We employ the sincGalerkin method to obtain approximate solutions of spacefractional order partial differential equations (FPDEs) with variable coefficients. The fractional derivatives are used in the Caputo sense. The method is applied to three different problems and the obtained solutions are compared with the exact solutions of the problems. These comparisons demonstrate that the sincGalerkin method is a very efficient tool in solving spacefractional partial differential equations.
Fractional calculus, which might be considered as an extension of classical calculus, is as old as the classical calculus and fractional differential equations have been often used to describe many scientific phenomena in biomedical engineering, image processing, earthquake engineering, signal processing, physics, statistics, electrochemistry, and control theory.
Because finding the exact or analytical solutions of fractional order differential equations is not an easy task, several different numerical solution techniques have been developed for the approximate solutions of these types of equations. Some of the wellknown numerical techniques might be listed as generalized differential transform method [
In this paper we propose a new solution technique for approximate solution of spacefractional order partial differential equations (FPDEs) with variable coefficients and boundary conditions by using the sincGalerkin method that has almost not been employed for the spacefractional order partial differential equations in the form
The paper is organized as follows. Section
In this section, we present the definitions of the fractional RiemannLiouville derivative and the Caputo of fractional derivatives. By using these definitions, we give the definition of the integration by parts of fractional order.
Let
the left and right RiemannLiouville fractional derivatives of order
the left and right Caputo fractional derivatives of order
Now, we can write the definition of integration by parts of fractional order by using the relations given in (
If
In this section, we recall notations and definitions of the sinc function state some known results and derive some useful formulas to be used in the next sections of the present paper.
The function which defined all
Let
The domains
Let
Let
Let
If there exist positive constants
Consider fractional boundary value problem
Let
The following relations hold:
See [
For
The inner product with sinc basis elements of
Using Definition
We will use the sinc quadrature rule given with (
Replacing each term of (
If the assumed approximate solution of the boundaryvalue problem (
To obtain the approximate solution equation (
In this section, the present method will be tested on three different problems.
Consider fractional boundary value problem
Numerical results for


Exact sol.  Num. sol.  Error 

0.03  0.3  0.00000160  0.000069824  0.000068223 
0.6  0.00000365  0.000557026  0.000553368  
0.9  0.00000205  0.000286490  0.000284433  


0.06  0.3  0.00001202  −0.000160475  0.000172499 
0.6  0.00002748  0.000688434  0.000660950  
0.9  0.00001545  0.000405155  0.000389695  


0.09  0.3  0.00003803  −0.000317899  0.000355931 
0.6  0.00008693  0.000879725  0.000792795  
0.9  0.00004889  0.000577303  0.000528404 
Numerical results for


Exact sol.  Num. sol.  Error 

0.03  0.3  0.00000160  0.00000376 

0.6  0.00000365  0.00001237 


0.9  0.00000205  0.00000667 




0.06  0.3  0.00001202  0.00001645 

0.6  0.00002748  0.00004476 


0.9  0.00001545  0.00002455 




0.09  0.3  0.00003803  0.00004497 

0.6  0.00008693  0.00011262 


0.9  0.00004889  0.00006228 

Graphs of approximate solutions for different values.
Graph of exact solution.
Consider fractional boundary value problem
Numerical results for


Exact sol.  Num. sol.  Error 

0.03  0.3  0.0059288  0.0071562  0.00122741 
0.6  0.0135516  0.0149881  0.00143653  
0.9  0.0076227  0.0083725  0.00074978  


0.06  0.3  0.0118050  0.0131991  0.00139409 
0.6  0.0269829  0.0254147  0.00156823  
0.9  0.0151779  0.0145261  0.00065176  


0.09  0.3  0.0175764  0.0193179  0.00174151 
0.6  0.0401747  0.0362484  0.00392631  
0.9  0.0225983  0.0208906  0.00170767 
Numerical results for


Exact sol.  Num. sol.  Error 

0.03  0.3  0.0059288  0.0059287 

0.6  0.0135516  0.0135515 


0.9  0.0076227  0.0076227 




0.06  0.3  0.0118050  0.0118048 

0.6  0.0269829  0.0269826 


0.9  0.0151779  0.0151778 




0.09  0.3  0.0175764  0.0175761 

0.6  0.0401747  0.0401744 


0.9  0.0225983  0.0225983 

Graphs of approximate solutions for different values.
Graphs of exact solution.
Consider the fractional convectiondiffusion equation
The problem has the following exact solution:
Numerical results for


Exact sol.  Num. sol.  Error 

0.03  0.3  0.0073366  0.0043780  0.00295855 
0.6  0.0111718  0.0092590  0.00191284  
0.9  0.0048740  0.0045355  0.00033852  


0.06  0.3  0.0144134  0.0107858  0.00362752 
0.6  0.0219479  0.0228301  0.00088218  
0.9  0.0095754  0.0110942  0.00151873  


0.09  0.3  0.0209795  0.0167664  0.00421307 
0.6  0.0319465  0.0354700  0.00352348  
0.9  0.0139377  0.0172001  0.00326239 
Numerical results for


Exact sol.  Num. sol.  Error 

0.03  0.3  0.0073366  0.0073365 

0.6  0.0111718  0.0111718 


0.9  0.0048740  0.0048740 




0.06  0.3  0.0144134  0.0144139 

0.6  0.0219479  0.0219489 


0.9  0.0095754  0.0095759 




0.09  0.3  0.0209795  0.0209786 

0.6  0.0319465  0.0319454 


0.9  0.0139377  0.0139373 

Graphs of approximate solutions for different values.
Graphs of exact solution.
In this study, we use the sincGalerkin method to obtain approximate solutions of boundary value problems for spacefractional partial differential equations with variable coefficients. In order to illustrate the efficiency and accuracy of the present method, the method is applied to three examples in the literature and the obtained results are compared with exact solutions. As a result, it is shown that sincGalerkin method is very effective and accurate for obtaining approximate solutions of spacefractional differential equations with variable coefficients. In the future, we plan to extend the present numerical solution technique to nonlinear spacefractional partial differential equations.
The authors declare that there is no conflict of interests regarding the publication of this paper.