Two new algorithms for discrete boundary value problems
Journal of Applied Mathematics and Stochastic Analysis
Two New Algorithms for Discrete Boundary Value Problems* Ravi P. Agarwal and Tara R. Nanda
Ravi P. Agarwal 0
Tara R. Nanda 0
0 AMS Subject Classification: 65Q05 , 65L10
We propose two new methods of constructing the solutions of linear multipoint discrete boundary value problems. These methods are applied to solve some continuous twopoint boundary value problems which are known to be numerically unstable. The purpose of this paper is to provide two new algorithms to compute the solution of the linear discrete system u(k + 1) = A(k)u(k) + b(k), k e N(k1, kr) r where A(k) is a given nonsingular n x n matrix with elements aqp(k), 1 q, p< n; b(k) is a given n x 1 vector with components bq(k), 1 <_ q n; u(k) is an finknown n x 1 vector with components uq(k), 1 < q _< n; 0 _< k 1 < k2 < ...< kr (r > 2 ) where each k is a positive integer, N(k1, kr) is the discrete interval [k1, kl+ 1,...,kr]; Oqp ,tq, 1 <q,p<n, 1 _< i <_ r are given constants.
Adjoint Identities; Discrete Systems; Multipoint Boundary Value Problems

1.
INTRODUCTION
satisfying the multipoint boundary conditions
(1.1)
(1.2)
1 _< i _< r _< (2 _< r _< n),
Motivated by the work of Angel and Kalaba [
4
] on twopoint boundary value
problems for difference equations, recently in [
1, 3, 7, 12
], we have discussed several
new methods of constructing the solutions of linear as well as nonlinear multipoint discrete
boundary value problems. In this paper we shall develop adjoint identifies which are in
terms of solutions of (1.1) and its adjoint system. These identifies lead to the
backwardforward and forwardbackward methods, which seem to be new processes for computing
the solutions of (1.1) and (1.2). However, the application of adjoint equations to solve
discrete problems is not new, e.g., Clenshaw [6] used adjoint difference equations to sum
the Chebyshev series. To demonstrate the usefulness of the proposed methods we solve
some continuous twopoint boundary value problems which are known to be unstable.
BACKWARDFORWARD AND FORWARDBACKWARD METHODS
The adjoint system of the difference system (1.1) is defined as
v(k) = AT(k) v(k + I), k
N(kI, kr)
(2.1)
where AT(k) is an n x n matrix with elements aqp(k), 1 _< p, q < n and v(k) is an n x 1
vector with components vq(k), 1 < q _< n.
obtain
We multiply the qth equation of (1.1) by vq(k + 1) and sum over all n equations to
n n n n
Z uq(k + l)vq(k + I) = Z vq(k + I) Z aqp(k)up(k) + Z bq(k)vq(k + I).
q=l q=l p=l q=l
Next we multiply the qth equation of (2.1) by uq(k) and sum over all n equations to get
n
ql
n
ql
n
11
uq(k)vq(k)=
uq(k)
aqp(k)vp(k+ 1).
On subtracting (2.3) from (2.2), we find
Let k0 e N(k1, kr) be fixed. In (2.4) letting k = t and summing from k0 to k 1
results in
n n
Z [uq(k + 1)vq(k + 1) uq(k)vq(k)] = Z bq(k)vq(k + 1).
q=l q=l
n k n
Z [uq(k)vq(k) uq(kOlVq(kO)] = Y. Z Vq(t) bq(t 11,
q=l t=k0+l q=l
for all k0 <_ k
N(kl, kr)
(2.2)
(2.3)
(2.4)
N(k1, kr)
(2.5/
and, similarly
n
Z [uq(k)vq(k) uq(k0)vq(k0)]
ql
n
X X Vq(t) bq(t
t=k+1 q=l
I),
(2.6)
for all k0>_k
N(kl, kr).
p(i) (ki)
Vq
q
Equations (2.5) and (2.6) will be referred to as adjoint identities. We compute backward
solutions once for each uq(ki), 2 _< i _< r appearing in (1.2) with the conditions
into the adjoint identity (2.5) with k0 = k1, we obtain
where v(i) (ki) is the qth component at k for the pth backward solution. Substituting (2.7)
n
10;tN uq(ki).
n
q=l
vqp(i) (kl) uq(kl) =
ki
t=kl+1
n
ql
vpq(i) (t)bq(t
1), 2 < i _< r.
(2.8)
2 _<i_< r, 1 < p, q _< n
(2.7)
Summing (r 1) equations (2.8) and making use of (1.2), we get
n r r ki
X [xt + X V(i) (kl)]uq(kl)=tp X X
q=l i=2 i=2 t=kl+l
q=l
vPq(i)(t)bq(t 1),
l_<p_<n.
(2.9)
r
i2
ki
Z
t=k+l
If the matrix [:ilN +
v(i) (kl)] is nonsingular, then the system (2.9) provides
the unknowns uq(kl), 1 _< q < n. The solution of the problem (1.1) and (1.2) is obtained
by computing the solution of (1.1) with these values of uq(kl), 1 < q < n. However, to
evaluate the summation term in (2.9) we need to store the solutions of (2.1). This can be
avoided at the cost of solving another (r 1) system. For this, we denote
Wp(i)(k) =
Zv(i)(t)bq(t 1); l<p<n, 2<i<r,
q=l
n
Wp(i)(k)= E v.,t.tj)(k + 1) bq(k) + Wp(i)(k + 1)
q=l
Wp(i)(ki) = 0; 1 _< p
n, 2
i _< r.
(2.10)
(2.11)
(2.12)
Thus, at the point ki, 2 < i _< r, we solve a system of order 2n given by (2.1) and (2.10)
subject to the conditions (2.7) and (2.11).
With this adjustment system (2.9) takes the form
X [orq + X vqi) (kl)] uq(k1) = tp + X Wp(i)(kl), 1 <_ p _< n.
n n r
q=l i=2 i=2
This method of constructing the solution of (1.1) and (1.2) is called the
backwardforward process and requires (r 1)n backward solutions of the adjoint system (2.1)
satisfying (2.7), (r 1) backward solutions of (2.10) satisfying (2.11), and 1 forward
solution of (1.1) with the obtained values of u(k 1), 1 _< q _< n, from the system (2.12),
i.e., a total of (r 1) (n + 1) + 1 solutions of ntfi order systems. In particular, if r=2 then
we need (n + 2) solutions. Similar to the backwardforward process we have the
forwardbackward process. For this we solve (2.1) forward once for each uq(ki), 1 < i < r 1,
appearing in (1.2) with the conditions
v i)
(ki) = lXl
l<i<rl,l<p,q_<n,
(2.13)
where v)P. (ki) is t_he qth component at ki for the pth forward solution.
Substituting (2.13) into the adjoint identity (2.6) with k0 = kr, we obtain
n
Y. O;pq
uq(ki)q=l
n
q=l
,,ti)
vt? (kr) uq(kr) =
kr
t=ki+l
q=l
vP_q(i) (t)bq(t
1),
Summing (r 1) equations (2.14) and making use of (1.2), we get
l_<i<r 1.
(2.14)
n
q=l
(2.15)
We introduce
W.p(i)(k) =
t=ki+ 1
q=l
v
(t)bq(t
1); 1 _< p _< n,1
i _< r 1,
n
q=l
W.p(i)(k) =
vqp(i) (k + 1) bq(k) + W.p(i)(k + 1)
W.p(i)(ki) = O; 1 < p <_ n, 1 <_ i _< r 1.
(2.16)
(2.17)
Thus, the system (2.15) is the same as
n
E [a q+
r1 r1
Z v.p) (kr)] uq(kr) = tp + Z Wp(i)(kr), 1 <_ p < n.
i=l i=l
The solution of the problem (1.1) and (1.2) is obtained by solving backward
the system (1.1) with the obtained values of uq(kr), 1 < q < n from the system (2.18).
Next we shall consider the system (1.1) together with the implicit separated
conditions (1.3). We compute (n I]1) solutions of (2.1) backward with the
conditions
where Viq(Si)(ki) is the qth component at k for the sith backward solution.
Substituting (2.19) into (2.5) with k0 = k1 and using (1.3), we obtain
vSi)(ki) = cti(si),q; 2 < i < r, 1 < s _< [I
1 < q < n,
(2.19)
n i(si)
Z *q
q=l
(kl)Uq(kl) = ti,i(si)
ki
Z
t=kl +I
n
X viq(Si) (t)bq(t 1)"
1
2_<i <r, 1 < si< i.
(2.20)
We introduce
ki
Wi(si)(k) = Z
t=k+l
n "i(si)
Z "q
q=l
(t)bq(t I); 2<i_<r, 1
si_< i
Thus, the system (2.20) can be written as
n
viq(Si) (kl)Uq(kl) = 1.i,i(si) + Wi(si)(kl); 2
i <_ r, 1 < s _< i.
System (2.23) together with (1.3) for i = 1, i.e.,
n
O;l(Sl), q uq(kl) = tl,l(s1), 1 _< s < i,
forms a system of n equations in n unknowns uq(kl), 1 _< q <_ n. The solution of (1.1)
and (1.3) is obtained by solving forward the system (1.1) with these values of uq(kl),
l_<q_<n.
In practice we couple the adjoint system (2.1) with the equation (2.21) and solve
this system of (n + 1) equations from the point ki, 2 < i _< r to k1 with the conditions (2.19)
and (2.22).
Similarly, in the forwardbackward process for (1.1) and (1.3), the unknowns
uq(kr), 1 _< q _< n are computed from the system
n
qE=l vi(qSi) (kr)uq(kr) = ti,i(si) + wi(si)(kr); 1 _< i < r 1 1 < s < i,
n
Z (XrCsr) q uq(kr) = tr,rCsr), 1 < Sr< r,
ql
where vi,Si,t (k) is the qth component of the sith forward solution from the point k of the
adjoint system (2.1) satisfying
i(si) (ki) = Oq(si), q; 1 < i _< r 1, 1
V_q
s <_ li, 1 < q _< n
and w.i(si)(k) is the forward solution of the initial value problem
w.i(si)(k) =
n
vi(Si) (k + 1)bq(k) + w.i(si)(k + 1)
w.i(si)(ki) = O; 1 < i _< r 1, 1
s _< i.
(2.21)
(2.22)
(2.23)
(2.24)
(2.25)
(2.26)
(2.27)
(2.28)
The solution of (1.1) and (1.3) is obtained by solving backward the system (1.1)
with the obtained values of uq(kr), 1 _< q < r.
We shall construct an appproximate solution of the continuous boundary value problem
3.
NUMERICAL EXAMPLES
y"= f(t)y + g(t)
y(cc) = A, Y(I) = B
by employing the discrete variable method due to Nomerov. Let h =K+i? tk = cc + kh,
k N(O,K+I), fk = f(tk), gk = g(tk), and u(k) be the approximation to the true solution of
y(t) at tk, satisfies the second order difference equation,
( 1 + 1h2fk. 1)u(k 1) + (2 + 1i0h.2fk)u(k) + ( 1 + l_h2fk+ 1)u(k+ 1)
=
l?h22(gk1+ 10gk + gk+l), k
N(1,K)
together with the boundary conditions
u(0) = A, u(K+ 1) = B.
Theorem 3.1. (see [
12
])The discrete boundary value problem (3.2) and (3.3) has a unique
solution provided
(a) f(t) >_ 0 on [ct,13] and 1h22fk < 1, k
N(0,K+I); or
4sin 2K+I
max If(t)l < 1
a<t<l]
[
12
]
if l_h22 maax_f<(tt) < 1 then in system form the problem (3.2) and (3.3) can be written
as
c2(k) 1
2
u 1 (k + 1) = u2(k)
Ul (0) = A, u2(K) = B
(3.1)
(3.3)
(3.4)
(3.5)
d(k)  (k) x
Applying the backwardforward process for the boundary value problem (3.4) and (3.5),
we note that (2.1), (2.21), (2.19) and (2.22) reduce to
Further, the system (2.23) and (2.24) takes the form
which easily determines
v22(k) = v21(k + 1)+
c (k)
c2(k)
v22(k + l)
w2(k) = v22(k + 1) d(k) + w2(k + 1)
v(K) = O, v(K)= 1, w2(K)= 0.
v21(0)u1(0) + v22(0)u2(0) = B + w2(0)
u(0) =A,
UI(0) = A, u2(0) =
B + w2(0)
v](0)A
v(0)
The solution of (3.4) and (3.5) is obtained by recursing forward the system (S.4) with the
initial values (S.8).
Similarly, applying the forwardbackward process we f?md that (2.1), (2.28), (2.27)
and (2.29) reduce to
v.: (k + 1)= v.21(k)+c vll (k)
Cl(k]
1
* +
c2(k)
Vl
W.l(k + 1) = W.l(k) + v.2(k + 1)d(k)
Further, the system (2.25) and (2.26) becomes
v.11 (K)Ul (K) + v.2I(K)u2(K) = A + w. I(K)
u2(K) = B
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
Two New Algorithms for Discrete Boundary Value Problms: Agarwal & Nanda
which gives
u 1 (K) =
A + W.l(K) v.](K)B
vI (K)
u2(K) = B.
(3.1 I)
The solution of (3.4) and (3.5) is obtained by recursing backward the system (3.4) with the
final values (3.11).
2. y(t)
y,, = 2
t
both the methods discussed in this section work equally well. The errors obtained, as
1
with h = 25? are presented in Table 1.
calculated from the exact solution y(t) = 6 (19t 5t2
0 ) and approximate solution U l(k)
Example 3.2. For the discrete analogue (3.4) and (3.5) of the boundary value problem
y" = 400y; y(0) = 1, y(5) = e100,
we apply both the methods. The errors obtained, as calculated from the exact solution
5
y(t) = e20t and approximate solution Ul(k) with h = 1024? are presented in Table 2.
0.3125
0.6250
0.9375
1.2500
1.5625
1.8750
2.1875
2.5000
2.1825
3.1250
3.4375
3.7500
..........4,.0625
.,..,,,
4.6875
Example 3.3. The boundary value problem
y"= (2m + 1 + t2)y; y(0) = 13, y(oo) = 0,
(3.12)
where rn >_ 0 and 13 are known constants, is known as Holt?s problem. This problem is a
typical example wherein usual shooting methods fail [
2,5,7,911
]. Replacing the boundary
condition y(,,o) = 0 by y(T) = 0 (T finite) Holt [8] used f?mite dif,ference methods (however,
for rn = 0, 13 = 1, T = 12; m = 1, 13 = 1/2, T = 8; rn = 2, 13 = T = 8, the results are
unsatisfactory [
8,11
]), whereas Osborne [9] used a multiple shooting method and Roberts
and Shipman [
10
] used a multipoint approach. In [
2
] we have formulated a new shooting
method which gives accurate solutions of (3.12) for several different values of rn and 13 up
to T = 18. (This value of T has been chosen in view of restricted computer capabilities.)
For the same and several other values of rn and 13 accurate solutions of (3.12) up to T = 18
have also been obtained in [
5
]. Here the error estimates in the solution of (3.12) when
approximating y(.o) = 0 by an appropriate boundary condition at T are also available. For
,1
the discrete analogue (3.4) and (3.5) of (3.12) (replacing y(,o) = 0 by y(18) = 0) with rn =
0, = 1 and h = we apply both the methods of this section. The numerical solution
u 1 (k) is shown in Table 3.
7.0
$.0
iO.?O
?ii?.0?
i?.
13.0
14.0
15.0
16.0
17.0
18.0
Example 3.4. For the discrete analogue (3.4) and (3.5) of the boundary value problem
y" = (sin2t)y + cos2t; y( 1) = y(1) = 0,
1
we apply both the methods. The numerical solution Ul(k) for h = 54 is shown in Table 4.
t
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