Chapter

Modern Analysis and Applications

Volume 191 of the series Operator Theory: Advances and Applications pp 229-252

High-accuracy Stable Difference Schemes for Well-posed NBVP

  • Allaberen AshyralyevAffiliated withDepartment of Mathematics, Fatih University

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Abstract

The single step difference schemes of the high order of accuracy for the approximate solution of the nonlocal boundary value problem (NBVP)
$$ v'\left( t \right) + Av\left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant 1} \right),v\left( 0 \right) = v\left( \lambda \right) + \mu , 0 < \lambda \leqslant 1 $$
for the differential equation in an arbitrary Banach space E with the strongly positive operator A are presented. The construction of these difference schemes is based on the Padé difference schemes for the solutions of the initial-value problem for the abstract parabolic equation and the high order approximation formula for \( v\left( 0 \right) = v\left( \lambda \right) + \mu \). The stability, the almost coercive stability and coercive stability of these difference schemes are established.

Mathematics Subject Classification (2000)

Primary 65N12 Secondary 47D06

Keywords

Parabolic equation nonlocal boundary value problem Padé difference schemes high order of accuracy well-posedness coercive inequalities