Mathematical modelling ppt
Download as PPTX, PDF3 likes8,899 views
This document discusses mathematical modeling. It begins by defining mathematical modeling as translating beliefs about how the world functions into mathematical language. This allows problems to be precisely formulated and leverage mathematical results and computing. The document then provides examples of problems that can be addressed through modeling, like estimating population growth. It outlines principles of modeling like identifying needed parameters and assumptions. Advanced modeling can use statistics, decision making, and geometry. Mathematical models are important in science, allowing theories to be expressed through equations. In conclusion, modeling has been successfully used in many domains and is a valuable problem-solving approach.
Mathematical modelling ppt
- 1. Mathematical Modelling Presented By : Sanjeev Kumar Prajapati sanjeevkumarprajapati2000@gmail.com
- 2. Content Introduction Why Mathematical Modelling Principles of Mathematical Modelling Advanced Mathematical Modelling Significance in the natural sciences Conclusion
- 3. What distinguishes a mathematical model from , says a poem ,a song , a portrait or any other kind of model , is that the mathematical model is an image or picture of reality painted with logicals symbols instead of with words , sounds or watercolors. -John L.Casti
- 4. Introduction Models describe our beliefs about how the world functions. In mathematical modelling , we translate those beliefs into the language of mathematics. This has many advantages: 1. Mathematics is a very precise language. This helps us to formulate ideas and identify underlying assumptions. 2. All the results that mathematicians have proved over hundreds of years are at our disposal. 3. Computers can be used to perform numerical calculations.
- 5. Mathematical modelling is the study of some part (or form) of some real- life problems in mathematical terms , i.e. the conversion of a physical situation into mathematics using some suitable conditions. Mathematical modelling is the process of using various mathematical structures such as graphs, equations ,diagrams, scatterplots, tree diagram and so forth-to represent real worlds situations. Mathematical modelling is useful for a variety of reasons . Foremost, models represent the mathematical core of situations without extraneour informations .The equation a=b+6, for instance , removes all of the unnecessary words from the original statement “Ram is 6 years older than shyam”.
- 6. Why Mathematical Modelling Sometimes we solve the problems without going in to the physical insight of the situational problems. Situational problems need physical insight that is introduction of physical laws and some symbols to compare the mathematical results obtained with practical values. To solve many problems faced by us , we need a technique and this is what is known as mathematical modelling . Let us consider the following problems :
- 7. 1. To find the width of a river ( particularly , when it is difficult to cross the river). 2. To find the height of a tower ( particularly , when it is not possible to reach the top of the tower). 3. To find the temperature at the surface of the sun. 4. To find the mass of the earth. 5. Estimate the population of India in 2022 (a person is not allowed to wait till then). 6. To find the optimal angle in case of shot-put (by considering the variables such as : the height of a thrower , resistance of media , acceleration due to gravity etc.).
- 8. 7 To find the volume of blood inside the body of a person (a person is not allowed to bleed completely). 8. Estimate the yield of pulses in India from the crops ( a person is not allowed to cut it all). 9. Why heart patients are not allow to use lifts (without knowing the physiology of a human being).
- 9. Principles of Mathematical Modelling Mathematical modelling is a principled activity and so it has some principles behind it. These principles are almost philosophical in nature. Some of the basic principles of mathematical modelling are listed below in terms of following instructions: Identify the need for the model( for what we are looking for ) List the parameters/variables which are required for the model.
- 10. Identify the available relevant data(what is given) Identify the circumstances that can be applied.(assumptions) Identify the governing physical principles. Identify : 1. the equations that will be used. 2. the calculations that will be made. 3. the solutions which will follow. Identify tests that can check the 1.consistency of the model 2. unity of the model
- 11. The above principles of mathematical modelling lead to the following: steps for mathematical modelling, which can be viewed from the following diagram
- 13. Lets consider an example Ex. Find the height of the given tower using mathematical modelling.
- 14. Solution : Step1: Given physical situation is “to find the height of given tower”. Step2: Let AB be the given tower (figure). Let PQ be an observer measuring the height of the tower with his eyes at P. Let PQ=h and let height of tower be H. Let α be the angle of elevation from the eye of the observer to the top of the tower. PC=QB=l tanα=AC/PC=(H-h)/l l tanα=H-h H=h+l tanα …………………………(1) Note that the values of the parameters h,l and α ( using sextant) are known to observer and so (1) gives the solution of the problem. Step4: In case ,if the foot of the tower is not accessible ,i.e., when l is not known to the observer , let β be the angle of deviation from P to the foot B of the tower. So from PQB , we have,
- 15. tanβ=PQ/QB tanβ=h/l l=h/tanβ or l=hcotβ Step 5 is not required in this situation as exact values of the parameters h , l , α and β are known.
- 16. Advanced Mathematical Modelling Mathematical modelling include the analysis of information using statistical methods and probability ,modeling change and mathematical relationships , mathematical decision making in finance ,and spatial and geometric modelling for decision-making. Students will learn to become crucial consumers of the quantitative data that surround them everyday, knowledgeable decision makers who use logical reasoning and mathematical thinkers who can use their quantitative skills to solve problems related to a wide range of situations. Mathematical modelling helps students to develop a mathematical proficiency in a developmentally-appropriate progressions of standards.
- 17. Significance in the natural sciences Mathematical models are of great importance in the natural sciences , particularly in Physics. Physical theories are almost invariably expressed using mathematical models. Throughout history , more and more accurate mathematical models describe many everyday phenomena , but at certain limits relativity theory and quantum mechanics must be used ;even these do not apply to all situations and need further refinement .
- 18. The laws of physics are represented with simple equations such as Newton’s laws , Maxwell’s equations and the Schrodinger equation. These laws are such as a basis for making mathematical models of real situations . Many real situations are very complex and thus modeled approximate on a computer , a model that is computationally feasible to compute is made from the basic laws.
- 19. Conclusion Till today many mathematical models have been developed and applied successfully to understand and get an insight into thousands of situations. Some of the subjects like mathematical physics , mathematical economics , operations research , bio-mathematics etc. are almost synonyms with mathematical modelling.
- 20. Finally it can be concluded that mathematical modelling is a good concept which helps us in various realistic life experiment and phenomenon to convert in to mathematical models in the form of equations , graphs , histogram , scatterplots , tree diagram , etc. and then solve them.
- 21. It saves time , money and efforts of performing results in laboratory with experimental tools. It has widespread applications in every aspects of life as well as in various branches such as engineering , medical , mathematical economics , medicine ,biology and several other interdisciplinary areas. Thus it is an advanced concept and have bright future in upcoming time.
- 22. Thank you