=Paper=
{{Paper
|id=Vol-3687/Short2.pdf
|storemode=property
|title=Adomian Decomposition Method in The Theory of Nonlinear Periodic Boundary Value Problems with Delay
|pdfUrl=https://ceur-ws.org/Vol-3687/Short_2.pdf
|volume=Vol-3687
|authors=Peter Benner,Sergey Chuiko,Viktor Chuiko
|dblpUrl=https://dblp.org/rec/conf/dsmsi/BennerCC23
}}
==Adomian Decomposition Method in The Theory of Nonlinear Periodic Boundary Value Problems with Delay==
https://ceur-ws.org/Vol-3687/Short_2.pdf
Adomian Decomposition Method in The Theory of Nonlinear
Periodic Boundary Value Problems with Delay
Peter Benner 1, Sergey Chuiko ,1,2 and Viktor Chuiko 3
1
Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstrasse, 1, Magdeburg, 39106,
Germany;
2
Donbass State Pedagogical University, Donetsk region, Slavyansk, st. General Batyuk, 19, Slaviansk, 84116,
Ukraine;
3
Taras Shevchenko National University of Kyiv, Volodymyrska St, 64/13, Kyiv, 01601, Ukraine.
Abstract
Among numerous studies of functional-differential equations, research on periodic boundary
value problems for differential equations with concentrated delay holds a special place. This
is primarily due to the wide application of periodic boundary value problems for differential
equations with concentrated delay in physics, economics [3], biology [4], and mechanics [5].
By applying the Adomian decomposition method, we have derived the necessary and
sufficient conditions for the existence of solutions to the weakly nonlinear periodic boundary
value problem for a system of differential equations with concentrated delay in the critical
case.
Keywords 1
Functional-differential equations, differential equations with concentrated delay, periodic
boundary value problems, weakly nonlinear boundary value problems, Adomian
decomposition method.
1. Introduction
We studied the problem of constructing approximations to the π-periodic solution [1, 2]
π§(π‘, π) βΆ π§(β, π) β πΆΒΉ[0, π], π§(π‘,β) β πΆ[0, πβ]
of a system of differential equations with concentrated delay
ππ§(π‘, π)/ππ‘ = π΄(π‘)π§(π‘, π) + π΅(π‘)π§(π‘ β β, π) + π(π‘) + π π(π§(π‘, π), π§(π‘ β β, π), π‘, π). (1)
The solution of the periodic problem for equation (1) is sought in a small neighborhood of the T-
periodic solution
π§β(t) β β1 [0, π]
of the generating system
ππ§β/ππ‘ = π΄(π‘)π§β(π‘) + π΅(π‘)π§β(π‘ β β) + π(π‘), β β βΒΉ. (2)
Where π΄(π‘), π΅(π‘) are continuous π-periodic (n Γ n)-matrices, π(π‘) is continuous π-periodic vector-
function, π(π§(π‘, π), π§(π‘ β β, π), π‘, π) is nonlinear vector function, which is analytic in a small
neighborhood of the generating problem (2), continuous and π-periodic with the respect to the
variable t, and also analytic with respect to the small parameter Ξ΅ on the interval [0, πβ]. As is known,
in the critical case [2], specifically, in the presence of π-periodic solutions
Dynamical System Modeling and Stability Investigation (DSMSI-2023), December 19-21, 2023, Kyiv, Ukraine
EMAIL: benner@mpi-magdeburg.mpg.de (A. 1); chujko-slav@ukr.net (A. 2); vitya.chuyko@gmail.com (A. 3)
ORCID: 0000-0003-3362-4103 (A. 1); 0000-0001-7186-0129 (A. 2); 0009-0000-8450-9979 (A. 3)
Β©οΈ 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
Workshop
ceur-ws.org
ISSN 1613-0073
98
Proceedings
π§0 π‘, ππ = ππ π‘ ππ , ππ β βπ
of the homogeneous part
ππ§0 /ππ‘ = π΄(π‘)π§0 (π‘) + π΅ (π‘)π§0 (π‘ β Ξ) (3)
of system (2), and in the case of constant matrices π΄(π‘) β‘ π΄ and π΅(π‘) β‘ π΅, with the presence of
purely imaginary roots
ππ = Β±πππ π, π = β1, π β β
of the characteristic equation
det π΄ + π΅π βπΞ β ππΌπ = 0,
the generating periodic problem for equation (2) is solvable not for all vector functions π(π‘). In the
critical case, the adjoint system
ππ¦ π‘ /ππ‘ = βπ΄β π‘ π¦ π‘ β π΅β π‘ π¦(π‘ + Ξ)
has a family of π-periodic solutions of the form
π¦ π‘, ππ = π»π π‘ ππ , ππ β βπ .
Periodic problem for the equation (2) is solvable iff
π
π»πβ π π π ππ = 0. (4)
0
Here π»π (π‘) is (n Γ r)- matrix formed by r linearly independent π-periodic solutions of the adjoint
system. Let us assume condition (4) is satisfied; in this case, the general solution of the generating π-
periodic problem for equation (2) has the form
π§0 π‘, ππ = ππ π‘ ππ + πΊ π π π‘ , ππ β βπ ,
where πΊ[π (π )](π‘) is a particular solution of the generating π-periodic problem for equation (2), ππ (π‘)
is (n Γ r)- matrix formed by r linearly independent π-periodic solutions of the system (2). To construct
a particular solution πΊ[π(π )](π‘) of the generating π-periodic problem for equation (2), provided its
solvability, the method of least squares [6] is applicable.
2. The necessary and sufficient conditions for solvability
Similarly to [2], we obtain the necessary condition for the solvability of the π-periodic problem for
equation (2).
Lemma. Let us assume that for the generating periodic problem for equation (2), a critical case
occurs, and the solvability condition (4) is satisfied. In this case, the periodic problem for equation
(2) has a family of T-periodic solutions in the form
π§0 (π‘, ππ ) = ππ (π‘)ππ + πΊ[π (π )](π‘), ππ β βπ .
Let us also assume that the T-periodic problem for equation (1) has a T-periodic solution
π§(π‘, π) βΆ π§(β, π) β βΒΉ[0, π], π§(π‘,β) β β[0, πβ],
which, at π = 0, transforms unto the generating solution
π§(π‘, 0) = π§0 (π‘, ππβ ).
Under these conditions, the vector ππβ β βπ satisfies the equation for the generating amplitudes
π
πΉ (ππβ ) βΆ= β« π»πβ (π )π(π§0 (π , ππβ ), π§0 (π β Ξ, ππβ ), π , π)ππ = 0. (5)
0
99
We will refer to equation (5) as the equation for the generating amplitudes of the nonlinear periodic
boundary value problem for equation (1). The roots ππβ β βπ of equation (5) determine the generating
solutions π§0 (π‘, ππβ ), in a neighborhood of which the sought solutions to the original nonlinear T-
periodic boundary value problem for equation (1) may exist. However, if equation (5) has no real
solutions for ππβ β βπ , then the original nonlinear π-periodic boundary value problem for equation
(1) has no sought-after solutions.
Let us denote an (r Γ r)- matrix
T
B0 β β« Hrβ (s)[A1,0 (s)Xr (s) + A 0,1 (s)Xr (s β Ξ)]ds;
0
here
βZ(z(t, Ξ΅), z(t β Ξ, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβ )
A1,0 (t) = |
βz(t, Ξ΅) z(t β Ξ, Ξ΅) = z0 (t β Ξ, crβ )
and
βZ(z(t, Ξ΅), z(t β Ξ, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβ )
A0,1 (t) = |
βz(t β Ξ, Ξ΅) z(t β Ξ, Ξ΅) = z0 (t β Ξ, crβ )
Are (n Γ n)- matrices. The traditional solvability condition for the nonlinear periodic boundary value
problem for equation (1) in a small neighborhood of the generating solution z0 (t, crβ ) is the
requirement for the simplicity of the roots [2, 6]
det B0 β 0
of equation (5) for the generating amplitudes. We will demonstrate that the requirement for the
simplicity of the roots of equation (5) for the generating amplitudes is a sufficient condition for the
solvability of the nonlinear periodic boundary value problem for equation (1) in a small neighborhood
of the generating solution
π§0 (π‘, ππβ ) = ππ ππβ + πΊ[π (π )](π‘), ππβ β βπ .
In the article [6], we found constructive necessary and sufficient conditions for solvability, along with
a scheme for constructing solutions of the nonlinear π-periodic boundary value problem for equation
(1). Based on the method of simple iterations, we developed a convergent iterative scheme to find
approximations to the solutions of this problem. However, in the process of constructing solutions to
the nonlinear π-periodic boundary value problem for equation (1) using the least squares method, the
issue of impossibility of finding solutions in terms of elementary functions arises, which, in turn,
leads to significant errors in solving nonlinear boundary value problems.
Furthermore, the construction of solutions for nonlinear boundary value problems using the method of
simple iterations [2] and the least squares method is significantly complicated by the computation of
derivatives of nonlinearities. Given this, simplifying the computation of nonlinear derivatives and the
potential to find solutions for nonlinear boundary value problems, including periodic boundary value
problems, in elementary functions can be achieved using the Adomian decomposition method [7, 8].
Additionally, the use of the Adomian decomposition method significantly simplifies the proof of
convergence of iterative schemes for constructing solutions to nonlinear boundary value problems. An
example of such simplification will be provided below. Thus, the purpose of this article is to find
constructive solvability conditions and a scheme for constructing solutions to the nonlinear π-periodic
boundary value problem for equation (1) using the Adomian decomposition method.
Fixing one of the solutions of equation (5), we approach the problem of finding analytical
solutions for the nonlinear π-periodic boundary value problem for equation (1) in a small
neighborhood of the generating solution π§0 (π‘, ππβ ). We seek the solution of the nonlinear π-periodic
boundary value problem for equation (1) in the critical case in the form
π§(π‘, π) βΆ= π§0 (π‘, ππ ) + π’1 (π‘, π) + β¦ + π’π (π‘, π) + β¦ .
100
The nonlinear vector function π(π§(π‘, π), π§(π‘ β Ξ, π), π‘, π) is analytic in a small neighborhood of the
generating solution of problem (2); therefore, in the specified neighborhood, occurs an expansion in
the form [7, p. 502]
π(π§(π‘, π), π§(π‘ β Ξ, π), π‘, π)
= π(π§0 (π‘, ππβ ), π§0 (π‘ β Ξ, ππβ ), π‘, 0)
+ π1 (π§0 (π‘, ππβ ), π’1 (π‘, π), π§0 (π‘ β Ξ, ππβ ), π’1 (π‘ β Ξ, π), π‘, π) + β¦ +
+ ππ (π§0 (π‘, ππ ), π’1 (π‘, π), β¦ , π’π (π‘, π), π§0 (π‘ β Ξ, ππβ ), π’1 (π‘ β Ξ, π), β¦ , π’π (π‘ β Ξ, π), π‘, π) + β¦ .
β
First approximation to solution of nonlinear π- periodic boundary value problem for equation (1) in
critical case
π§1 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π), π’1 (π‘, π) = ππ (π‘)π1 (π) + (π‘, π), π1 (π) β βπ
determines the solution of the π- periodic boundary value problem for equation
ππ’1 (π‘, π)
= π΄(π‘)π’1 (π‘, π) + π΅(π‘)π’1 (π‘ β Ξ, π) + ππ(π§0 (π‘, ππβ ), π§0 (π‘ β Ξ, ππβ ), π‘, 0);
ππ‘
Here
(1)
π’1 (π‘, π) = ππΊ[π(π§0 (π , ππβ ), π§0 (π β Ξ, ππβ ), π , 0)](π‘)
is a particular solution of this problem. The solvability of the π-periodic boundary value problem in
the first approximation is guaranteed by choosing the root ππβ of equation (5) for the generating
amplitudes of the nonlinear periodic boundary value problem for equation (1). The second
approximation to the solution of the nonlinear π-periodic boundary value problem for equation (1) in
the critical case
(1)
π§2 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π) + π’2 (π‘, π), π’2 (π‘, π) = ππ (π‘)π2 (π) + π’2 (π‘, π), π2 (π) β βπ
determines the solution of the π-periodic boundary value problem for the equation
ππ’2 (π‘, π)
= π΄(π‘)π’2 (π‘, π) + π΅(π‘)π’2 (π‘ β Ξ, π) +
ππ‘
+ππ1 (π§0 (π‘, ππβ ), π’1 (π‘, π), π§0 (π‘ β Ξ, ππβ ), π’1 (π‘ β Ξ, π), π‘, π);
Here
(1)
π’2 (π‘, π) = ππΊ[π1 (π§0 (π , ππβ ), π’1 (π , π), π§0 (π β Ξ, ππβ ), π’1 (π β Ξ, π), π , π)](π‘)
is a particular solution of this problem. The solvability of the π-periodic boundary value problem in
the second approximation guarantees the solvability of the equation
π
πΉ1 (π1 (π)) β β«0 π»πβ (π )π1 (π§0 (π , ππβ ), π’1 (π , π), π§0 (π β Ξ, ππβ ), π’1 (π β Ξ, π), π , π)ππ = 0.
Unlike the equation for the generating amplitudes, the last equation is linear:
πΉ1 (π1 (π)) = π΅0 π1 (π) + πΏ1 (ππβ , π) = 0,
and also solvable, provided the roots of the equation for the generating amplitudes are simple;
here
π΅0 = πΉ1β² (π1 (π)) β βπΓπ , πΏ1 (ππβ ) β πΉ1 (π1 (π)) β π΅0 π1 (π).
In order to prove this, let us denote the vector function [8, 9]
π£(π‘, π) β π§0 (π‘, ππβ ) + ππ’1 (π‘, π) + β¦ + ππ π’π (π‘, π) + β¦ ;
in this case
π
πΉ1 (π1 (π)) β β«0 π»πβ (π )π1 (π§0 (π , ππβ ), π’1 (π , π), π§0 (π β Ξ, ππβ ), π , 0, π’1 (π β Ξ, π), π)ππ β
π
= β«0 π»πβ (π )ππβ² (π£(π , π), π£ (π β Ξ, π), π , π)ππ | π=0=
π
=β«0 π»πβ (π )[π΄1,0 (π )π’1 (π , π) + π΄0,1 (π )π’1 (π β Ξ, π)] ππ ,
and thus
101
π
(1) (1)
π΅0 = πΉ1β² (π1 (π)), πΏ1 (ππβ , π) = β« π»πβ (π ) [π΄1,0 (π )π’1 (π , π) + π΄0,1 (π )π’1 (π β Ξ, π)] ππ .
0
Therefore, assuming the simplicity of the roots of the equation for the generating amplitudes (5), we
obtain a unique solution to the boundary value problem in the first approximation
(1)
π’1 (π‘, π) = ππ (π‘)π1 (π) + π’1 (π‘, π),
π
(1) (1)
π1 (π) = βπ΅0β1 β« π»πβ (π ) [π΄1,0 (π )π’1 (π , π) + π΄0,1 (π )π’1 (π β Ξ, π)] ππ .
0
Third approximation to solution of nonlinear π- periodic boundary value problem for equation (1) in
critical case
π§3 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π) + π’2 (π‘, π) + π’3 (π‘, π),
(1)
π’3 (π‘, π) = ππ (π‘)π3 (π) + π’3 (π‘, π), π3 (π) β βπ
determines the solution of the π-periodic boundary value problem for the equation
ππ’3 (π‘, π)
= π΄(π‘)π’3 (π‘, π) + π΅(π‘)π’3 (π‘ β Ξ, π) +
ππ‘
+ππ2 (π§0 (π‘, ππ ), π’1 (π‘, π), π’2 (π‘, π), π§0 (π‘ β Ξ, ππβ ), π’1 (π‘ β Ξ, π), π’2 (π‘ β Ξ, π), π‘, π);
β
Here
(1)
π’3 (π‘, π) =
= ππΊ[π2 (π§0 (π , ππβ ), π’1 (π , π), π’2 (π , π), π§0 (π β Ξ, ππβ ), π’1 (π β Ξ, π), π’2 (π β Ξ, π), π , π)](π‘)
is a particular solution of this problem. The solvability of the π-periodic boundary value problem in
the second approximation guarantees the solvability of the equation
π
πΉ2 (π2 (π), π) β β« π»πβ (π ) Γ
0
Γ π2 (π§0 (π , ππβ ), π’1 (π , π), π’2 (π , π), π§0 (π β Ξ, ππβ ), π’1 (π β Ξ, π), π’2 (π β Ξ, π), π , π)ππ = 0.
Unlike the equation for the generating amplitudes, the last equation is linear:
πΉ2 (π2 (π)) = π΅0 π2 (π) + πΏ2 (ππβ , π1 (π), π) = 0,
and also solvable, provided that the roots of the equation for the generating amplitudes are simple.
Here
π΅0 = πΉ2β² (π2 (π)) β βπΓπ , πΏ2 (ππβ , π1 (π), π) β πΉ2 (π2 (π)) β π΅0 π2 (π).
Denote (n Γ n) β matrices
A2,0 (t, u1 (π‘, π)) β
π βZ(z(t, Ξ΅), z(t β Ξ, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβ )
= [ π’1 (π‘, π)]|
ππ§(π‘, π) βz(t, Ξ΅) z(t β Ξ, Ξ΅) = z0 (t β Ξ, crβ ),
A1,1 (t, u1 (π‘ β Ξ, π)) β
π βZ(z(t, Ξ΅), z(t β Ξ, Ξ΅), t, Ξ΅) z(t, Ξ΅) = z0 (t, crβ )
= [ π’1 (π‘ β Ξ, π)]|
ππ§(π‘, π) βz(t β Ξ, Ξ΅) z(t β Ξ, Ξ΅) = z0 (t β Ξ, crβ ),
and
A0,2 (t, u1 (π‘ β Ξ, π)) β
π βZ z t, Ξ΅ , z t β Ξ, Ξ΅), t, Ξ΅)
( ( ) ( z(t, Ξ΅) = z0 (t, crβ )
= [ π’1 (π‘ β Ξ, π)]|
ππ§(π‘ β Ξ, π) βz(t β Ξ, Ξ΅) z(t β Ξ, Ξ΅) = z0 (t β Ξ, crβ ),
Indeed,
1 π β
πΉ2 (π2 (π), π) = β« π» (π )ππβ²β²2 (π£(π , π), π£(π β Ξ, π), π , π)ππ | =
2! 0 π π=0
π
= β« π»πβ (π )[π΄1,0 (π )π’2 (π , π) + π΄0,1 (π )π’2 (π β Ξ, π)]ππ +
0
102
1 π β
+ β« π» (π )[π΄2,0 (π , π’1 (π , π)) + 2π΄1,1 (π , π’1 (π , π))π’1 (π β Ξ, π) +
2! 0 π
+π΄0,2 (π , π’1 (π β Ξ, π))π’1 (π β Ξ, π)]ππ ,
thus
π΅0 = πΉ2β² (π2 (π), π),
furthermore
π
(1) (1)
πΏ2 (ππβ , π1 (π), π) = β« π»πβ (π ) [π΄1,0 (π )π’2 (π , π) + π΄0,1 (π )π’2 (π β Ξ, π)] ππ +
0
1 π
+ β« π»πβ (π )[π΄2,0 (π , π’1 (π , π))π’1 (π , π) + 2π΄1,1 (π , π’1 (π , π))π’1 (π β Ξ, π) +
2! 0
π΄0,2 (π , π’1 (π β Ξ, π))π’1 (π β Ξ, π)]ππ .
Thus, provided that the roots of the equation for the generating amplitudes (5) are simple, we obtain a
unique solution to the boundary value problem in the second approximation
(1)
π’2 (π‘, π) = ππ (π‘)π2 (π) + π’2 (π‘, π),
π
(1) (1)
π2 (π) = βπ΅0β1 β« π»πβ (π ) [π΄1,0 (π )π’2 (π , π) + π΄0,1 (π )π’2 (π β Ξ, π)] ππ β
0
π
1
β π΅0β1 β« π»πβ (π )[π΄2,0 (π , π’1 (π , π))π’1 (π , π) + 2π΄1,1 (π , π’1 (π , π))π’1 (π β Ξ, π) +
2! 0
+π΄0,2 (π , π’1 (π β Ξ, π))π’1 (π β Ξ, π)]ππ .
Solvability of the π-periodic boundary value problem in π + 1 approximation guarantees the
solvability of the equation
π
πΉπ+1 (ππ (π)) β β« π»πβ (π )ππ (π§0 (π , ππβ ), π’1 (π , π), β¦ , π’π+1 (π , π),
0
π§0 (π β Ξ, ππβ ), π’1 (π β Ξ, π), β¦ , π’π+1 (π β Ξ, π), π , π) ππ = 0.
The sequence of approximations to the solution of the nonlinear π-periodic boundary value problem
for equation (1) in the critical case is determined by the iterative scheme
(1)
π§1 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π), π’1 (π‘) = ππ (π‘)π1 (π) + π’1 (π‘, π),
(1)
u1 (π‘, π) = ππΊ[π(π§0 (π , ππβ ), π§0 (π β Ξ, ππβ ), π , 0)](π‘),
π
(1) (1)
π1 (π) = βπ΅0β1 β« π»πβ (π ) [π΄1,0 (π )π’1 (π , π) + π΄0,1 (π )π’1 (π β Ξ, π)] ππ ,
0
(1)
π§2 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π) + π’2 (π‘, π), π’2 (π‘, π) = ππ (π‘)π2 (π) + π’2 (π‘, π),
(1)
π’2 (π‘, π) = ππΊ[π1 (π§0 (π , ππβ ), π’1 (π , π), π§0 (π β Ξ, ππβ ), π’1 (π β Ξ, π), π , π))](π‘),
π
(1) (1)
π2 (π) = βπ΅0β1 β« π»πβ (π ) [π΄1,0 (π )π’2 (π , π) + π΄0,1 (π )π’2 (π β Ξ, π)] ππ β
0
π
1
β π΅0β1 β« π»πβ (π )[π΄2,0 (π , π’1 (π , π))π’1 (π , π) + 2π΄1,1 (π , π’1 (π , π))π’1 (π β Ξ, π) +
2! 0
+π΄0,2 (π , π’1 (π β Ξ, π))π’1 (π β Ξ, π)] ππ , β¦,
π§π+1 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π) + β¦ + π’π+1 (π‘, π), (6)
103
(1)
π’π+1 (π‘, π) = ππ (π‘)ππ+1 (π) + π’π+1 (π‘, π), πΏπ+1 (ππβ , π1 (π), π) = πΉπ+1 (π2 (π)) β π΅0 ππ+1 (π),
(1)
π’π+1 (π‘, π) = ππΊ[ππ (π§0 (π , ππβ ), π’1 (π , π), β¦ , π’π (π , π),
π§0 (π β Ξ, ππβ ), π , π, π’1 (π β Ξ, π), β¦ , π’π (π β Ξ, π))](π‘),
ππ+1 (π) = βπ΅0β1 πΏπ+1 (ππβ , ππ+1 (π), π), π = 1,2, β¦ .
Theorem. In the critical case, the periodic problem for equation (2) with concentrated delay, under
condition (4), has an r-parametric family of solutions
π§0 (π‘, ππβ ) = ππ (π‘)ππ + πΊ[π(π )](π‘), ππ β βπ .
Assuming det π΅0 β 0 the simplicity of the roots of the equation (5) for the generating amplitudes in a
small neighborhood of the generating solution π§0 (π‘, ππβ ), the nonlinear periodic boundary value
problem for equation (1) has a unique solution
π§(π‘, π) βΆ π§(β, π) β πΆΒΉ[0, π], π§(π‘,β) β πΆ[0, πβ].
The sequence of approximations to the solution of the nonlinear periodic boundary value problem for
equation (1) is determined by the iterative scheme (6). If there exists a constant 0 < πΎ < 1, for which
the inequalities
βπ’1 (π‘, π)β β€ πΎβπ§0 (π‘, ππβ )β, βπ’π+1 (π‘, π)β β€ πΎβπ’π (π‘, π)β, π = 1,2, β¦ (7)
hold, then the iterative scheme (6) converges to the solution of the nonlinear periodic boundary value
problem for equation (1) with concentrated delay.
3. Finding approximations to the periodic solution of the equation modeling
a non-isothermal chemical reaction
Let us apply the iterative scheme (6) in order to find approximations to the periodic solution of the
equation with concentrated delay, which models a non-isothermal chemical reaction [10, 11].
Example. The conditions of the proven theorem hold in the case of a 2π-periodic boundary value
problem with concentrated delay
ππ§(π‘, π)/ππ‘ = π΄(π‘)π§(π‘, π) + π΅(π‘)π§(π‘ β Ξ, π) + π (π‘) + ππ(π§(π‘, π), π§(π‘ β Ξ, π), π‘, π); (8)
here
0 1 cos π‘ π
π΄(π‘) β ( ) , π΅(π‘) β 0, π (π‘) β ( ), Ξ β ,
0 0 0 2
and also
π
β 1 π₯(π‘, π)
π(π§(π‘, π), π§(π‘ β Ξ, π), π) β (1 + π₯ (π‘, π))π 1+π¦(π‘βΞ,π) ( ) , π§(π‘, π) β ( ).
1 π¦(π‘, π)
For the generating periodic problem for equation (8), there is a critical case [2, 12], and condition (4)
is satisfied, therefore, it is solvable:
π§0 (π‘, ππ ) = ππ (π‘)ππ + πΊ[π (π )](π‘), ππ β β1 ;
here
1 sin π‘
ππ (π‘) = ( ) , πΊ[π (π )](π‘) = ( ).
0 0
104
The equation for the generating amplitudes (5) in the case of a problem of finding a periodic solution
for equation (8) has a simple root ππβ = 1, which determines the generating solution
1 + sin π‘
π§0 (π‘, ππβ ) = β ( ).
0
The matrix π΅0 = 2π is non-singular, so according to the proven theorem, the 2π-periodic problem for
equation (8) with concentrated delay is uniquely solvable. Thus, we obtain the first approximation
(1)
π§1 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π), π’1 (π‘, π) = ππ (π‘)π1 (π) + π’1 (π‘, π), π1 (π) = βπ
to the solution of the periodic problem for equation (8); here
(1) 1 β sin π‘ β cos π‘
π’1 (π‘, π) = ππΊ[π(π§0 (π , ππβ ), π§0 (π β Ξ, ππβ ), π , 0)](π‘) = π ( ),
β cos π‘
and also
1
π(π§0 (π‘, ππβ ), π§0 (π‘ β Ξ, ππβ ), π‘, 0) = (1 + ππβ + sin π‘ ) ( ) .
1
Similarly, we obtain the second approximation to the solution of the nonlinear periodic boundary
value problem for equation (8) in the critical case
(1)
π§2 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π) + π’2 (π‘, π), π’2 (π‘, π) = ππ (π‘)π2 (π) + π’2 (π‘, π);
here
(1)
π’2 (π‘, π) = ππΊ[π1 (π§0 (π , ππβ ), π’1 (π , π), π§0 (π β Ξ, ππβ )), π’1 (π β Ξ, π), π , π)](π‘) =
sin π‘ + 3 cos π‘ β 3 7π 2
= π2 ( ), π2 (π) = .
2 cos π‘ β sin π‘ 2
and also
π1 (π§0 (π‘, ππβ ), π’1 (π‘, π), π§0 (π‘ β Ξ, ππβ ), π’1 (π‘ β Ξ, π), π‘, π) =
1
= (1 + π1 (π) β 2 sin π‘ β cos π‘ ) ( ).
1
In the same way, we obtain the third approximation to the solution of the nonlinear periodic boundary
value problem for equation (8) in the critical case
65π 3
π§3 (π‘, π) β π§0 (π‘, ππβ ) + π’1 (π‘, π) + π’2 (π‘, π) + π’3 (π‘, π), π3 (π) = β ,
8
(1) (1)
π’3 (π‘, π) = ππ (π‘)π3 (π) + π’3 (π‘, π), π’3 (π‘, π) =
= ππΊ[π2 (π§0 (π , ππβ ), π’1 (π , π), π’2 (π , π), π§0 (π β Ξ, ππβ )), π’1 (π β Ξ, π), π’2 (π β Ξ, π), π , π)](π‘) =
π 3 53 β 52 cos π‘ β cos 2π‘ + 12 sin π‘ + 2 sin 2π‘
= ( );
8 2(16 sin π‘ + sin 2π‘ β 10 cos π‘)
and also
π2 (π§0 (π‘, ππβ ), π’1 (π‘, π), π’2 (π‘, π), π§0 (π‘ β Ξ, ππβ ), π’1 (π‘ β Ξ, π), π’2 (π‘ β Ξ, π), π‘, π) =
π2 1
= (7 β 2π2 (π) β 8 cos π‘ β cos 2π‘ β 5 sin π‘) ( ).
2 1
For the obtained approximations to the periodic solution of equation (8), the inequalities
βπ’1 (π‘, π)β β€ πΎ βπ§0 (π‘, ππβ )β, βπ’π+1 (π‘, π)β β€ πΎ βπ’π (π‘, π)β, πΎ β 0, 131 256, π = 0,1,2,
hold, indicating the practical convergence of the obtained approximations to the periodic solution of
the equation (8) for
π β [0, π0 ], π0 β 0, 25.
The accuracy of the obtained approximations to the periodic solution of equation (8) is determined by
the residuals
Ξπ (π) β ||ππ§π (π‘, π)/ππ‘ β π΄(π‘)π§π (π‘, π) β π΅ (π‘)π§π (π‘ β Ξ, π) β π (π‘) β
βππ(π§π (π‘, π), π§π (π‘ β Ξ, π), π‘, π)|| , π = 0, 1, 2, 3.
β[0; 2π]
In particular,
105
Ξ0 (0, 1) β 0, 0904 837, Ξ1 (0,1) β 0, 0213 474,
Ξ2 (0, 1) β 0, 00 469 105, Ξ3 (0,1) β 0, 00 112 528;
Ξ0 (0, 01) β 0, 0099 005, Ξ1 (0,01) β 0, 000 222 616,
Ξ2 (0, 01) β 4, 97 520 Γ 10β6 , Ξ3 (0,01) β 1, 21 626 Γ 10β7 .
The research scheme proposed in the article for investigating solvability conditions and constructing
approximations to the periodic solution of equation (1) can be transferred to matrix boundary value
problems, including those with concentrated delay [13 - 16].
The authors of the article express their sincere gratitude to the Managing Director of the Max
Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Professor Peter Benner,
for his support and discussion of the obtained results.
4. References
[1] J. K. Hale, Theory of Functional Differential Equations, Springer β Verlag. York, Heidelberg,
Berlin, 1975.
[2] A. A. Boichuk, A.M. Samoilenko, Generalized inverse operators and Fredholm boundary-value
problems, Utrecht, Boston: VSP, 2004.
[3] R. Bellman, K. L .Cooke, Differential-Difference Equations. Mathematics in Science and
Engineering, New York, Academic Press, 1963.
[4] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population
Dynamics. Mathematics and Its Applications, Dordrecht: Kluwer Academic Publishers, 1992.
[5] Th. Erneux, Applied Delay Differential Equations. Surveys and Tutorials in the Applied
Mathematical Sciences, Vol. 3, Springer Science+Business Media, New York, 2009.
[6] S.M. Chuiko, A.S. Chuiko, On the approximate solution of periodic boundary value problems
with delay by the least-squares method in the critical case, Nonlinear Oscillations (N.Y.) 14, 3
(2012), pp. 445-460.
[7] G. Adomian, A review of the decomposition method in applied mathematics, Journ. of Math.
Math. Anal. and Appl., 135 (1988), pp. 501-544.
[8] G. Adomian, R. Rach, Nonlinear Stochastic Differential Delay Equations, Journal of Math.
analysis and appl., 91 (1983), pp. 94-101.
[9] M. Mac, C. S. Leung, T. Harko, A brief introducion to the Adomian decomposition method,
Romanian Astron. Journ., 1, (2019), pp. 1-41.
[10] P. Benner, A. Seidel-Morgenstern, A. Zuyev, Periodic switching strategies for an isoperimetric
control problem with application to nonlinear chemical reactions, Applied Mathematical
Modelling, 69 (2019), pp. 287-300.
[11] A. Zuyev, A. Seidel-Morgenstern and P. Benner, An isoperimetric optimal control problem for a
non-isothermal chemical reactor with periodic inputs, Chemical Engineering Science 161 (2017),
206-214..
[12] Π.Π. Boichuk and A.M. Samoilenko, Generalized inverse operators and Fredholm boundary-
value problems, Boston: De Gruyter, 2016.
[13] A.A. Boichuk, S.A. Krivosheya, Criterion of the solvability of matrix equations of the Lyapunov
type, Ukrainian Mathematical Journal, 50, 8 (1998), β 8, 1162-1169.
[14] P. Lancaster, M. Tismenetsky, The Theory of Matrices, New York, Academic Press,1985.
[15] S.Π. Chuiko, Nonlinear matrix differential-algebraic boundary value problem, Lobachevskii
Journal of Mathematics, 38 (2018), 236β244.
[16] A.A. Boichuk, J. Diblik, D. Ya. Khusainov, M. Ruzickova, Fredholmβs boundary-value problems
for differential systems with a single delay, Nonlinear Anal., 72 (2010), pp. 2251-2258.
106