Parabolic Partial Differential Equations as Inverse Moments Problem

Abstract

We considerer parabolic partial differential equations

w<SUB>t</SUB> − (w<SUB>x</SUB>)<SUB>x</SUB> = r (x,t)

under the conditions

w<SUB>x</SUB> (a<SUB>1</SUB>, t) = k<SUB>1</SUB> (t) w<SUB>x</SUB> (b<SUB>1</SUB>, t) = k<SUB>2</SUB> (t) w (x, a<SUB>2</SUB>) = h<SUB>1</SUB> (t) w (x, b<SUB>2</SUB>) = h<SUB>2</SUB> (t)

on a region E = (a<SUB>1</SUB>, b<SUB>1</SUB>) (a<SUB>2</SUB>, b<SUB>2</SUB>). We will see that we can write the equation in partial derivatives as an Fredholm integral equation of first kind and will solve this latter with the techniques of inverse moments problem. We will find an approximated solution and bounds for the error of the estimated solution using the techniques on moments problem. Also we consider the one-dimensional one-phase inverse Stefan problem.