Chapter 0: An Introduction And First Order Partial Di erential Equationsff

0.1 Introduction

0.1.1. Brief History of Partial Di erential Equation and itsff

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 di erential ffequations 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∂ x

=ux ,

∂u∂ y

=u y ,

∂u∂ z

=uz , ⋯

∂2u∂ x ∂ y

=uxy ,

∂2u∂ y ∂ x

=u yx ,

∂2u∂ x ∂ x

=uxx ,

∂2u∂ y ∂ y

=u yy , ⋯

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∂3u∂ x ∂ y ∂ z

=uxyz , ⋯

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 di erential equation in independent variables ff x, y, … and one dependent variable u is

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

Example 1. The equation x

∂u∂ x + y

∂u∂ y = cos(xy) is a partial di erential ff

equation for the unknown function u(x,y).

Example 2. The system

u

∂u∂ x + v

∂v∂ y = x−y

u

∂u∂ y + v

∂ v∂ x = x + y

is a system of partial di erential equations for the unknown functions ffu(x,y) and v(x,y).

Definition. A solution of a partial di erential equation ff

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

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

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Example 3.

(i) Show that u( x , y )= x2 y3

6+ ln (√ x+5)

is a solution of the PDE

uxy=xy 2

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

partial differential equation

∂2u∂ x2

-

∂2 u∂ y2

= 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 di erential equation with initial conditions is called ffan initial value problem ( IVP )and a partial di erential equation with ffboundary conditions is called a boundary value problem ( BVP ).

0.1.4. Classification of partial di erential equationsffClassification of partial di erential equations plays a central role in ffstudying partial di erential equations. There exist several classifications. ffOne 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 di erential equationff is the order of the highest partial derivative that appears in the PDE.

Example 4

(a) x

∂u∂ x + y

∂u∂ y =0 is a first-order PDE equation in two

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variables with variable coefficients.

(b) a

∂u∂ x + b

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

constants, is PDE of first order with constant coefficients.

(c)

∂u∂ x +

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

(d) a(x)

∂2u∂ x2

+2b(x)

∂2 u∂ x ∂ y +c(x)

∂2u∂ y2

= f (x,y,u, ∂u∂ x ,

∂u∂ y )

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

x ,y ,u, ∂u∂ x and

∂u∂ y , is a PDE of second order.

(e) (∂ u∂ x )

2

+ (∂u∂ y )

2

= 1 is a PDE of first-order and second degree.

(f) ∂2u∂ x2 + 2y

∂2 u∂ x ∂ y + 3x

∂2 u∂ y2 = 4 sin x is a PDE of second order and

degree one.

(g) uxxy+ xuyy+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

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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 coe cients are functions offfi

the independent variables only.

Example 7.

The general form of first order almost linear PDE is

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

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The equation

A∂2u∂ x2 + B

∂2 u∂ x ∂ y +C

∂2 u∂ y2

= f(x, y, u, ∂u∂ x ,

∂u∂ y ), 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).

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,

where p = ∂u∂ x and q =

∂u∂ y ,

Example10.

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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.

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.

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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 coe cientsffi

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 coe cientsffiRemember first that the general form of the linear first order PDE with non constant coe cients is ffi 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 ,

dxP ( x , y ,u )

= dyQ (x , y ,u )

= duR (x , y ,u )

This system is called subsidiary equations and it is equivalent to the system

dydx

=QP and

dudx

= RP

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

of multipliers . Recall that if ab= c

d , then

λa+μcλb+μd

=ab= c

d for arbitrary values of multipliers λ,µ.

Examples Solve the PDE

(1) uux + yuy = x

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(2) (y−x)ux + (y + x)uy = x2+ y2

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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