Chaos-free numerical solutions of reaction-diffusion equations

Abstract
Two numerical methods, which do not bring contrived chaos into the solution, are proposed for the solution of the Riccati (logistic) equation. Though implicit in nature, with the resulting improvements in stability, the methods are applied explicitly. When extended to the numerical solution of Fisher's equation, in which the quadratic polynomial representing the derivative in the Riccati equation appears as the reaction term, the solution is found by solving a linear system of algebraic equations at each time step, as opposed to solving a nonlinear system which frequently happens when solving nonlinear partial differential equations. The approaches adopted are extended to an ordinary differential equation in which the derivative is expressed as a cubic polynomial in the dependent variable. The solution of this initial-value problem is not available in closed form for finite values of the independent variable t. Under the conditions stated, numerical solutions are seen to converge to the correct steady-state solution. A nonlinear partial differential equation which governs the conduction of electrical impulses along a nerve axon and which has the aforementioned cubic polynomial as its reaction term, is solved by applying the numerical methods developed for solving the ordinary differential equation. The solution to this nonlinear reaction-diffusion equation is determined by solving a linear algebraic system at each time step.

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