Mathematical Modeling: Definition, Classifications, Applications

Why do we need mathematical models? A simple answer to this question is that we need mathematical modeling to understand the world and how it works. Mathematical modeling is useful in real-world applications. It simply signifies the world in simple models and shapes. These models should work with engineers and scientists to solve real-world problems using simple math. It will also involve equations. 

Therefore, mathematicians call this mathematical process modeling. One should understand how it works. So, let us see some explanations and theories behind it. While doing so, let us also see how we can describe its advantages and disadvantages.

Common knowledge behind Mathematical Modeling 

Mathematical modeling is described as illustrating real-life problems in mathematical terms and expressions. It is usually simplified in the form of equations. Furthermore, we could easily identify answers to those problems by utilizing such equations. It will also help us discover a whole new variety of features concerning the problems.

While modeling, the observer’s perspective is very important. One should be able to see the models through their mind. Engineers and scientists use such techniques to model and design future technologies. Along with this process, prototypes are commonly used. A prototype is a compact model of an actual working model. Prototypes are used on all occasions where there is a need for testing or analyzing a model without affecting or damaging the actual one.

Applications of Mathematical Modeling

Moreover, using models, one can see the future of our climatic conditions, understand how atoms and particles work, and design and manufacture anything in this world, starting from toy cars to actual cars. It will not just stop there; there are more applications.

Let us see a common example of mathematical modeling: weather forecasting. Given below are the steps:

1. First comes the laws of physics.
2. Now, encode them as differential equations, especially the Navier-Stokes equations.
3. Take weather stations and satellite data to demonstrate accurate weather for the day.
4. Consider that as an initial condition. Now, use a supercomputer to solve the 24 hours equations. It will provide tomorrow’s weather.
5. Update the forecast constantly.
6. Finally, present these results in an understandable form.

In addition, people claim this method is a failure, although it is not. Despite rumors, this process works fine. It may not predict weather conditions for a long-term period. However, it works well in the short term. Moreover, it is a special case method. It is mainly useful for general processes of mathematical modeling.

It can be described as follows:

1. Talk to the people and identify the problem.
2. Clarify the scientific method involved.
3. Formulate it mathematically.
4. With the help of a computer, try solving the math.
5. Start concluding.
6. Explains the results.

Apart from weather forecasting, there are several applications as well. The following are some of them:

• Design and develop any vehicle models, starting from cars to spaceships.
• Develop new medicines.
• Controlling electricity supply network.
• It helps in establishing the cause of any natural or manmade disaster.
• Provides pressure and temperature conditions in the workplace, etc.

Creating a Simulation

Simulation is very different from a model. Under this topic, we will discuss mathematical simulations. In the simulation, one tends to find the accuracy of any working model mainly using various types of software. One will try to obtain more details using it.

For example, we could gather information regarding how long a car will last. It is possible to test the car without damaging it. It only requires software. Some of the most common ones are ANSYS, Fusion 360, etc. The most interesting part is that one can analyze buildings and bridges without even constructing them. It is also possible by various means of computer simulation. 

Moreover, mathematical simulations are useful in airplanes. It is especially used to train pilots in aircraft simulators. These simulations are created so that they are almost the same as reality. The pilots can obtain much-needed training even before entering the cockpit with its assistance. 

Computer games are other common applications. Simulators play a vital role in video games. Many educational applications use these simulators. For example, a car driving simulator is a common one. 

Disadvantages of Simulators

Everything in this world has its disadvantages. It goes the same for simulators. Even though they are useful in numerous places, there are a few disadvantages.

Given below are a few:

• It will require a lot of space. As a result, it slows the computers down.
• Sometimes, the equations are hard to solve. So, supercomputers might be necessary. It is because larger computers contain higher graphics.
• It takes a lot of time to get the result. It also consumes energy because it is conditioned to produce larger data. It will be hard to categorize what is needed and what is not.
• Doing even simpler simulations can take a lot of time.
• It is only used in places with the involvement of basic science. The main reason is that it involves mathematical calculations to perform. Therefore, one should be careful while buying a simulator. They should be aware of all these things.

Classifications of Mathematical Modeling

There are different types of math models present. Some of them are listed below:

• Static vs. Dynamic
• Linear vs. Nonlinear
• Explicit vs. Implicit
• Deterministic vs. Probabilistic (or stochastic)
• Discrete vs. Continuous
• Floating, Inductive, or Deductive

1. Static vs. Dynamic:

Another name for the static model is the steady-state model. This model helps calculate the system in equilibrium. Therefore, it is time-invariant. In contrast, dynamic models are mainly used for time-dependent changes in a system. Hence, their representation is by difference equations or differential equations.

2. Linear vs. Nonlinear

Linear mathematical modeling is obtained when all the operators exhibit linearity. On the contrary, it is considered a nonlinear model when an object does not exhibit linearity. The descriptions of linearity and nonlinearity are mainly based on context. A linear object might have some nonlinear expressions in it. 

For instance, take a linear statistical model. Its relationship is linear in some of the parameters. However, it is nonlinear in the predictor variables. Likewise, a differential equation contains nonlinear expressions in it. But, it tends to be linear while writing with linear differential operators.

A mathematical programming model is considered linear if the constraints and objective functions are described fully by linear equations. The model is regarded as a nonlinear one if there are many constraints and objective functions described with a nonlinear equation.

3. Explicit vs. Implicit

The model is considered explicit if the overall model’s input models are known. One can calculate output parameters in an explicit function by a finite computation series. In contrast, it is called an implicit model if outputs are already shown. Two examples of implicit functions are Broyden’s method and Newton’s method. 

Let us look at an example to understand these models properly. Consider the physical properties of a jet engine. Nozzle throat and turbine areas are explicitly calculated using a design thermodynamic cycle, which uses area model math for derivation. It involves data regarding temperatures, pressures, and fuel and airflow rates. It is only calculated at a specific power setting and flight conditions. But for other flight conditions, it may not be possible to calculate explicitly. In such cases, we use implicit models. 

4. Deterministic vs. Probabilistic (or stochastic)

Parameters in a deterministic model separately define all the variable states. It involves a collection of previous states of these variables. So, it is common for this model to perform the same way for initial conditions. 

In the probabilistic model, randomness is present. Variable states are not determined using unique values. Instead, they are done by probability distributions. Other names of this model are the statistical model and the stochastic model.

5. Discrete vs. Continuous

In discrete models, objects are treated as discrete, namely states in a statistical model or particles in a molecular model.

In continuous models, objects are represented continuously. For example, stresses and temperatures in solids and the application of electric fields over an entire model continuously are some of them.

6. Deductive, Indicative, or Floating

Deductive models are usually logical structures. They are based on a theory. Indicative models come from empirical findings of deductive models. On the other hand, floating models rest on neither observation nor theory. They are purely based on the invocation of the expected structure. 

Conclusion

To conclude, mathematical models are useful in all domains. Many applications, starting from furniture to spaceships, can be done using mathematical modeling. It has a huge effect on our lives. Models with mathematics mainly contain equations and theories, which will help solve real-time problems. They are used in engineering and sciences to perform various applications.

A mathematical model is used to maximize output and enhance productivity. Apart from that, we have also discussed everything related to simulations in mathematics and how they differ from models. We have also seen its benefits and flaws. Finally, we have classified mathematical models and explained their subtopics in detail. All this information provided in this article will help the students to obtain full knowledge.

Frequently Asked Questions

1. What does modeling with mathematics mean?

Ans. Modeling with mathematics means that we use math to describe a system and then use that description to predict how the system will behave in the future.

2. What is an example of a math model?

Ans. An example of a math model is the Pythagorean Theorem. It’s a very simple formula that you can use to find the length of any side of a right triangle.

3. What are the types of math models?

Ans. There are many different types of math models, but they can be broken down into three main categories:

–Linear Models

-Non-Linear Models

-Dynamic Models

4. What is a model in math?

Ans. A model in math is a simplified version of a real-world system. It’s meant to help you understand how a system works so that you can predict the future or design solutions for it.

5. What are some examples of mathematical models?

Ans. Some examples of mathematical models include:

Fluid dynamics, which is used to describe the movement of liquids and gases,

The Navier-Stokes equations, which are used to describe the motion of fluids,

The Black-Scholes equation, which describes the behavior of financial markets,