Question

Students will be asked to formulate, define, and interpret mathematical modeling (particularly ordinary differential equation) which involves real engineering applications and related to their majoring (E.g. Newton’s law cooling/warming, mixture problem, radioactive decay, spring-mass system, series circuit, deflection of the beam, etc.).. The selected model should be solved analytically using any methods that have been learnt in the mathematic lecture. just give me an example with related topic and how to solve it using math

Answer 1

What is mathematical modeling?

Mathematical model is a representation in mathematical terms of the behavior of real devices and objects. In an elementary picture of the scientific method , we identify a “real world” and a “conceptual world.” The external world is the one we call real; here we observe various phenomena and behaviors, whether natural in origin or produced by artifacts. The conceptual world is the world of the mind—where we live when we try to understand what is going on in that real, external world. The conceptual world can be viewed as having three stages: observation, modeling, and prediction.

Many of the principles in science and engineering concern relationships between changing quantities. Since rates of change are represented mathematically by derivatives, it should not be surprising that such principles are often expressed in terms of differential equations.

Examples of Mathematics modelling with differential equations:

1. A MODEL OFFREE-FALL MOTION RETARDED BY AIR RESISTANCE:

In this we considered the free-fall model of an object moving along a vertical axis near the surface of the Earth. It is assumed in that model that there is no air resistance and that the only force acting on the object is the Earth’s gravity. Our goal here is to find a model that takes air resistance into account.

2. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS:

POPULATION GROWTH: One of the simplest models of population growth is based on the observation that when populations (people, plants, bacteria, and fruit flies, for example) are not constrained by environmental limitations, they tend to grow at a rate that is proportional to the size of the population—the larger the population, the more rapidly it grows. To translate this principle into a mathematical model, suppose that   denotes the population at time t. At each point in time, the rate of increase of the population with respect to time is

so the assumption that the rate of growth is proportional to the population is described by the differential equation

where is a positive constant of proportionality that can usually be determined experimentally. Thus, if the population is known at some point in time, say at time  , then a general formula for the population can be obtained by solving the initial-value problem

3. CARBON DATING When the nitrogen in the Earth’s upper atmosphere is bombarded by cosmic radiation, the radioactive element carbon-14 is produced. This carbon-14 combines with oxygen to form carbon dioxide, which is ingested by plants, which in turn are eaten by animals. In this way all living plants and animals absorb quantities of radioactive carbon-14. In 1947 the American nuclear scientist W. F. Libby proposed the theory that the percentage of carbon-14 in the atmosphere and in living tissues of plants is the same. When a plant or animal dies, the carbon-14 in the tissue begins to decay. Thus, the age of an artifact that contains plant or animal material can be estimated by determining what percentage of its original carbon-14 content remains. Various procedures, called carbon dating or carbon-14 dating, have been developed for measuring this percentage.

4. VIBRATIONS WITH STRING: Modelling with second order differential equations.

Here in this we consider a block of mass M that is suspended from a vertical spring and allowed to settle into an equilibrium position. we assume that the block is then set into vertical vibratory motion by pulling or pushing on it and releasing it at time t = 0. We will be interested in finding a mathematical model that describes the vibratory motion of the block over time.