Oscillatory Spherical Traveling-Wave–Type Solutions to the Cardiac BVP

Abstract

This study demonstrates how the assumption of a spherical traveling wave form for the general solution to the reaction–diffusion simplified cardiac system of PDEs can lead to a semi-analytical nonlinear iterative particular solution to an associated two-point boundary value problem (BVP). We demonstrate that a conventional traveling wave solution to this BVP does not exist in spherical geometry. Satisfaction, however, of a certain asymptotic parametric stability condition by the solution to the cardiac BVP is shown to be possible only when the spherical traveling wave speed is purely imaginary. The pertaining fundamentally oscillatory and bifurcative spherical traveling-wave–type solution yields further a direct iterative relation for the space–time phase of some associated scroll-type waves.

Keywords:

• Iterative solution
• Oscillatory spherical traveling waves
• Scroll-type waves
• Systems of nonlinear Volterra integral equations
• Three-dimensional cardiac reaction diffusion equations
• Two-point nonhomogeneous boundary value problems

AMS Subject Classification:

• 34A34
• 34A45
• 34B99

ACKNOWLEDGMENTS

I am grateful to the editorial board of this journal for their critical reading of an earlier version of this paper.

This research has been partially supported by a grant from the Lebanese National Council for Scientific Research, Beirut, Lebanon.