−𝛼𝜕𝑡^𝑚𝑚×^𝑡₀(𝑡−𝜏)^{𝑚−𝛼−1}(𝑢()(𝜕𝑥,𝜏−𝑢𝑥,0))𝑑𝜏,for𝑚−1<𝛼<𝑚,_𝑚𝑢(𝑥,𝑡)𝜕𝑡^𝑚,for𝛼=𝑚∈𝑁, (14) where 𝑚−1<𝛼<𝑚 with 𝑚∈𝑁.

To obtain the nine unknowns 𝑑_𝑙𝑙0(𝑙,𝑙₀=1,2,3) in (7), we solve the following linear equations: ³³_{𝑙=1𝑙0=1}𝑑_𝑙𝑙0𝜆_𝑙0𝑥^𝑙−𝑗!𝑡𝑙−𝑗^{1+𝑙0−𝛼}𝐻𝑙−𝑗𝐻𝑙₀−𝑗₀^𝑥0₀^𝑡0₀=𝜕^{𝑗+(𝑗0+𝛼)}𝑢(𝑥,𝑡)𝜕𝑥^𝑗𝜕𝑡^{𝑗0+𝛼𝑥0}₀^𝑡0₀, (15) where 𝑗=0,1,2, 𝑗₀=0,1,2, and 𝜆_𝑙0=((−1)^2𝑙0)(Γ(𝛼−(𝑙₀+2))⋅((𝑙₀+2)−𝛼)/Γ(1−𝛼)Γ(𝛼−1)).

Then, we arrange the order of equations as 𝑗=0, 𝑗₀=0,1,2; 𝑗=1, 𝑗₀=0,1,2; and 𝑗=2, 𝑗₀=0,1,2; we can have the nine unknowns 𝑑₁₁, 𝑑₁₂, 𝑑₁₃, 𝑑₂₁, 𝑑₂₂, 𝑑₂₃, 𝑑₃₁, 𝑑₃₂, and 𝑑₃₃ in a vector form. Then, we solve this coefficient in the matrix form. By solving the matrix, we can determine the coefficients 𝑑_𝑙𝑙0(𝑙=𝑙₀=1,2,3).

5. Numerical Results

Problem 1 (see [2]). We consider the linear time-fractional wave equation 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=12𝑥²𝜕²𝑢(𝑥,𝑡)𝜕𝑥²𝑢()(),1<𝛼≤2,𝑡>0𝑥,0=𝑥,𝜕𝑢𝑥,0𝜕𝑡=𝑥². (16)
The exact solution for the case 𝛼=2 is given by 𝑢(𝑥,𝑡)=𝑥+𝑥²sinh𝑡 (see Table 1).
For comparison between fractional corrected Fourier series (FCFS) and variational iteration method (VIM) where we take 𝑡=0.2. (See Tables 2 and 3.)
To be cleared here, for 𝛼=2, we solve it by using original corrected Fourier series.

Table 1 

Table 2 

Table 3 

Problem 2 (see [2]). Consider 𝜕^𝛼𝑢=𝜕𝜕𝑡²𝑢𝜕𝑥²,0<𝑡<1,0<𝑥<1,0<𝛼<1, (17) where 𝑢(𝑥,0)=sin(𝑥). For 𝑡=0.2, see Tables 4 and 5.

Table 4 

Table 5 

Problem 3 (see [8]). Let us first consider the following time-fractional differential equation as follows: 𝜕^𝛼𝑢(𝑥,𝑡)𝜕𝑡^𝛼=12𝑥²𝜕²𝑢(𝑥,𝑡)𝜕𝑥²𝑢(),0<𝛼≤1,𝑡>0𝑥,0=𝑥²()𝑢(),𝑢0,𝑡=0,1,𝑡=𝑒^𝑡. (18)
The exact solution for the case 𝛼=1 is given by 𝑢(𝑥,𝑡)=𝑥²𝑒^𝑡 (see Table 6).
For comparison between corrected Fourier series and other methods see Tables 7 and 8.

Table 6 

Table 7 

Table 8 

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 Tables 1–8. It is shown that there is smaller error in between CFS method and other methods.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme having Project no. 5527068.