Chapter 0: An Introduction And First Order Partial

Differential Equations
0.1 Introduction
0.1.1. Brief History of Partial Differential Equation and its
applications?
A partial differential equation (PDE) is a differential equation that
contains unknown multivariable functions and their partial derivatives. (A
special case is ordinary differential equations (ODEs), which deal
with functions of a single variable and their derivatives.) PDEs are used
to formulate problems involving functions of several variables, and are
either solved by hand, or used to create a relevant computer model.
The study of partial differential equations(PDE's) started in the 18th
century in the work of Euler, d'Alembert, Lagrange and Laplace as a
central tool in the description of mechanics and more generally, as the
principal mode of analytical study of models in the physical science. The
analysis of physical models has remained to the present day one of the
fundamental concerns of the development of PDE's.

Now PDE's appear frequently in all areas of physics and engineering.

Moreover, in recent years we have seen a use of partial differential
equations in areas such as biology, chemistry, computer sciences and in
economics (finance).

0.1.2. Basic Concepts and Definitions.

If u = u(x,y,z, …), where:

 x, y, z, … are independent variables and

 u is a dependent variable ( unknown function), then we will use
the following notations:

∂u ∂u ∂u
=u , =u , =u , ⋯
∂x x ∂y y ∂z z

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 ∂2 u ∂2 u 2
∂u
=u , =u , =u ,
∂ x∂ y xy ∂ y ∂ x yx ∂ x∂ x xx
∂2 u
=u , ⋯
∂ y ∂ y yy
∂3 u
=u , ⋯
∂ x∂ y∂ z xyz
Remark. The unknown functions always depend on more than one
variable.

Definition. An equation containing the dependent ( unknown function)

and independent variables and one or more partial derivatives of the
dependent variable is called a partial differential equation. The general
form of a partial differential equation in independent variables x, y, …
and one dependent variable u is

F (x,y,.....,u,ux, uy,......,uxx, uyy,......)= 0.

∂u ∂u
Example 1. The equation x ∂ x + y ∂ y = cos(xy) is a partial
differential equation for the unknown function u(x,y).

Example 2. The system

∂u ∂v
u ∂ x + v ∂ y = x−y

∂u ∂v
u ∂ y + v ∂x = x + y

is a system of partial differential equations for the unknown functions

u(x,y) and v(x,y).

Definition. A solution of a partial differential equation

F (x,y,.....,u,ux, uy,......,uxx, uyy,......)= 0.

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is a function u(x,y,...) with continuous partial derivatives of all orders that
appear in the equation and that satisfies the differential equation at every
point of its domain of definition.

Example 3.

x2 y3
u( x , y )= + ln( √ x +5 )
(i) Show that 6 is a solution of the PDE

2
u xy =xy

(ii) Show that u(x,y)=(x+y)3 and u(x,y)=sin (x-y) are solutions of the

partial differential equation

2 2
∂u ∂u
∂ x 2 - ∂ y 2 = 0.

Sol.

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0.1.3. Initial and boundary conditions
PDEs have in general infinitely many solutions. In order to obtain a
unique solution one must supplement the equation with additional
conditions. There are two types of conditions commonly associated with
PDEs:

(1) Initial conditions which give information about the solution u

(or its derivative) at a given initial time t0.
(2) Boundary conditions which give information about the
behavior of the solution u (or its derivatives) at the boundary of
the domain under consideration.

The conditions associated to a PDE depend on the type of the PDE under
consideration.

Definition. A partial differential equation with initial conditions is called

an initial value problem ( IVP )and a partial differential equation with
boundary conditions is called a boundary value problem ( BVP ).

0.1.4. Classification of partial differential equations

Classification of partial differential equations plays a central role in
studying partial differential equations. There exist several classifications.
One classification is according to the order of the equation and another
classifications is according to linearity. Other important classifications
will be described in later chapters.

Definition. The order of a partial differential equation is the order of

the highest partial derivative that appears in the PDE.
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Example 4

∂u ∂u
(a) x ∂ x + y ∂ y =0 is a first-order PDE equation in two

variables with variable coefficients.

∂u ∂u
(b) a ∂ x + b ∂ y =c, where x,y are independent variables, a and b

are constants, is PDE of first order with constant coefficients.

∂u ∂u
(c) ∂ x + ∂ y -(x+y) u = 0 is a PDE of first-order.

∂2u ∂2 u ∂2 u ∂u ∂u
2 2
(d) a(x) ∂ x +2b(x) ∂x∂ y +c(x) ∂ y = f (x,y,u, ∂ x , ∂ y )

where a(x), b(x) and c(x) are functions of x and f is a function of

∂u ∂u
x ,y ,u, ∂ x and ∂ y , is a PDE of second order.

2 2
∂u
(e)
∂u
( )
∂x +
( )
∂y = 1 is a PDE of first-order and second degree.

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∂2 u ∂u ∂2 u
(f)
2
∂ x + 2y ∂x∂ y + 3x ∂ y 2 = 4 sin x is a PDE of second order

and degree one.

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(g)
u xxy +xu yy +8 u=7 y is a third order PDE.

Definition. A PDE is said to be linear if the unknown function u(x,y,...)

and all its partial derivatives appear in an algebraically linear form, 'that

is, of the first degree and are not multiplied together. The PDE which is

not linear is called non linear.

Example 5.

 The general form of first order linear PDE is

A(x,y) ux + B(x,y) uy + C(x,y) u =f(x,y).

 The equation Auxx+Buxy+Cuyy+Dux+Euy+Fu = f, where the

coefficients A,B,C,D,E, F, and the function f are functions of x

and y only, is a general form of the second order linear PDE in the

continuous unknown function u(x,y).

Example 6

 (x+2y) ux +x2uy = sin (x2+y2) is a first order linear PDE.

 uux+x2uy=0 is a non linear PDE of first order.

 xuxx +yuxy+uyy= u2 is a non linear PDE of second order.

Definition. A PDE is called almost linear or semi-linear if the highest-

order derivatives have degree one and their coefficients are functions of

the independent variables only.

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Example 7.

 The general form of first order almost linear PDE is

A(x,y) ux + B(x,y) uy = C(x,y,u)

 The equation
2 2
∂2u ∂u ∂u ∂u ∂u
2
A ∂ x + B ∂x∂ y +C ∂ y = f(x, y, u, ∂ x , ∂ y ),
2
where

A,B and C are functions of x and y only, is a general form of the

second order almost linear PDE in the continuous unknown

function u(x, y).

Example 8.

 uxyy +uyy = sin u is a third order almost linear PDE.

 uux+x2uxy=0 is a second order almost linear PDE.

 xuxx +yuxy+uuyy= 0 is a not almost linear PDE of second order.

Definition. A PDE is said to be quasilinear if it is linear in all the

highest-order derivatives of the dependent variable.

Example 9.

 The general form of first order quasi linear PDE is

A(x,y,u) ux + B(x,y,u) uy = C(x,y,u).

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  The general form of a quasi linear second order PDE is

A(x,y,u,p,q) uxx + B(x,y,u,p,q) uxy + C(x,y,u,p,q) uyy +f(x,y,u,p,q)=0,

∂u ∂u
where p = ∂ x and q = ∂ y ,

Example10.

 uyxuxyy +uyy = sin u is a third order quasi linear PDE.

 uyxuyy +x2uxy=0 is a second order not quasi linear PDE.

 xuxx +yuxy+uuyy= 0 is a quasi linear PDE of second order.

Remark

Every linear PDE is almost linear and every almost linear PDE is quasi

linear.

The following figure gives the relation between linear PDE, almost linear

PDE and quasi linear PDE.

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Definition. A PDE is called homogeneous if the equation does not

contain a term independent of the unknown function and its derivatives.

Example 10.

 ut + cux= 0, is linear, 1st Order, homogeneous PDE.

 uxx+uyy=xy, is linear, 2nd Order, non homogeneous PDE.

 α(x, y)uxx+2uxy+3x2uyy=4ex, is linear, 2nd Order, non

homogeneous PDE.

 uxuxx+(uy)2 = 0, is non linear, nor almost linear, but quasi linear,

2nd order, homogeneous PDE.

 yuxx+(uy)2 = sin x, is non linear, but almost linear 2nd order, non

homogeneous PDE.

Definition The general solution G.S. of the nonhomogeneous PDEs is

u = uh+up where uh is the solution of the corresponding homogeneous

equation and up is a particular solution of the nonhomogeneous equation.

Example 11 Find the G. S. of the PDE ux + u = y.

Sol.

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0.2. General Solution of Linear first order PDE with

constant coefficients

A PDE of the form Aux + Buy + Cu = f(x,y), where A, B and C

are constants Making the change of variables r = x, s = Bx−Ay, then the

PDE becomes Aur + Cu = f(r,s), which is ordinary DE in r keeping s

constant.

Examples Find the G.S. of the following equations.

(1) u + ux + uy = 0

Sol.

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(2) 2ux + 3uy = x2

Sol.

(3) 3ux + 6uy + u = x + 2ey

Sol.

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0.3. General Solution of Linear first order PDE with non
constant coefficients
Remember first that the general form of the linear first order PDE with
non constant coefficients is A(x,y)ux + B(x,y)uy + C(x,y)u = f(x,y).

Let r = x, s satisfies the solution of ODE dy/dx = B /A in which the G.S. is

s(x,y) = constant, where A ¿ 0. Then the transformed equation is

Aur + Cu = h(r,s), which is ODE in r keeping s constant, solving this

equation we get the solution of the PDE.

Examples: Find the G.S. of the following equations:

(1) x2ux −xyuy + yu = 0.

Sol.

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(2) xux + yuy = 2u + x2

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0.4 Quasilinear PDEs
Remember that the general form of quasilinear PDE of first order is

P(x,y,u)ux + Q(x,y,u)uy = R(x,y,u) .

First: Method of Lagrange:

Consider the system of first order ODE ,

dx dy du
= =
P( x , y, u) Q( x , y , u) R( x , y , u)
This system is called subsidiary equations and it is equivalent to the
system

dy Q du R
= =
dx P and dx P

where x is the independent variable . The G.S. of this system has the form
y = y(x,c1,c2), u = u(x,c1,c2) where c1,c2 are arbitrary constants . If these
equations are solved for c1,c2 , then the G.S. of the subsidiary equations
can be written in the form v(x,y,u) = c1 , w(x,y,u) = c2 and the G.S. of the
PDE is F(v,w) = 0 or w = f(v) or v = g(w) where F, f, g are arbitrary
functions .

Examples Find the G.S. of

(1) xuux + yuuy = −(x2 + y2)

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(2) xux + yuy + u = 0

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Second: Method of Multipliers

A useful technique for integrating a system of first order equations is that

a c λa+μc a c
= = =
of multipliers . Recall that if b d , then λb+μd b d for
arbitrary values of multipliers λ,µ.

Examples Solve the PDE

(1) uux + yuy = x

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 2 2
x +y
(2) (y−x)ux + (y + x)uy = u

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(3) x2ux + y2uy = xy

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0.5 Cauchy problem for Quasilinear PDES
'
In ODE to determine a solution of first order equations y =f (x , y )
which passes through a point in xy − plane under a general condition I.C.

A unique solution to the problem exists .

In the study of first order PDE in two independent variables (x,y) we

determine an integral surface such that the surface passes through a curve
in xyu space . Such a problem is called Cauchy problem. Before
proceeding with the general case , consider the following examples.

Examples Solve the Cauchy problems

(a) yux −xuy = 0, u(x,0) = x4

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(b) ux + uy = u, u(x,0) = sinx

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Cauchy problem with parameterized curve :
Let Γ be a given smooth curve defined perimetrically by

x = f(t), y = g(t), u = h(t), a<t<b

To construct an integral surface of PDE which contains Γ , we proceed as

follows : Let v(x,y,u) = c1, w(x,y,u) = c2 be two independent integrals
of the subsidiary equations . Wright v(f(t),g(t),h(t)) = c1 and
w(f(t),g(t),h(t)) = c2 . Eliminate t from these equations to derive the
relation , F(c1,c2) = 0. Then the solution of the Cauchy problem is
F(v,w) = 0.

Examples

(a) Find an integral surface of (y + xu)ux + (x + yu)uy = u2 −1 which

passes through the parabola x = t, y = 1, u = t2.

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(b) Solve the Cauchy problem uux + uuy = y + x, x = 1, y = t, u = t2.

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