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

## Abstract

*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 *j*th 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.

## Keywords

Integral Equation Olfactory System Tikhonov Regularization Fredholm Integral Equation Odor Stimulus## Preview

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## References

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