Engineering Mathematics Exam > Engineering Mathematics Notes > Differential Equation and Mathematical Modeling-II > Lecture 7 - First Order Partial Differential Equations
Lecture 7 - First Order Partial Differential Equations
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Page 1
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
Page 2
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
Page 3
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning Outcomes:
After reading this chapter, learner will be able to understand
? How to construct a first order PDE
? How to solve a first order PDE using the method of characteristics
? How to transform a PDE of first order in canonical form.
? How to solve PDE of first order using the method separation of
variables.
2. Introduction:
Page 4
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning Outcomes:
After reading this chapter, learner will be able to understand
? How to construct a first order PDE
? How to solve a first order PDE using the method of characteristics
? How to transform a PDE of first order in canonical form.
? How to solve PDE of first order using the method separation of
variables.
2. Introduction:
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 4
An equation is called a partial differential equation if it contains one or more
partial derivatives. In the 18th century, Lagrange, Laplace D'Alembert and
Euler started the study of partial differential equations (PDE's) as the
principal mode of analytical study of models in the physical science. In the
middle of the 19
th
century, PDE's also became a necessary tool in other
branches of mathematics.
3. Classification of partial differential equations of order
one:
The most general form of the partial differential equations of order one in
two independent variables and is given by
(
)
Or
( ) (1)
Where is a function of two independent variables , and dependent
variable and its partial derivatives
.
Similarly, if there are three independent variables then the partial
differential equations of order one can be written as
(
) (2)
3.1. Linear partial differential equations:
A partial differential equation ( ) is said to be linear if the
degree of the dependent variable and all its partial derivatives i.e. is
one.
i.e. if it is of the form
( )
( )
( ) ( )
Where ( ) ( ) ( ) and ( ) are functions of only.
3.1.1. Examples of linear partial differential equations:
(a)
( )
Page 5
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Math
Lesson: First Order Partial Differential Equations
Course Developer: Manoj Kumar
College/Department: Hans Raj College (D.U.)
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter : First Order Partial Differential Equations
? 1: Learning Outcomes
? 2: Introduction
? 3: Classification of partial differential equations of order one
? 3.1: Linear partial differential equations
? 3.2: Semi linear partial differential equations
? 3.3: Quasi linear partial differential equations
? 3.4: Non linear partial differential equations
? 4: Construction of partial differential equations of first order
? 4.1: Elimination of arbitrary constants
o Exercises
? 4.2: By elimination of arbitrary functions
o Exercises
? 5: Classification of Integrals or Solutions
? 6: Geometrical interpretation of partial differential equation of
first order
? 7: Method of Characteristics
? 8: The Cauchy problem for partial differential equation of first
order
o Exercises
? 9: Canonical Forms of Linear Equations of First Order
o Exercise
? 10: Method of Separation of Variables
o Exercises
? Summary
? Glossary
? References/ Further Reading
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 3
1. Learning Outcomes:
After reading this chapter, learner will be able to understand
? How to construct a first order PDE
? How to solve a first order PDE using the method of characteristics
? How to transform a PDE of first order in canonical form.
? How to solve PDE of first order using the method separation of
variables.
2. Introduction:
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 4
An equation is called a partial differential equation if it contains one or more
partial derivatives. In the 18th century, Lagrange, Laplace D'Alembert and
Euler started the study of partial differential equations (PDE's) as the
principal mode of analytical study of models in the physical science. In the
middle of the 19
th
century, PDE's also became a necessary tool in other
branches of mathematics.
3. Classification of partial differential equations of order
one:
The most general form of the partial differential equations of order one in
two independent variables and is given by
(
)
Or
( ) (1)
Where is a function of two independent variables , and dependent
variable and its partial derivatives
.
Similarly, if there are three independent variables then the partial
differential equations of order one can be written as
(
) (2)
3.1. Linear partial differential equations:
A partial differential equation ( ) is said to be linear if the
degree of the dependent variable and all its partial derivatives i.e. is
one.
i.e. if it is of the form
( )
( )
( ) ( )
Where ( ) ( ) ( ) and ( ) are functions of only.
3.1.1. Examples of linear partial differential equations:
(a)
( )
First Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi pg. 5
(b)
(c) (
)
(
)
( ) ( )
3.2. Semi linear partial differential equations:
A partial differential equation of order one ( ) is said to be semi
linear if it is of the form
( )
( )
( )
Where and are the functions of only.
3.2.1. Examples of semi linear partial differential equations:
(a)
(b) ( )
( )
( )
(c)
( )
3.3. Quasi linear partial differential equations:
A partial differential equation ( ) is said to be quasi linear if the
degree of the all partial derivatives i.e. is one. i.e. if it is of the form
( )
( )
( )
Where and are the functions of .
3.3.1. Examples of quasi linear partial differential equations:
(a) (
)
(b) (
)
(c) (
)
(
)
3.4. Non linear partial differential equations:
A partial differential equation ( ) is said to be non linear if the
degree of the partial derivatives i.e. is not one.
3.4.1. Examples of non-linear partial differential equations:
(a)
(b)
(
)
(c)
Read MoreFAQs on Lecture 7 - First Order Partial Differential Equations
| 1. What are first-order partial differential equations? |  |
Ans. First-order partial differential equations are mathematical equations that involve partial derivatives of an unknown function with respect to one or more independent variables. They are called "first-order" because they involve only first derivatives and not higher-order derivatives.
| 2. What is the significance of first-order partial differential equations in engineering mathematics? | |
Ans. First-order partial differential equations have significant applications in engineering mathematics. They are used to model various physical phenomena, such as heat transfer, fluid flow, and electromagnetic fields. Solving these equations helps engineers understand and predict the behavior of these systems, enabling them to design better solutions.
| 3. How do we solve first-order partial differential equations? | |
Ans. The solution to a first-order partial differential equation involves finding a function that satisfies the equation. Different techniques can be used depending on the specific equation. Common methods include the method of characteristics, separation of variables, and the use of integral transforms such as the Laplace transform or Fourier transform.
| 4. Can you provide an example of a first-order partial differential equation in engineering mathematics? | |
Ans. Certainly! One example of a first-order partial differential equation is the advection equation, which describes the transport of a quantity by a moving medium. It can be written as ∂u/∂t + c∂u/∂x = 0, where u is the unknown quantity, t is time, x is the spatial variable, and c is the velocity of the medium.
| 5. Are there any real-life applications of first-order partial differential equations in engineering? | |
Ans. Yes, first-order partial differential equations find numerous applications in engineering. For example, they are used to model the flow of fluids in pipes, analyze heat conduction in solid materials, simulate electrical circuits, and study the propagation of sound waves. These equations allow engineers to optimize designs, predict behavior, and solve practical problems in various fields of engineering.
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