MATHS PROJECT WORK

MATHEMATICAL MODELLING

What is mathematical modelling?

Conversion of physical situation into mathematics with some suitable conditions is known as mathematical modeling. Mathematical modeling is nothing but a technique and the pedagogy taken from fine arts and not from the basic sciences.

Describe real-world phenomena Investigate important questions about

the observed world Explain real-world phenomena Test ideas Make predictions about the real world

Mathematical modeling is the use of mathematics to:-

Process of mathematical modelling

ModelReal world data

Predictions/Explanations

Mathematical conclusions

Formulation

An

alysis

Interpretation

Test

Describing a simple model Example: how much space is

inside this cardboard box? We know three measurements:• h (height),• w (width), and• l (length), and (luckily!) the formula for

the volume of a cuboids is:• Volume = h × w × l

Conclusion

So we have a very simple model here through which we are solving a real world problem

Mathematical modeling in meteorology and weather forecasting

How it is done??Weather forecasts are

made by collecting quantitative - numerical - data about the current state of the atmosphere. This data is then put into a mathematical model, which will predict the weather based on current conditions.

Mathematical modelling cycleStart

Formulation

Assumptions/Axioms

Solution

Satisfied

Stop

Interpretation

Validation

YesNO

General formulation Mathematical formulation of atmospheric

models used for weather forecasting is based on the equation of mechanics of a compressible fluid, which stem from three fundamental laws: the laws of the momentum and mass conservation and the first law of thermodynamics. These three laws give rise to equations of motion, equation of continuity, and thermodynamic energy equation respectively.

A mathematical model to predict future

dV = -1 grad p-2ω xv - grad U + D

dt ρ

where v is the velocity vector, d/dt is symbol of the total derivative with respect to time, ρis density, p is pressure, ω is the vector of angular velocity of earth’s rotation, U is potential of the gravity force, and D is the vector representing dissipation force.

How it is processed?In mathematical models, raw

data is entered into a computer. A series of calculations is performed on the raw data on it to determine how it will change over time. Normally, mathematical modeling is done by computers, which can carry out many calculations per second.

In the case of weather models, data such as rain fall, temperature and wind speed are fed into a computer. The computer performs calculations on - models - this data, using equations produced from our scientific understanding of atmospheric processes, such as fluid dynamics and thermodynamic equations. These models allow forecasters to predict how the conditions in the atmosphere will evolve. The more sophisticated and up-to-date your model is, the more accurate your forecast should be. Powerful supercomputers are needed to perform the complex calculations in mathematical weather models.

Conclusion

It's thanks to models that we can process raw weather data into sophisticated and accurate forecasts. The model used to create the unique 1 kilometer radius forecast on Weather Labs is the most sophisticated model available.

Even the most sophisticated model can't be right all the time.

Prepared By-MD. SARWAR AZAD

XI-B23

Thank you