Malaysian Journal of Mathematical Sciences, August 2012, Vol. 6(S)
Special Issue: International Workshop on Mathematical Analysis (IWOMA)
Well-Posedness of Parabolic Differential and Difference Equations with the Fractional Differential Operator
Allaberen Ashyralyev
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
The stable difference scheme for the approximate solution of the initial value problem \(\frac{du(t)}{dt}+D_{t}^{\frac{1}{2}}u(t)+Au(t)=f(t)\), \(0< t< 1\), \(u(0)=0\) for the differential equation in a Banach space \(E\) with the strongly positive operator \(A\) and fractional operator \(D_{t}^{\frac{1}{2}}\) is presented. The well-posedness of the difference scheme in difference analogues of spaces of smooth functions is established. In practice, the coercive stability estimates for the solution of difference schemes for the 2m-th order multi-dimensional
fractional parabolic equation and the one-dimensional fractional parabolic equation with nonlocal boundary conditions in space variable are obtained.
Keywords: Fractional parabolic equation, basset problems, well-posedness, coercive stability, difference scheme
