An initial-value method for the solution of certain nonlinear diffusion equations in biology

Abstract

The steady-state one-dimensional diffusion equation with a nonlinear source term is a class of differential equations governing the behavior of many biological systems. As with other types of nonlinear differential equations, exact analytical solutions exist only in some very special cases. Previously, analytical solutions could be obtained only by a linearization process; moreover, the analytical solutions thus obtained approach the exact solution in a very limited range of some physical parameters. On the other hand, numerical solutions obtained by using digital computers, although exact, usually require an iteration process due to the two-point nature of the boundary conditions in such problems.In this article a method of transformation is introduced that makes it possible to transform the governing differential equation from a boundary-value to an initial-value problem. As a result, exact numerical solutions to this class of equations can be obtained in a single step. Numerical solutions of the concentration profiles in an enzyme system are presented as an illustration of the method.