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Page 1
PARTIAL DIFFERENTIAL EQUATIONS
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either
by eliminating the arbitrary constants or by eliminating the arbitrary functions from the given
relation.
The Physical system contains arbitrary constants or arbitrary functions or both.
Equations which contain one or more partial derivatives are called Partial Differential Equations.
Therefore, there must be atleast two independent variables and one dependent variable.
Let us take to be two independent variables and to be dependent variable.
Order: The Order of a partial differential equation is the order of the highest partial derivative in
the equation.
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE
Notations
, , , ,
Formation of Partial Differential Equation
Formation of PDE by elimination of Arbitrary Constants
Formation of PDE by elimination of Arbitrary Functions
Solution of a Partial Differential Equation
Let us consider a Partial Differential Equation of the form 1
If it is Linear in and , it is called a Linear Partial Differential Equation
(i.e. Order and Degree of and is one)
If it is Not Linear in and , it is called as nonlinear Partial Differential Equation
(i.e. Order and Degree of and is other than one)
Consider a relation of the type
By eliminating the arbitrary constants and from this equation, we get ,
which is called a complete integral or complete solution of the PDE.
A solution of obtained by giving particular values to and in the complete
Integral is called a particular Integral.
Page 2
PARTIAL DIFFERENTIAL EQUATIONS
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either
by eliminating the arbitrary constants or by eliminating the arbitrary functions from the given
relation.
The Physical system contains arbitrary constants or arbitrary functions or both.
Equations which contain one or more partial derivatives are called Partial Differential Equations.
Therefore, there must be atleast two independent variables and one dependent variable.
Let us take to be two independent variables and to be dependent variable.
Order: The Order of a partial differential equation is the order of the highest partial derivative in
the equation.
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE
Notations
, , , ,
Formation of Partial Differential Equation
Formation of PDE by elimination of Arbitrary Constants
Formation of PDE by elimination of Arbitrary Functions
Solution of a Partial Differential Equation
Let us consider a Partial Differential Equation of the form 1
If it is Linear in and , it is called a Linear Partial Differential Equation
(i.e. Order and Degree of and is one)
If it is Not Linear in and , it is called as nonlinear Partial Differential Equation
(i.e. Order and Degree of and is other than one)
Consider a relation of the type
By eliminating the arbitrary constants and from this equation, we get ,
which is called a complete integral or complete solution of the PDE.
A solution of obtained by giving particular values to and in the complete
Integral is called a particular Integral.
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER
A Differential Equation which involves partial derivatives and only and no higher order
derivatives is called a first order equation. If and have the degree one, it is called a linear
partial differential equation of first order; otherwise it is called a non-linear partial equation of
first order.
Ex: 1) is a linear Partial Differential Equation.
2) is a non-linear Partial Differential Equation.
LAGRANGE’S LINEAR EQUATION
A linear Partial Differential Equation of order one, involving a dependent variable and two
independent variables and , and is of the form , where are functions of
is called Lagrange’s Linear Equation.
Solution of the Linear Equation
Consider
Now,
Case 1: If it is possible to separate variables then, consider any two equations, solve them by
integrating. Let the solutions of these equations are
is the required solution of given equation.
Case 2: If it is not possible to separate variables then
To solve above type of problems we have following methods
Method of grouping: In some problems, it is possible to solve any two of the equations
(or) (or)
In such cases, solve the differential equation, get the solution and then substitute in the
other differential equation
Method of Multiplier: consider
In this, we have to choose so that denominator=0. That will give us solution by
integrating .
Page 3
PARTIAL DIFFERENTIAL EQUATIONS
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either
by eliminating the arbitrary constants or by eliminating the arbitrary functions from the given
relation.
The Physical system contains arbitrary constants or arbitrary functions or both.
Equations which contain one or more partial derivatives are called Partial Differential Equations.
Therefore, there must be atleast two independent variables and one dependent variable.
Let us take to be two independent variables and to be dependent variable.
Order: The Order of a partial differential equation is the order of the highest partial derivative in
the equation.
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE
Notations
, , , ,
Formation of Partial Differential Equation
Formation of PDE by elimination of Arbitrary Constants
Formation of PDE by elimination of Arbitrary Functions
Solution of a Partial Differential Equation
Let us consider a Partial Differential Equation of the form 1
If it is Linear in and , it is called a Linear Partial Differential Equation
(i.e. Order and Degree of and is one)
If it is Not Linear in and , it is called as nonlinear Partial Differential Equation
(i.e. Order and Degree of and is other than one)
Consider a relation of the type
By eliminating the arbitrary constants and from this equation, we get ,
which is called a complete integral or complete solution of the PDE.
A solution of obtained by giving particular values to and in the complete
Integral is called a particular Integral.
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER
A Differential Equation which involves partial derivatives and only and no higher order
derivatives is called a first order equation. If and have the degree one, it is called a linear
partial differential equation of first order; otherwise it is called a non-linear partial equation of
first order.
Ex: 1) is a linear Partial Differential Equation.
2) is a non-linear Partial Differential Equation.
LAGRANGE’S LINEAR EQUATION
A linear Partial Differential Equation of order one, involving a dependent variable and two
independent variables and , and is of the form , where are functions of
is called Lagrange’s Linear Equation.
Solution of the Linear Equation
Consider
Now,
Case 1: If it is possible to separate variables then, consider any two equations, solve them by
integrating. Let the solutions of these equations are
is the required solution of given equation.
Case 2: If it is not possible to separate variables then
To solve above type of problems we have following methods
Method of grouping: In some problems, it is possible to solve any two of the equations
(or) (or)
In such cases, solve the differential equation, get the solution and then substitute in the
other differential equation
Method of Multiplier: consider
In this, we have to choose so that denominator=0. That will give us solution by
integrating .
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER
A partial differential equation which involves first order partial derivatives and with degree
higher than one and the products of and is called a non-linear partial differential equation.
There are six types of non-linear partial differential equations of first order as given below.
Type I:
Type II:
Type III: (variable separable method)
Type IV: Clairaut’s Form
Equation reducible to standard forms and and
and
CHARPIT’S METHOD
Let us see in detail about these types.
Type I:
Equations of the type i.e. equations containing and only
Let the required solution be
and
Substituting these values in , we get
From this, we can obtain in terms of (or) in terms of
Let , then the required solution is
Note: Since, the given equation contains two first order partial derivatives , the final
solution should contain only two constants.
Type II:
Let us consider the Equations of the type 1
Let is a function of and
i.e. and
Now, .1
. a a
1 is the 1
st
order differential equation in terms of dependent variable
and independent variable .
Solve this differential equation and finally substitute gives the required solution.
Page 4
PARTIAL DIFFERENTIAL EQUATIONS
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either
by eliminating the arbitrary constants or by eliminating the arbitrary functions from the given
relation.
The Physical system contains arbitrary constants or arbitrary functions or both.
Equations which contain one or more partial derivatives are called Partial Differential Equations.
Therefore, there must be atleast two independent variables and one dependent variable.
Let us take to be two independent variables and to be dependent variable.
Order: The Order of a partial differential equation is the order of the highest partial derivative in
the equation.
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE
Notations
, , , ,
Formation of Partial Differential Equation
Formation of PDE by elimination of Arbitrary Constants
Formation of PDE by elimination of Arbitrary Functions
Solution of a Partial Differential Equation
Let us consider a Partial Differential Equation of the form 1
If it is Linear in and , it is called a Linear Partial Differential Equation
(i.e. Order and Degree of and is one)
If it is Not Linear in and , it is called as nonlinear Partial Differential Equation
(i.e. Order and Degree of and is other than one)
Consider a relation of the type
By eliminating the arbitrary constants and from this equation, we get ,
which is called a complete integral or complete solution of the PDE.
A solution of obtained by giving particular values to and in the complete
Integral is called a particular Integral.
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER
A Differential Equation which involves partial derivatives and only and no higher order
derivatives is called a first order equation. If and have the degree one, it is called a linear
partial differential equation of first order; otherwise it is called a non-linear partial equation of
first order.
Ex: 1) is a linear Partial Differential Equation.
2) is a non-linear Partial Differential Equation.
LAGRANGE’S LINEAR EQUATION
A linear Partial Differential Equation of order one, involving a dependent variable and two
independent variables and , and is of the form , where are functions of
is called Lagrange’s Linear Equation.
Solution of the Linear Equation
Consider
Now,
Case 1: If it is possible to separate variables then, consider any two equations, solve them by
integrating. Let the solutions of these equations are
is the required solution of given equation.
Case 2: If it is not possible to separate variables then
To solve above type of problems we have following methods
Method of grouping: In some problems, it is possible to solve any two of the equations
(or) (or)
In such cases, solve the differential equation, get the solution and then substitute in the
other differential equation
Method of Multiplier: consider
In this, we have to choose so that denominator=0. That will give us solution by
integrating .
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER
A partial differential equation which involves first order partial derivatives and with degree
higher than one and the products of and is called a non-linear partial differential equation.
There are six types of non-linear partial differential equations of first order as given below.
Type I:
Type II:
Type III: (variable separable method)
Type IV: Clairaut’s Form
Equation reducible to standard forms and and
and
CHARPIT’S METHOD
Let us see in detail about these types.
Type I:
Equations of the type i.e. equations containing and only
Let the required solution be
and
Substituting these values in , we get
From this, we can obtain in terms of (or) in terms of
Let , then the required solution is
Note: Since, the given equation contains two first order partial derivatives , the final
solution should contain only two constants.
Type II:
Let us consider the Equations of the type 1
Let is a function of and
i.e. and
Now, .1
. a a
1 is the 1
st
order differential equation in terms of dependent variable
and independent variable .
Solve this differential equation and finally substitute gives the required solution.
Type III: (variable separable method)
Let us consider the differential equation is of the form
Let (say)
Now (I.e. writing in terms of )
(I.e. writing in terms of )
Now,
By Integrating this, we get the required solution.
Note: This method is used only when it is possible to separate variables.
i.e. on one side and on other side.
Type IV: Clairaut’s Form
Equations of the form
Let the required solution be , then
and
Required solution is
i.e. Directly substitute in place of and in place of in the given equation.
Equations Reducible to Standard Forms
Equations of the type , where and are constants.
Now, let us transform above equation to the form (Type-I)
Case-I: If and
Put and , then
( and )
( and )
Substituting these values in the given equation, we get
which is in the form of (Type-I)
Solve this, get the result which will be in terms of and and the substitute and
, which is the required solution.
Page 5
PARTIAL DIFFERENTIAL EQUATIONS
The Partial Differential Equation (PDE) corresponding to a physical system can be formed, either
by eliminating the arbitrary constants or by eliminating the arbitrary functions from the given
relation.
The Physical system contains arbitrary constants or arbitrary functions or both.
Equations which contain one or more partial derivatives are called Partial Differential Equations.
Therefore, there must be atleast two independent variables and one dependent variable.
Let us take to be two independent variables and to be dependent variable.
Order: The Order of a partial differential equation is the order of the highest partial derivative in
the equation.
Degree: The degree of the highest partial derivative in the equation is the Degree of the PDE
Notations
, , , ,
Formation of Partial Differential Equation
Formation of PDE by elimination of Arbitrary Constants
Formation of PDE by elimination of Arbitrary Functions
Solution of a Partial Differential Equation
Let us consider a Partial Differential Equation of the form 1
If it is Linear in and , it is called a Linear Partial Differential Equation
(i.e. Order and Degree of and is one)
If it is Not Linear in and , it is called as nonlinear Partial Differential Equation
(i.e. Order and Degree of and is other than one)
Consider a relation of the type
By eliminating the arbitrary constants and from this equation, we get ,
which is called a complete integral or complete solution of the PDE.
A solution of obtained by giving particular values to and in the complete
Integral is called a particular Integral.
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER
A Differential Equation which involves partial derivatives and only and no higher order
derivatives is called a first order equation. If and have the degree one, it is called a linear
partial differential equation of first order; otherwise it is called a non-linear partial equation of
first order.
Ex: 1) is a linear Partial Differential Equation.
2) is a non-linear Partial Differential Equation.
LAGRANGE’S LINEAR EQUATION
A linear Partial Differential Equation of order one, involving a dependent variable and two
independent variables and , and is of the form , where are functions of
is called Lagrange’s Linear Equation.
Solution of the Linear Equation
Consider
Now,
Case 1: If it is possible to separate variables then, consider any two equations, solve them by
integrating. Let the solutions of these equations are
is the required solution of given equation.
Case 2: If it is not possible to separate variables then
To solve above type of problems we have following methods
Method of grouping: In some problems, it is possible to solve any two of the equations
(or) (or)
In such cases, solve the differential equation, get the solution and then substitute in the
other differential equation
Method of Multiplier: consider
In this, we have to choose so that denominator=0. That will give us solution by
integrating .
NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER
A partial differential equation which involves first order partial derivatives and with degree
higher than one and the products of and is called a non-linear partial differential equation.
There are six types of non-linear partial differential equations of first order as given below.
Type I:
Type II:
Type III: (variable separable method)
Type IV: Clairaut’s Form
Equation reducible to standard forms and and
and
CHARPIT’S METHOD
Let us see in detail about these types.
Type I:
Equations of the type i.e. equations containing and only
Let the required solution be
and
Substituting these values in , we get
From this, we can obtain in terms of (or) in terms of
Let , then the required solution is
Note: Since, the given equation contains two first order partial derivatives , the final
solution should contain only two constants.
Type II:
Let us consider the Equations of the type 1
Let is a function of and
i.e. and
Now, .1
. a a
1 is the 1
st
order differential equation in terms of dependent variable
and independent variable .
Solve this differential equation and finally substitute gives the required solution.
Type III: (variable separable method)
Let us consider the differential equation is of the form
Let (say)
Now (I.e. writing in terms of )
(I.e. writing in terms of )
Now,
By Integrating this, we get the required solution.
Note: This method is used only when it is possible to separate variables.
i.e. on one side and on other side.
Type IV: Clairaut’s Form
Equations of the form
Let the required solution be , then
and
Required solution is
i.e. Directly substitute in place of and in place of in the given equation.
Equations Reducible to Standard Forms
Equations of the type , where and are constants.
Now, let us transform above equation to the form (Type-I)
Case-I: If and
Put and , then
( and )
( and )
Substituting these values in the given equation, we get
which is in the form of (Type-I)
Solve this, get the result which will be in terms of and and the substitute and
, which is the required solution.
Case-II: If and
Put and , then
( and )
( and )
Substituting these values in the given equation, we get
(Type-I)
Solve this, get the result which will be in terms of and and the substitute and
, which is the required solution.
? Equations of the type , where and are constants
This equation can be reduced in to (Type-II) by taking above substitutions.
Equations of the type , where is a constant
In order to convert into the form , we have to take the following substitutions
Put
? Equations of the type where is a constant.
These type of equations can be reduced to the form (Type-I) (or)
by taking above substitutions given for the equation
CHARPIT’S METHOD
This is a general method to find the complete integral of the non-linear PDE of the form
Now Auxillary Equations are given by
Here we have to take the terms whose integrals are easily calculated, so that it may be easier to
solve and .
Finally substitute in the equation
Integrate it, we get the required solution.
Note that all the above (TYPES) problems can be solved in this method.
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FAQs on PPT: Partial Differential Equations - Engineering Mathematics - Civil Engineering (CE)
| 1. What are partial differential equations? |  |
Ans. Partial differential equations (PDEs) are mathematical equations that involve multiple variables and their partial derivatives. They describe the relationships between multiple unknown functions and their partial derivatives with respect to the variables. PDEs are widely used in various fields of science and engineering to model and analyze physical phenomena.
| 2. How are partial differential equations different from ordinary differential equations? | |
Ans. The main difference between partial differential equations (PDEs) and ordinary differential equations (ODEs) is that PDEs involve multiple independent variables, while ODEs involve only one independent variable. PDEs describe functions that depend on multiple variables and their partial derivatives, whereas ODEs describe functions that depend only on one variable and its derivatives.
| 3. What are some common methods for solving partial differential equations? | |
Ans. There are several methods for solving partial differential equations (PDEs), depending on the type of equation and its properties. Some common methods include separation of variables, method of characteristics, Fourier series, Laplace transform, numerical methods (such as finite difference, finite element, and finite volume methods), and Green's functions. The choice of method depends on the specific problem at hand and the desired level of accuracy.
| 4. Can you provide an example of a practical application of partial differential equations? | |
Ans. Sure! One practical application of partial differential equations is in heat conduction analysis. The heat equation, which is a type of PDE, is used to model the distribution of heat in a solid material over time. By solving the heat equation, engineers can predict how heat will propagate through different materials, which is crucial for designing efficient heating or cooling systems, understanding thermal behavior in electronic devices, and optimizing energy consumption.
| 5. What is the importance of partial differential equations in the field of physics? | |
Ans. Partial differential equations (PDEs) play a fundamental role in the field of physics. They are used to describe and analyze physical phenomena, such as fluid flow, electromagnetic fields, quantum mechanics, and general relativity. PDEs provide mathematical models that help physicists understand and predict the behavior of complex physical systems. They are essential tools for formulating and solving problems in various branches of physics, including classical mechanics, fluid dynamics, electromagnetism, and quantum physics.
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