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Page 1
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 1
Lesson: Classification of Second Order Partial Differential
Equations
Course Developer: Sada Nand Prasad
College/Department: Acharya Narendra Dev College
Page 2
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 1
Lesson: Classification of Second Order Partial Differential
Equations
Course Developer: Sada Nand Prasad
College/Department: Acharya Narendra Dev College
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 2
Table of Contents:
Chapter : Classification of Second Order Partial Differential
Equations
? 1. Learning Outcomes
? 2. Introduction
? 3. Second – Order Equation in Two Independent Variables
? 4. Canonical Forms
? 5. Equations with Constant Coefficients
? 6. General Solutions
? 7. Further Simplification
? 8. The Cauchy Problem
? Summary
? Exercises
? Glossary
? References/ Further Reading
1. Learning Outcomes:
After studying this chapter, you will be able to
? classify linear second order PDEs into elliptic, parabolic and
hyperbolic types;
? reduce linear second order PDEs into canonical form;
? classify linear equation with constant coefficient and Euler equation;
? obtain the general solution of linear second order PDEs;
? further Simplification of the reduced linear second order PDEs by
introducing the new dependent variable;
? derive the Cauchy problem;
Page 3
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 1
Lesson: Classification of Second Order Partial Differential
Equations
Course Developer: Sada Nand Prasad
College/Department: Acharya Narendra Dev College
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 2
Table of Contents:
Chapter : Classification of Second Order Partial Differential
Equations
? 1. Learning Outcomes
? 2. Introduction
? 3. Second – Order Equation in Two Independent Variables
? 4. Canonical Forms
? 5. Equations with Constant Coefficients
? 6. General Solutions
? 7. Further Simplification
? 8. The Cauchy Problem
? Summary
? Exercises
? Glossary
? References/ Further Reading
1. Learning Outcomes:
After studying this chapter, you will be able to
? classify linear second order PDEs into elliptic, parabolic and
hyperbolic types;
? reduce linear second order PDEs into canonical form;
? classify linear equation with constant coefficient and Euler equation;
? obtain the general solution of linear second order PDEs;
? further Simplification of the reduced linear second order PDEs by
introducing the new dependent variable;
? derive the Cauchy problem;
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 3
2. Introduction:
In the last chapter we have derived the three fundamental equations of
mathematical physics, namely one and two - dimensional wave equation,
one - dimensional heat conduction equation/diffusion equation and
Laplace’s equation and mentioned that these equations are of hyperbolic,
parabolic and elliptic type. Here, in this chapter we will explain in detail,
about the classification of these second order partial differential equations
with variable and constant coefficients and will find the general solution
after reducing these equations to their respective canonical form.
To begin with, we have in this chapter described the second order partial
differential equations (PDEs) in two independent variables and classified
linear PDEs of second order into elliptic, parabolic and hyperbolic types.
3. Second – Order Partial Differential Equation in Two
Independent Variables :
It is seen that a large number of PDEs arising in the study of applied
mathematics with special reference to biological, physical and engineering
applications, can be treated as a particular case of the most general form
of a linear, second order PDE in two independents of the form
2 2 2
22
,
z z z z z
a b c d e f z g
x x y y x y
? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
or,
,
xx xy yy x y
az bz cz d z ez f z g ? ? ? ? ? ? (1)
where a, b, c, d, e, f, and g are functions of the independent variables x
and y and do not vanish simultaneously.
Value Addition: Do you know?
From coordinate geometry, we know that general equation of second
degree in two variables
22
0, ax bxy cy d x ey f ? ? ? ? ? ?
represents, ellipse if b
2
– 4 a c < 0,
parabola if b
2
– 4 a c = 0 or, (2)
hyperbola if b
2
– 4 a c > 0.
The classification of second order partial differential equations (3.1), is
suggested by the classification of the above equation (2), and based upon
the possibilities of transforming equation (1) to canonical form at any
point (x
0
, y
0
) by suitable coordinate transformation. Therefore, an
equation at any point (x
0
, y
0
) is called
elliptic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) < 0,
parabolic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) = 0 or, (3)
hyperbolic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) > 0.
Page 4
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 1
Lesson: Classification of Second Order Partial Differential
Equations
Course Developer: Sada Nand Prasad
College/Department: Acharya Narendra Dev College
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 2
Table of Contents:
Chapter : Classification of Second Order Partial Differential
Equations
? 1. Learning Outcomes
? 2. Introduction
? 3. Second – Order Equation in Two Independent Variables
? 4. Canonical Forms
? 5. Equations with Constant Coefficients
? 6. General Solutions
? 7. Further Simplification
? 8. The Cauchy Problem
? Summary
? Exercises
? Glossary
? References/ Further Reading
1. Learning Outcomes:
After studying this chapter, you will be able to
? classify linear second order PDEs into elliptic, parabolic and
hyperbolic types;
? reduce linear second order PDEs into canonical form;
? classify linear equation with constant coefficient and Euler equation;
? obtain the general solution of linear second order PDEs;
? further Simplification of the reduced linear second order PDEs by
introducing the new dependent variable;
? derive the Cauchy problem;
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 3
2. Introduction:
In the last chapter we have derived the three fundamental equations of
mathematical physics, namely one and two - dimensional wave equation,
one - dimensional heat conduction equation/diffusion equation and
Laplace’s equation and mentioned that these equations are of hyperbolic,
parabolic and elliptic type. Here, in this chapter we will explain in detail,
about the classification of these second order partial differential equations
with variable and constant coefficients and will find the general solution
after reducing these equations to their respective canonical form.
To begin with, we have in this chapter described the second order partial
differential equations (PDEs) in two independent variables and classified
linear PDEs of second order into elliptic, parabolic and hyperbolic types.
3. Second – Order Partial Differential Equation in Two
Independent Variables :
It is seen that a large number of PDEs arising in the study of applied
mathematics with special reference to biological, physical and engineering
applications, can be treated as a particular case of the most general form
of a linear, second order PDE in two independents of the form
2 2 2
22
,
z z z z z
a b c d e f z g
x x y y x y
? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
or,
,
xx xy yy x y
az bz cz d z ez f z g ? ? ? ? ? ? (1)
where a, b, c, d, e, f, and g are functions of the independent variables x
and y and do not vanish simultaneously.
Value Addition: Do you know?
From coordinate geometry, we know that general equation of second
degree in two variables
22
0, ax bxy cy d x ey f ? ? ? ? ? ?
represents, ellipse if b
2
– 4 a c < 0,
parabola if b
2
– 4 a c = 0 or, (2)
hyperbola if b
2
– 4 a c > 0.
The classification of second order partial differential equations (3.1), is
suggested by the classification of the above equation (2), and based upon
the possibilities of transforming equation (1) to canonical form at any
point (x
0
, y
0
) by suitable coordinate transformation. Therefore, an
equation at any point (x
0
, y
0
) is called
elliptic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) < 0,
parabolic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) = 0 or, (3)
hyperbolic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) > 0.
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 4
Example 1. Consider the equation,
2
0,
xx yy
z x z ?? , comparing this equation
with equation (1), we get, a = 1, b = 0, c = x
2
, and therefore, b
2
- 4ac <
0. Thus, the given equation is elliptic. Similarly, we can say that the
Laplace’s equation derived in the last chapter 0,
xx yy
uu ?? is elliptic.
Example 2. Consider the equation, 2 0,
xx xy yy
z z z ? ? ? , comparing this
equation with equation (1), we get, a = 1, b = 2, c = 1, and therefore,
b
2
- 4ac = 0. Thus, the given equation is parabolic. Similarly, we can say
that the one – dimensional heat equation / diffusion equation derived in
the last chapter
t xx
T KT ? is hyperbolic.
Example 3. Consider the equation,
2
xx yy
z x z ? , comparing this equation
with equation (1), we get, a = 1, b = 0, c = x
2
, and therefore, b
2
- 4ac =
4x
2
> 0. Thus, the given equation is hyperbolic. Similarly, we can say
that the one - dimensional wave equation derived in the last chapter
2
tt xx
u c u ? is hyperbolic.
It may also be remarked here that an equation can be of mixed type,
depending upon its coefficients.
Example 4. The equation given by
2
,
xx yy
xz z x ?? can be classified as
elliptic if, x > 0, parabolic, if x = 0, or hyperbolic if x < 0 as b
2
- 4ac = - 4
x.
Now, we shall show that by a suitable change in the independent
variables, we can reduce any equation of type (1) to one of the three
standard or canonical forms. Let us suppose we change the variables x
and y to u and v, respectively where
u = u (x, y), v = v(x, y). (4)
We will also assume that u and v are twice continuously differentiable and
in the region under consideration the Jacobian
0.
xy
xy
uu
vv
?
Then for the system (4), we can determine x and y uniquely. Suppose x
and y are twice continuously differentiable functions of u and v. Then, we
have,
Page 5
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 1
Lesson: Classification of Second Order Partial Differential
Equations
Course Developer: Sada Nand Prasad
College/Department: Acharya Narendra Dev College
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 2
Table of Contents:
Chapter : Classification of Second Order Partial Differential
Equations
? 1. Learning Outcomes
? 2. Introduction
? 3. Second – Order Equation in Two Independent Variables
? 4. Canonical Forms
? 5. Equations with Constant Coefficients
? 6. General Solutions
? 7. Further Simplification
? 8. The Cauchy Problem
? Summary
? Exercises
? Glossary
? References/ Further Reading
1. Learning Outcomes:
After studying this chapter, you will be able to
? classify linear second order PDEs into elliptic, parabolic and
hyperbolic types;
? reduce linear second order PDEs into canonical form;
? classify linear equation with constant coefficient and Euler equation;
? obtain the general solution of linear second order PDEs;
? further Simplification of the reduced linear second order PDEs by
introducing the new dependent variable;
? derive the Cauchy problem;
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 3
2. Introduction:
In the last chapter we have derived the three fundamental equations of
mathematical physics, namely one and two - dimensional wave equation,
one - dimensional heat conduction equation/diffusion equation and
Laplace’s equation and mentioned that these equations are of hyperbolic,
parabolic and elliptic type. Here, in this chapter we will explain in detail,
about the classification of these second order partial differential equations
with variable and constant coefficients and will find the general solution
after reducing these equations to their respective canonical form.
To begin with, we have in this chapter described the second order partial
differential equations (PDEs) in two independent variables and classified
linear PDEs of second order into elliptic, parabolic and hyperbolic types.
3. Second – Order Partial Differential Equation in Two
Independent Variables :
It is seen that a large number of PDEs arising in the study of applied
mathematics with special reference to biological, physical and engineering
applications, can be treated as a particular case of the most general form
of a linear, second order PDE in two independents of the form
2 2 2
22
,
z z z z z
a b c d e f z g
x x y y x y
? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
or,
,
xx xy yy x y
az bz cz d z ez f z g ? ? ? ? ? ? (1)
where a, b, c, d, e, f, and g are functions of the independent variables x
and y and do not vanish simultaneously.
Value Addition: Do you know?
From coordinate geometry, we know that general equation of second
degree in two variables
22
0, ax bxy cy d x ey f ? ? ? ? ? ?
represents, ellipse if b
2
– 4 a c < 0,
parabola if b
2
– 4 a c = 0 or, (2)
hyperbola if b
2
– 4 a c > 0.
The classification of second order partial differential equations (3.1), is
suggested by the classification of the above equation (2), and based upon
the possibilities of transforming equation (1) to canonical form at any
point (x
0
, y
0
) by suitable coordinate transformation. Therefore, an
equation at any point (x
0
, y
0
) is called
elliptic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) < 0,
parabolic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) = 0 or, (3)
hyperbolic if b
2
(x
0
, y
0
)– 4 a(x
0
, y
0
) c(x
0
, y
0
) > 0.
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 4
Example 1. Consider the equation,
2
0,
xx yy
z x z ?? , comparing this equation
with equation (1), we get, a = 1, b = 0, c = x
2
, and therefore, b
2
- 4ac <
0. Thus, the given equation is elliptic. Similarly, we can say that the
Laplace’s equation derived in the last chapter 0,
xx yy
uu ?? is elliptic.
Example 2. Consider the equation, 2 0,
xx xy yy
z z z ? ? ? , comparing this
equation with equation (1), we get, a = 1, b = 2, c = 1, and therefore,
b
2
- 4ac = 0. Thus, the given equation is parabolic. Similarly, we can say
that the one – dimensional heat equation / diffusion equation derived in
the last chapter
t xx
T KT ? is hyperbolic.
Example 3. Consider the equation,
2
xx yy
z x z ? , comparing this equation
with equation (1), we get, a = 1, b = 0, c = x
2
, and therefore, b
2
- 4ac =
4x
2
> 0. Thus, the given equation is hyperbolic. Similarly, we can say
that the one - dimensional wave equation derived in the last chapter
2
tt xx
u c u ? is hyperbolic.
It may also be remarked here that an equation can be of mixed type,
depending upon its coefficients.
Example 4. The equation given by
2
,
xx yy
xz z x ?? can be classified as
elliptic if, x > 0, parabolic, if x = 0, or hyperbolic if x < 0 as b
2
- 4ac = - 4
x.
Now, we shall show that by a suitable change in the independent
variables, we can reduce any equation of type (1) to one of the three
standard or canonical forms. Let us suppose we change the variables x
and y to u and v, respectively where
u = u (x, y), v = v(x, y). (4)
We will also assume that u and v are twice continuously differentiable and
in the region under consideration the Jacobian
0.
xy
xy
uu
vv
?
Then for the system (4), we can determine x and y uniquely. Suppose x
and y are twice continuously differentiable functions of u and v. Then, we
have,
Classification of Second Order Partial Differential Equations
Institute of Lifelong Learning, University of Delhi 5
. . . .
. . . .
2
. . . .
2
z z u z v
z z u z v
x u x v x
x u x v x
z z u z v
z z u z v
y u y v y
y u y v y
z z u v z u z v
z
xx
x x u x v x u x v x
x
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?
22
2 2 2 2 2
2
2 . 2 2 2
2 2 2 2
2
2
. . . .
2
z u z u v z v z u z v
x u v x x x u v
u v x x
z u z u v z v z u z v
uu x uv x x vv x u x v x
z z u v z u z v
z
xy
x y x y u x v x u y v y
z
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ??
? ? ? ? ??
?? ? ? ? ?
?? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ?
?
?
?
2 2 2 2
2 . 2
2
. . . .
2
u u z u v u v z v v z u z v
x y u v x y y x x y u y x v y x
uv
z u u z u v u v z v v z u z v
uu x y uv x y y x vv x y u xy v xy
z z u v z u z v
z
yy
y y u y v y u y v
y
??
??
??
??
??
??
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
?
? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ? ? ?
?
22
2 2 2 2 2
2
2 . 2 2 2
2 2 2 2
2
y
z u z u v z v z u z v
y u v y y y u v
u v y y
z u z u v z v z u z v
uu y uv y y vv y u y v y
??
??
??
??
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ?
? ? ? ? ? ? ? ?
? ? ? ?
? ? ? ? ?
(5)
Substituting all these values in equation (1), we get,
* * * * * * *
uu uv vv u v
a z b z c z d z e z f z g ? ? ? ? ? ?
where
? ?
* 2 2
*
* 2 2
*
*
*
*
,
2 2 ,
,
,
,
,
.
x x y y
x x x y y x y y
x x y y
xx xy yy x y
xx xy yy x y
a au bu u cu
b au v b u v u v cu v
c av bv v cv
d au bu cu du eu
e av bv cv dv ev
ff
gg
? ? ?
? ? ? ?
? ? ?
? ? ? ? ?
? ? ? ? ?
?
?
(6)
The classification of the second order PDE (1) depends on the coefficients
a(x, y), b(x, y), and c(x, y) at any point (x, y). So, we can write equation
(1) and equation (2) as
? ?
, , , ,
xx xy yy x y
az bz cz h x y z z z ? ? ? (7)
and
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FAQs on Lecture 8 - Classification of Second Order Partial Differential Equations - Differential Equation and Mathematical Modeling-II - Engineering Mathematics
| 1. What is the classification of second order partial differential equations in engineering mathematics? |  |
Ans. Second order partial differential equations in engineering mathematics can be classified into three main types: elliptic equations, parabolic equations, and hyperbolic equations. The classification is based on the nature of the equation's solutions and the behavior of the equation in different regions of the domain.
| 2. What are elliptic equations in the context of second order partial differential equations? | |
Ans. Elliptic equations are second order partial differential equations that describe phenomena with steady-state behavior. They have solutions that are smooth and do not change significantly over time. Examples of elliptic equations in engineering mathematics include Laplace's equation and Poisson's equation.
| 3. How are parabolic equations different from elliptic equations in second order partial differential equations? | |
Ans. Parabolic equations in second order partial differential equations describe phenomena with time-dependent behavior. The solutions of parabolic equations change significantly over time, and they often involve initial conditions that specify the behavior of the system at a given starting point. Heat conduction and diffusion equations are examples of parabolic equations.
| 4. What are hyperbolic equations and how do they differ from elliptic and parabolic equations in second order partial differential equations? | |
Ans. Hyperbolic equations in second order partial differential equations describe phenomena with wave-like behavior. They have solutions that propagate in space and time, such as waves or vibrations. Hyperbolic equations involve both initial conditions and boundary conditions to fully specify the behavior of the system. Examples of hyperbolic equations include the wave equation and the telegraph equation.
| 5. How is the classification of second order partial differential equations useful in engineering mathematics? | |
Ans. The classification of second order partial differential equations helps engineers and scientists understand the behavior of physical systems and design appropriate mathematical models. By categorizing equations into elliptic, parabolic, or hyperbolic types, engineers can apply specific solution techniques and boundary conditions that are relevant to the behavior of the system. This classification also provides insights into the fundamental properties and characteristics of the equations, aiding in the analysis and interpretation of real-world problems.
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their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Lecture 8 - Classification of Second Order Partial Differential Equations on the App
Students of Engineering Mathematics can study Lecture 8 - Classification of Second Order Partial Differential Equations alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Lecture 8 - Classification of Second Order Partial Differential Equations,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Lecture 8 - Classification of Second Order Partial Differential Equations is prepared as per the latest Engineering Mathematics syllabus.
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