Chapter 9 Eigenvalue Problems - msharpmath€¦ · [101] 009 Numerical Analysis Using M#math, 1 "msharpmath" revised on 2014.09.01 Numerical Analysis Using M#math Chapter 9 Eigenvalue

[101] 009 Numerical Analysis Using M#math, www.msharpmath.com

1

"msharpmath" revised on 2014.09.01

Numerical Analysis Using M#math

Chapter 9 Eigenvalue Problems

9-1 Existence of Solution and Eigenvalues

9-2 Sturm-Liouville Problems

9-3 Eigenvalues on Stability and Convergence Rate

9-4 Exercises ==============================

9-5 Higher-order ODE and Eigenvalue Problems (optional)

Eigenvalue problems occurring in electromagnetics, vibration, structure

analysis and so on are related to the boundary value problems of ordinary

differential equations. The eigenvalue-finding methods discussed in Chapter 4

provide the necessary platform to solve the eigenvalue problems of ordinary

differential equations. Nevertheless, eigenvalues for relevant to an

matrix consist of at most eigenvalues. This is in contrast to the

fact that the eigenvalue problems are associated with an infinite number of

eigenvalues. Therefore, there occurs an intrinsic limitation in obtaining

eigenvalues for eigenvalue problems such as the well-known Sturm-Liouville

problems. However, fortunately the fundamental eigenvalue that has the

smallest absolute value is the most important in applications. And engineering

importance diminishes as the absolute value of eigenvalues increases. In this

regard, the methods treated in this chapter are of practical importance even

though only a finite number of eigenvalues can be determined.

First of all, discussed is the situation where an infinite number of solutions

are existing. Next, the eigenvalue problems are properly defined. Then,

numerical approaches to obtain eigenvalues are discussed.

In doing this, the generalized nonlinear differential operator adopted earlier


2

is exploited again and the eigenvalue problems are expressed in the following

compact form

In the last section, the role of eigenvalues is explored in the initial value

problems and in the iteration methods.

//======================================================================

// syntax in M#math

//----------------------------------------------------------------------

Umbrella ‘bvp’ is designed to solve BVPs the general forms of which are

cz*{f''''}

+cw*{f'''} +cfw*{ff'''} +cuw*{f'f'''} + cvw*{f''f'''} + cww*{f'''f'''}

+cv*{f''} +cfv*{ff''} +cuv*{f'f''} + cvv*{f''f''}

+cu*{f'} +cfu*{ff'} +cuu*{f'f'}

+cf*{f} +cff*{ff} = cs

bw*{f'''} + bv*{f''} + bu*{f'} + bf*{f} = bs

Syntax

bvp .x[n=101,g=1](a,b) ( DE, [ BCs ] ) .Spoke.Spoke ...

x = (a,b).span(n,g=1); bvp [x] ( DE, [BCs] ) .Spoke.Spoke ...

Spokes

.peep // peep the iteration procedure

.relax(relax=0.5) // under-relaxation factor

.tol/abstol(abstol=1.e-5) // absolute tolerance

.reltol(reltol=1.e-5) // relative tolerance

.maxiter(maxiter=500) // maximum iteration count

.shift(scalar) // eigenvalue shift

.plot // plot default data

.plot(f,f',f'',...) // plot user-defined data

.return (f,f',f'',...) // return user-defined data


3

.togo (Y=y,Y1=y',...) // takeout each field

Section 9-1 Existence of Solution and Eigenvalues

In this section, the existence of an infinite number of solutions are

discussed focusing on second-order ODEs, and the eigenvalue problems are

defined.

■ Existence of an infinite number of solutions. As was discussed in

Chapter 8, solutions to boundary value problems are either ① unique, ②

nonexisting, or ③ of an infinite number. Nevertheless, despite the nonexistence

of mathematical solution, a meaningless numerical solution is always obtained

due to the inherent limitation of computing with finite number of digits, as was

confirmed in Example 8-5.

As a beginning step, let us investigate the case where an infinite number of

solutions are existing. For a particular second-order ODE , there are

two types of boundary conditions with which an infinite number of solutions are

obtained.

○ infinite number of solutions with multiple integration constants

(1)

○ infinite number of solutions with a single integration constant

(2)

(3)

For the cases of equations (1) and (2), only a trivial solution is obtained

numerically. This is because both the ODE and boundary conditions are

homogeneous.

■ Homogeneous conditions and eigenvalue problems. As was discussed

above, the ODE with non-homogeneous boundary conditions cannot have a

trivial solution . In this case, only one non-trivial solution is


4

numerically obtained even though there exist an infinite number of solutions

[the case of equation (3)].

The eigenvalue problems for the ODE over an interval treat

only homogeneous boundary conditions

(4)

and only nontrivial solutions . In the above, are not zero

simultaneously. The homogeneous boundary condition guarantees that a

constant multiple of an eigenfunction is also an eigenfunction. Of course, the

ODE relevant to the eigenvalue problem is type of homogeneous differential

equation.

■ Eigenvalue problems, definition and property. Especially for the second-

order ODE, the eigenvalue problems are generalized as the Sturm-Liouville

problem the detail of which will be discussed in the next section. Here, the

characteristics of eigenvalue problems are addressed briefly.

Let us find the mathematical solution to the following second-order ODE

with homogeneous differential equation and boundary conditions

(5)

First for the case of , the general solution is given as

(6)

so that from the boundary condition . And from the boundary

condition , we must have . However, since , it

follows that . In other words, a trivial solution is the only

possible solution. Next for the case of , from the general solution

and from the boundary conditions, the only possible solution is

found to be the trivial solution .

Last for the case of , the general solution

(7)


5

is utilized to find from the boundary condition . This means

that . And the remaining boundary condition enforces

that

(8)

which can be satisfied by . At this point, the corresponding

value is called the eigenvalue, and the corresponding solution

is called the eigenfunction. Meanwhile, it is conventional that the

integration constant is set to be unity, unless otherwise specified.

From the above simple example, the eigenvalue problems are understood

to exhibit the following characteristics.

• homogeneous ODE

• homogeneous boundary condition

• infinite number of eigenvalues

• eigenfunction corresponding to an eigenvalue

• , a constant multiple of , is also an eigenfunction

• the fundamental eigenvalue of smallest absolute value

Next, let us investigate the trend of eigenfunctions in response to the increase of

their absolute values. The graphical solution to the above-studied eigenvalue

problem

(9)

is shown in Figure 9-1.


6

Figure 9-1 Eigenfunctions corresponding to

The eigenfunction corresponding to the smallest absolute eigenvalue is

especially termed to be the fundamental eigenfunction. As shown in Figure 9-1,

the fundamental eigenfunction does not have any root that satisfies

inside the domain. Similarly, the second, third and fourth eigenfunctions have

one, two and three roots of , respectively, inside the domain. This

property is valid for other types of boundary conditions. For example, the

profiles of eigenfunctions corresponding to the boundary conditions

are displayed in Figure 9-2. It can be easily seen that the number

of roots of is the same as that shown in Figure 9-1.

Figure 9-2 Eigenfunctions corresponding to

Section 9-2 Sturm-Liouville Problems

The Sturm-Liouville problem, the most famous eigenvalue problem of

second-order ODE, is studied in this section due to its practical importance in

engineering and science.


7

★ Self-adjoint form. A standard second-order linear differential equation

written as

(10)

can be recast into the so-called self-adjoint form by multiplying

to both sides of the equation

(11)

The above self-adjoint form serves as the generic form producing an infinite

number of orthogonal functions, and is used to define the Sturm-Liouville

problem.

Table 9-1 Sturm-Liouville Problem and eigenfunction expansion

• differential equation:

• boundary conditions: at each side

or at both sides

(periodic)

• for , is a real continuous function

• for an interval , ,

• eigenvalue: all the eigenvalues are real and of infinite number

• eigenfunction: two distinct eigenfunctions are orthogonal,

• eigenfunction expansion:

■ Sturm-Liouville problem. For the interval , the Sturm-Liouville

equation is defined by the following self-adjoint form

(12)


8

where a parameter is called the eigenvalue. In the above, are all

real-valued and continuous over an interval . Especially

over an interval .

The self-adjoint form in equation (12) and the boundary conditions as

given in Table 9-1 consist of the boundary value problem called the Sturm-

Liouville problem. Frequently, the conditions ①, ②, ③ in Table 9-1 are

combined together as

(13)

where and are not zeros simultaneously.

The condition ④ in Table 9-1 is related to the singular Sturm-Liouville

problem, and the condition ⑤ to the periodic Sturm-Liouville problem. In

contrast, the regular Sturm-Liouville problem is defined by the conditions ①,

② and ③.

■ Discretization of Sturm-Liouville equation. The following Sturm-

Liouville equation

(14)

can be discretized for the internal grids such that

(15)

where the central difference approximation over the interval shown in Figure 9-

3 is employed. By rearranging, we have

(16)

which is further standardized to represent a typical eigenvalue problem

(17)


9

Then, by utilizing the tridiagonal form of matrix , eigenvalues can be easily

found by the inverse power method.

Figure 9-3 Grid configuration for discretization of Sturm-Liouville equation

(Example 1) For a Sturm-Liouville problem , ,

, find the fundamental eigenvalue and the fundamental eigenfunction

by numerical treatment.

The given ODE belongs to the Sturm-Liouville problem and is characterized

by . Re-writing equation (16)

and inserting gives

(18)

where 5 uniform subintervals are used, i.e. . And the

boundary condition at can be discretized from equation

(8-47) such that

or

(19)


10

Also, the boundary condition at is simply written as

(20)

These two boundary conditions are inserted into equation (18), and the

following linear equations are obtained.

(21)

By defining the matrix

(22)

and by using the Fadeev-Leverrier method, the characteristic equation is derived

as (refer to the note ☑)

(23)

the solution of which gives

(24)

Indeed, for the given matrix , the same result can be obtained by the QR

iteration method also. However, the above method is frequently used only for

small matrices, and does not utilize the tridiagonal form. Therefore, it is

desirable to use the inverse power method and the eigenvalue shifting. In order

to exploit the inverse power method, by putting and

solving , we have


11

(25)

The above-obtained is again substituted into the right-hand side of

and is solved to give

(26)

This procedure is repeated to obtain the following result

iter 1/lambda f[1] f[2] f[3] f[4]

1

2

3

4

5

6

7

8

9

10

11

1.899266

1.696450

1.674655

1.672051

1.671722

1.671676

1.671668

1.671667

1.671667

1.671667

1.671667

0.917610

0.922965

0.926777

0.927945

0.928246

0.928319

0.928336

0.928340

0.928341

0.928341

0.928341

1.000000

1.000000

1.000000

1.000000

1.000000

1.000000

1.000000

1.000000

1.000000

1.000000

1.000000

0.874528

0.844323

0.837483

0.835947

0.835600

0.835521

0.835502

0.835498

0.835497

0.835497

0.835497

0.541195

0.485133

0.475924

0.474158

0.473787

0.473705

0.473687

0.473682

0.473682

0.473681

0.473681

from which the converged solution is found to be

(27)

The fundamental eigenvalue shows an error of 3.6%

compared with the exact value (see a note in ☑). And the

fundamental eigenfunction is shown as below [the boundary value is

determined by using equation (19)].


12

0.000000

0.628319

1.256637

1.884956

2.513274

3.141593

0.637443

0.928341

1.000000

0.835497

0.473681

0.000000

0.621057

0.921032

1.000000

0.839012

0.476699

0.000000

☑ The exact eigenvalue can be determined from and

the eigenfunction is determined to be where .

M#math to find the exact eigenvalue is as follows.

#> double ftn(x) = tan(x*pi)+x;

#> w = .bisect(0.51,0.99, ftn); lam = w^2; // eigenvalue

w = 0.78763729

lam = 0.62037251

#> bvp.x(0,pi) ( {y''}+{%y}=0, [ {y}-{y'}=0, {y}=0 ] ).togo(u = y);

ans = 0.62142265

#> x = (0,pi).span(6).tr;

#> [ x, u.tr.rowskip(-20), sin(w*(pi-x)), sin(w*(pi-x))/0.996293 ];

ans =

[ 0 0.617094 0.618755 0.621057 ]

[ 0.628319 0.916981 0.917618 0.921032 ]

[ 1.25664 0.996459 0.996293 1 ]

[ 1.88496 0.836422 0.835902 0.839012 ]

[ 2.51327 0.475338 0.474932 0.476699 ]

[ 3.14159 0 0 0 ]

■ Use of generalized differential operator. Here, the following generalized

differential operator

(28)

or the linearized differential operator as in equation (8-65)


13

(29)

is employed to express the eigenvalue problem as

(30)

Meanwhile, the differential operator is thought to be discretized to be

(31)

as was discussed in Chapter 8. Therefore, the eigenvalue problem

(32)

can be discretized as above. At this point, it is observed that the term in the

right-hand side of equation (32) contributes nothing to the discretization of the

differential operator . The eigenvalues are obtained in sequence of smaller

magnitudes by using the inverse power method and eigenvalue shifting.

For example, the Sturm-Liouville equation can be expanded and modified

as

(33)

where the coefficients of the differential operator are defined to be

(34)

★(Example 2) The Bessel equation is

transformed into the Sturm-Liouville problem as below

For the case of and , find the three smallest eigenvalues.

The boundary condition is given as . Use the form for

numerical solution.


14

Re-writing the equation in the form with , we have

(35)

By the way, it is actually unnecessary to specify the boundary condition at

due to the nature of singular Sturm-Liouville problem. Indeed, in order

to exploit the compact form, it is required to specify the boundary

condition at . Therefore, an additional condition is imposed

considering the symmetry (in reality, corresponds to the center of the

cylinder). M#math is written as below

#> double eiglam(s) {

Lam = bvp.x[201](0,1) ( {y''} +1/x*{y'}-1/x^2*{y}+{%y} = 0,

[ {y'}=0, {y}=0 ] ) .shift(s);;

return sqrt(Lam);

}

#> eiglam(0); eiglam.Lam = 14.681712 ans = 3.8316723



then the result is

1

2

3

3.831672

7.015327

10.172594

3.831706

7.015587

10.173468

It can be seen that use of 200 subintervals gives very accurate eigenvalues.

☑ The exact eigenvalues are obtained from the following M#math.

#> solve.x ( .J_1(x) = 0 ).span(0.01,11);

ans =

[ 3.83171 -1.40363e-008 ]

[ 7.01559 4.56504e-009 ]

[ 10.1735 3.24155e-008 ]


15

Section 9-3 Eigenvalues on Stability and Convergence Rate

In contrast to the situations where the main objective is to find eigenvalues,

it also happens that eigenvalues are not required to be sought even though they

play an underlying serious effect. This can be illustrated with stiff differential

equations and iteration methods on matrix equations.

■ Eigenvalues associated with initial value problems. It was already

discussed in section 7-8 that the stiffness of the differential equations is

determined from the eigenvalues of the Jacobian matrix given in equation (7-

91). In solving initial value problems, eigenvalues of the Jacobian matrix have

the following nature.

◦ greatest eigenvalue determines step sizes due to stability

◦ smallest eigenvalue determines the number of steps for integration

Therefore, there occurs a severe difficulty if a ratio of the greatest to smallest

eigenvalues differ by several orders of magnitudes. An example is presented

below to discuss this behavior.

(Example 3) When a numerical solution to the following coupled linear

equations is obtained by the Runge-Kutta method with ,

discuss the numerical stability.

First, the coupled linear equations are written in the following matrix form

(36)

and M#math is written as

#> double y1e(x) = 2*exp(-2*x)-exp(-1000*x);

#> double y2e(x) = -exp(-2*x)+exp(-1000*x);

#> matrix odesol(double h) {


16

x = (0,h).march(6);

return ode[x] ( y1' = 996*y1+1996*y2, y2' = -998*y1-1998*y2,

y1 = 1, y2 = 0 )

.return( x, y1/y1e(x), y2/y2e(x) );

}

#> odesol(0.01);

ans =

[ 0 1 -1.#IND ]

[ 0.01 343.05 685.116 ]

[ 0.02 -233985 -467970 ]

[ 0.03 1.60066e+008 3.20132e+008 ]

[ 0.04 -1.09499e+011 -2.18997e+011 ]

[ 0.05 7.49062e+013 1.49812e+014 ]

#> odesol(0.05);

ans =

[ 0 1 -1.#IND ]

[ 0.005 9.28357 17.6239 ]

[ 0.01 -135.193 -271.393 ]

[ 0.015 2248.68 4496.36 ]

[ 0.02 -37094.6 -74190.3 ]

[ 0.025 612226 1.22445e+006 ]

#> odesol(0.001);

ans =

[ 0 1 -1.#IND ]

[ 0.001 1.00153 1.00396 ]

[ 0.002 1.00099 1.00213 ]

[ 0.003 1.00052 1.00107 ]

[ 0.004 1.00025 1.00051 ]

[ 0.005 1.00011 1.00023 ]

From the above results, it can be seen that solutions with

diverge, whereas a considerably accurate solution is obtained with the smallest

step size of . The reason why these results are appearing is that the

eigenvalues associated with the corresponding matrix

(37)

differ in two orders of magnitude, i.e. .

The greatest eigenvalue limits the step size through the

Euler's stability condition


17

(38)

The convergence of numerical solutions depends on whether the stability

condition is satisfied or not.

☑ The exact solution to (Example 3) is given below.

A proper use of M#math is utilizing Spoke '.stiff(tol=0.01)'

#> ode.x[1000](0,1) ( y1' = 996*y1+1996*y2, y2' = -998*y1-1998*y2,

y1=1, y2=0 ) .stiff .return(x,y1,y2).skip(10).plot;

■ Eigenvalues occurring in iteration methods. When the Gauss-Seidel

iteration method is used to solve coupled linear equations, the diagonal

dominance should be determined a priori as was discussed in section 3-6. The

convergence characteristics of iteration methods is greatly affected by

distribution of the eigenvalues of associated matrix.

(Example 4) Solve the following coupled linear equation by the

Gauss-Seidel iteration method until the error falls below .

The diagonal elements of the given matrix are greater than the off-diagonal

elements, and therefore the diagonal dominance is satisfied. This implies that

solution can be obtained by iteration. To apply the Gauss-Seidel iteration

method, the given equations are modified as

(39)


18

By initializing and performing the first iteration, we have

(40)

Subsequent iterations can be performed by using the following M#math

#> (a,b,c,d) = (3.2,3,5,5.2);; (x,y) = (0,0);; a=3.2;

#> for.i(1,1000) {

r1 = 7-b*y;; // a*x + b*y = 7

r2 = -2-c*x;; // c*x + d*y = -2

x = r1/a;;

y = r2/d;;

err = |7-a*x-b*y|+|-2-c*x-d*y| ;;

[ i, x, y, err ];

if( err < 1.e-7) break; // 354 iterations are required

}

ans = [ 1 2.1875 -0.384615 12.0913 ]

ans = [ 2 2.54808 -2.48798 8.11298 ]

ans = [ 3 4.51998 -2.83469 10.8997 ]

...

ans = [ 353 25.8537 -25.2439 1.41759e-007 ]

ans = [ 354 25.8537 -25.2439 9.51168e-008 ]

Then, a final solution can be obtained by a total of 354 iterations. Even though

the given matrix is just a simple matrix, it requires more than 300

iterations, the reason of which stems from the underlying effect of the

eigenvalue of the given matrix. This will be discussed in detail below.

■ Role of eigenvalues in iteration methods. In order to examine the

iteration method from the standpoint of eigenvalues, matrices , are

introduced as

(41)

then, the iteration given in equation (39) can be written as

(42)


19

where is given as an initial guess at the start of iteration. Meanwhile, the

error at the th iteration

(43)

is defined, and equation (42) is recast into

(44)

or

(45)

By utilizing the above expression, the error at the th iteration becomes

(46)

or

(47)

If the matrix is selected to be , then . This means that the

solution is obtained by the direct method since iteration is not required.

Otherwise the error at the -th itration is calculated recursively from equation

(47) such that

(48)

where the matrix is a key factor determining the convergence rate.

In the case of (Example 4), the matrix is

(49)


20

the eigenvalues of which are and their

absolute values are very close to the unity (see the note in ☑).

If the absolute value of eigenvalues of matrix exceeds the unity, the

error diverges as iterations continue. In other words, by letting

be the absolute eigenvalue close to the unity, the magnitude of error

diminishes as

(50)

Therefore, convergence is very slow when , whereas convergence is

considerably accelerated when . For example, for the following case

(51)

the matrix is calculated to be

(52)

the eigenvalues of which are . As a result, the error diminishes rapidly

as iterations continue. Actual computation shows that

iter x y residue

1

2

3

4

5

6

7

8

0.233333

0.237333

0.239667

0.239707

0.239730

0.239730

0.239731

0.239731

-0.040000

-0.063333

-0.064733

-0.063967

-0.063971

-0.063973

-0.063973

-0.063973

1.286667

0.090000

0.012867

0.000900

0.000129

0.000009

0.000001

0.000000

where the rate of residue decreasing is about a factor multiplied by ,

i.e. a decreasing rate of one-tenth per iteration is found.

☑ M#math for equation (52)

#> I = .I(2); A = [ 30,3; 5,50 ]; B = [ 30; 0,50 ];

#> (I-A*B.inv).eig;


21

ans =

[ 0.1 ]

[ -0.1 ]

Section 9-4 Exercises ===========================

【1】For an interval of , the Sturm-Liouville problem is given as

. Find the fundamental eigenvalue for the following

homeogenous boundary conditions.

(a)

(b)

(c)

【2】For an interval of , the Sturm-Liouville equation known as the

Bessel function is given as

Find the fundamental eigenvalue for the following boundary conditions

(a)

(b)

(c)

【3】When an axial load of is applied, the curvature of the slender rod can

be determined from

where . The final result becomes an eigenvalue problem written as

For the two cases shown in the accompanying figure, find the axial road

where values are given as .


22

【4】An axisymmetric vibration of a circular membrane is governed by the

following differential equation

For the case of , find the fundamental eigenvalue.

【5】The most well-known Sturm-Liouville problems are as follows

(a) Bessel equation

(b) Legendre equation

(c) Tchebyshev equation

(d) Hermite equation


23

(e) Laguerre equation

//----------------------------------------------------------------------

// end of Lectures

//----------------------------------------------------------------------

Section 9-5 Higher-order ODE and Eigenvalue Problems

(optional)

Eigenvalue problems associated with higher-order ODEs can be

solved in a similar way. In this section, eigenvalue related to the bending and

vibration of a beam is considered which is characterized by the fourth-order

ODEs.

■ Bending of a beam . When a distributed load is applied on a beam,

the vertical displacement of the beam is governed by the following fourth-order

differential equation

(53)

where is the modulus of elasticity, is the moment of inertia of the cross

section. Especially, the product of is termed the flexural rigidity. The

boundary conditions for the beam shown in Figure 9-4 are

(54)

(55)

By converting the ODE in equation (53) into the form , we have

(56)


24

where in general.

Figure 9-4 Bending of a beam with a fixed end

☑ However, the boundary condition at takes a different form other than

equation (55), depending on the state in which the right end of the beam is

located.

(Example 5) Find the displacement of a beam when a distributed load

is applied to the beam. Here, boundary conditions are given

as , and ,

(units are omitted for convenience). Find a numerical solution with 100

subintervals.

Since is a constant, solution can be found easily by using equation (53)

with the following program

#> double ye(x) {

L = 8;

Lpi = L/pi;

kEI = 0.001;

a2 = 0.5*L/pi^3;

return kEI*( Lpi^3*(Lpi*sin(x/Lpi)-x) + a2*x^2*(3*L-x) );

}

#> kEI = 0.001; L = 8;

#> bvp.x(0,L) ( {y''''}-kEI*sin(pi*x/L) = 0,

[ {y}=0, {y'}=0, {y}=0, {y''}=0 ] )

.return( x, ye(x), y/ye(x) ) .rowskip(-10);

A use of 100 subintervals produces a fairly exact solution as below.


25

kEI = 0.001

L = 8

ans =

[ 0 0 1.#INF ]

[ 0.8 0.00169925 0.999101 ]

[ 1.6 0.00569333 0.999442 ]

[ 2.4 0.0104385 0.999555 ]

[ 3.2 0.0146278 0.999611 ]

[ 4 0.0172803 0.999643 ]

[ 4.8 0.0177982 0.999663 ]

[ 5.6 0.0159868 0.999675 ]

[ 6.4 0.0120342 0.999684 ]

[ 7.2 0.00645494 0.999691 ]

[ 8 0 1.#INF ]

☑ The exact solution is given as

■ Vibration of beam. Now, the dynamic behavior related to the vibration of

a beam is expressed by the following partial differential equation

(57)

where is the distributed load on the beam, is the mass per length.

Consider the case where the displacement is given as

(58)

from which equation (57) becomes ( )

(59)

If were constants, the exact solution to the above equation can be found

to be


26

(60)

from the boundary conditions [equations (54) and (55)].

(Example 6) For the case of , find the smallest three

eigenvalues for the eigenvalue problem given in equation (59). Numerical

solutions should be obtained with from 10 to 100 subintervals by adding 10

subintervals after each run. Find the effect of grid density on the eigenvalue

obtained numerically.

Exact solutions are found to be , and

(see the note ☑ below). Since is a constant, the form

of differential operator is very simple. M#math is written as

#> L = 8; EIm = 1000; s = 0; // change s for other eigenvalues

#> row = [];

#> for.nx(10,100, 10) {

Lam = bvp.x[nx+1](0,L) ( -EIm*{f''''}+{%f} = 0,

[ {f} = 0, {f'} = 0, {f''}=0, {f'''}=0 ] ).shift(s);

row _= [ nx, sqrt(Lam)];; // vertical concatenation

}

#> row;

row =

[ 10 1.75116 ]

[ 20 1.74066 ]

[ 30 1.73877 ]

[ 40 1.73812 ]

[ 50 1.73782 ]

[ 60 1.73765 ]

[ 70 1.73755 ]

[ 80 1.73749 ]

[ 90 1.73745 ]

[ 100 1.73742 ]

Comparing with the exact solution , the result corresponding to

100 subintervals gives a consistency up to 4 significant digits. Finding other

eigenvalues can be carried out by changing the shifintg value in the above

M#math code.


27

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Chapter 9 Eigenvalue Problems - msharpmath€¦ · [101] 009 Numerical Analysis Using M#math, 1 "msharpmath" revised on 2014.09.01 Numerical Analysis Using M#math Chapter 9 Eigenvalue