The new implicit finite difference scheme for two-sided space-time fractional partial differential equation

Subject Areas : Statistics

Hamid Reza  Khodabandehlo ¹ (Department of Applied Mathematics, Imam Khomeini International University, Qazvin, Iran)
Elyas  Shivanian ² (Department of Applied Mathematics, Imam Khomeini International University, Qazvin, Iran)
Shaaban  Mostafaee ³ (Department of Applied Mathematics, Imam Khomeini International University, Qazvin, Iran)

Received: 2019-07-08 Accepted : 2020-04-19 Published : 2020-08-22

Keywords: فرمول گرانوالد لتینکو انتقال یافته, تقریب تفاضلات متناهی ضمنی, معادلات دیفرانسیل جزئی کسری عددی, معادلات دیفرانسیل جزئی مرتبه کسری زمان- مکان دوطرفه, تحلیل پایداری,

Abstract :

Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical methods to solve a class of initial- boundary value fractional partial differential equations with variable coefficients on a finite domain. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional method based on the shifted Grunwald formula is unconditionally stable. This study concerns both theoretical and numerical aspects, where we deal with the construction and convergence analysis of the discretization schemes. A numerical example is presented and compared with exact solution for its order of convergence./////////Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical methods to solve a class of initial- boundary value fractional partial differential equations with variable coefficients on a finite domain. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional method based on the shifted Grunwald formula is unconditionally stable. This study concerns both theoretical and numerical aspects, where we deal with the construction and convergence analysis of the discretization schemes. A numerical example is presented and compared with exact solution for its order of convergence.

References:

 

[1] Goreno  R., Mainardi  F., Scalas  E., Raberto  M., Fractional calculus and continuous-time finance. III, The diffusion limit. Mathematical finance (Konstanz, 2000), Trends in Math., Birkhuser, Basel, 2001, pp. 171-180.

 

[2] Lubich  C., Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986) 704719. 13.

 

[3] Meerschaert  M.M. , Tadjeran  C ., Finite difference approximations for fractional advection - diffusion flow

equations, J.comput. Appl. Numer. Math. 172 (2004) 6577.

 

[4] Podlubny  I., Fractional Differential Equations, Academic Press, New York, 1999.

 

[5] Samko  S., Kilbas  A., Marichev  O., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, 1993.

 

[6] Oldham  K.B., Spanier  J., The Fractional Calculus, Academic Press, New York, 1974.

 

[7] Tadjeran  C.,  Meerschaert  M.M., H.P. Scheffer, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006) 205-213.

 

[8] Miller  K., Ross  B., An Introduction to the Fractional Calculus and Fractional Differential, Wiley, New York, 1993.

 

[9] zhang  Y., A Finite difference method for fractional partial differential equation, Appl. Math. comput. 215 (20009) 524-529.

 

[10] Shivanian. E, Khodabandehlo. H.R.,  A second-order accurate numerical approximation for fractional advection-dispersion flow equations, J. Sci. I. A. U (JSIAU), Vol. 23, No. 90.2, Winter 2014.

 

[11] Hilfer. R., Applications of Fractional Calculus in Physics, World Scientific, Singapore,2000.

 

[12] Magin. R.L., Fractional Calculus in Bioengineering, Begell House Publishers, 2006.