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Next: Special inverse problems Up: Numerical Solution of Bilinear Previous: Introduction

Linerarization

We linearize (1.1)-(1.2) about a solution tex2html_wrap_inline1470 of (1.1). Replacing tex2html_wrap_inline1376 by tex2html_wrap_inline1474 , f by f + h, (1.1) reads

displaymath1480

Neglecting higher order terms and using that tex2html_wrap_inline1376 , f solve (1.1) we obtain

  equation98

Similarly, neglecting higher order terms in (1.2) yields

displaymath1486

Thus the operator tex2html_wrap_inline1488 is given by

  equation102

where tex2html_wrap_inline1490 is the solution of (2.1).

In the application to concrete problems, the bulk of the analytical work consists in determining the adjoint operator tex2html_wrap_inline1492 . While an explicit formula for this operator in the functional analytic framework is easy to derive, this formula is not very useful in practice. Thus we derive tex2html_wrap_inline1494 for specific cases in the next section.



Frank Wuebbeling
Mon Sep 7 11:40:50 MET DST 1998