A comparison of fourier solutions for three sobolev approximations to initial value heat problems

Abstract

In earlier papers, the author examined transmutation relations between the solutions of three abstract Sobolev problems and a corresponding abstract heat problem. In this note, we examine these relationships in the classical setting by means of Fourier series and integral solution representations. One of the three Sobolev problems has received an extensive treatment in the literature. Of the remaining two, the solution of one involves an approximation of the exponential function exp(−n²π²t) by the function , while the solution of the other exhibits a wave type behavior.

Keywords:

• Heat problem
• Sobolev approximations
• semigroups
• continuous groups
• Fourier sine representations
• wave behavior

AMS:

• 35O20
• 34C10