Solving Fractional Partial Differential Equations with Corrected Fourier Series Method

and Applied Analysis 3 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 0.609679 applying the Fourier projection on the basis function em, we solved for a 1m , a 2m , and a 3m : a 3m = 1 x 0 (F 2 ⟨ ∂ 2 u ∂x 2 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 x 0


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.

Basic Definitions
The Riemann-Liouville fractional integral is the most popular definition that we always find in the study of fractional calculus.

Corrected Fourier Series
The CFS is described in the form of where   = 2/ 0 and   = 2/ 0 .Due to the periodicity of either     or     , 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 (, ) and its partial derivative, we obtain the following linear equations: where  = 0, 1, 2 and  0 = 0, 1, 2.
Next, we want to determine the coefficients  1 ,  2 , and  3 .With respect to , the endpoints effect of (, ) and its partial derivatives yields Again, due to the periodicity of     , the first and third terms of ( 7) and its partial derivatives are identically zero.Then, by Similar in the case with respect to , we have (11)

Fractional Corrected Fourier Series
In this paper, we consider the following general form of the linear time-fractional equation: and subject to the initial conditions  (, 0) =  () , (, 0) =  () ,  (, 0)  =  () , For the case of  = 1, the fractional equation reduces to the classical linear PDE and is similar to the case of  = 2. Definition 6.For  to be the smallest integer that exceeds , the modified Riemann-Liouville time-fractional derivative operator of order  > 0 is defined as where  − 1 <  <  with  ∈ .

Numerical Results
Problem 7 (see [2]).We consider the linear time-fractional wave equation The exact solution for the case  = 2 is given by (, ) =  +  2 sinh  (see Table 1).
For comparison between fractional corrected Fourier series (FCFS) and variational iteration method (VIM) where we take  = 0.2.(See Tables 2 and 3.) To be cleared here, for  = 2, we solve it by using original corrected Fourier series.
For comparison between corrected Fourier series and other methods see Tables 7 and 8.

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.

Table 4 𝛼
Fourier projection on the basis function     , we solved for  1 ,  2 , and  3 :