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 and the initial concentration of phytoplankton as
, find the concentration distribution of phytoplankton in
at any time t.
b) When an aeroplane ascends from take-off to an altitude of , by how much does the gravitational attraction acting on it decrease?
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 , ____ __________ _____ _____ _____ ______.
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State two real-world problems where you think that mathematical modelling is the only approach to find the solution of the problem. Give 4 essentials for each of the problems. Why do you think that there is no other scientific alternative for the treatment of these problems.
b) Classify the following into linear and non-linear models, justifying your classification.
i) Simple harmonic motion for small amplitude of oscillation.
ii) Population growth model given by
iii) Equation for velocity v of a particle at any time t, moving with a constant acceleration a, and initial velocity u.
iv) Equation describing dynamic stability of market equilibrium price given by
$
and pt is the price in period t.
लघुदोलन आयाम वाली सरल आवर्त गति
A monopolist sets a price 'p' per unit and the quantity demanded 'q' is given by the following relation:
Let there be a fixed cost of Rs.9 and a marginal cost of Rs.1 per unit.
i) Write the profit function of monopolist.
ii) For maximum profit, find the number 'x' of units produced. Also find the maximum profit.
iii) A potential entrant enters into the business of the monopolist. He believes that the monopolist will go on making 'x' units. Write the profit function of the entrant.
iv) For maximum profit of entrant, find the number 'z' of units produced.
v) Find the maximum profit of the entrant. Explain whether he should enter into the business or not.
vi) Find the profit made by the monopolist after the entrant has entered into business.
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 and the initial concentration of phytoplankton as
, find the concentration distribution of phytoplankton in
at any time t.
b) When an aeroplane ascends from take-off to an altitude of , by how much does the gravitational attraction acting on it decrease?
एक ट्यूमर पर रसायन औषधि के प्रभाव जबकि इन्हें एक दी हुई अवधि तक रोगी को दिया गया हो।
एक ट्यूमर पर विकिरण (रेडिएशन) उपचार का प्रभाव जबकि उपचार थोड़े समय के लिए परंतु नियमित अंतराल पर किया गया हो।
For a given set of securities, all their portfolios lie on or within the boundary of the region shown in Fig.1.
In the feasible region, find a portfolio which has maximum return. Also, find a portfolio in this region which has minimum risk.
b) Explain the method of delineating the efficient frontier of a feasible region.
c) Given all the portfolios of n securities what criterion would an investor use to select a good portfolio?