Mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling.
Mathematical modeling Mathematical models are used in the natural sciences (such as physics, biology, Earth science, meteorology) and engineering disciplines (such as computer science, artificial intelligence), as well as in the social sciences (such as economics, psychology, sociology, political science). Physicists, engineers, statisticians, operations research analysts, and economists use mathematical models most extensively. A model may help to explain a system and to study the effects of different components, and to make predictions about behavior.
Elements of a mathematical model In the physical sciences, the traditional mathematical model contains four major elements. These are Governing equations Defining equations Constitutive equations Constraints
Steps for Mathematical modeling 1: Identify the physical situation 2: Convert the physical situation into a mathematical model by introducing parameters / variables and using various known physical laws and symbols 3: Find the solution of the mathematical problem 4: Interpret the result in terms of the original and compare the result with observations or experiments 5: If the result is in good agreement, then accept the model. Otherwise modify the hypothesis / assumptions according to the physical situation and go to step 2
Steps for Mathematical modeling Physical Situation Mathematical Modeling Mathematical solution Solution of Original Problem Accept or Modify
Examples Find the height of a given tower using mathematical modeling Step1 : Given physical situation is to find the height of a given tower Step2 : Let AB be the given tower. Let PQ be an observer measuring the height of the tower with his eye at P. Let PQ=h and let height of tower be H. Let x be the angle of elevation from the eye of the observer to the top of the tower.
Examples Let l=PC=QB Now tan x = AC / PC = H – h / l H = h + l tan x ………… (1) Step3 : Note that the values of the parameters h, l, and x (using secant) are known to be the observer and so (1) gives the solution of the problem.
Examples Step4 : In case, if the foot of the tower is not accessible. (i.e) when l is not known to the observer, let y be the angle of depression from P to the foot B of the tower. So from triangle PQB, we have, tan y = PQ / QB = h / l or l = h cot y Step5 : is not required in this situation as exact values of the parameters h, l, x, y are known.
In the electric power industry, models and modeling have played a significant role in studying the regimes of electric power systems in predicting the consumption of electrical energy. The elements of the electric power system are themselves modeled by substitution schemes and their parameters. The replacement circuit serves to enable the calculation of all electrical quantities inherent in the real element.
Apps Computer mathematics systems such as Maple, Mathematica, Mathcad, MATLAB, VisSim and others are designed to support mathematical modeling. They allow you to create formal and block models of both simple and complex processes and devices and easily change the parameters of models during the simulation. Block models are represented by blocks (most often graphic), the set and connection of which are specified by the diagram of the model.