In this paper, the method of generalized quasilinearization has been extended to reaction diffusion equations. The extension includes earlier known results as special cases. The earlier results developed are when (i) the right-hand side function is the sum of a convex and concave function, and (ii) the right-hand function can be made convex by adding a convex function. In our present result, if the monotone iterates are mildly nonlinear, we establish the quadratic convergence as in the quasilinearization method. If the iterates are totally linear then the iterates converge semi-quadratically.

Recent efforts of one of us (HML) have been directed toward demonstrating that line inductance must be included in any mathematical formulation of the electrical properties of axons, for nerve and muscle as well as pacemaker and receptor cells. This is in order that such a formulation will yield the required propagation behavior, free of numerical instability, and in order that the results obtained therefrom will be in agreement with certain empirical facts that seem to be indispensable in describing highly organized life. These efforts have yielded unquestionable, if somewhat indirect, evidence of the occurrence of magnetic fields that accompany every action potential, and of surprisingly large inductance characteristics for any axon which is thin enough to be regarded as a transmission line. Meanwhile, what appear to be relatively large magnetic fields propagating with action potentials have actually been detected on axons! Moreover, the continuing efforts of two schools working independently for approximately a decade have been directed toward removing any possibility that these findings are due to secondary electrostatic effects occuring in the detection equipment, and have left workers convinced that their results are essentially correct. The last statement is based on a private communication from J. A. Gengerelli, University of California, Los Angeles, May 11, 1969. Reflecting on the manner in which current is known to travel along an axon and through its membrane, we see that the current accompanying an action potential circulates in tightly wound toroidal sheets about the perimeter of a cross section which itself moves with the action potential. Maxwell's equations are used here to show that the line inductance of such toroidal sheets of current is inversely proportional to the cross-sectional area of the axon. For any superthreshold phenomena on an axon, the inductance is therefore quite large, undoubtedly large enough to give what is known as a “Heavified” or “Pupinized” line in linear approximation. For subthreshold phenomena, the induced voltage would be small even if the inductance coefficient were large. It is speculated here that highly concentrated magnetic fields are thus produced by an action potential and focused on the membrane in front of it, there either to serve directly as motive power in gating for entry and exit of sodium and potassium ions, respectively, or to serve as electromechanical chemical agents which trigger in the membrane molecular level activities that have the same effect.