IGNOU MTE 14 2026 - Consider a one-dimensional growth c(x, t) of phytoplankton in a water

Question

Consider a one-dimensional growth c(x, t) of phytoplankton in a water mass. Formulate the model describing the dynamics of growth taking into account the following: D, its diffusion coefficient, r its rate of growth, R its mortality rate due to sinking. Fixing the area of interest as $0 \le x \le 2$ and the initial concentration of phytoplankton as $30 \text{ moles/cm}^3$ , find the concentration distribution of phytoplankton in $0 \le x \le 2$ at any time t.
b) When an aeroplane ascends from take-off to an altitude of $10\text{ km}$ , by how much does the gravitational attraction acting on it decrease?

09 Jan 2026

Answer :

Word Count : 258

Let (c(x,t)) denote the concentration of phytoplankton at position (x) and time (t). The processes acting on the population are horizontal diffusion with coefficient (D), biological growth with rate (r), and loss due to sinking with rate (R). Combining these effects, the one-dimensional model is [ \frac{\partial c}{\partial t} = D,\frac{\partial^2 c}{\partial x^2} + r,c - R,c = D,\frac{\partial^2 c}{\partial x^2} + (r-R)c , \quad 0\le x\le 2,; t>0. ] The initial condition is [ c(x,0)=30 \quad

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