FDM for fractional parabolic equations with the Neumann condition
© Ashyralyev and Cakir; licensee Springer 2013
Received: 2 January 2013
Accepted: 11 April 2013
Published: 25 April 2013
In the present study, the first and second order of accuracy stable difference schemes for the numerical solution of the initial boundary value problem for the fractional parabolic equation with the Neumann boundary condition are presented. Almost coercive stability estimates for the solution of these difference schemes are obtained. The method is illustrated by numerical examples.
MSC:34K37, 35R11, 35B35, 39A14, 47B48.
Keywordsfractional parabolic equations Neumann condition difference schemes stability
Mathematical modeling of fluid mechanics (dynamics, elasticity) and other areas of physics lead to fractional partial differential equations. Numerical methods and theory of solutions of the problems for fractional differential equations have been studied extensively by many researchers (see, e.g., [1–31] and the references given therein).
for the fractional differential equation in a Banach space E with the strongly positive operator A was investigated. This fractional differential equation corresponds to the Basset problem . It represents a classical problem in fluid dynamics where the unsteady motion of a particle accelerates in a viscous fluid due to the gravity of force. Here is the standard Riemann-Liouville’s derivative of order .
The well-posedness of (1.1) in spaces of smooth functions was established. The coercive stability estimates for the solution of the 2m th order multidimensional fractional parabolic equation and the one-dimensional fractional parabolic equation with nonlocal boundary conditions in space variable were obtained.
The well-posedness of (1.2) in difference analogues of spaces of smooth functions was established. Namely, we have the following theorems.
Here, and in future, positive constants, which can differ in time (hence: not a subject of precision) will be indicated with an M. On the other hand is used to focus on the fact that the constant depends only on .
Finally, the coercive stability and almost coercive stability estimates for the solution of difference schemes the first order of approximation in t for the 2m th order multidimensional fractional parabolic equation and the one-dimensional fractional parabolic equation with nonlocal boundary conditions in space variable were obtained.
with boundary S, , and () and (, ) are given smooth functions and , and is the normal vector to S.
The first and second order of accuracy difference schemes for the approximate solution of problem (1.8) are presented. The almost coercive stability estimates for the solution of these difference schemes are established. The theoretical statements for the solution of these difference schemes for one-dimensional fractional parabolic equations are supported by numerical examples.
2 The well-posedness of difference scheme
holds for the solution of problem (1.7). Here, and .
Estimate (2.2) follows from estimates (2.9) and (2.10). Theorem 2.1 is proved. □
Estimate (2.11) follows from estimates (2.13) and (2.14). Theorem 2.2 is proved. □
for a finite system of ordinary fractional differential equations.
for the approximate solution of problem (1.8).
for the approximate solution of problem (1.8).
Theorem 3.2 
The proof of Theorem 3.3 is based on the abstract Theorem 2.2 and on estimate (3.4) and on the positivity of the operator in and on Theorem 3.2 on the almost coercivity inequality for the solution of the elliptic difference equation in .
Note that one has not been able to get a sharp estimate for the constants figuring in the almost coercive stability estimates of Theorems 3.1 and 3.3. Therefore, our interest in the present paper is studying the difference schemes (3.2) and (3.3) by numerical experiments. Applying these difference schemes, the numerical methods are proposed in the following section for solving the one-dimensional fractional parabolic partial differential equation. The method is illustrated by numerical experiments.
4 Numerical results
for the one-dimensional fractional parabolic partial differential equation is considered. The exact solution of problem (4.1) is .
4.1 First order of accuracy difference scheme
where , is the identity matrix and is the zero matrix.
4.2 Second order of accuracy difference scheme
4.3 Error analysis
where represents the exact solution and represents the numerical solutions of these difference schemes at .
Comparison of errors
N = M = 30
N = M = 60
N = M = 120
1st order difference scheme
2nd order difference scheme
Thus, the results show that, by using the Crank-Nicholson difference scheme increases faster then the first order of accuracy difference scheme.
for semilinear fractional parabolic partial differential equations with smooth and .
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