(The full article is here: https://app.box.com/s/bjb63h4nt12rsvdke6yl. The part I would like to ask is on page 995-996))(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to understand the idea behind the author's method in solving the differential equation

[itex]

I'(t) = -D(t) F \big (e^{\int_0^t \theta (u) du} I(t) \big )- \theta (t) I(t) \qquad (1.5)

[/itex]

where [itex] D(t) [/itex] is positive and [itex] \theta(t) [/itex] is nonnegative for [itex] 0 < t < H [/itex].

{start: How is this part relevant? --------------

Let

[itex]

\mu(x,y) = \int_x^y D(t) e^{\int_0^t \theta (u) du} dt

[/itex]

where [itex] F(v) [/itex] is positive for [itex] 0 < v < \ell [/itex] and [itex] F \in C(0,\ell) [/itex],

for some number [itex] \ell [/itex] for which

[itex]

\mu(0,H) \leq \int_0^\ell \frac{dv}{F(v)} < \infty.

[/itex]

end:-----------}

Equation(1.5)holds for [itex] t_{j-1} \leq t < t_j [/itex], with the boundary condition

[itex]

I(t) \rightarrow 0 \quad \text{as} \quad t \uparrow t_j \qquad (1.8)

[/itex]

for [itex] j = 1, 2, \ldots n [/itex]. To solve the equation subject to this condition, we change the dependent variable to [itex] J(t) = e^{\int_0^t \theta (u) du} I(t) [/itex].

With this as the unknown,(1.5)becomes

[itex]

J'(t) = D(t) e^{\int_0^t \theta (u) du} F \big ( J(t) \big ).

[/itex]

Subsequently. by separation of variables and imposition of(1.8)we find

[itex]

\int_0^{J(t)} \frac{dv}{F(v)} = \mu(t,t_j)

[/itex]

for [itex] t_{j-1} \leq t < t_j [/itex].

{start: This is the bit that I don't understand. Why need to define such

integral?}

Hence, if we define [itex] \psi [/itex] via

[itex]

z = \int_0^{\psi(z)} \frac{dv}{F(v)}

\quad \text{for} \quad

0 \leq z \leq \mu(0,H),

[/itex]

end:-----------}

we have

[itex]

J(t) = \psi \big ( \mu(t,t_j) \big )

\quad \text{for} \quad

t_{j-1} \leq t < t_j .

[/itex]

This gives

[itex]

I(t) = \kappa(t,t_j) \quad \text{for} \quad t_{j-1} \leq t < t_j

[/itex]

where

[itex]

\kappa(x,y) = e^{-\int_0^t \theta (u) du} \psi \big ( \mu(x,y) \big ).

[/itex]

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# Help needed in understanding the author method of solving a DE

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