Lecture 9 - Mathematical Models | Differential Equation and Mathematical Modeling-II - Engineering Mathematics PDF Download

 Page 1


Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Differential Equation and Mathematical 
Modeling-I 
Lesson: Mathematical Models  
Lesson Developer: Sada Nand Prasad 
Department/ College: A.N.D. College (D.U.)  
 
Page 2


Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Differential Equation and Mathematical 
Modeling-I 
Lesson: Mathematical Models  
Lesson Developer: Sada Nand Prasad 
Department/ College: A.N.D. College (D.U.)  
 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 2 
 
Table of Contents: 
 Chapter : Mathematical Models 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Mathematical modelling of the vibrating string 
? 3.1. Mathematical modelling of The Vibrating Membrane 
? 4. Conduction of Heat in Solid 
? 4.1. Derivation of heat equation 
? 5. The Gravitational Potential 
? 6. Conservation Law 
? 7. The Burgers Equation 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
Page 3


Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Differential Equation and Mathematical 
Modeling-I 
Lesson: Mathematical Models  
Lesson Developer: Sada Nand Prasad 
Department/ College: A.N.D. College (D.U.)  
 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 2 
 
Table of Contents: 
 Chapter : Mathematical Models 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Mathematical modelling of the vibrating string 
? 3.1. Mathematical modelling of The Vibrating Membrane 
? 4. Conduction of Heat in Solid 
? 4.1. Derivation of heat equation 
? 5. The Gravitational Potential 
? 6. Conservation Law 
? 7. The Burgers Equation 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 3 
 
 
1. Learning Outcomes: 
After studying this lesson, reader should be able to 
? derive one - dimensional wave equation; 
? derive two - dimensional wave equation; 
? derive  one - dimensional heat conduction equation; 
? derive Laplace's equation;  
? state conservation law and derive Burgers equation; 
 
 
 
 
 
 
2. Introduction: 
In our day to day life, we face problems arising from different disciplines - 
physics, chemistry, biology, sociology, management, finance etc. 
Mathematical modelling consists of simplifying the real world problems 
and representing them in mathematical language, as well as solving the 
mathematical problems and interpretation of these solutions in the real 
world language. At some point of time, while studying mathematics, we 
must have solved problems around us. So far the physical systems have 
been primarily studied by ordinary differential equations. Now we are 
interested in all those phenomena (or physical process) that requires 
partial derivatives in the describing equation. Partial differential equations 
are frequently used to formulate the laws of nature and to study the 
physical, chemical and biological models. In this chapter, we will study 
the partial differential equations representing the  mathematical models of 
physical problems in detail.  
Page 4


Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Differential Equation and Mathematical 
Modeling-I 
Lesson: Mathematical Models  
Lesson Developer: Sada Nand Prasad 
Department/ College: A.N.D. College (D.U.)  
 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 2 
 
Table of Contents: 
 Chapter : Mathematical Models 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Mathematical modelling of the vibrating string 
? 3.1. Mathematical modelling of The Vibrating Membrane 
? 4. Conduction of Heat in Solid 
? 4.1. Derivation of heat equation 
? 5. The Gravitational Potential 
? 6. Conservation Law 
? 7. The Burgers Equation 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 3 
 
 
1. Learning Outcomes: 
After studying this lesson, reader should be able to 
? derive one - dimensional wave equation; 
? derive two - dimensional wave equation; 
? derive  one - dimensional heat conduction equation; 
? derive Laplace's equation;  
? state conservation law and derive Burgers equation; 
 
 
 
 
 
 
2. Introduction: 
In our day to day life, we face problems arising from different disciplines - 
physics, chemistry, biology, sociology, management, finance etc. 
Mathematical modelling consists of simplifying the real world problems 
and representing them in mathematical language, as well as solving the 
mathematical problems and interpretation of these solutions in the real 
world language. At some point of time, while studying mathematics, we 
must have solved problems around us. So far the physical systems have 
been primarily studied by ordinary differential equations. Now we are 
interested in all those phenomena (or physical process) that requires 
partial derivatives in the describing equation. Partial differential equations 
are frequently used to formulate the laws of nature and to study the 
physical, chemical and biological models. In this chapter, we will study 
the partial differential equations representing the  mathematical models of 
physical problems in detail.  
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 4 
 
 
3. Mathematical modelling of the vibrating string:  
The motion of vibration of a tightly stretched, flexible string is the most 
interesting and important problems in applied mathematics and 
mathematical physics. It was modelled approximately 250 years ago and 
still widely used as an excellent introductory example.  
 
 
 
 
 
 
Fig 1(a) Deformed, flexible string of length l at an instant t. 
Let the length of the stretched string, which is fastened at each end, is l. 
we wish to get a describing equation for the deflection u of the string for 
any position x and for any time t. Consider a differential element of the 
string at a particular instant enlarged in Fig 1(b). 
 
 
 
 
 
 
 
 
 
 
 
Fig 1(b) Small element of vertically displaced string 
 
b a 
0 l x 
u 
T 
T 
1
?  
2
?  
b 
a 
0 xx ??  x 
u 
x 
Page 5


Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Differential Equation and Mathematical 
Modeling-I 
Lesson: Mathematical Models  
Lesson Developer: Sada Nand Prasad 
Department/ College: A.N.D. College (D.U.)  
 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 2 
 
Table of Contents: 
 Chapter : Mathematical Models 
? 1. Learning Outcomes 
? 2. Introduction 
? 3. Mathematical modelling of the vibrating string 
? 3.1. Mathematical modelling of The Vibrating Membrane 
? 4. Conduction of Heat in Solid 
? 4.1. Derivation of heat equation 
? 5. The Gravitational Potential 
? 6. Conservation Law 
? 7. The Burgers Equation 
? Summary 
? Exercises 
? Glossary 
? References/ Further Reading 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 3 
 
 
1. Learning Outcomes: 
After studying this lesson, reader should be able to 
? derive one - dimensional wave equation; 
? derive two - dimensional wave equation; 
? derive  one - dimensional heat conduction equation; 
? derive Laplace's equation;  
? state conservation law and derive Burgers equation; 
 
 
 
 
 
 
2. Introduction: 
In our day to day life, we face problems arising from different disciplines - 
physics, chemistry, biology, sociology, management, finance etc. 
Mathematical modelling consists of simplifying the real world problems 
and representing them in mathematical language, as well as solving the 
mathematical problems and interpretation of these solutions in the real 
world language. At some point of time, while studying mathematics, we 
must have solved problems around us. So far the physical systems have 
been primarily studied by ordinary differential equations. Now we are 
interested in all those phenomena (or physical process) that requires 
partial derivatives in the describing equation. Partial differential equations 
are frequently used to formulate the laws of nature and to study the 
physical, chemical and biological models. In this chapter, we will study 
the partial differential equations representing the  mathematical models of 
physical problems in detail.  
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 4 
 
 
3. Mathematical modelling of the vibrating string:  
The motion of vibration of a tightly stretched, flexible string is the most 
interesting and important problems in applied mathematics and 
mathematical physics. It was modelled approximately 250 years ago and 
still widely used as an excellent introductory example.  
 
 
 
 
 
 
Fig 1(a) Deformed, flexible string of length l at an instant t. 
Let the length of the stretched string, which is fastened at each end, is l. 
we wish to get a describing equation for the deflection u of the string for 
any position x and for any time t. Consider a differential element of the 
string at a particular instant enlarged in Fig 1(b). 
 
 
 
 
 
 
 
 
 
 
 
Fig 1(b) Small element of vertically displaced string 
 
b a 
0 l x 
u 
T 
T 
1
?  
2
?  
b 
a 
0 xx ??  x 
u 
x 
Mathematical Models 
Institute of Lifelong Learning, University of Delhi 
pg. 5 
 
We make the following assumptions to obtain a simple equation 
describing the vibration of the stretched string. 
 
1. The string is elastic and flexible and therefore offers no resistance 
to bending so that no shearing force exists on a surface normal to 
the string. 
2. The tension is so large that the weight of the string is negligible. 
3. There is no elongation of a single segment of the string and hence 
the tension is constant ( Hooke’s Law ). 
4. The slope of deflection curve is small. So if ? is the inclination angle 
of the tangent to the deflection curve then we can replace sin? by 
tan?. 
5. The deflection is negligible as compared to the string's length so 
that the resulting change in length of the string has no effect upon 
the tension. 
6. There is only pure transverse vibration, i.e., the motion takes place 
entirely in one plane and every particle moves at right angles to the 
equilibrium position of the string in this plane. 
 
 
 
 
 
 
 
Fig 1(c) Vector representation of tension at x 
Let the tension at the end points is T as shown in Fig 1(b). The forces 
acting, in the vertical direction, on the element of the string are  
T sin ?
2 
– T sin ?
1
 
using the Newton’s second law of motion we know that F = m a, and 
therefore 
? ? , ( , ) x u x t
  
?
  
cos
x
TT ? ??
  
sin
u
TT ? ??
  
T 
sin
u
TT ? ??
  
?  
? ? , ( , ) x x u x x t ? ? ? ?
 
cos
x
TT ? ??
  
T