Articolo G.A. Partial Differential Equations and Boundary Value Problems with Maple V

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68 Chapter 1

with q(0) = 200 coulombs, and the differential equation is



q(t)



q(t)

= e(t)

1.21. Solve Exercise 1.20 where the impressed voltage is e(t) = 10 te

−t

volts and the initial

charge is q(0) = 0 coulombs. Find the time at which the charge in the circuit is a

maximum.

Consider the following second-order linear differential equations with constant coefﬁcients.

Evaluate two basis vectors y1(t) and y2(t) and, from these, evaluate the second-order Green’s

function G2(t, s). With the Green’s function, evaluate a particular solution. When possible, use

the Maple dsolve command to verify the solution.

1.22.

y(t) +4



y(t)



+3y(t) = 3e

1.23.

y(t) −5



y(t)



+4y(t) = 10 sinh(t)

1.24.

y(t) +2



y(t)



−3y(t) = 2 +3e

1.25.

y(t) +4



y(t)



+13y(t) = sin(t)

Consider the following Cauchy-Euler second-order differential equations with variable

coefﬁcients. Evaluate two basis vectors y1(t) and y2(t) and, from these, evaluate the

second-order Green’s function G2(t, s).

1.26.



y(t)



+7t



y(t)



+9y(t) = f(t)

1.27.



y(t)





y(t)



+y(t) = f(t)

Ordinar y Linear Differential Equations 69

1.28.



y(t)



−4t



y(t)



+6y(t) = f(t)

1.29.



y(t)



+5t



y(t)



+y(t) = f(t)

1.30. In Exercise 1.26, let f(t) = ln(t). Use the evaluated Green’s function to ﬁnd a particular

solution. Use the Maple dsolve command to verify the solution.

1.31. In Exercise 1.27, let f(t) = t. Use the evaluated Green’s function to ﬁnd a particular

solution. Use the Maple dsolve command to verify the solution.

1.32. In Exercise 1.28, let f(t) = t

. Use the evaluated Green’s function to ﬁnd a particular

solution. Use the Maple dsolve command to verify the solution.

1.33. In Exercise 1.29, let f(t) = t

−t. Use the evaluated Green’s function to ﬁnd a

particular solution. Use the Maple dsolve command to verify the solution.

1.34. The simple spring-mass system consists of a mass m that is attached to an ideal spring

with a spring constant k. The system is immersed in a viscous damping medium where

γ is the damping coefﬁcient. The external applied force acting on the system is denoted

f(t), and the displacement of the mass from its equilibrium position is denoted y(t). The

differential equation that describes the system is



y(t)



+γ



y(t)



+ky(t) = f(t)

We look at the case where m = 1 kg, γ = 0 N/m/sec, and k = 9 N/m. (a) Evaluate a set

of system basis vectors y1(t) and y2(t), and evaluate the second-order Green’s function

G2(t, s). (b) If the initial conditions are y(0) = 2m,v(0) = 10 m/sec, and the driving

force is f(t) = 10 sin(t) N, evaluate the time-dependent solution y(t) for the problem.

simple second-order initial value problem.

1.35. Do Exercise 1.34 for the case where m = 1 kg, γ = 1 N/m/sec, k = 37/4 N/m, and the

driving force is f(t) = 20 te

−t

N. All other conditions are the same. Develop all

graphics and animation.

1.36. Do Exercise 1.34 for the case where m = 1 kg, γ = 1 N/m/sec, k = 82/4 N/m, and the

driving force is f(t) = 10 Heaviside(t) N. [Note: Heaviside(t) denotes the unit step

function.] Develop all graphics and animation.

70 Chapter 1

1.37. The series RLC circuit consists of a resistor R, an inductor L, and a capacitor C

connected in series. If the electric charge on the capacitor in the system is denoted as

q(t), and the driving voltage is e(t), then the differential equation that describes the time

dependence of the charge is given as



q(t)





q(t)



q(t)

= e(t) (1.199)

We consider the case where L = 1 henry, R = 1 ohms, and C = 4/50 farads and the

driving voltage is e(t) = 10 te

−t

volts. (a) Evaluate the system basis vectors q1(t) and

q2(t) and the second-order Green’s function G2(t, s). (b) If the initial charge q(0) = 0

coulombs and the initial current i(0) = 0 amps, evaluate the time dependence of the

charge q(t) on the capacitor. (c) Graph the solution and develop the animation as done

in Example 1.8.1. This is a simple second-order initial value problem.

1.38. Do Exercise 1.37 for the case where L = 1 henry, R = 1 ohms, and C = 4/10 farads,

and the driving voltage is e(t) = 10 sin(t) volts. All other conditions are the same.

Develop all graphics and animation.

For the following differential equations, you are given one basis vector y1(t). You are asked to

evaluate a second basis vector y2(t) using the method of reduction of order. You are then asked

to evaluate the second-order Green’s function G2(t, s).

1.39. Given one basis vector y1(t) = e

3 t

, ﬁnd the second-order Green’s function G2(t, s).

y(t) −6



y(t)



+9y(t) = f(t)

1.40. Given one basis vector y1(t) = t, ﬁnd the second-order Green’s function G2(t, s).



y(t)



−t



y(t)



+y(t) = f(t)

1.41. Given one basis vector y1(t) = t −1, ﬁnd the second-order Green’s function G2(t, s).



2 t −t





y(t)



+2(t −1)



y(t)



−2y(t) = f(t)

1.42. Given one basis vector y1(t) = t

, ﬁnd the second-order Green’s function G2(t, s).



y(t)



−6y(t) = f(t)

Ordinar y Linear Differential Equations 71

1.43. Given one basis vector y1(t) = t, ﬁnd the second-order Green’s function G2(t, s).



y(t)





y(t)



−y(t) = f(t) (1.200)

1.44. Let f(t) = te

3 t

in Exercise 1.39. Use the Green’s function that you evaluated to ﬁnd a

particular solution. Check your answer by substituting the solution back into the

differential equation.

1.45. Let f(t) =

in Exercise 1.40. Use the Green’s function that you evaluated to ﬁnd a

particular solution. Use the Maple dsolve command to verify the solution.

1.46. Let f(t) = t

ln(t) in Exercise 1.42. Use the Green’s function that you evaluated to ﬁnd

a particular solution. Use the Maple dsolve command to verify the solution.

1.47. Let f(t) = t in Exercise 1.43. Use the Green’s function that you evaluated to ﬁnd a

particular solution. Use the Maple dsolve command to verify the solution.

The following linear differential equations are to be solved by using the method of Frobenius

for ﬁnding a series solution about the origin (x = 0). For the Exercises 1.48 through 1.51,

(a) evaluate the indicial equations and the corresponding recursion formula. (b) From the

indicial equation, obtain a set of basis vectors from different choices of roots of the indicial

equation. If you are able to obtain only one solution, then use the method of reduction of order

to ﬁnd a second basis vector.

1.48.

y(x) +x



y(x)



+y(x) = 0

1.49.

y(x) −x



y(x)



−x

y(x) = 0

1.50.



y(x)





y(x)



+xy(x) = 0

1.51.



y(x)





y(x)



−y(x) = 0

72 Chapter 1

1.52. The Bessel differential equation solved in Section 1.11 comes about when one

considers the diffusion and wave equations in the cylindrical coordinate system. For an

integer m, the Bessel function of the ﬁrst kind of order m reads as

J(m, x) =

∞



n=0

(−1)

m+2 n

n!(m +n)!

From the preceding series, verify the following

(a)

J(0, 0) = 1

(b)

J(0, −x) = J(0,x)

(c)

J(0,x)=−J(1,x)

(d)



∂

∂x

J(m, x)



=−mJ(m, x) +xJ(m −1,x)

(e)



J(m −1,x)dx = x

J(m, x)

(f)

2 mJ(m, x) = x(J(m −1,x)+J(m +1, x))

CHAPTER 2

Sturm-Liouville Eigenvalue

Problems and Generalized

Fourier Series

2.1 Introduction

In Chapter 1 we examined both ﬁrst- and second-order linear homogeneous and

nonhomogeneous differential equations. We established the signiﬁcance of the dimension of

the solution space and the basis vectors. With a set of basis vectors, we could span the entire

solution space of the homogeneous equation by way of a linear superposition of these basis

vectors. We also showed how the method of variation of parameters utilized the basis

vectors in the construction of the Green’s function. With the Green’s function in hand, we

were then able to evaluate the solution to the corresponding nonhomogeneous differential

equation.

2.2 The Regular Sturm-Liouville Eigenvalue Problem

We will now focus on second-order differential equations, over ﬁnite intervals, when the

independent variable is the spatial variable x. These problems will play a prominent role in

the type of partial differential equations that we will look at later.

The problems that we examine here fall under the general category of what are called “regular”

Sturm-Liouville eigenvalue problems. These are basically second-order linear homogeneous

differential equations whose constraints are in the form of two-point, homogeneous, nonmixed

boundary conditions.

To begin, we ﬁrst deﬁne the generalized Sturm-Liouville differential operator L acting on

some function y(x) over a ﬁnite interval I ={x |a<x<b} as follows:

L(y) = D(p(x)D(y)) +q(x)y (2.1)

74 Chapter 2

Here, D is the standard differentiation operator deﬁned as

D(y) =

y(x)

The “regular” Sturm-Liouville eigenvalue problem consists of two parts:

1. The homogeneous Sturm-Liouville differential equation

L(ϕ) +λ w(x)ϕ = 0

2. The “regular” boundary conditions that are homogeneous, nonmixed (single point,

spatially separated) conditions at the end points

ϕ(a) +κ

(a) = 0

and

ϕ(b) +κ

(b) = 0

The coefﬁcients κ

,κ

are all real constants. Restrictive conditions on the system are

that over the ﬁnite interval I, the functions p(x), p



(x), q(x), and w(x) are real continuous

functions and that p(x) and w(x) are positive over this interval I. In some situations, the

preceding conditions can be relaxed and modiﬁed somewhat and the problem becomes

“singular.” We shall consider singular problems later.

The Sturm-Liouville differential equation given earlier is linear and homogeneous, and the

boundary conditions are homogeneous and nonmixed. The allowed values of λ

for which the

preceding differential equation satisﬁes the boundary conditions are called the “eigenvalues,”

and the corresponding solutions ϕ

(x) are called the “eigenfunctions.”

We state, without proof, some of the important features of “regular” Sturm-Liouville problems:

1. There exist an inﬁnite number of eigenvalues λ

that can be ordered in magnitude and that

can be indexed by the positive integers n = 0, 1, 2, 3,....

2. All eigenvalues are real.

3. Corresponding to each eigenvalue λ

there exists a unique eigenfunction ϕ

(x).

4. The eigenfunctions form a “complete” set with respect to any piecewise smooth function

f(x) over the ﬁnite interval I ={x |a<x<b}. This means that, over this interval, the

function can be represented as a generalized Fourier series expansion in terms of the

eigenfunctions shown here:

f(x) =

∞



n=0

F(n)ϕ

(x)

where the terms F(n) are the properly evaluated Fourier coefﬁcients.

Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 75

5. The inﬁnite series given converges to the average of the left- and right-hand limits of the

function at any point in the interval; that is,

∞



n=0

F(n)ϕ

(x) =

f(x +0) +f(x −0)

6. The eigenfunctions corresponding to different eigenvalues are orthogonal relative to the

“weight function” w(x) over the interval I. If the orthogonal eigenfunctions are further

normalized, then we can express the statement of orthonormality of the eigenfunctions in

terms of the inner product (see Chapter 0), with respect to the weight function w(x),as



(x)ϕ

(x)w(x)dx = δ(n, m)

where we have used the Kronecker delta function δ

n,m

, which, by deﬁnition, has the value

1 for n = m and the value 0 for n = m.

2.3 Green’s Formula and the Statement of Orthonormality

In terms of the standard differentiation operator D, the Sturm-Liouville operator L acting on

some function y(x) reads

L(y) = D(p(x)D(y)) +q(x)y

It should be remarked that any second-order linear differential operator

a2(x)D

+a1(x)D +a0(x)

can be written in Sturm-Liouville form (see Exercise 2.1) by multiplying the operator by the

integrating factor

r(x) =



a1(x)

a2(x)

We now provide a formal proof of the statement of orthogonality of the eigenfunctions. We

consider the Sturm-Liouville operator acting on the two different functions u and v:

L(u) = D(p(x)D(u)) +q(x)u

and

L(v) = D(p(x)D(v)) +q(x)v

76 Chapter 2

We evaluate the Green’s formula, which is shown next for the two functions u and v.



(uL(v) −vL(u))dx =



(u(Dp(x)D(v) +q(x)v) −v(Dp(x)D(u) +q(x)u))dx

Expansion and simpliﬁcation of the integrand on the right-hand side yields



(uL(v) −vL(u))dx =



D(p(x)(uD(v) −vD(u)))dx

Since the integrand on the right is an exact differential, a simple integration yields



(uL(v) −vL(u))dx = p(b)(u(b)v

(b) −v(b)u

(b)) −p(a)(u(a)v

(a) −v(a)u

(a))

If we impose the regular Sturm-Liouville boundary conditions from Section 2.2 on the

functions u and v, then the right-hand term in the preceding Green’s formula vanishes, and we

obtain



uL(v)dx =



vL(u)dx

When the preceding statement holds, we say the operator L, for the appropriate boundary

conditions, is “self-adjoint.”

If we now write the Sturm-Liouville differential equation for the two different eigenfunctions

(x) and ϕ

(x), we have

(

)

+λ

w(x)ϕ

= 0

and

(

)

+λ

w(x)ϕ

= 0

Rearranging these equations and integrating over the interval yields



(

)

−ϕ

(

))

dx =

(

−λ

)

⎛

⎝



(x)ϕ

(x)w(x)dx

⎞

⎠

Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 77

If the eigenfunctions satisfy the regular Sturm-Liouville boundary conditions from Section 2.2,

then Green’s formula shows the left-hand side to be zero and, thus, the right-hand side is also

zero. This yields the statement of orthogonality of the eigenfunctions:

(

−λ

)

⎛

⎝



(x)ϕ

(x)w(x)dx

⎞

⎠

The integral here denotes the inner product (see Chapter 0) of the two different eigenfunctions

with respect to the weight function w(x) over the interval I ={x |a<x<b}. This result shows

that if the eigenvalues are not equal—that is, λ

= λ

—then the eigenfunctions,

corresponding to the different eigenvalues, are orthogonal with respect to the weight function

w(x) over the interval I.

The Statement of Or thonormality

To normalize a vector, we simply divide the vector by its length or norm. From Chapter 0,

recall that the norm is the square root of the inner product of the vector with itself with respect

to the weight function w(x)—that is,

norm =









(x)

w(x)dx

If we normalize the eigenfunctions—that is, divide each by its length (norm)—then the

preceding condition of orthogonality yields an equivalent “statement of orthonormality,”

which, in terms of the Kronecker delta function, reads



(x)ϕ

(x)w(x)dx = δ(n, m) (2.2)

This statement indicates the two different eigenfunctions to be orthonormal (a combination

of being orthogonal and normalized) with respect to the weight function w(x) over the

interval I. This result will play a signiﬁcant role in the development of the generalized Fourier

series.

Types of Boundary Conditions

Before we consider different examples of Sturm-Liouville operators with various types of

nonmixed homogeneous boundary conditions, we ﬁrst classify the regular boundary conditions

that we will look at in three types. Let point a be the boundary point.