Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers

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1.5 Superposition Principle 21

Now let us consider the problem

L [u]=G,

[u]=g

, (1.5.3)

[u]=g

By virtue of the linearity of the equation and the supplementar y condi-

tions, we may divide problem (1.5.3) into a series of problems as follows:

L [u

]=G,

]=0,

]=0, (1.5.4)

]=0,

L [u

]=0,

]=g

]=0, (1.5.5)

]=0,

L [u

]=0,

]=0, (1.5.6)

]=g

Then the solution of problem (1.5.3) is gi ven by

u =



i=1

. (1.5. 7)

Let us consider one of the subproblems, say, (1.5.5). Suppose we ﬁnd a

sequence of functions φ

,φ

, ..., which may be ﬁnite or inﬁnite, satisfying

the homogeneous system

22 1 Introduction

L [φ

]=0,

[φ

]=0, (1.5.8)

[φ

]=0,i=1, 2, 3,...

and suppose we can express g

in terms of the series

= c

[φ

]+c

[φ

]+.... (1.5.9)

Then the linear combination

= c

+ c

+ ..., (1.5.10)

is the solution of problem (1.5.5). In the case of an inﬁnite number of terms

in the linear combination (1.5.10), we require that the inﬁnite series be

uniformly convergent and suﬃciently diﬀerentiable, and that all the series

)whereN

= L, N

= M

for j =1,2,..., n convergence unif ormly.

1.6 Exercises

1. For each of the following, state whether the p arti al diﬀerential equa-

tion is linear, quasi-linear or nonlinear. If it is linear, state whether it

is homogeneous or nonhomogeneous, and gives its order.

(a) u

+ xu

= y,(b)uu

− 2xyu

=0,

+ uu

=1, (d)u

xxxx

+2u

xxyy

+ u

yyyy

=0,

(e) u

+2u

+ u

=sinx,(f)u

xxx

+ u

xyy

+logu =0,

(g) u

+ u

+sinu = e

,(h)u

+ uu

+ u

xxx

=0.

2. Verify that the functions

u (x , y)=x

− y

u (x , y)=e

sin y

u (x , y)=2xy

are the solutions of the equation

+ u

=0.

3. Show that u = f (xy), where f is an arbitrary diﬀerentiable function

satisﬁes

1.6 Exercises 23

− yu

and verify that the functions sin (xy), cos (xy), log (xy ), e

,and(xy)

are solutions.

4. Show that u = f (x) g (y)wheref and g are arbitrary twice diﬀeren-

tiable functions satisﬁes

− u

=0.

5. Determine the general solution of the diﬀerential equation

+ u =0.

6. Find the general solution of

+ u

=0,

by setting u

= v.

7. Find the general solution of

− 4u

+3u

=0,

by assuming the solution to be in the form u (x, y)=f (λx + y), where

λ is an unknown parameter.

8. Find the general solution of

− u

=0.

9. Show that the general solution of

∂

∂t

− c

∂

∂x

=0,

is u (x, t)=f (x − ct)+g (x + ct), where f and g are arbitrary twice

diﬀerentiable functions.

10. Verify that the fun ction

u = φ (xy)+xψ





is the general solution of the equation

− y

=0.

11. If u

= v

and v

= −u

, show that both u and v satisfy the Laplace

equations

∇

u = 0 and ∇

v =0.

24 1 Introduction

12. If u (x, y) is a h omogeneous function of degree n, show that u satisﬁes

the ﬁrst-order equation

+ yu

= nu.

13. Verify that

u (x , y, t)=A cos (kx) cos (ly ) cos (nct)+B sin (kx)sin(ly)sin(nct) ,

where k

+ l

= n

, is a solution of the equation

= c

+ u

) .

14. Show that

u (x , y; k)=e

−ky

sin (kx) ,x∈ R,y>0,

is a solution of the equation

∇

u ≡ u

+ u

for any real parameter k.Verifythat

u (x , y)=



∞

c (k) e

−ky

sin (kx) dk

is also a solution of the above equation.

15. Show, by diﬀerentiation that,

u (x , t)=

√

4πkt

exp



−

4kt



,x∈ R,t>0,

is a solution of the diﬀusion equation

= ku

where k is a constant.

16. (a) Verify that

u (x , y)=log





+ y



satisﬁes the equation

+ u

for all (x, y) =(0, 0).

(b) Show that

1.6 Exercises 25

u (x , y, z)=



+ y

+ z



−

is a solution of the Laplace equation

+ u

except at the origin.

u (r)=ar

satisﬁes the equation

′′

+2ru

′

− n (n +1)u =0.

17. Show that

(r, θ)=r

cos (nθ)andu

(r, θ)=r

sin (nθ) ,n=0, 1, 2, 3, ···

are solutions of the Laplace equation

∇

u ≡ u

θθ

=0.

18. Verify by diﬀerentiation that u (x, y)=cosx cosh y satisﬁes the Laplace

equation

+ u

=0.

19. Show that u (x, y)=f



2y + x



+ g



2y − x



is a general solution of

the equation

−

− x

=0.

20. If u satisﬁes the Laplace equation ∇

u ≡ u

= 0, show that both

xu and yu satisfy the biharmonic equation

∇

⎛

⎝

⎞

⎠

=0,

but xu and yu will not satisfy the Laplace equation.

21. Show that

u (x , y, t)=f (x + iky − iωt)+g (x − iky − iωt)

is a general solution of the wave equation

= c

+ u

) ,

where f and g are arbitrary twice diﬀerentiable functions, and ω



− 1



, k, ω, c are constants.

26 1 Introduction

22. Verify that

u (x , y)=x

+ y

+ e

(cos x sin y cosh y − sin x cos y sinh y)

is a classical solution of the Poisson equation

+ u

=(6x +2).

23. Show that

u (x , y)=exp



−



f (ax − by)

satisﬁes the equation

+ au

+ u =0.

24. Show that

− c

+2bu

has solutions of the form

u (x , t)=(A cos kx + B sin kx) V (t) ,

where c, b, A and B are constants.

25. Show that





− u

has solutions of the form

u (r, t)=

V (r)

cos (nct) ,n=0, 1, 2,....

Find a diﬀerential equation for V (r).

First-Order, Quasi-Linear Equations and

Method of Characteristics

“As long as a branch of knowledge oﬀers an abundance of problems, it is

full of vitality.”

David Hilbert

“Since a general soluti on must be judged impossible from want of analysis,

we must be content with the knowledge of some special cases, and that all

the more, since the development of various cases seems to be the only way

to bringing us at last to a more perfect knowledge.”

Leonhard Euler

2.1 Introduction

Many problems in mathematical, physical, and engineering sciences deal

with the formulation and the solution of ﬁrst-order partial diﬀerential equa-

tions. From a mathematical point of view, ﬁrst-order equations have the

advantage of providing a conceptual basis that can be utilized for second-,

third-, and higher-order equations.

This chapter is concerned with ﬁr st-order, quasi-linear and linear partial

diﬀerential equations and their solution by using the Lagrange method of

characteristics and its generalizations.

2.2 Classiﬁcation of First-Order Equations

The most general, ﬁrst-order, partial diﬀerential equation in two indepen-

dent variables x and y is of the form

28 2 First-Order, Quasi-Linear Equations and Method of Characteristics

F (x, y, u, u

)=0, (x, y) ∈ D ⊂ R

, (2.2.1)

where F is a given function of its arguments, and u = u (x, y) is an unknown

function of the independent variables x and y which lie in some given do-

main D in R

, u

∂u

∂x

and u

∂u

∂y

. Equation (2.2.1) is often written in

terms of standard notation p = u

and q = u

so that (2.2.1) takes the

form

F (x, y, u, p, q)=0. (2.2.2)

Similarly, the most general, ﬁrst-order, partial diﬀerential equation in

three independent variables x, y, z canbewrittenas

F (x, y, z, u, u

)=0. (2.2.3)

Equation (2.2.1) or (2.2.2) is called a quasi-linear partial diﬀerential

equation if it is linear in ﬁrst-partial derivatives of the unknown f unction

u (x , y). So, the most general quasi-linear equation must be of the form

a (x, y, u) u

+ b (x, y, u) u

= c (x, y, u) , (2.2.4)

where its coeﬃcients a, b,andc are functions of x, y,andu.

The following are examples of quasi-linear equations:



+ u



− y



+ u





− y



u, (2.2.5)

+ u

+ nu

=0, (2.2.6)



− u



− xy u

= xu. (2.2.7)

Equation (2.2.4) is called a semilinear partial diﬀerential equation if its

coeﬃcients a and b are independent of u, and hence, the semilinear equation

can be expressed in the f orm

a (x, y) u

+ b (x, y) u

= c (x, y, u) . (2.2.8)

Examples of semilinear equations are

+ yu

= u

+ x

, (2.2.9)

(x +1)

+(y − 1)

=(x + y) u

, (2.2.10)

+ au

+ u

=0, (2.2.11)

where a is a constant.

Equation (2.2.1) is said to be linear if F is linear in each of the variables

u, u

,andu

, and the coeﬃcients of these variables are functions only of the

independent var iables x and y. The most general, ﬁrst-order, linear partial

diﬀerential equation has the form

a (x, y) u

+ b (x, y) u

+ c (x, y) u = d (x, y) , (2.2.12)

2.3 Construction of a First-Order Equation 29

where the coeﬃcients a, b,andc, in general, are functions of x and y and

d (x , y) is a given function. Unless stated otherwise, these functions are

assumed to be continuously diﬀerentiable. Equations of the form (2.2.12)

are called homogeneous if d (x, y) ≡ 0ornonhomogeneous if d (x, y) =0.

Obviously, linear equations are a special kind of the quasi-linear equa-

tion (2.2.4) if a, b are independent of u and c is a linear function in u.

Similarly, semilinear equation (2.2.8) redu ces to a linear equation if c is

linear in u.

Examples of linear equations are

+ yu

− nu =0, (2.2.13)

+(x + y) u

− u = e

, (2.2.14)

+ xu

= xy, (2.2.15)

(y − z) u

+(z − x) u

+(x − y) u

=0. (2.2. 16)

An equation which is not linear is often called a nonlinear equation.So,

ﬁrst-order equations are often classiﬁed as linear and nonlinear.

2.3 Construction of a First-Order Equation

We consider a system of geometrical surfaces described by the equation

f (x, y, z, a , b)=0, (2.3.1)

where a and b are arbitrary parameters. We diﬀerentiate (2.3.1) with respect

to x and y to obtain

+ pf

=0,f

+ qf

=0, (2.3.2)

where p =

∂z

∂x

and q =

∂z

∂y

The set of three equations (2.3.1) and (2.3.2) involves two arbitrary

parameters a and b. In general, these two parameters can be eliminated

from this set to obtain a ﬁrst-order equation of the form

F (x, y, z, p, q)=0. (2.3.3)

Thus the system of surfaces (2.3.1) gives rise to a ﬁrst-order partial dif-

ferential equation (2.3.3). In other words, an equation of the form (2.3.1)

containing two arbitrary parameters is called a complete solution or a com-

plete integral of equation (2.3.3). Its role is somewhat similar to that of a

general solution for the case of an ordin ary diﬀerential equ ation.

On the other hand, any relationship of the form

f (φ, ψ)=0, (2.3.4)

30 2 First-Order, Quasi-Linear Equations and Method of Characteristics

which involves an arbitrary function f of two known functions φ = φ (x, y, z)

and ψ = ψ (x, y, z) and provides a solution of a ﬁrst-order partial diﬀerential

equation is called a general solution or general integral of this equation.

Clearly, the general solution of a ﬁrst-order partial diﬀerential equation

depends on an arbitrary function. This is in striking contrast to the situation

for ordinary diﬀerential equations where the general solution of a ﬁrst-

order ordinary diﬀerential equation depends on one arbitrary constant. The

general solution of a partial diﬀerential equati on can be obtained from

its complete integral. We obtain the general solution of (2.3.3) from its

complete integral (2.3.1) as follows.

First, we prescribe the second parameter b as an arbitrary function of

the ﬁrst parameter a in the complete solution (2.3.1) of (2.3.3), that is,

b = b (a). We then consider the envelope of the one-parameter family of

solutions so deﬁned. This envelope is represented by the two simultaneous

equations

f (x, y, z, a , b (a)) = 0, (2.3.5)

(x, y, z, a, b (a)) + f

(x, y, z, b (a)) b

′

(a)=0, (2.3.6)

where the second equation (2.3.6) is obtained from the ﬁrst equation (2.3.5)

by partial diﬀerentiation with respect to a. In principle, equation (2.3.5) can

be solved for a = a (x, y, z) as a function of x, y,andz. We substitute this

result back in (2.3.5) to obtain

f {x, y, z, a (x, y, z) ,b(a (x, y, z))} =0, (2.3.7)

where b is an arbitrary function. Indeed, th e two equations (2.3.5) and

(2.3.6) together deﬁne the general solution of (2.3.3). When a deﬁnite b (a)

is prescribed, we obtain a particular solution from the general solution.

Since the general solution depends on an arbitrary function, there are in-

ﬁnitely many solutions. In practice, only one solution satisfying prescribed

conditions is required for a physical problem. Such a solution may be called

a particular solution.

In addition to the general and particular solutions of (2.3.3) , if the enve-

lope of the two-parameter system (2.3.1) of surfaces exists, it also represents

a solution of the given equation (2.3.3); the envelope is called the singular

solution of equation (2.3.3). The singular solution can easily be constructed

from the complete solution (2.3.1) representing a two-parameter family of

surfaces. The envelope of this family is given by the system of three equa-

tions

f (x, y, z, a , b)=0,f

(x, y, z, a, b)=0,f

(x, y, z, a, b)=0. (2.3.8)

In general, it is p ossible to eliminate a and b from (2.3.8) to obtain

the equation of the envelope which gives the singular solution. It may b e

pointed out that the singular solution cannot be obtained from the general