Exact Analytic Solution of the Simplified Telegraph Model of Propagation and Dissipation of Excitation Fronts

The telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion in several branches of sciences (E. Ahmed, H. A. Abdusalam, and E. S. Fahmy, 2001, Int. J. Mod. Phys. C 12(5), 717; E. Ahmed and H. A. Abdusalam, 2004, Chaos, Solitons and Fractals 22, 583; H. A. Abdusalam and E. S. Fahmy, 2003, Chaos, Solitons and Fractals 18, 259; Abdusalam, 2004 Appl. Math. Comp. 157, 515). An excitation wave in cardiac tissue fails to propagate if the transmembrane voltage at its front rises too slow and does not excite the tissue ahead of it. Then the sharp voltage profile of the front will dissipate, and subsequent spread of voltage will be purely diffusive. This mechanism is impossible in FitzHugh–Nagumo type system (V. N. Biktashev, 2003, Int. J. Bifarcation and Chaos 13(12), 3605). Biktashev suggested a simplified mathematical model for this mechanism and in the present work we generalize this model to telegraph system. Our generalized telegraph model has exact traveling front solutions and we show the effect of the time delay on the velocity and we show that, the post-front voltage depends on two parameters in which one of them is the time delay.