Chapter 3
Partial Di!erential Equations
A partial di!erential equation (PDE) is a di!erential equation that involves partial deriva-
tives of one or more dependent variables with respect to one or more independent variables.
In general, a partial di!erential equation of variables x, y can be written as
f (x, y, u, ux , uy , uxx , uxy , uyy , ...) = 0 Implicit form .
where u = u(x, y) is the solution.
ωu
Example 3.1. Consider = x2 + y 2 . The solution is
ωx
!
u = (x2 + y 2 ) dx + g(y)
. u(x( y)
a xyz + g(y)
=
+
Remark 3.2. Order of a partial di!erential equation is the same as that of the order of
the highest di!erential co-e”cient in it.
Remark 3.3. If the dependent variable and all its partial derivatives occur linearly in
any PDE, then such an equation is called linear PDE otherwise a non-linear PDE.
Remark 3.4. A PDE is called a quasi–linear PDE if all the terms with highest order
derivatives of dependent variables occur linearly. That is, the coe”cients of such terms
are functions of only lower-order derivatives of the dependent variables.
Remark 3.5. If all the terms of a PDE contain the dependent variable or its partial
derivatives, then such a PDE is called a homogeneous PDE or non-homogeneous oth-
erwise.
Example 3.6. The following are some partial di!erential equations involving the inde-
pendent variables x, y.
21
, Remark 3 2 .
Ex :
U autu o
-
+ = order an
xy'sin + au = Su => Order 2
Ouzy
xyzciu ase =>
+ Order 3
Remark 3 2 .
22 + u = -1 - Not linear
8x
Gu +u = Not liner e
Remark 3 4
.
2Ou
2x2
+u = 3 Quasi-linera
m
linear respect to highest order terms.
22+ u = - 1 E Not
quasi-linear either .
2x2
Remark 3 .
5
xc + 90
by
+ u = 0 Homogeneous a
xc Ou + - => Non-Homogeneous .
T
Since this does not contain dependent variable
any or
it's derivatives .
, ω 2u ω 2u
a) + = 0, (linear, homogeneous)
ωx2 ωy 2
b) uxx + uyy = 0 (linear, homogeneous)
! "2 ! "2
ω 2u 2 ω 2u ω 2u ωu ωu
c) ux 2 + u xy + uy 2 + + + u2 = 0 (non-linear but quasi-
ωx ωxωy ωy ωx ωy
linear, homogeneous) linear with respect to higher order terms.
d) uxx + uyy = [(ux )2 + (uy )2 ] u (quasi-linear)
ωu ω 2u
e) (x2 + y 2 ) + → 3u = 0 (linear, 2nd order PDE)
ωx ωxωy
3.1 Classification of PDE
The classification of PDEs is an important concept because the general theory and solving
methods usually apply only to a given class of equations.
The most general linear partial di!erential equation of second order with two
independent variables has the form
ω 2u ω 2u ω 2u ωu ωu
A 2
+ B + C 2
+D +E + Fu + G = 0 (3.1)
ωx ωyωx ωy ωx ωy
where A, B, C, D, E, F and G are functions of x, y and constant terms. The equation (3.1)
may be written in the form
principal 2 Auxx + Buxy + Cuyy + f (x, y, ux , uy , u) = 0 (3.2)
part .
Assume that A, B, and C are continuous functions of x and y processing continuous
partial derivatives of high orders as necessary.
The classification of PDE is motivated by the classification of second-order algebraic
equations in two variables. In other words, the nature of equation (3.1) is determined by
the principal part containing the highest partial derivatives. i.e.
Lu = Auxx + Buxy + Cuyy (3.3)
The classification of equation (3.1) is done as follows.
1. If B 2 → 4AC < 0, the equation is said to be elliptic.
2. If B 2 → 4AC > 0, the equation is said to be hyperbolic.
3. If B 2 → 4AC = 0, the equation is said to be parabolic.
22
, The above classification of (3.1) is still valid if the coe!cients A, B, C, D, E, and F
depend on x, y. In this case,
1. If B 2 (x, y) → 4A(x, y)C(x, y) < 0, the equation is said to be elliptic at (x, y).
2. If B 2 (x, y) → 4A(x, y)C(x, y) > 0, the equation is said to be hyperbolic at (x, y).
3. If B 2 (x, y) → 4A(x, y)C(x, y) = 0, the equation is said to be parabolic at (x, y).
Example 3.7. Consider the following PD equations.
1. uxx + 2uyy = 1 is elliptic.
2. uxx → uyy = 1 is hyperbolic.
3. uxx + 3uyy → 2ux + 24Uy + 5u = 0 is elliptic.
4. uxx + uyy = 0 (Laplace equation) is an elliptic.
5. ut = uxx (Heat equation) is of parabolic type.
6. utt → uxx = 0 (Wave equation) is of hyperbolic type.
7. uxx + xuyy = 0, x ↑= 0 (Tricomi equation) is hyperbolic for x < 0 and elliptic for
x > 0. This example shows that equations with variable coe!cients can change form
in the di”erent regions of the domain.
In general, a partial di”erential equation of order n has a solution that contains at
most n arbitrary functions. Therefore the general solution can be written as the linear
combination of n arbitrary functions. This general solution can be particularized to
a unique solution if appropriate extra conditions are provided. These are classified as
boundary conditions. The kind of boundary conditions we need to specify depends on
the nature of the problem.
3.2 Techniques for Solving PDEs
Di”erent types of equations usually require di”erent solving techniques. However, there
are some methods that work for most linear partial di”erential equations with appropriate
boundary conditions on a regular domain. These methods include the separation of
variables, series expansions, similarity solutions, hybrid methods, and integral transform
methods.
23
Partial Di!erential Equations
A partial di!erential equation (PDE) is a di!erential equation that involves partial deriva-
tives of one or more dependent variables with respect to one or more independent variables.
In general, a partial di!erential equation of variables x, y can be written as
f (x, y, u, ux , uy , uxx , uxy , uyy , ...) = 0 Implicit form .
where u = u(x, y) is the solution.
ωu
Example 3.1. Consider = x2 + y 2 . The solution is
ωx
!
u = (x2 + y 2 ) dx + g(y)
. u(x( y)
a xyz + g(y)
=
+
Remark 3.2. Order of a partial di!erential equation is the same as that of the order of
the highest di!erential co-e”cient in it.
Remark 3.3. If the dependent variable and all its partial derivatives occur linearly in
any PDE, then such an equation is called linear PDE otherwise a non-linear PDE.
Remark 3.4. A PDE is called a quasi–linear PDE if all the terms with highest order
derivatives of dependent variables occur linearly. That is, the coe”cients of such terms
are functions of only lower-order derivatives of the dependent variables.
Remark 3.5. If all the terms of a PDE contain the dependent variable or its partial
derivatives, then such a PDE is called a homogeneous PDE or non-homogeneous oth-
erwise.
Example 3.6. The following are some partial di!erential equations involving the inde-
pendent variables x, y.
21
, Remark 3 2 .
Ex :
U autu o
-
+ = order an
xy'sin + au = Su => Order 2
Ouzy
xyzciu ase =>
+ Order 3
Remark 3 2 .
22 + u = -1 - Not linear
8x
Gu +u = Not liner e
Remark 3 4
.
2Ou
2x2
+u = 3 Quasi-linera
m
linear respect to highest order terms.
22+ u = - 1 E Not
quasi-linear either .
2x2
Remark 3 .
5
xc + 90
by
+ u = 0 Homogeneous a
xc Ou + - => Non-Homogeneous .
T
Since this does not contain dependent variable
any or
it's derivatives .
, ω 2u ω 2u
a) + = 0, (linear, homogeneous)
ωx2 ωy 2
b) uxx + uyy = 0 (linear, homogeneous)
! "2 ! "2
ω 2u 2 ω 2u ω 2u ωu ωu
c) ux 2 + u xy + uy 2 + + + u2 = 0 (non-linear but quasi-
ωx ωxωy ωy ωx ωy
linear, homogeneous) linear with respect to higher order terms.
d) uxx + uyy = [(ux )2 + (uy )2 ] u (quasi-linear)
ωu ω 2u
e) (x2 + y 2 ) + → 3u = 0 (linear, 2nd order PDE)
ωx ωxωy
3.1 Classification of PDE
The classification of PDEs is an important concept because the general theory and solving
methods usually apply only to a given class of equations.
The most general linear partial di!erential equation of second order with two
independent variables has the form
ω 2u ω 2u ω 2u ωu ωu
A 2
+ B + C 2
+D +E + Fu + G = 0 (3.1)
ωx ωyωx ωy ωx ωy
where A, B, C, D, E, F and G are functions of x, y and constant terms. The equation (3.1)
may be written in the form
principal 2 Auxx + Buxy + Cuyy + f (x, y, ux , uy , u) = 0 (3.2)
part .
Assume that A, B, and C are continuous functions of x and y processing continuous
partial derivatives of high orders as necessary.
The classification of PDE is motivated by the classification of second-order algebraic
equations in two variables. In other words, the nature of equation (3.1) is determined by
the principal part containing the highest partial derivatives. i.e.
Lu = Auxx + Buxy + Cuyy (3.3)
The classification of equation (3.1) is done as follows.
1. If B 2 → 4AC < 0, the equation is said to be elliptic.
2. If B 2 → 4AC > 0, the equation is said to be hyperbolic.
3. If B 2 → 4AC = 0, the equation is said to be parabolic.
22
, The above classification of (3.1) is still valid if the coe!cients A, B, C, D, E, and F
depend on x, y. In this case,
1. If B 2 (x, y) → 4A(x, y)C(x, y) < 0, the equation is said to be elliptic at (x, y).
2. If B 2 (x, y) → 4A(x, y)C(x, y) > 0, the equation is said to be hyperbolic at (x, y).
3. If B 2 (x, y) → 4A(x, y)C(x, y) = 0, the equation is said to be parabolic at (x, y).
Example 3.7. Consider the following PD equations.
1. uxx + 2uyy = 1 is elliptic.
2. uxx → uyy = 1 is hyperbolic.
3. uxx + 3uyy → 2ux + 24Uy + 5u = 0 is elliptic.
4. uxx + uyy = 0 (Laplace equation) is an elliptic.
5. ut = uxx (Heat equation) is of parabolic type.
6. utt → uxx = 0 (Wave equation) is of hyperbolic type.
7. uxx + xuyy = 0, x ↑= 0 (Tricomi equation) is hyperbolic for x < 0 and elliptic for
x > 0. This example shows that equations with variable coe!cients can change form
in the di”erent regions of the domain.
In general, a partial di”erential equation of order n has a solution that contains at
most n arbitrary functions. Therefore the general solution can be written as the linear
combination of n arbitrary functions. This general solution can be particularized to
a unique solution if appropriate extra conditions are provided. These are classified as
boundary conditions. The kind of boundary conditions we need to specify depends on
the nature of the problem.
3.2 Techniques for Solving PDEs
Di”erent types of equations usually require di”erent solving techniques. However, there
are some methods that work for most linear partial di”erential equations with appropriate
boundary conditions on a regular domain. These methods include the separation of
variables, series expansions, similarity solutions, hybrid methods, and integral transform
methods.
23
