Mathematical Methods in Engineering pp 13-22 | Cite as

# Fractional spaces generated by the positive differential and difference operators in a Banach space

Conference paper

## Abstract

The structure of the fractional spaces
with

*E*_{α,q},(*L*_{q}[0, 1],*A*^{x}) generated by the positive differential operator*A*^{x}defined by the formula*A*^{x}*u*= −*a*(*x*)*d*^{2}*u*/*dx*^{2}+*δu*, with domain*D*(*A*^{x}) = {*u*∈*C*^{(2)}[0, 1] :*u*(0) =*u*(1),*u*′(0) =*u*′(1)} is investigated. It is established that for any 0 <*α*< 1/2 the norms in the spaces*E*_{α,q}(*L*_{q}[0, 1],*A*^{x}) and*W*_{q}^{2α }[0, 1] are equivalent. The positivity of the differential operator*A*^{x}in*W*_{q}^{2α}[0, 1](0 ≤*α*< 1/2) is established. The discrete analogy of these results for the positive difference operator*A*_{ h }^{x}a second order of approximation of the differential operator*A*^{x}, defined by the formula$$
A_h^x u^h = \left\{ { - a\left( {x_k } \right)\frac{{u_{k + 1} - 2u_k + u_{k - 1} }}
{{h^2 }} + \delta u_k } \right\}_1^{M - 1} ,u_h = \left\{ {u_k } \right\}_0^M ,Mh = 1
$$

*u*_{0}=*u*_{M}and −*u*_{2}+ 4*u*_{1}− 3*u*_{0}=*u*_{ M−2}− 4*u*_{ M−1}+ 3*u*_{M}is established. In applications, the coercive inequalities for the solutions of the nonlocal boundary-value problem for two-dimensional elliptic equation and of the second order of accuracy difference schemes for the numerical solution of this problem are obtained.## Keywords

Banach Space Positive Operator Discrete Analogy Fractional Space Nonlocal Boundary Condition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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