Partial Differential Equations - Lecture 1
Classification and Formation of Partial
Differential Eqautions
Dr. Waheed Abdelwahab
May 6, 2025
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Introduction
Partial differential equations (PDEs) play a fundamental role in describing a
wide range of phenomena in mathematical physics, engineering, ecology, fluid
dynamics, quantum mechanics, and other scientific fields. From heat flow and
wave propagation to population dynamics and chemical dispersion, PDEs provide essential mathematical models for these natural processes.
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Definition of a PDE:
A partial differential equation (PDE) is an equation that contains the dependent
variable (the unknown function), and its partial derivatives. It is known that
in the ordinary differential equations (ODEs), the dependent variable u = u(x)
depends only on one independent variable x. Unlike the ODEs, the dependent
variable in the PDEs, such as u = u(x, t) or u = u(x, y, t), must depend on more
than one independent variable. If u = u(x, t), then the function u depends on
the independent variable x, and on the time variable t. However, if u = u(x, y, t),
then the function u depends on the space variables x, y, and on the time variable
t. In general form the partial differential equation it may be written in the form:
f (x, y, . . . , u, ux , uy , . . . , uxx , uxy , . . .) = 0,
(1)
which involve several independent variables x, y, . . . , dependent variable (an
unknown function) u as function of theses variables, and partial derivatives
ux , uy , . . . , uxx , uxy , . . . , of the dependent variable. The subscripts on dependent
variables denote differentiation , e.g.,
ux =
∂u
∂u
, uxy =
∂x
∂xy
The equation 1 is considered or defined in suitable domain D of n dimensional
space such the (x, y, . . .) ∈ D ⊆ Rn . Therefore, we look for a function u(x, y, . . .)
which satisfy equation 1 identically in the given domain D. In case, such functions u(x, y, . . .) exist, are called solutions of the given partial differential equations 1.
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Following are such examples of the partial differential equations:
ut = Kuxx ,
(2)
ut = K(uxx + uyy ),
(3)
ut = K(uxx + uyy + uzz ),
(4)
equation 2 describe the heat flow in one dimensional space, equation 3 describe
the heat flow in two dimensional space, and equation 4 describe the heat flow
in three dimensional space.
The following partial differential equations use to describe the wave propagation
in the considered dimensional space:
utt = C 2 uxx ,
(5)
utt = C 2 (uxx + uyy ),
(6)
2
utt = C (uxx + uyy + uzz ),
(7)
While the well known Laplace partial differential equation can be described as
follow
uxx + uyy = 0,
(8)
uxx + uyy + uzz = 0,
(9)
1
1
(10)
urr + ur + 2 uθθ = 0,
r
r
where equation 8 is the Laplace equation in two dimensional form and equation
9 is the Laplace partial differential equation in three dimensional form and both
of them are given in the Cartesian coordinate form while the equation 10 is
describing the Laplace equation in the Polar coordinate form where in this case
u=u(r,θ).
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Order of Partial Differential Equations:
The order of a partial differential equation is the order of the highest-ordered
partial derivatives appearing in the given partial differential equation. For example, the given previous partial differential equations 2,3,4, 5,6, 7, 8, 9, 10 are
all of second order partial differential equation since the highest order partial
derivatives in all of them is two. While the following:
Boussinesq equation given by
utt − uxx + 3(u2 )xx − uxxx = 0,
(11)
the Korteweg de-Vries (KdV) equation given by
ut + auux + buxxx = 0,
(12)
and the modified KdV equation (mKdV) given by
ut − 6u2 ux + buxxx = 0,
(13)
are of third order partial differential equation since the highest order partial
derivatives on them is three.
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Linear and Nonlinear Partial Differential Equation:
A partial differential equation is said to be linear if it is linear in the dependent
variable and all its partial derivatives with coefficients depend only on the independent variables. Therefore, the following two conditions necessarily to be
satisfied in the partial differential equation to be linear:
1. the power of the dependent variable and each partial derivative contained
in the equation is one, and
2. the coefficient of the dependent variable and the coefficient of each partial
differential derivatives are constants or of independent variable only. In
other case, any of this condition is not satisfied the partial differential
equation is said to nonlinear.
Example: Classify the following PDEs as linear or nonlinear:
xuxx + yuyy = 0,
(14)
uut + xux = 2,
(15)
utt 2 + uxxx = 0,
1
1
urr + ur + 2 uθθ = 0,
r
r √
ux + u = x.
(16)
(17)
(18)
Solutions:
PDE 14: Is linear since the power of each partial derivatives uxx and uyy is one.
In addition, the coefficients of the partial derivatives are the independent
variables x and y respectively.
PDE 15: Is nonlinear since the coefficient of the partial derivatives ut is the
dependent variable u.
PDE 16: Complete....
PDE 17: Complete....
PDE 18: Complete....
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Homogenous and Inhomogenous PDEs:
Partial differential equations are also classified as homogeneous and inhomogeneous. A partial differential equation of any order is said to be homogeneous if
every term of the PDE contains the dependent variable u or any of its derivatives, otherwise, it is said to be inhomogenous.
Example: Classify the following partial differential equations as homogeneous
or inhomogeneous:
ut = 4uxx ,
(19)
ut = uxx + x,
(20)
uxx + uyy = 0.
(21)
Solutions:
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PDE 19: The PDE is homogeneous since the terms of the equation are contain
partial derivatives of u only.
PDE 20: The PDE is inhomogeneous since a term contains the independent
variable x only.
PDE 21: The PDE is homogeneous since each terms of its contains partial
derivatives of u.
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Solution of a PDE:
A solution to a partial differential equation (PDE) is a function u that satisfies
the equation itself while also fulfills the associated conditions. In other words,
when u is substituted into the PDE, both sides of the equation must be equal,
confirming that the equation is satisfied.
Example: Show that u(x, t) = sin xe−4t is a solution of the following PDE
ut = 4uxx .
(22)
Solutions:
Left Hand Side(LHS) of Eq. 22:
ut =
∂
∂u(x, t)
= (sin xe−4t ) = −4 sin xe−4t
∂t
∂t
Right Hand Side of Eq. 22:
4uxx = 4
∂ 2 u(x, t)
∂2
= 4 2 (sin xe−4t ) = −4 sin xe−4t = LHS
2
∂x
∂x
Hence, u(x, t) = sin xe−4t is a solution to the given PDE 22.
Example: Show that
1. u(x, y) = xy,
2. u(x, y) = x2 y 2 ,
3. u(x, y) = sin(xy),
are solutions of the equation
xux − yuy = 0.
(23)
Solution:
(1)
u = xy, s. t. ux = y, uy = x ⇒ LHS = xy − yx = 0 = RHS
(2)
u = x2 y 2 , s. t. ux = 2xy 2 , uy = 2x2 y ⇒ LHS = 2x2 y 2 −2x2 y 2 = 0 = RHS
(3)
u = sin(xy), s. t. ux = y cos(xy), uy = x cos(xy) ⇒ LHS = xy cos(xy)−xy cos(xy) = 0 = RHS
This leads us to that the genral solution of the given PDE is of the form:
u = f (xy).
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Remarks on the Concept of a Solution to a PDE
Several key observations can be made regarding the nature of solutions to partial
differential equations (PDEs):
1. Superposition Principle:
In the case of a linear homogeneous ordinary differential equation (ODE),
it is well known that if u1 , u2 , u3 , . . . , un are individual solutions, then any
linear combination of these functions—given by
u = c1 u1 + c2 u2 + c3 u3 + . . . + cn un
Is also a valid solution. This idea is known as the superposition principle.
Interestingly, the same principle applies to linear homogeneous PDEs
within a given domain.
2. General Solution and Arbitrary Functions:
For linear ODEs, the general solution typically involves arbitrary constants. In contrast, the general solution of a linear PDE involves arbitrary
functions. This distinction is evident in the PDE:
ux + uy = 0,
Whose general solution is
u = f (x–y),
Where f is an arbitrary differentiable function. This implies that the
solution may take various forms, such as:
u = x − y, u = ex−y , u = sinh(x − y), u = ln(x − y),
or any function of the form f (x − y).
However, the general solution alone is often insufficient in practical scenarios. To completely solve a PDE, a particular solution must be determined
that satisfies the given initial or boundary conditions.
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Boundary Conditions:
Boundary conditions (BCs) in partial differential equations (PDEs) are additional constraints that specify the behavior of the solution on the boundary of
the domain where the PDE is defined. They are essential to ensure a unique
and physically meaningful solution.
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Types of Boundary Conditions:
Dirichlet Boundary Conditions: In such cases, the function u is typically
specified along the boundary of the defined domain. For example, in a rod
of length L (where 0 < x < L), the boundary conditions might be given
as u(0) = α and u(L) = β, where α and β are constants. Similarly, for a
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rectangular plate defined over 0 < x < L1 and 0 < y < L2 , the boundary
values u(0, y), u(L1 , y), u(x, 0), and u(x, L2 ) are generally specified. These
conditions are referred to as homogeneous if the value of u is zero at all
boundary points; otherwise, they are known as inhomogeneous boundary
conditions.
Neumann Boundary Conditions: In this type of condition, the normal deriva∂u
(i.e., the rate of change of u in the direction perpendicular to the
tive ∂n
boundary) is specified. For example, in the case of a rod of length L,
Neumann boundary conditions take the form:
ux (0, t) = α, ux (L, t) = β,
Where α and β are given constants.
Mixed Boundary Conditions: Here, the boundary condition involves a lin∂u
ear combination of the function u and its normal derivative ∂n
. This type
of condition generalizes both Dirichlet and Neumann types.
It’s also important to note that the domain does not always have to be
bounded. In some problems, parts of the boundary may extend to infinity.
Such cases will be explored in upcoming chapters.
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Initial Conditions:
As previously mentioned, partial differential equations (PDEs) often arise in
modeling physical phenomena such as heat conduction, wave motion, and quantum mechanics. Many of these equations—like the heat (diffusion) equation and
the wave equation—are time-dependent, involving the variable t. Therefore, it
is necessary to specify initial conditions, which define the state of the system at
the initial time t = 0.
In the case of the heat equation, the initial temperature distribution u(t = 0)
must be provided. For the wave equation, two initial conditions are typically
required: the initial displacement u(t = 0) and the initial velocity ut (t = 0).
These initial conditions are essential for determining a unique solution to timedependent PDEs.
References:
1. Wazwaz, A.-M. (2009). Partial differential equations and solitary waves
theory. Springer Berlin Heidelberg.
2. Neta, B. (2002). Partial Differential Equations MA 3132 Lecture Notes.
Department of Mathematics, Naval Postgraduate School. Monterey, CA.
3. Finan, M. B. (2009). A First Course of Partial Differential Equations in
Physical Sciences and Engineering. Arkansas Tech University.
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