Heat Conduction in a Finite Bar with a Nonlinear Source

Abstract

A model reaction-diffusion equation for temperature with a nonlinear source term is introduced which is an extension of the linear source one treated in Chapter 5. This is equivalent to the interaction-diffusion equation for population density originally analyzed by Wollkind et al. (SIAM Rev 36:176–214, 1994, [142]) through terms of third-order in its supercritical parameter range. That analysis is extended through terms of fifth-order to examine the behavior in its subcritical regime. It is shown that under the proper conditions the two subcritical cases behave in exactly the same manner as the two supercritical ones unlike the outcome for the truncated system. Further there also exists a region of metastability allowing for the possibility of population outbreaks discussed in Chapter . These results are then used to offer an explanation for the occurrence of isolated vegetative patches and sparse homogeneous distributions in the relevant ecological parameter range where there is subcriticality for a plant-ground water model to be treated in Chapter . Finally these results are discussed in the context of Matkowsky’s (Bull Amer Math Soc 76:646–649, 1970, [78]) two-time nonlinear stability analysis. The problem applies this nonlinear stability analysis through terms of third-order to a particular reaction-long range diffusion model equation  (Wollkind et al, SIAM Rev 36:176–214, 1994, [142]).

Problem

16.1.

Consider the model interaction–long-range diffusion equation [142] for $f = f(s,\tau )$ with $s\equiv$ the one-dimensional spatial variable and $\tau \equiv$ time:

$$\begin{aligned} \frac{\partial f}{\partial \tau } + \frac{\partial }{\partial s}\left[ D(f)\frac{\partial f}{\partial s}\right] + D_2\frac{\partial ^4 f}{\partial s^4} + r_0 f_e\sinh \left( \frac{f}{f_e}\right) = 0, \, |s| < \infty ; \end{aligned}$$

(P1.1a)

where

$$\begin{aligned} D(f) = D_0 + D_1 f \end{aligned}$$

(P1.1b)

and

$$\begin{aligned} f \text { remains bounded as } s^2 \rightarrow \infty . \end{aligned}$$

(P1.1c)

Here f may be interpreted as the deviation from a homogeneous distribution $f_e$ for the density of a population over an unbounded flat environment where $r_0$ is the reference scale of its interaction rate while $D_2$ and D(f) represent its long- and short-range diffusion effects, respectively. In addition an affine-type density dependent constitutive relation of (P1.1b) has been assumed for the latter with intercept $D_0 > 0$ and slope $D_1$. The sign of this intercept implies that when (P1.1a) is placed in the normal interaction–diffusion equation canonical form its diffusion coefficient is negative. For an activation-inhibition type system the possibility of such an occurrence depends on its lateral inhibition mechanism being much more severe and acting over a much longer range than is usual as surveyed in detail by Murray [84]. Finally, the interaction source term is also opposite in sign from that contained in this canonical interaction–diffusion equation form.

• (a) Introducing the nondimensional variables and parameters

  $$\begin{aligned} x = \frac{s}{\root 4 \of {\frac{D_2}{r_0}}}, \, t = r_0\tau , \, \zeta = \frac{f}{f_e}, \, \mathcal {R} = \frac{D_0}{\sqrt{D_2r_0}}, \, \alpha = \frac{f_e D_1}{D_0}; \end{aligned}$$

  (P1.2)

  transform (P1.1) into the following equation for $\zeta = \zeta (x, t)$:

  $$\begin{aligned} \frac{\partial \zeta }{\partial t} + \mathcal {R}\frac{\partial }{\partial x} \left[ \mathcal {D}(\zeta )\frac{\partial \zeta }{\partial x}\right] + \frac{\partial ^4 \zeta }{\partial x^4} + \sinh (\zeta ) = 0, \, -\infty< x < \infty ; \end{aligned}$$

  (P1.3a)

  where

  $$\begin{aligned} \mathcal {D}(\zeta ) = 1 + \alpha \zeta \end{aligned}$$

  (P1.3b)

  and

  $$\begin{aligned} \zeta \text { remains bounded as } x \rightarrow \pm \infty . \end{aligned}$$

  (P1.3c)

• (b) Show that

  $$\begin{aligned} \zeta \equiv \zeta _e = 0, \end{aligned}$$

  (P1.4)

  which represents a homogeneous density distribution, is a solution to equation (P1.3).

• (c) Recalling

  $$\begin{aligned} \cos (y \pm z) = \cos (y)\cos (z) \mp \sin (y)\sin (z), \end{aligned}$$

  (P1.5a)

  deduce that

  $$\begin{aligned} \cos (y)\cos (z) = \frac{1}{2}[\cos (y + z) + \cos (y-z)] \end{aligned}$$

  (P1.5b)

  and

  $$\begin{aligned} \sin (y)\sin (z) = \frac{1}{2}[\cos (y-z) - \cos (y+z)]. \end{aligned}$$

  (P1.5c)

• (d) In particular, use (P1.5b) to conclude that

  $$\begin{aligned} \cos ^2(\omega x) = \frac{1}{2}[1 + \cos (2\omega x)] \end{aligned}$$

  (P1.6a)
  $$\begin{aligned} \cos (\omega x)\cos (2\omega x) = \frac{1}{2}[\cos (\omega x) + \cos (3\omega x)]; \end{aligned}$$

  (P1.6b)

  and, then by employing (P1.6a, P1.6b), that

  $$\begin{aligned} \cos ^3(\omega x) = \frac{1}{4}[\cos (\omega x) + \cos (3\omega x)]. \end{aligned}$$

  (P1.6c)

• (e) Similarly use (P1.5c) to conclude that

  $$\begin{aligned} \sin ^2(\omega x) = \frac{1}{2}[1 - \cos (2\omega x)], \end{aligned}$$

  (P1.7a)
  $$\begin{aligned} \sin (\omega x)\sin (2\omega x) = \frac{1}{2}[\cos (\omega x) - \cos (3\omega x)]. \end{aligned}$$

  (P1.7b)

• (f) Examine the stability of the solution (P1.4) by considering a Stuart-Watson cosine expansion of equation (P1.3) of the form

  $$\begin{aligned}&\zeta (x, t) \sim A(t)\cos (\omega x) + A^2(t)[\zeta _{20} + \zeta _{22}\cos (2\omega x)] \nonumber \\&\quad + A^3(t)[\zeta _{31}\cos (\omega x) + \zeta _{33}\cos (3\omega x)] \end{aligned}$$

  (P1.8a)

  where

  $$\begin{aligned} \dot{A}(t)\equiv \frac{dA(t)}{dt} \sim \sigma A(t) - a_1A^3(t) \end{aligned}$$

  (P1.8b)

  and $\omega = 2\pi /\lambda$, $\lambda$ being the wavelength of the class of periodic perturbations under investigation. Substituting (P1.8) into (P1.3), noting that $\sinh (\zeta ) \sim \zeta + \zeta ^3/6$ while $\partial [\mathcal {D}(\zeta )\partial \zeta /\partial x]/\partial x = (1 + \alpha \zeta )\partial ^2 \zeta /\partial x^2 + \alpha (\partial \zeta /\partial x)^2$, and employing the identities of (P1.6) and (P1.7), obtain a sequence of problems, one for each pair of values of n and m which corresponds to a term of the form $A^n(t)\cos (m\omega x)$ appearing explicitly in (P1.8a).

• (g) From the linear stability problem for $n = m = 1$, arrive at the secular equation

  $$\begin{aligned} \sigma = \sigma _{\omega ^2}(\mathcal {R}) = [\mathcal {R} - \mathcal {R}_0(\omega ^2)]\omega ^2 \text { where } \mathcal {R}_0(\omega ^2) = \omega ^2 + \frac{1}{\omega ^2}. \end{aligned}$$

  (P1.10)

• (h) Plot the marginal stability curve for (P1.10) in the $\omega ^2$-R plane

  $$\begin{aligned} \mathcal {R} = \mathcal {R}_0(\omega ^2) \end{aligned}$$

  (P1.11a)

  and conclude that it has a minimum value of $\mathcal {R}_c$ at $\omega ^2 = \omega _c^2$ where

  $$\begin{aligned} \omega _c^2 = 1 \text { and } \mathcal {R}_c = \mathcal {R}_0(\omega _c^2) = 2. \end{aligned}$$

  (P1.11b)

• (i) After DiPrima et al. [28], take $\omega \equiv \omega _c > 0$ in (P1.8a). Then show (P1.10) becomes

  $$\begin{aligned} \sigma = \sigma _1(\mathcal {R}) = \mathcal {R} - 2 \end{aligned}$$

  (P1.12)

  and hence observe that the homogenous solution is linearly stable or unstable depending upon whether $\mathcal {R} < \mathcal {R}_c = 2$ or $\mathcal {R} > \mathcal {R}_c = 2$, respectively.

• (j) Under this condition solve the problems for $\zeta _{20}$, $\zeta _{22}$, and $\zeta _{33}$ in a straightforward manner to obtain that

  $$\begin{aligned} \zeta _{20}(\mathcal {R},\alpha ) \equiv 0, \end{aligned}$$

  (P1.13a)
  $$\begin{aligned} \zeta _{22}(\mathcal {R},\alpha ) = \frac{\alpha \mathcal {R}}{13 - 2\mathcal {R}}, \end{aligned}$$

  (P1.13b)
  $$\begin{aligned} \zeta _{33}(\mathcal {R},\alpha ) = \frac{9\alpha \mathcal {R}\zeta _{22}(\mathcal {R},\alpha ) - 1/12}{4(28 -3 \mathcal {R})}. \end{aligned}$$

  (P1.13c)

• (k) Demonstrate that the problem for $\zeta _{31}$ now gives rise to the following relation

  $$\begin{aligned} a_1(\alpha ) = 2\sigma _1(\mathcal {R})\zeta _{31}(\mathcal {R},\alpha ) + \frac{1}{8} - \frac{\alpha \mathcal {R}\omega _c^2 \zeta _{22}(\mathcal {R},\alpha )}{2}. \end{aligned}$$

  (P1.14)

• (l) Taking the limit of (P1.14) as $\mathcal {R} \rightarrow \mathcal {R}_c = 2$ deduce the solvability condition

  $$\begin{aligned} a_1(\alpha ) = \frac{1}{8}\left[ 1 - \left( \frac{4\alpha }{3}\right) ^2\right] \, \Rightarrow \, \zeta _{31}(\mathcal {R},\alpha ) = \frac{\alpha ^2(9\mathcal {R} + 26)}{36(13 - 2\mathcal {R})}. \end{aligned}$$

  (P1.15)

• (m) Finally defining $\beta = 1/\mathcal {R}$ which implies $\sigma _1(1/\beta ) = (1-2\beta )/\beta )$ plot its horizontal marginal stability line $\beta = \beta _c = 1/2$ and the loci $\alpha = \alpha _0^\pm = \pm 3/4$ in the $\alpha$-$\beta$ plane identifying those regions corresponding to the four cases depicted in Fig. 16.1 by examining the signs of $\sigma _1(1/\beta )$ and $a_1(\alpha )$.