Difference schemes of high order of accuracy for ordinary differential equations with a regular singularity

DIFFERENCE SCHEMES OF HIGH ORDER OF ACCURACY FOR ORDINARY DIFFERENTIALEQUATIONS WITH A REGULAR SINGULARITY* G.I. BAGMUT Moscow 19 June

(Received

THIS paper studies difference boundary value problem Lzu = -f(x), and the Sturm-Liouville

1967:

revised

version 25 September

1968)

schemes of a high order of accuracy for the

lu(O)l -=c009

O
u(1) = G

(1)

problem

Lqu + hu = 0,

lu(O)( <

00,

u(l) = 0,

(2)

with the operator

go > 0, r(x) is a piecewise

continuous function,

06 r(xl 6 c,.

The use of the net method is based on the replacement of the differential equation by an approximate difference equation. The difference equation which replaces the original differential equation may be constructed by several methods. The simplest method consists of approximating the derivatives occurring in the equation by corresponding difference relations. Another method of constructing the difference equation consists of establishing a connection between the values of the unknown functions at several nodes of the net by combining these values represented by Taylor’s formula, using the differential equation. In this paper difference schemes of a high order of accuracy are constructed for problems (1) *Zh. vychisl.

Mat. mat. Fiz. 9, 1, 221-226,

300

1969.

Difference

schemes

301

of high order of accuracy

and (2) by the method developed in [l-21. The difference between this method and those mentioned above is that it enables a difference scheme of any order of accuracy to be constructed and imposes weaker requirements on the smoothness of the coefficients of the equation. 1. Boundary

value problem

It is known that a bounded solution of the homogeneous equation (1) can be represented

in the form u(z) = s’cp(s), cp(0) ;f 0, 1cp(x) 1 G positive root of the equation p(p - 1) -

PO =

C,

where p is a

0.

Any other solution, linearly independent of the bounded solution has a at zero. This remark enables the singularity u(z) = rP(z) / xfl-‘, q(O) P 0 condition of boundedness in problems (l), (2) to be replaced by the condition U(0) = 0. We will now construct a three point difference scheme giving an exact solution of problem (1) in the class of piecewise continuous coefficients. In the segment [O, 11 we consider the uniform net ah = {xi= ih, i = 0, 1,. . . , N; h = 1 /N}.

In the interval (xtl,

xi+ 1) we pass to a local coordinate

system

attached to the point xi, for which we put x = xi + sh, -1~ s 6 1. Equation (1) assumes the form

-

hzf’,

where U*(s) = u(xi + sh), and so on. Following 111, we obtain for i # 1 the relation

u* (s)

(Li(S, h)

-_ ___

u*(l)+

a(l, h) where ai(s, h) and pi(s,

L',Ui

=

0,

-!Y9 ___ h, Bi(-- 1, h)

u*(--l)+

h2yi(s,

h),

h) are solutions of the problems

ai(-

1, h)=

0,

XCdai

1, h)=

1,

i # 1;

(4)

302

G. I. Bagmut

L'qpi

0,

=

pi(l, h)=

dpf $1, h) =

0,

(4 - 1,

h’ y i(s, h) is the solution of the inhomogeneous equation (3) with the homogeneous boundary conditions ~(-1, h) = ~(1, h) = 0.

We construct a relation similar to (4) for the case i = 1. We define the function a, (s, h) as the solution of the homogeneous equation Lc* a, = 0 with the boundary conditions ai(-I, h) = 0, ~(4, h) = az(0, h). Using the remark about the behaviour at zero of the solution of the homogeneous equation (11, we arrive at the relation al(s, h)

= ~

d(s)

af(l,

u’(l)+

h2Yl (St h).

(6)

h)

Putting s = 0 in (4) and (61, we obtain a system of linear algebraic equations for the unknowns ylr yZ, . . . , yN_r, generated by the difference boundary problem which we will write in the form

(ay,) % -

;

Y -7 -

cp,

y;; = r.

YO=o,

Using Green’s formulas we find the following expressions of the exact scheme:

’

d*=D,[q*(s)I=~~_~~+~ds+~~

0

1

i ai=-,

‘pi

=

~~

[j*(s)]

i#

ai (0)

=

1;

a, =

(0) --i

(i+s)2

for the coefficients

ds

’

i #

i;

di = 0.50, [q’(s)];

-L i .j’(S)ai(s)ds+--ai

!f(s)Pi(s)

(7)

1

’ s

j’ (s) pi (3) ds.

Pi(O) o

Therefore, the solution of the difference boundary value problem (7) with coefficients defined by (8) is the same as the solution of the original problem (1) at the nodal points of the arbitrary uniform net ah. We consider the following approximate method of finding the coefficients of the scheme. The solution of problems (5) will be sought as a series of

(8)

Difference

schemes

303

of high order of accuracy

powers of Ia’:

a(~, h)=

;

cdR)(s)hzk,

jjfi”‘(s)hzk.

fi(s, h) =

k-0

h-0

Taking as pattern functions a(.~, h) and PCs, h), respectively,

P(m)(s,

r,

a(h)(s)h2k,

h) =

tp)(S,

h) =

k=O

i

the polynomials

w,

p’“‘(s)

A-O

we find by formulas (8) the coefficients a ( m), d m), q5( m, of a truncated scheme of rank m. For the coefficients of the expansion of the pattern functions it is easy to obtain the explicit expressions

2(s) =

Ti(S, -I),

a:O) (s) =

i # 1;

ai(k’(S) = dk)[r*(s)] =i

1 -

2’-W

21.1_l

(l+sP1

i # 1;

&*-‘)(+jt,

r*(t)Ti(S,

--1

$Ik’ (s)=

f$:h’[r*(s)]=

(l-ts)p) 2p

(,)-!?!? j,.*(t);;) (t)a;k-‘)(t)&a(;)(l) t

a:“‘(s) _a(01,(1)

F:)(S)=Ti(1,

S),

Ti(U, U)=

s I

d”’(S)=

r*(t) pi(O’(t)ai(‘-‘jt)dt,

Si*‘[r*

1

(s)] =

(i+u)p

-

2p-1

(9);

-1

t

(i+u)‘-’

_

S

r* (t)Ti (t, S)$-‘)(t)dt_

sti+u)P (i+u)p-t

>’

Using the apparatus of Green’s difference functions and elementary estimates of the pattern functions and polynomials [l, 21, we conf?rm the validity of the following theorem.

Theorem 1 A truncated difference has order of accuracy

scheme of the m-th rank, if h < Iho= v [ (2~ + 1) / 4co]

(2m+ 2) in the class of piecewise continuous coefficients

of equation (l), so that the inequality

IIu- y(m)~(~ G MF”z+~,

is always satisfied,

where M is a positive constant independent of the step h of the net.

Let u* ‘m’ fs) be the completion of the net function, defined as follows

the constant M does not depend on the netThe proof follows from the comparison of t6*‘d kd with (4) and (61. Therefore, the proposed methodof ~nte~olation at two points enables a solution to be found at my pttint of the segment 16, 11 with the same depee of accuracy in Izas at the nodes of the net 133. Renaark. The restriction h & h, on the step of the net arose frumthe requirementof ~undedness downwardsof Pltrnf(s, h) for na3 1; with the obviws generalization of the scheme to the case of a non-uniformnet the restriction etionds only to the first step of the net h, I x, < h,. 2. The Sturm-Liouville

problem

We will first of all explain the conditions for which an exact scheme exists for problem (21. After transformationby the shiti x = xi + staequation (2) assumes the form

The pattern functions are defined by K9. We choose the step h of the net of such a value that the distance 6 between two successive zeros of any solution

Difference

schemes

305

of high order of accuracy

of equation (1) will satisfy the inequality 6 > 2. This will be satisfied (see [41, p. 134-139),

if hZ(h -

9)

< 1.

(11)

The initial conditions for the pattern functions together with (11) ensure the inequalities u(s, h) >Ofor

--l
If condition (11) is satisfied, as for problem (1).

1,

fi(s, h) > 0 for --1 G s < 1.

exact scheme is constructed in the same way

Therefore, we arrive at the difference problem

(~Y&-~Y+~py=*, Yo = y,= 0, 9

where the coefficients

ai, di aredetermined from (8), pi

= Fi RI.

The coefficients

of the truncated scheme of rank m aredetermined in the same way as for problem (l), onlythe expressions

for the coefficients

of the expansion of the pattern

functions are changed:

a:) (s)

=

Rik)[r*

(S)-

A],

By’ (s) =

Sy’ [r*(s) -

A].

Each of the functions a ( k, (s) and /3 ( k, (s) is a polynomial of degree k in A. Therefore, the truncated schemes of rank m, with the exception of the scheme of rank zero, are non-linear with respect to the eigenvalues Ah, which complicates the discovery of a solution. The following recurrence process of linearization

L51is

used to overcome this difficulty.

The problem is first solved for the linear

scheme of zero rank. We then find the solution for the scheme of the first rank, in the coefficient substituted.

of which the eigenvalue found in the preceding scheme is

Continuing the process,

after m steps we find a condition for the

scheme of rank m the coefficients of which are calculated by using the eigenvalue found in the linear scheme of rank (m - 1). For the solvability of the truncated difference schemes it is important to clarify the question of the underboundedness of the pattern polynomials. It is easy to prove the validity of the following lemma.

Lemma With the restrictions

h’(X - r*) < 1 on the step of the net, for h >/co, 2h’c, < 1,

306

G. 1. Bagmut

if h < c, the following inequalities are satisfied: C’(S) > (1 + s)cos(o.5S*+ 2s + 1.5),

I# i,

Ey(s) 2 -ti+w

--i
2P

drn’(s) > (1 -

co91.5,

s)cos(0.532- 2s + i.5),

--i
OGS
Therefore, for A & c, the sufficient conditions of solvability of the exact scheme and of the truncated schemes of rank m arethe same. The satisfaction of these conditions is attained by the choice of a sufficiently fine net h 6 h,, where h, * l/k depends on the ordinal numberk of the eigenvalues. The following theorems are proved by a slight change of the methodof working [61. Theorem 2 If the difference scheme (12) has zero rank, the solution (A,h, y,) of the problem

W) converges at the rate 00~9 to the corresponding solution (A,, U”(X)) of problem (2). normalized as follows: (pT, una)= 1, where P,,, is the coefficient of the exact scheme. The order of accuracy of the recurrentprocess of linearization described above is established. Theorem 3 If the coefficients of the truncated scheme of rank m depend on h(h*m-l) the eigenvalue of the scheme of rank m - 1, the uniformestimate

is valid if h& ho, is small enough where the constant M is independentof the choice of the net. Remark.

In Theorems 2, 3 the solutions of the difference problems converge

Difference

schemes

of high order

307

of accuracy

to the exact solution normalized to the special form: (p,, u’) = 1. Let the problem be posed of finding the eigenfunction satisfying the normalization condition

s

u2(z)dz = i,

0

then the statements of Theorems 2, 3 are preserved, if the completion of the net function y (m) @m--L))

p(m)(s, --__,

Pm)(i,

h(h.m-0)

U*(m) (8) =

Q-)(8, w---l))

!I:,“,‘+. Pcm)(q

(_)

i #

I/i-l,

i,

A(h.rn-I))

P(mI(.?, w--I)) P(“)(l,

as follows from section 1,

(m)

,Y2

I

I =

i,

p.m-I))

is normalized as follows: # _ (u’“)(z))*

c Q In conclusion

dz =

1.

some results of a numerical experiment to solve the problem 0.75 u” - Zt u + ku = 0,

u(O)=

u(i)=

0

by schemes of the second, fourth and sixth orders of accuracy are presented. The exact solution has the form

where J, and J, are Bessel functions of the first kind, and /.L,,are the roots of the equation J,$,) = 0. For the third eigenvalue A, = 103.49945 on the net h = 0.1 for schemes of the second, fourth and sixth orders of accuracy the values Athpo) = 96.1, Athpl) = 104.3, h(h,2) = 103.466, and on the net h = 0.05 we have correspondingly, AthPo) = 101.6, Ath,‘) = 103.512, and Athe’) = 103.499. The discussion has been carried out for a boundary value problem of the first kind. However, the method used in this paper enables the results obtained to be extended to problems with other boundary conditions.

308

G. I. Bagmut

I take this opportunity to thank A. A. Samarskii for supervision and assistance, and A. A. Abramova and V. B. Andreeva for a number of editorial comments. Translated

by J. Berry

REFERENCES 1.

TIKHONOV, A. N. and SAMARSKII, A. A. Homogeneous difference schemes of a high order of accuracy on non-uniform nets. Zh. vj%hisl. Mat. mat. Fiz. 1,3, 425440. 1961.

2.

TIKHONOV, A. N. and SAMARSKII, A. A. Homogeneous difference schemes of a high order of accuracy on non-uniform nets. Zh. vjkhisl. Mat. mat. Fiz. 3, 1, 99-108, 1963,

3.

BAGMUT,

G. I.

4, 873-878,

Interpolation 1968.

for special

functions.

equation (Differentsial’nye

Zh. v3hisZ.

Mat. mat. Fiz. 8,

4.

TRICOMI, F. G. Differential Moscow, 1962.

5.

PRIKASCHIKOV, V. G. Homogeneous difference schemes of the fourth order of accuracy for the Sturm-Liouville problem. In: Computing Methods and Programming Wychisl. metody i programmirovanie), Izd-vo MGU, Moscow, ‘1965. 232-236.

6.

TIKHONOV,

A. N. and SAMARSKII,

Zh. vychisl.

A. A.

uravneniya),

The Sturm-Liouville 1962.

Mat. mat. Fiz. 2, 5, 784-805.

Izd-vo

difference

in. lit.,

problem.