The document defines and provides examples of partial differential equations (PDEs). It discusses key concepts like order, linearity, homogeneity, and boundary conditions for PDEs. Specifically, it defines a PDE as an equation with partial derivatives of a dependent variable with respect to two or more independent variables. It provides examples of first, second, and third order PDEs. It also defines linear PDEs as those where the dependent variable and its partial derivatives are of degree one with no products or transcendental terms. The document outlines methods for solving PDEs and representing them using subscript or conventional notation. It describes Dirichlet, Neumann, and other boundary conditions which specify the dependent variable or its derivatives on the boundary

1. Historical background of partial differential equation:

2. Historical background of partial differential equation:
Definition:

3. Historical background of partial differential equation:
Definition:

4. Historical background of partial differential equation:
Definition:
Partial differential equation:
An equation which consists
of partial derivatives of the
dependent variable with
respect to more than one
independent variables is
called partial differential
equation.

5. Examples:

6. Examples:
 x
𝜕𝑧
𝜕𝑥
+ 𝑦
𝜕𝑧
𝜕𝑦
=𝑛𝑥

7. Examples: 
𝜕𝑧
𝜕𝑥
+
𝜕𝑧
𝜕𝑦
=2𝑥
 x
𝜕𝑧
𝜕𝑥
+ 𝑦
𝜕𝑧
𝜕𝑦
=𝑛𝑥

8. Examples: 
𝜕𝑧
𝜕𝑥
+
𝜕𝑧
𝜕𝑦
=2𝑥
 x
𝜕𝑧
𝜕𝑥
+ 𝑦
𝜕𝑧
𝜕𝑦
=𝑛𝑥

𝜕2
𝑧
𝜕2
𝑥
+
𝜕2
𝑧
𝜕𝑦2=0

9. Examples: 
𝜕𝑧
𝜕𝑥
+
𝜕𝑧
𝜕𝑦
=2𝑥
 x
𝜕𝑧
𝜕𝑥
+ 𝑦
𝜕𝑧
𝜕𝑦
=𝑛𝑥

𝜕2
𝑧
𝜕2
𝑥
+
𝜕2
𝑧
𝜕𝑦2=0
Are partial
differential
equations.

10. Historical background of partial
differential equation:

11. Historical background of partial
differential equation:
 Frist of all laplace introduced the
concept of partial differential equation
by giving an equation 𝑢xx+𝑢yy=0 which
he named laplace equation.

12. Historical background of partial
differential equation:
 After that brook taylor (1685-1721)
discovered the wave equation
𝑢tt=c2 𝑢xx where c is a constant and is
the velocity of wave.

13. Historical background of partial
differential equation:
 The telegraph equation
𝑢xx=a𝑢tt+𝑏𝑢+c𝑢 is given by Maxwell,
where, 𝑎,𝑏 and c are constants.

14. Some important terms in partial
differential equation:

15. Some important terms in partial
differential equation:

16. Some important terms in partial
differential equation:
Definition: The order of the
partial differential equation
is the order of the highest
order partial derivative that
appears in the equation.

17. Example:

18. Example: 𝑢xx=𝑢xxx+𝑢+1

19. Example: 𝑢xx=𝑢xxx+𝑢+1 𝑢x-𝑢y=0

20. Example: 𝑢xx=𝑢xxx+𝑢+1 𝑢x-𝑢y=0 𝑢tt=𝑢xx+𝑢yy+𝑢zz

21. Example: 𝑢xx=𝑢xxx+𝑢+1 𝑢x-𝑢y=0 𝑢tt=𝑢xx+𝑢yy+𝑢zz
Are of the
order three,
one, and two
respectively.

22. Definition: The part of Partial differential equation containing derivatives of
order equal to the order of the equation is called principle part of the equation.

23. Definition: The part of Partial differential equation containing derivatives of
order equal to the order of the equation is called principle part of the equation.
Uxx=uxxx+u+1 has uxxx principle part.

24. Definition: The part of Partial differential equation containing derivatives of
order equal to the order of the equation is called principle part of the equation.
Uxx=uxxx+u+1 has uxxx principle part.
Similarly, utt=uxx+uyy+uzz has uxx+uyy+uzz - utt

25. Definition: The part of Partial differential equation containing derivatives of
order equal to the order of the equation is called principle part of the equation.
Uxx=uxxx+u+1 has uxxx principle part.
Similarly, utt=uxx+uyy+uzz has uxx+uyy+uzz - utt
Principle part.

26. Linear Parital
differential
equation:

27. Linear Parital
differential
equation:
A partial
differential
equation is said
to linear if it
satisfies the
following
conditions.

28. Linear Parital
differential
equation:
A partial
differential
equation is said
to linear if it
satisfies the
following
conditions.
The dependent
variable and its
partial derivatives
are all of degree
one.

29. Linear Parital
differential
equation:
A partial
differential
equation is said
to linear if it
satisfies the
following
conditions.
The dependent
variable and its
partial derivatives
are all of degree
one.
No, products of
dependent variable
or any of its partial
derivatives appear.

30. Linear Parital
differential
equation:
A partial
differential
equation is said
to linear if it
satisfies the
following
conditions.
The dependent
variable and its
partial derivatives
are all of degree
one.
No, products of
dependent variable
or any of its partial
derivatives appear.
No transcendental
function of
dependent variable
or its partial
derivative occur.

31. Linear Parital
differential
equation:
A partial
differential
equation is said
to linear if it
satisfies the
following
conditions.
The dependent
variable and its
partial derivatives
are all of degree
one.
No, products of
dependent variable
or any of its partial
derivatives appear.
No transcendental
function of
dependent variable
or its partial
derivative occur.
𝑢xx=𝑢xxx+𝑢+1 is a
linear partial
differential
equation.

32. Note: The
equation which
is not linear is
called non-
linear partial
differential
equation.

33. Note: The
equation which
is not linear is
called non-
linear partial
differential
equation.
𝑢2+𝑢yy+𝑢zz - 𝑢tt ,
+𝑢zz - 𝑢𝑢tt=0 are
non-linear
partial
differential
equations.

34. Note: The
equation which
is not linear is
called non-
linear partial
differential
equation.
𝑢2+𝑢yy+𝑢zz - 𝑢tt ,
+𝑢zz - 𝑢𝑢tt=0 are
non-linear
partial
differential
equations.
Note: The case
of linearity is
the same for
both PDE and
ODE.

35. Homogenous and inhomogeneous
partial differential equations:

36. Homogenous and inhomogeneous
partial differential equations:
A partial differential equations of
any order is called homogenous if
every term of the equation
contains the dependent variable or
one of its derivatives.𝑢t=4𝑢xx ,
𝑢xx+𝑢yy+𝑢=0 are both
homogenous equations.

37. Homogenous and inhomogeneous
partial differential equations:
A partial differential equations of
any order is called homogenous if
every term of the equation
contains the dependent variable or
one of its derivatives.𝑢t=4𝑢xx ,
𝑢xx+𝑢yy+𝑢=0 are both
homogenous equations.
A partial differential equation of
any order is called inhomogeneous
if any term of the equationdoes
not contain the dependent variable
or one of its derivatives. 𝑢t=4𝑢xx+2,
𝑢xx+𝑢yy+𝑢=4x , 𝑢x+𝑢y=𝑢+4 are
inhomogeneous.

38. Solution of the partial
differential equations:

39. Solution of the partial
differential equations:
The solution of a partial
differential equation is the
function “𝑢” such that the
function “𝑢” and derivatives
obtain from it satisfy the
given PDE.

40. Example: show that 𝑢(𝑥, 𝑡)=sin𝑥e4t is a solution of the PDE. 𝑢t=4𝑢xx
Solution:
𝑢(𝑥, 𝑡)=sinxe4t ----->(1)
𝑢t=-4𝑢xx -----> (2)
partially differentiate (1) w.r.to “t”
𝑢t= 4sinxe4t
Now,
partially differentiate (1) w.r.to “x”
𝑢x=cosxe4t
again partially differentiate (1) w.r.to “x”
𝑢xx= -sinxe4t
Equation (2) becomes
𝑢t = -4𝑢xx
4sinxe4t= 4sinxe4t
Thus,𝑢(𝑥, 𝑡)=sinxe4t is the solution of 𝑢 t=-4𝑢xx.

41. Notations for PDE:

42. Notations for PDE:
The partial
differential equation
can be represented in
two ways,

43. Notations for PDE:
The partial
differential equation
can be represented in
two ways,
𝜕𝑢
𝜕𝑥
+
𝜕𝑢
𝜕𝑦
=0 is the
conventional
representation.

44. Notations for PDE:
The partial
differential equation
can be represented in
two ways,
𝜕𝑢
𝜕𝑥
+
𝜕𝑢
𝜕𝑦
=0 is the
conventional
representation.
𝑢xx= -sinxe4t is the
subscript notation.

45. Boundary conditions:

46. Boundary conditions:
The dependent variable “u” is usually
prescribed at the boundary of the
domain D.
The boundary data is called boundary
conditions, The general solution of a
PDE is of little use a particular solution
is frequently required that satisfy
given conditions,

47. Boundary conditions:
The dependent variable “u” is usually
prescribed at the boundary of the
domain D.
The boundary data is called boundary
conditions, The general solution of a
PDE is of little use a particular solution
is frequently required that satisfy
given conditions,
The boundary conditions are of the
following three types.

48. Dirichlet
boundary
conditions:

49. In this case
the function
“u” is
prescribed on
the boundary
of the domain.
Dirichlet
boundary
conditions:

50. In this case
the function
“u” is
prescribed on
the boundary
of the domain.
• For a rode of length “L” where 0<x<L the
boundary conditions are,
Dirichlet
boundary
conditions:

51. In this case
the function
“u” is
prescribed on
the boundary
of the domain.
• For a rode of length “L” where 0<x<L the
boundary conditions are,
𝑢(0)=β ; 𝑢 (L)= µ where β and µ are given
constants.
Dirichlet
boundary
conditions:

52. Neumann boundary condition:

53. Neumann boundary condition:
The Neumann boundary
conditions consists of
derivatives at the boundary
for example for a rod of
length “L” the Neumann
conditions are, 𝑢x(𝐿, 𝑡)= β
and 𝑢x(0, 𝑡)= µ

54. Presentation of
Partial differential
equations.
Presented by
Aman ullah
(MSc
mathematics )
University of Baluchistan
department of Mathematics.
Presented to Dr.
Saleem Iqbal Saab
(chair person department
of Mathematics)
Dated:08/04/2019