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We examine some finite difference numerical methods to solve the fractional partial differential equation of the form

# Finite difference approximations for two-sided space-fractional partial differential equations

Applied Numerical Mathematics, no. 1 (2006): 80-90

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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 eq...More

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Introduction
• The authors examine some finite difference numerical methods to solve the fractional partial differential equation (FPDE) of the form

M.M.
• The authors examine some finite difference numerical methods to solve the fractional partial differential equation (FPDE) of the form
• If the differential equation (9) is discretized in time using an explicit (Euler) method, one obtains u(x, tn+1)
Highlights
• We examine some finite difference numerical methods to solve the fractional partial differential equation (FPDE) of the form
• If the fractional derivatives in (10) is discretized by the standard Grünwald estimates resulting from (6), we obtain a finite difference approximations to Eq (9) which stability analysis  shows to be unstable, the numerical solution does not converge to the exact solution
• We only examine and refer to the stability of the numerical methods discussed in this paper
• We examine the implicit Euler approximation to the two-sided fractional PDE (1)
• The explicit Euler method approximation using the shifted Grünwald estimates to the fractional partial differential equation (1) with 1 < α 2 is stable if t hα α(c+ max + c− max)
Results
• If the fractional derivatives in (10) is discretized by the standard Grünwald estimates resulting from (6), the authors obtain a finite difference approximations to Eq (9) which stability analysis  shows to be unstable, the numerical solution does not converge to the exact solution.
• To obtain a stable explicit Euler method when 1 α 2, the authors define the shifted Grünwald formula dαf (x) dxα gk · f x − (k − 1)h k=0 which defines the following shifted Grünwald estimate to the left-handed fractional derivative
• With the spectral radius so bounded, the numerical errors do not grow, and the explicit Euler method defined above is conditionally stable.
• For the classical parabolic PDE, that is α = 2 in (9), the resulting explicit Euler method from (13) is the classical finite difference equation given by (g0 = 1, g1 = −2, g2 = 1, and g3 = g4 = · · · = 0)
• For the classical hyperbolic PDE, that is α = 1 in (9), the resulting explicit Euler method is the classical finite difference equation given by (g0 = 1, g1 = −1, and g2 = g3 = g4 = · · · = 0)
• The implicit Euler method approximation defined by (16) to the fractional partial differential equation (1) with 1 α 2 is unconditionally stable.
• The explicit Euler method approximation using the shifted Grünwald estimates to the fractional partial differential equation (1) with 1 < α 2 is stable if t hα α(c+ max + c− max)
• It may be surprising to note that the addition of a ‘balancing’ right-handed derivative does not improve the instability of the Euler methods with the standard Grünwald estimates.
Conclusion
• To examine the performance of this finite difference method for this example problem, the maximum numerical error at time t = 1.0 was computed starting with t = 0.1 and x = h = 0.2.
• The implicit Euler method, based on a modified Grünwald approximation to the fractional derivative, is consistent and unconditionally stable.
• The explicit Euler method, using the shifted Grünwald method to solve the two-sided fractional PDEs, is conditionally stable.
Funding
• The first author was supported by NSF grants DMS-0139927 and DMS-0417869 from Marsden Foundation in New Zealand
• The second author was supported by NSF grant DMS-0139927
Reference
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