Numerical Solution of a Class of Integral Equations Arising in a Biological Laboratory Procedure

Abstract

We discuss a numerical method for certain integral equations of the form

$$I(t) = \int^L_0 k(x, t)\rho(x)dx, \quad 0\leq t\leq T,$$

(15.1)

where I is a smooth increasing function with I(0) = 0, L and T are positive constants, and the kernel k(·, ·) satisfies the following assumptions:

(i) k(·, ·) is continuous and bounded on [0, L] × [0, T] ? {(0, 0)}, positive on (0, L] × (0, T], and k(·, ·) ∈ C ¹((0, L) × (0, T)).

(ii) k(x, 0) = 0 for x > 0 and there is a positive constant k with k(0, t) ≥ k for t ∈ (0, T].

(iii) ∂₁ k < 0 < ∂₂ k on (0, L) × (0, T) (we use the notation ∂ _j to indicate the partial derivative with respect to the jth variable).

Our study of this class of Fredholm integral equations of the first kind is motivated by a mathematical model of an aspect of the olfactory system of frogs (see [FG06]). The function of the olfactory system is to transduce an odor stimulus into an electrical signal that is fed to the nervous system. This transduction is accomplished by a cascade of chemical processes that leads to an influx of ions through channels in very thin hair-like features, known as cilia, that reside in the nasal mucus. The potential difference across the membrane forming the lateral surface of the cilium resulting from this ion migration produces the electrical signal.