On One Algorithm To Find a Solution to a Linear Two-Point Boundary Value Problem

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ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2021, Vol. 42, No. 3, pp. 606–612. c
Pleiades Publishi ng, Ltd., 2021.
On One Algorithm To Find a Solution
to a Linear Two-Point Boundary Value Problem
S. M. Temesheva1*, D. S. Dzhumabaev2, and S. S. Kabdrakhova1**
(Submitted by T. K. Yuldashev)
1Department of Differential Equations,
Institute of Mathematics and Mathematical Modeling, Al-Farabi Kazakh National University,
Almaty, Kazakhstan
2Department of Differential Equations, Institute of Mathematics and Mathematical Modeling,
Almaty, Kazakhstan
Received June 14, 2020; revised July20, 2020; accepted July 31, 2020
Abstract—A two-parameter family of algorithms for finding an approximate solution to a linear
two-point boundary value problem for a system of ordinary differential equations is offered. The
convergence conditions for the algorithms are obtained. The necessary and sufficient coefficient
conditions for the well-posedness of considered problem are established.
DOI: 10.1134/S1995080221030173
Keywords and phrases: boundary value problem, system of ordinary differential equations,
algorithms, parametrization method, necessary and sufficient conditions.
1. INTRODUCTION
Consider a linear two-point boundary value problem
dx
dt =A(t)x+f(t),x∈Rn,t∈(0,T),(1)
Bx(0) + Cx(T)=d, (2)
where A(t)and f(t)are continuous on [0,T],Band Care the given (n×n)matrices, dis a given n
vector, ||x|| =max
i=1:n|xi|,and||A(t)|| =max
i=1:n
n

j=1 |aij (t)|≤α,α=const.
Denote by C([0,T],Rn)the space of continuous functions x:[0,T]→Rnwith the norm ||x||0=
max
t∈[0,T ]||x(t)||.
Solution to problem (1), (2) is a function x∗(t)∈C([0,T],Rn), continuously differentiable on (0,T)
and satisfying the differential equation (1) and boundary condition (2).
Boundary value problems for ordinary differential equations are widely used in many application
areas [1–12]. To research and solve boundary value problems for ordinary differential equations there
used various methods of qualitative theory of differential equations and integral equations. The review
and bibliography can be found in [2–8, 10, 12]. Based on these methods, the conditions for the
solvability of boundary value problems were established, and the ways for finding the solutions were
suggested. Note that under some appropriate substitutions, the ordinary differential equations of higher
orders lead to a system of ordinary differential equations [2, 5, 10, 12]. The approximate and numerical
methods gain a particular importance for constructing the solutions to boundary value problems for
*E-mail: temeshevasvetlana@gmail.com
**E-mail: s_kabdrachova@mail.ru
606

ON ONE ALGORITHM TO FIND 607
equations and systems of ordinary differential equations [9, 11, 13]. This article is focused on the
establishment of coefficient criteria for the unique solvability of two-point boundary value problems for a
system of ordinary differential equations and on the construction of a constructive method for finding an
approximate solution to problem (1), (2).
In [8], to investigate and solve a problem (1), (2) there is suggested a parametrization method.
Interval [0,T]in this method is divided into equal parts with step h>0:Nh =T(N∈N).Thenpa-
rameters λr,r=1:N, and new unknown functions ur(t),t∈[(r−1)h, rh),r=1:N, are introduced,
and hereafter the initial problem is reduced to the equivalent problem with parameters. To find a pair
system (λr,u
r(t)),r=1:N, the solution of latter problem, there is offered a two-parameter family of
algorithms, where h>0and the number of repeated integrals νused for the construction of matrix
Qν(h):RnN →RnN are the numerical parameters of this family. Each algorithm’s step consists of two
items: a)finding of parameters λr,r=1:N, from the system of linear algebraic equations, which is
compiled on the initial data of problem (1), (2); b)finding of function ur(t), which is the solution of
Cauchy problem on interval [(r−1)h, rh)at the given value of parameter λrfor all r=1:N.
Present article offers an other family of algorithms of parametrization method. Herein each algo-
rithm’s step also consists of two items. The difference of offering algorithms is in item b),wherethe
unknown functions ur(t),t∈[(r−1)h, rh),r=1:N, are to be computed by the recurrent formula.
The results are illustrated by an example, implemented in MathCAD software.
2. MODIFICATION OF ALGORITHM THE PARAMETRIZATION METHOD
Take some step h>0:Nh =T(N∈N) and make a partition of interval [0,T)according to this
step:
[0,T)=
N

r=1
[(r−1)h, rh).
We will denote by C([0,T],h,RnN )the Banach space of function systems x[t]=x1(t),x
2(t),...,
xN(t)with the norm ||x[·]||1=max
r=1:Nsup
t∈[(r−1)h,rh)||xr(t)||,where xr:[(r−1)h, rh)→Rnis continu-
ousandhasafinite limit lim
t→rh−0xr(t)for all r=1:N.
Let xr(t):xr(t)=x(t),t∈[(r−1)h, rh),r=1:N, denote the restriction of function x(t)to the
r-th interval [(r−1)h, rh)and reduce the problem (1), (2) to the equivalent multi-point boundary value
problem
dxr
dt =A(t)xr+f(t),t∈[(r−1)h, rh),r=1:N, (3)
Bx1(0) + Clim
t→Nh−0xN(t)=d, (4)
lim
t→sh−0xs(t)=xs+1(sh),s=1:(N−1),(5)
where (5) are the conditions bonding the solution of problem (1), (2) at interior points of [0,T]-interval’s
partition.
Suppose that P(t)is a square matrix continuous on [(r−1)h, rh)or a vector of dimension nand it
has a finite limit lim
t→rh−0P(t),r=1:N. Takeanumericν∈Nand denote by Eν,r(A(·),P(·),t)the sum
t

(r−1)h
P(τ1)dτ1+
t

(r−1)h
A(τ1)
τ2

(r−1)h
P(τ2)dτ2dτ1+...
+
t

(r−1)h
A(τ1)...
τν−2

(r−1)h
A(τν−1)
τν−1

(r−1)h
P(τν)dτνdτν−1...dτ
1,
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608 TEMESHEVA et al.
t∈[(r−1)h, rh),r=1:N.
Sum Eν,r(A(·),P(·),t)is continuous on [(r−1)h, rh)and has a finite limit
lim
t→rh−0Eν,r(A(·),P(·),t)=Eν,r(A(·),P(·),rh)
for all ν∈N,r=1:N. It is evident that E∗,r (A(·),P(·),t) = lim
ν→∞Eν,r(A(·),P(·),t)is a sum of
uniformly convergent series on [(r−1)h, rh),and the sum is continuous on [(r−1)h, rh)and has a
finite limit
lim
t→rh−0E∗,r(A(·),P(·),t)=E∗,r(A(·),P(·),rh),r=1:N.
Let λrdenote the value of function xr(t)at point t=(r−1)h. Making a replacement ur(t)=
xr(t)−λr,r=1:Non [(r−1)h, rh),we obtain a multi-point boundary value problem with parameters
dur
dt =A(t)(λr+ur)+f(t),t∈[(r−1)h, rh),r=1:N, (6)
ur((r−1)h)=0,r=1:N, (7)
Bλ1+CλN+Clim
t→Nh−0uN(t)=d, (8)
λs+ lim
t→sh−0us(t)=λs+1,s=1:(N−1).(9)
Pair λ∗,u
∗[t]with the elements λ∗=λ∗
1,λ
∗
2,...,λ
∗
N∈RnN ,u∗[t]=u∗
1(t),u
∗
2(t),...,u
∗
N(t)∈
C([0,T],h,RnN )is a solution to problem (6)–(9). Here the function u∗
r(t)isasolutiontoCauchy
problem (6), (7) with λr=λ∗
r,r=1:Nand, for λ∗
r,lim
t→rh−0u∗
r(t),r=1:N, equalities (8), (9) hold.
Problems (1), (2) and (6)–(9) are equivalent. If x∗(t)is a solution to problem (1), (2), then the
pair (λ∗,u
∗[t]) is a solution to problem (6)–(9). Here λ∗=(λ∗
1,λ
∗
2,...,λ
∗
N)∈RnN ,λ∗
r=x∗
r((r−
1)h),u∗[t]=(u∗
1(t),u
∗
2(t),...,u
∗
N(t)),u∗
r(t)=x∗
r(t)−x∗
r((r−1)h),t∈[(r−1)h, rh),andx∗
r(t)is a
restriction of function x∗(t)to [(r−1)h, rh),r=1:N. Conversely, if the pair (
λ, u[t]) with components

λ=(

λ1,
λ2,...,
λN)∈RnN ,u[t]=(u1(t),u2(t),...,uN(t)) is a solution to problem (6)–(9), then the
function x(t)defined by equalities
x(t)=
λr+ur(t),t∈[(r−1)h, rh),r=1:N, and x(T)=
λN+ lim
t→Nh−0uN(t)
is a solution to problem (1), (2). Problem with parameters differs from the boundary value problem
(3)–(5) by the presence of initial conditions (7) for the new unknown functions.
For a fixed value of the parameter λr,r=1:N, Cauchy problem (6), (7) is equivalent to the Volterra
integral equation [14] of second kind
ur(t)=
t

(r−1)h
A(τ)(λr+ur(τ))dτ +
t

(r−1)h
f(τ)dτ, t ∈[(r−1)h, rh),r=1:N. (10)
Substituting the right-hand side of (10) instead of ur(τ)and repeating the process ν(ν∈N)times, we
obtain the following presentment of function ur(t):
ur(t)=Dν,r(t)λr+Fν,r (t)+Gν,r(ur,t),t∈[(r−1)h, rh),r=1:N, (11)
where Dν,r(t)=Eν,r (A(·),A(·),t),Fν,r (t)=Eν,r(A(·),f(·),t),and
Gν,r(ur,t)=
t

(r−1)h
A(τ1)...
τν−1

(r−1)h
A(τν)ur(τν)dτν...dτ
1,t∈[(r−1)h, rh),r=1:N.
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ON ONE ALGORITHM TO FIND 609
Determine lim
t→rh−0ur(t),r=1:Nfrom formula (11). Substituting the appropriate expressions into
(8), (9) pre-multiplying (8) by h>0:Nh =T, we obtain a system of linear algebraic equations with
respect to parameters:
Qν(h)λ=−Fν(h)−Gν(u, h),λ∈RnN,
where
Qν(h)=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
hB O O ... O hC(I+Dν,N (Nh))
I+Dν,1(h)−I O ... O O
OI+Dν,2(2h)−I ... O O
... ... ... ... ... ...
O O O ... −IO
O O O ... I +Dν,N −1((N−1)h)−I
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
I:Rn→Rnis the identity matrix, O:Rn→Rnis a zero matrix,
Fν(h)=−hd +hCFν,N (Nh),F
ν,1(h),...,F
ν,N−1((N−1)h)∈RnN ,
Gν(u, h)=hCGν,N (uN,Nh),G
ν,1(u1,h),...,G
ν,N−1(uN−1,(N−1)h)∈RnN .
Find (λ, u[t]), the solution to the multi-point boundary value problem with parameters (6)–(9), where
λ=(λ1,λ
2,...,λ
N)∈RnN ,u[t]=(u1(t),u
2(t),...,u
N(t)), according to the next algorithm. Suppose
that given ν,hthe matrix Qν(h):RnN →RnN has an inverse one.
Step 0. a)Find the initial approximation on parameter λ(0) =(λ(0)
1,λ
(0)
2,..., λ
(0)
N)∈RnN solving
the system of functions Qν(h)λ=−Fν(h).
b)Determine the components of function system u(0)[t]=(u(0)
1(t),u
(0)
2(t),...,u
(0)
N(t)) according to
the formulas
u(0)
r(t)=Dν,r(t)λ(0)
r+Fν,r(t),t∈[(r−1)h, rh),r=1:N.
Step 1. a)Find the next approximation on parameter λ(1) =(λ(1)
1,λ
(1)
2,...,λ
(1)
N)∈RnN solving the
system of equations Qν(h)λ=−Fν(h)−Gν(u(0),h).
b)Determine the components of function system u(1)[t]=(u(1)
1(t),u
(1)
2(t),...,u
(1)
N(t)) according to
the formulas
u(1)
r(t)=Dν,r(t)λ(1)
r+Fν,r(t)+Gν,r(u(0)
r,t),t∈[(r−1)h, rh),r=1:N.
And so on. Continuing this process, at the k-th step of algorithm we obtain the pair (λ(k),u
(k)[t]),
k=0,1,....
3. RESULTS
Theorem 1. Suppose that for some h>0:Nh =T(N∈N),ν∈N, the matrix Qν(h):RnN →
RnN is invertible and the following inequalities
||(Qν(h))−1|| ≤ γν(h),q
ν(h)=(αh)ν
ν!⎛
⎝1+γν(h)max(1,h||C||)
ν

j=1
(αh)j
j!⎞
⎠<1
hold. Then at k→∞the sequence of pairs (λ(k),u
(k)[t]),k=0,1,...,defined by the algorithm
converges to (λ∗,u
∗[t]),a unique solution to the problem with parameters (6)–(9), and the
estimates
||u∗[·]−u(k)[·]||1≤qν(h)
1−qν(h)||u(k)[·]−u(k−1)[·]||1,k∈N,
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610 TEMESHEVA et al.
||λ∗−λ(k)|| ≤ γν(h)
1−qν(h)max(1,h||C||)(αh)ν
ν!||u(k)[·]−u(k−1)[·]||1,k∈N,
are true.
Definition 1. A problem (1), (2) is called well-posed if given any f(t)∈C([0,T],Rn),d∈Rn
the problem has a unique solution x∗(t).
In light of equivalence of problems (1), (2) and (6)–(9) the Theorem 1 leads to
Theorem 2. Given some h>0:Nh =T(N∈N),ν∈Nsuppose that the conditions of
Theorem 1 hold. Then the boundary value problem (1), (2) is uniquely solvable.
Next assertions demonstrate that the conditions of Theorem 2 are not only sufficient, but necessary
as well for a unique solvability of problem (1), (2).
Theorem 3. If the boundary value problem (1), (2) is uniquely solvable, thengiven any h>0:
Nh =T(N∈N),χ∈(0,1] there exists ν=ν(h, χ)(ν∈N)such that the matrix Qν(h):RnN →RnN
is invertible, and the inequalities
||(Qν(h))−1|| γν(h),(12)
qν(h)=(αh)ν
ν!1+γν(h)max(1,h||C||)
ν

j=1
(αh)j
j!<χ (13)
hold.
Theorem 4. If the boundary value problem (1), (2) is uniquely solvable, then given any
ν(ν∈N),χ∈(0,1] there exists a number 
h=
h(ν, χ)>0such that for all h∈(0,
h]:Nh =T
(N∈N)the matrix Qν(h):RnN →RnN is invertible and the inequalities (12), (13) hold.
4. EXAMPLE
Consider as an example the two-point boundary value problem for a system of two linear equations
of the first order on [0,1]
⎧
⎪
⎨
⎪
⎩
dx1
dt =x2,
dx2
dt =t2x1+tx2−t4+t3−2t2+t+2,
x1,x
2∈R1,(14)
x1(0) = 0,x
2(1) = 1.(15)
Solution to this problem is a vector function x∗(t)∈C([0,1],R2)with coordinates x∗
1(t)=t2−t,
x∗
2(t)=2t−1. For this problem, α=2 and at h=0.25,ν=3the matrix Qν(h)and vector Fν(h)
take the form
Q3(0.25) =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0.2500 0000000
0000000.2558 0.0700
1.0003 0.2527 −100000
0.0053 1.0327 0 −10000
001.0037 0.2611 −10 0 0
000.0381 1.1041 0 −10 0
00001.0113 0.2702 −10
00000.1069 1.1845 0 −1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
F3(0.25) = (0.0000,0.0128,0.0652,0.5327,0.0686,0.5591,0.0650,0.5264),det Q3(0.25) =0.Since
||(Q3(0.25))−1|| ≤ 13.6069,q
3(0.25) = 0.2039 <1,then, in accordance with Theorem 2, problem (14),
(15) is uniquely solvable.
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ON ONE ALGORITHM TO FIND 611
Solving the system of equations Q3(0.25)λ=−F3(0.25) on the 0-step of algorithm, we find λ(0) ∈
R8and compute the components of function system u(0) [t]∈C([0,1],0.25,R8):
λ(0) =λ(0)
11 ,λ
(0)
12 ,λ
(0)
21 ,λ
(0)
22 ,λ
(0)
31 ,λ
(0)
32 ,λ
(0)
41 ,λ
(0)
42 
=(0.0000,−0.9991,−0.1873,−0.4991,−0.2497,0.0000,−0.1872,0.5009),
u(0)[t]=u(0)
11 (t),u
(0)
12 (t),u
(0)
21 (t),u
(0)
22 (t),u
(0)
31 (t),u
(0)
32 (t),u
(0)
41 (t),u
(0)
42 (t):
u(0)
11 (t)=t2−t+O(t),u
(0)
12 (t)=2t+O(t2),t∈[0,0.25),
u(0)
21 (t)=t2−t+0.187271 + O(t),u
(0)
22 (t)=2t−0.500029 + O(t2),t∈[0.25,0.5),
u(0)
31 (t)=t2−t+0.249405 + O(t),u
(0)
32 (t)=2t−1−0.000127 + O(t2),t∈[0.5,0.75),
u(0)
41 (t)=t2−t+0.179727 + O(t),u
(0)
42 (t)=2t−1−0.502316 + O(t2),t∈[0.75,1).
Is defined by x(0)
i(t)=u(0)
ri (t)+λ(0)
ri ,t∈[0.25(r−1),0.25r),r=1:4,x
(0)
i(1) = lim
t→1−0u(0)
4i(t)+λ(0)
4i,
i=1,2,the vector function x(0)(t)=x(0)
1(t),x
(0)
2(t). The inequality ||x∗−x(0)||0<0.001 is true.
Solve the system of equations Q3(0.25)λ=−F3(0.25) −G3(u(0),0.25) on the 1st step.
Here G3(u(0),0.25) = (0,−0.000145,−0.000013,−0.000004,−0.000096,−0.000066,−0.000270,
−0.000290).Find λ(1) =λ(1)
11 ,λ
(1)
12 ,λ
(1)
21 ,λ
(1)
22 ,λ
(1)
31 ,λ
(1)
32 ,λ
(1)
41 ,λ
(1)
42 =(0.000000,−1.000001,
−0.187501,−0.500011,−0.250000,0.000000,−0.187501,0.499999) and compute the components of
function system u(1)[t]:
u(1)
11 (t)=t2−t+O(t),u
(1)
12 (t)=2t+O(t2),t∈[0,0.25),
u(1)
21 (t)=t2−t+0.1875027 + O(t),u
(1)
22 (t)=2t−0.4999997 + O(t2),t∈[0.25,0.5),
u(1)
31 (t)=t2−t+0.25 + O(t),u
(1)
32 (t)=2t−1+O(t2),t∈[0.5,0.75),
u(1)
41 (t)=t2−t+0.1875071 + O(t),u
(1)
42 (t)=2t−1.4999979 + O(t2),t∈[0.75,1).
Determine the vector function x(1) (t)=x(1)
1(t),x
(1)
2(t)by the equalities:
x(1)
i(t)=u(1)
ri (t)+λ(1)
ri ,t∈[0.25(r−1),0.25r),r=1:4,i=1,2,
x(1)
i(1) = lim
t→1−0u(1)
4i(t)+λ(1)
4i,i=1,2,
and obtain the estimate ||x∗−x(1) ||0≤0.0000122 <0.0001.
Thus, for the problem (14), (15) considered on [0,1],theoffered algorithm with ν=3,h=0.25 gives
on first step the approximate solution up to ε=0.0001.
Calculations are made in the mathematical package MathCAD 15.
FUNDING
This research has was funded by the Science Committee of the Ministry of Education and Science of
the Republic of Kazakhstan (grant no. AP08956612).
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LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 42 No. 3 2021