Method of Lines for Quasilinear Functional Differential Equations
We give a theorem on the estimation of error for approximate solutions to ordinary functional differential equations. The error is estimated by a solution of an initial problem for a nonlinear functional differential equation. We apply this general result to the investigation of convergence of the n...
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Institute of Mathematics, NAS of Ukraine
2013
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508421060558848 |
|---|---|
| author | Kamont, Z. Czernous, W. Камонт, З. Черноус, В. |
| author_facet | Kamont, Z. Czernous, W. Камонт, З. Черноус, В. |
| author_sort | Kamont, Z. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:17:19Z |
| description | We give a theorem on the estimation of error for approximate solutions to ordinary functional differential equations. The error is estimated by a solution of an initial problem for a nonlinear functional differential equation. We apply this general result to the investigation of convergence of the numerical method of lines for evolution functional differential equations. The initial boundary-value problems for quasilinear equations are transformed (by means of discretization in spatial variables) into systems of ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for given operators are assumed. Numerical examples are given. |
| first_indexed | 2026-03-24T02:24:56Z |
| format | Article |
| fulltext |
UDC 517.9
W. Czernous, Z. Kamont (Inst. Math. Univ. Gdańsk, Poland)
METHOD OF LINES FOR QUASILINEAR
FUNCTIONAL DIFFERENTIAL EQUATIONS
МЕТОД ЛIНIЙ ДЛЯ КВАЗIЛIНIЙНИХ
ФУНКЦIОНАЛЬНО-ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
We give a theorem on the error estimate of approximate solutions for ordinary functional differential equations. The error
is estimated by a solution of an initial problem for nonlinear functional differential equation. We apply this general result to
the investigation of the convergence of the numerical method of lines for evolution functional differential equations. Initial
boundary-value problems for quasilinear equations are transformed by discretization in spatial variables into systems of
ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for
given operators are assumed. Numerical examples are given.
Наведено теорему про оцiнку похибки наближених розв’язкiв звичайних диференцiальних рiвнянь. Похибка оцi-
нюється за допомогою розв’язку початкової задачi для нелiнiйного функцiонально-диференцiального рiвняння. Цей
загальний результат застосовується при дослiдженнi збiжностi числового методу лiнiй для еволюцiї функцiонально-
диференцiальних рiвнянь. За допомогою дискретизацiї по просторових змiнних початково-крайовi задачi для квазiлi-
нiйних рiвнянь зводяться до систем звичайних диференцiальних рiвнянь. Припускається справедливiсть нелiнiйних
оцiнок перронiвського типу вiдносно функцiональних змiнних для заданих операторiв. Наведено також чисельнi
приклади.
1. Introduction. The numerical method of lines for partial differential or functional differential equa-
tions consists in replacing derivatives with respect to spatial variables by difference expressions. This
leads to systems of ordinary differential or functional differential equations. They satisfy consistency
conditions on classical solutions of original problems. The main task in these considerations is to find
sufficient conditions for the stability of differential difference problems.
There is an ample literature on the method of lines. The classical papers are [7, 8, 22, 23] where
parabolic equations were considered. Existence results based on the method of lines can be found in
[3, 14, 19, 24, 25]. Parabolic problems and first order partial differential equations and boundary-value
problems for nonlinear elliptic equations were considered. The papers [1, 2, 4, 13, 29, 30] initiated
the theory of the method of lines for evolution functional differential equations. Initial problems on
the Haar pyramid for Hamilton – Jacobi-type equations and parabolic equations with initial or initial
boundary conditions of the Dirichlet-type were investigated. For further bibliographical informations
concerning the method of lines see [9, 11, 17, 21, 28].
Results on the method of lines for evolution functional differential equations are based on the
following ideas. Comparison theorems for differential difference inequalities generated by nonlinear
functional differential equations are obtained. These theorems state that functions satisfying differential
difference inequalities may be estimated by solutions of ordinary differential or functional differential
equations. Comparison theorems are used in proofs of the existence of approximate solutions. Results
on the convergence of sequences of approximate solutions are also based on comparison theorems
for differential difference inequalities. Theorems on the numerical method of lines for nonlinear first
partial functional differential equations [1, 2, 12, 29] and for parabolic problems [15, 30] were obtained
by using the above comparison methods.
c© W. CZERNOUS, Z. KAMONT , 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10 1363
1364 W. CZERNOUS, Z. KAMONT
The aim of the paper is to show a new method of investigations of the numerical method of lines
for evolution functional differential equations. We prove that results on the existence of approximate
solutions and theorems on the convergence of the methods are consequences of simple theorems on
ordinary functional differential equations.
The paper is organized as follows. Section 2 deals with ordinary functional differential equations.
Numerical methods of lines for functional differential equations lead to systems of equations consid-
ered in this section. We prove a theorem on the existence of solutions to initial problems and we give a
theorem on estimates of the difference between the exact an approximate solutions. The errors of ap-
proximate solutions are estimated by solutions to initial problems for nonlinear comparison equations.
We apply this general idea to investigations of the numerical method of lines for evolution func-
tional differential equations. Initial boundary-value problems for quasilinear first order partial func-
tional differential equations are considered in Section 3. In the next section we give results on the
numerical method of lines for quasilinear parabolic functional differential problems.
Two types of assumption are needed in theorems on the convergence of the numerical methods of
lines. The first type conditions concern regularity of given functions. The second type conditions con-
cern the mesh. The authors of the papers [2, 12, 15, 29, 30] have assumed that given functions satisfy
the Lipschitz condition or they satisfy nonlinear estimates of the Perron-type with respect to functional
variables and these conditions are global with respect to functional variables. Our assumptions on the
regularity of given functions are more general. We assume nonlinear estimates of the Perron-type and
suitable estimates are local with respect to functional variables. There are differential equations with
deviated variables and differential integral equations such that local estimates of the Perron-type hold
and global inequalities are not satisfied. We give suitable examples.
Results presented in [1, 2, 15, 22 – 24, 29, 30] are not applicable to quasilinear functional problems
considered in the paper. Theorems presented here are new also in the case of differential equations
without the functional dependence.
Now we formulate our functional differential problems. For any metric spaces X and Y we denote
by C(X,Y ) the class of all continuous functions from X into Y. We use vectorial inequalities with the
understanding that the same inequalities hold between their corresponding components. Write
E0 = [−b0, 0]× [−b, b], E = [0, a]× [−b, b], ∂0E = [0, a]×
(
[−b, b] \ (−b, b)
)
,
where b0 ∈ R+, R+ = [0,+∞), a > 0 and b = (b1, . . . , bn) ∈ Rn, and bi > 0 for 1 ≤ i ≤ n. For
each (t, x) ∈ E we define the set D[t, x] as follows:
D[t, x] =
{
(τ, y) ∈ R1+n : τ ≤ 0, (t+ τ, x+ y) ∈ E0 ∪ E
}
.
It is clear that D[t, x] = [−b0 − t, 0] × [−b − x, b − x]. For a function z : E0 ∪ E → R and for a
point (t, x) ∈ E we define a function z(t,x) : D[t, x] → R as follows: z(t,x)(τ, y) = z(t + τ, x + y),
(τ, y) ∈ D[t, x]. Then z(t,x) is the restriction of z to the set (E0 ∪ E) ∩ ([−b0, t] × Rn) and this
restriction is shifted to the set D[t, x].
Write r = b0 + a and B = [−r, 0] × [−2b, 2b]. Then D[t, x] ⊂ B for (t, x) ∈ E. Set Ω =
= E×C(B,R) and suppose that F : Ω→ Rn, F = (F1, . . . , Fn), G : Ω→ R and ψ : E0∪∂0E → R
are given functions. Let z be an unknown function of the variables (t, x), x = (x1, . . . , xn). We
consider the functional differential equation
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1365
∂tz(t, x) =
n∑
i=1
Fi(t, x, z(t,x))∂xiz(t, x) +G(t, x, z(t,x)) (1)
with the initial boundary condition
z(t, x) = ψ(t, x) on E0 ∪ ∂0E. (2)
We will say that F and G satisfy the condition (V ) if for each (t, x) ∈ E and for w, w̃ ∈ C(B,R)
such that w(τ, y) = w̃(τ, y) for (τ, y) ∈ D[t, x] we have F (t, x, w) = F (t, x, w̃) and Gt, x, w) =
= G(t, x, w̃). Condition (V ) means that the values of F and G at (t, x, w) ∈ Ω depend on (t, x) and
on the restriction of w to the set D[t, x] only. We assume that F and G satisfy condition (V ) and we
consider classical solutions to (1), (2).
Now we formulate an initial boundary-value problem for a parabolic functional differential equa-
tion. Let Mn×n be the class of all n × n matrices with real elements. Suppose that F : Ω → Mn×n,
F : Ω→ Rn, G : Ω→ R ψ : E0 ∪ ∂oE are given functions and
F =
[
Fij
]
i,j=1,...,n
, F = (F1, . . . , Fn).
We consider the functional differential equation
∂tz(t, x) =
n∑
i,j=1
Fij(t, x, z(t,x))∂xixjz(t, x) +
n∑
i=1
Fi(t, x, z(t,x))∂xiz(t, x) +G(t, x, z(t,x)) (3)
with the initial boundary condition (2).
We will say that F satisfies the condition (V ) if for each (t, x) ∈ E and for w, w̃ ∈ C(B,R) such
that w(τ, y) = w̃(τ, y) for (τ, y) ∈ D[t, x] we have F(t, x, w) = F(t, x, w̃). We assume that F, F and
G satisfy condition (V ) and we consider classical solutions to (3), (2).
Sufficient conditions for the existence and uniqueness of classical or generalized solutions of evo-
lution functional differential equations can be found in [3, 5, 6, 10, 16, 18, 20, 26].
Differential equations with deviated variables and differential integral equations can be derived
from (1) and (3) by specializing given operators. Information on applications of functional differential
equations can be found in [11, 27].
2. Approximate solutions of ordinary functional differential systems. For any spaces X and
Y we denote by F (X,Y ) the class of all functions defined on X and taking values in Y . Let N and
Z be the sets of natural numbers and integers respectively. We define a mesh with respect to spatial
variables in the following way. Let (h1, . . . , hn) = h, hi > 0 for 1 ≤ i ≤ n, stand for steps of the
mesh. For m = (m1, . . . ,mn) ∈ Zn we put
x(m) =
(
x
(m1)
1 , . . . , x(mn)
n
)
= (m1h1, . . . ,mnhn)
and
R1+n
t.h =
{
(t, x(m)) : t ∈ R, m ∈ Zn
}
.
Write
Eh = E ∩ R1+n
t.h , E0.h = E0 ∩ R1+n
t.h , ∂0Eh = ∂0E ∩ R1+n
t.h
and
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1366 W. CZERNOUS, Z. KAMONT
Bh = B ∩ R1+n
t.h , Dh[t,m] = D
[
t, x(m)
]
∩ R1+n
t.h .
For z : E0.h ∪Eh → R and w : Bh → R we write z(m)(t) = z(t, x(m)) on E0.h ∪Eh and w(m)(t) =
= w(t, x(m)) on Bh. Let us denote by H the set of all h = (h1, . . . , hn) satisfying the condition:
there is K = (K1, . . . ,Kn) ∈ Nn such that (K1h1, . . . ,Knhn) = b. Set
K = {m ∈ Zn : −K ≤ m ≤ K}, IntK = {m ∈ Zn : −K < m < K} .
For x ∈ Rn, A ∈Mn×n where x = (x1, . . . , xn), A =
[
aij
]
i,j=1,...,n
, we put
‖x‖ =
n∑
i=1
|xi| and ‖A‖n×n = max
n∑
j=1
|aij | : 1 ≤ i ≤ n
.
Let Fc(E0.h ∪Eh,R) be the class of all z : E0.h ∪Eh → R such that z( · , x(m)) ∈ C([−b0, a],R) for
m ∈ K. In a similar way we define the spaces Fc(Bh,R) and Fc(E0.h ∪ ∂0Eh,R).
For z : E0.h ∪ Eh → R and (t, x(m)) ∈ Eh we define a function z[t,m] : Dh[t,m] → R in the
following way: z[t,m](τ, y) = z(t+ τ, x(m) + y), (τ, y) ∈ Dh[t,m]. Write
Λ =
{
λ = (λ1, . . . , λn) : λi ∈ {−1, 0, 1} for 1 ≤ i ≤ n and ‖λ‖ ≤ 2
}
,
Λ′ = Λ \ { θ }, θ = (0, . . . , 0) ∈ Rn,
and κ = 1 + 2n2. Note that κ is the number of elements of Λ. Let π : Λ→ {1, . . . , κ} be a function
such that π(λ) 6= π(λ̃) for λ 6= λ̃. We assume that ≺ is an order in Λ defined in the following way:
λ ≺ λ̃ if π(λ) < π(λ̃). Elements of the space Rκ will be denoted by ξ = {ξλ}λ∈Λ. Write
Ah =
{
x(m) : m = (m1, . . . ,mn) ∈ Λ
}
.
For ζ : Ah → R we put ζ(m) = ζ(x(m)).
For z : E0.h ∪ Eh → R and (t, x(m)) ∈ Eh, m ∈ IntK, we define a function z〈t,m〉 : Ah → R in
the following way: z〈t,m〉(y) = z(t, x(m) + y), y ∈ Ah. Write Ωh = Eh × Fc(Bh,R) and suppose
that
Gh : Ωh → R, Fh : Ωh → Rκ, Fh = {Fh.λ}λ∈Λ,
are given functions. For (t, x, w) ∈ Ωh, ζ ∈ F (Ah,R) we put
Fh(t, x, w) ◦ ζ =
∑
λ∈Λ
Fh.λ(t, x, w) ζ(λ).
Set
Fh(t, x, w, ζ) = Gh(t, x, w) + Fh(t, x, w) ◦ ζ.
Given ψh : E0.h ∪ ∂0Eh → R, we consider the functional differential equations
d
dt
z(m)(t) = Fh
(
t, x(m), z[t,m], z〈t,m〉
)
, m ∈ IntK, (4)
with the initial boundary condition
z(m)(t) = ψ
(m)
h (t) on E0.h ∪ ∂0Eh. (5)
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1367
We will say that Fh and Gh satisfy the condition (Vh) if for each (t, x(m)) ∈ Eh and for w, w̃ ∈
∈ Fc(Bh,R) such that w(τ, y) = w̃(τ, y) for (τ, y) ∈ D[t,m] we have Fh(t, x(m), w) =
= Fh(t, x(m), w̃) and Gh(t, x(m), w) = Gh(t, x(m), w̃). We assume that Fh and Gh satisfy condition
(Vh) and we consider classical solutions to (4), (5).
Let Wh : Fc(Bh,R)→ C([−r, 0],R+) be an operator defined by
Wh[w](t) = max
{
|w(t, x(m))| : x(m) ∈ [−2b, 2b]
}
, t ∈ [−r, 0].
The maximum norms in C(B,R) and Fc(Bh,R) are denoted by ‖ · ‖B and ‖ · ‖Bh
respectively. For
ω : [−r, a] → R and t ∈ [0, a] we define ωt : [−r, 0] as follows: ωt(τ) = ω(t + τ), τ ∈ [−r, 0]. If
η, η̃ ∈ C([−r, 0],R) then the inequality η ≤ η̃ states that η(τ) ≤ η̃(τ) for τ ∈ [−r, 0].
Assumption H[Υ]. The function Υ: [0, a] × C([−r, 0],R+) → R+ is continuous and it is
nondecreasing with respect to the second variable and
1) for t ∈ [0, a] and for υ, υ̃ ∈ C([−r, 0],R+) such that υ(τ) = υ̃(τ) for τ ∈ [−b0 − t, 0] we
have Υ(t, υ) = Υ(t, υ̃),
2) for each µ ∈ C([−b0, 0],R+) the maximal solution of the Cauchy problem
ω′(t) = Υ(t, ωt), ω(t) = µ(t) for t ∈ [−b0, 0], (6)
is defined on [−b0, a]. By a maximal solution of a Cauchy problem we mean a solution which
majorizes any other solution of the same problem on the intersection of respective domains.
Remark 2.1. Condition 1 of Assumption H[Υ] states that the value of Υ at (t, υ) ∈ [0, a] ×
× C([−r, 0],R+ depends on t and on the restriction of υ to [−b0 − t, 0] only.
Assumption H[Fh, Gh, ψh]. The functions Fh : Ωh → Rκ, Gh : Ωh → R, ψh : E0.h∪∂0Eh →
→ R satisfy the conditions:
1) Fh and Gh are continuous and they satisfy condition (Vh),
2) there is Υ: [0, a]× C([−r, 0],R+)→ R+ such that Assumption H[Υ] is satisfied and∣∣Gh(t, x, w)
∣∣ ≤ Υ(t,Wh[w]) on Ωh,
3) for (t, x, w) ∈ Ωh we have
Fh.λ(t, x, w) ≥ 0 for λ ∈ Λ′ and
∑
λ∈Λ
Fh.λ(t, x, w) ≤ 0,
4) ψh ∈ Fc(E0.h∪∂0Eh,R) and ηh ∈ C([−r, 0],R+) satisfies the conditions:
∣∣ψ(m)
h (t)
∣∣ ≤ ηh(t)
on E0.h and ∣∣ψ(m)
h (t)
∣∣ ≤ ω(t, ηh) on ∂0Eh,
where ω( · , ηh) is the maximal solution to (6) with µ = ηh.
Lemma 2.1. If h ∈ H and Assumption H[Fh, Gh, ψh] is satisfied then there is a solution
zh : E0.h ∪ Eh → R to (4), (5) and∣∣z(m)
h (t)
∣∣ ≤ ω(t, ηh) on Eh. (7)
Proof. From classical theorems on functional differential equations it follows that there is ε̃ > 0
such that the solution zh to (4), (5) is defined on (E0.h ∪ Eh) ∩ ([−b0, ε̃)× Rn). Suppose that zh is
defined on (E0.h ∪ Eh) ∩ ([−b0, ã)× Rn) and it is non continuable. We prove that∣∣z(m)
h (t)
∣∣ ≤ ω(t, ηh) on Eh ∩ ([0, ã)× Rn). (8)
For ε > 0 we denote by ω( · , ηh, ε) the maximal solution of the Cauchy problem
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1368 W. CZERNOUS, Z. KAMONT
ω′(t) = Υ(t, ωt) + ε, ω(t) = ηh(t) + ε for t ∈ [−b0, 0].
There is ε̃ > 0 such that for 0 < ε < ε̃ the solution ω( · , ηh, ε) is defined on [−b0, ã] and
lim
ε→0
ω(t, ηh, ε) = ω(t, ηh) uniformly on [−b0, ã].
We prove that
|z(m)
h (t)|∞ < ω(t, ηh, ε) on Eh ∩ ([0, ã)× Rn), (9)
where 0 < ε < ε̃. Set
ωh(t) = max
{
|z(m)
h (t)| : m ∈ K
}
, t ∈ [−b0, ã).
It is clear that ωh(t) < ω(t, ηh, ε) for t ∈ [−b0, 0]. Suppose by contradiction that (9) fails to be true.
Then there is t ∈ (0, ã) such that
ωh(τ) < ω(τ, ηh, ε) for τ ∈ [−b0, t) and ωh(t) = ω(t, ηh, ε).
This gives
D−ωh(t) ≥ ω′(t, ηh, ε), (10)
where D− is the left-hand lower Dini derivative. There is m ∈ K such that ωh(t) = z
(m)
h (t)
or ωh(t) = −z(m)
h (t). Let us consider the first case. We deduce from condition 4 of Assumption
H[Fh, Gh, ψh] that m ∈ IntK. Then we have
D−ωh(t) ≤ d
dt
z
(m)
h (t) = Gh(t, x(m), (zh)[t,m]) + Fh(t, x(m), (zh)[t,m]) ◦ (zh)〈t,m〉 ≤
≤ Υ(t, ωt( · , ηh, ε)) + ωh(t)
∑
λ∈Λ
Fh.λ(t, x(m), (zh)[t,m]) < Υ(t, ωt( · , ηh, ε)) + ε = ω′(t, ηh, ε),
which contradicts (10). The case ωh(t) = −z(m)
h (t) can be treated in a similar way. Hence, the proof
of (9) is completed. From (9) we obtain in the limit, letting ε tend to 0, inequality (8).
We prove that there are the limits limt→ã z
(m)
h (t) for m ∈ Int K. Write
ω̂(t, t̃) = max
{∣∣z(m)
h (t)− z(m)
h (t̃ )
∣∣ : m ∈ IntK
}
,
where t, t̃ ∈ [0, ã). We prove that
ω̂(t, t̃) ≤
∣∣ω(t, ηh)− ω(t̃, ηh)
∣∣ for t, t̃ ∈ [0, ã). (11)
Suppose that t ≥ t̃. There is m ∈ IntK such that ω̂(t, t̃) = z
(m)
h (t)−z(m)
h (t̃) or ω̂(t, t̃ ) = −
[
z
(m)
h (t)−
− z(m)
h (t̃)
]
. Let us consider the first case. Then we have
ω̂(t, t̃) =
t∫
t̃
[
Gh(τ, x(m), (zh)[τ,m]) + Fh(τ, x(m), (zh)[τ,m]) ◦ (zh)〈τ,m〉
]
dτ ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1369
≤
t∫
t̃
Υ(τ, ωτ ( · , ηh)) dτ + ω(t, ηh)
t∫
t̃
∑
λ∈Λ
Fh.λ(τ, x(m), (zh)[τ,m]) dτ ≤ ω(t, ηh)− ω(t̃, ηh).
The case ω̂(t, t̃) = −
[
z
(m)
h (t)− z(m)
h (t̃)
]
can be treated in a similar way. Then (11) is proved.
It follows from (11) that there are the limits
lim
t→ã
z
(m)
h (t) = z
(m)
h (ã), m ∈ Int K.
Then the solution zh is defined on (E0.h ∪Eh) ∩
(
[−b0, ã]× Rn
)
. If ã < a then there is ā > ã such
that zh is defined on (E0.h ∪ Eh) ∩
(
[−b0, ā] × Rn
)
. This contradicts our assumption that zh given
on (E0.h ∪Eh) ∩
(
[−b0, ã)×Rn
)
is non continuable. It follows that zh is defined on E0.h ∪Eh and
estimate (7) is satisfied.
Lemma 2.1 is proved.
We will consider approximate solutions to (4), (5). Let Xh ⊂ Fc(Bh,R) and Yh ⊂ F (Ah,R) be
fixed subspaces. Suppose that z̃h ∈ Fc(E0.h ∪Eh,R) and there exists
d
dt
z̃
(m)
h (t) on Eh and there are
ϑ, γ : H → R+ such that∣∣∣∣ ddt z̃(m)
h (t)− Fh
(
t, x(m), (z̃h)[t,m], (z̃h)〈t,m〉
)∣∣∣∣ ≤ γ(h), m ∈ IntK, (12)
∣∣∣z̃(m)
h (t)− ψ(m)
h (t)
∣∣∣ ≤ ϑ(h) for (t, x(m)) ∈ E0.h ∪ ∂0Eh, (13)
lim
h→0
γ(h) = 0, lim
h→0
ϑ(h) = 0, (14)
and (
(z̃h)[t,m], (z̃h)〈t,m〉
)
∈ Xh × Yh for (t, x(m)) ∈ Eh. (15)
The function z̃h satisfying the above relations is treated as an approximate solution to (4), (5).
It is important in our considerations that we look for approximate solutions to (4), (5) such that
condition (15) is satisfied with a fixed subspaces Xh × Yh ⊂ Fc(Bh,R) × F (Ah,R). Remarks 2.2
and 3.3 contain additional comments on (15).
We give a theorem on the estimate of the difference between the exact and approximate solutions
to (4), (5).
Assumption H[σ]. The function σ : [0, a]× C([−r, 0],R+)→ R+ satisfies the conditions:
1) σ is continuous and for each t ∈ [0, a] the function σ(t, · ) : C([−r, 0],R+) → R+ is nonde-
creasing;
2) for t ∈ [0, a] and for υ, υ̃ ∈ C([−r, 0],R+) such that υ(τ) = υ̃(τ) for τ ∈ [−b0 − t, 0] we
have σ(t, υ) = σ(t, υ̃),
3) for each c ≥ 1 the maximal solution of the Cauchy problem
ω′(t) = c σ(t, ωt), ω(t) = 0 for t ∈ [−b0, 0],
is ω̃(t) = 0 for t ∈ [−b0, a].
Theorem 2.1. Suppose that
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1370 W. CZERNOUS, Z. KAMONT
1) h ∈ H and Assumption H[Fh, Gh, ψh] is satisfied and zh : E0.h ∪ Eh → R is the solution to
(4), (5) and
(zh)[t,m] ∈ Xh for (t, x(m)) ∈ Eh,
2) z̃h ∈ Fc(E0.h ∪ Eh,R), the derivatives
d
dt
z̃
(m)
h (t) exist on Eh and there are ϑ, γ : H → R+
such that condition (12) – (15) are satisfied,
3) there exists σ : [0, a] × C([−r, 0],R+) → R+ such that Assumption H[σ] is satisfied and for
w, w̃ ∈ Xh, ζ ∈ Yh we have∣∣Fh(t, x, w, ζ)− Fh(t, x, w̃, ζ)
∣∣ ≤ σ(t,Wh[w − w̃]),
where (t, x) ∈ Eh,
4) the maximal solution ω( · , γ, υ) of the Cauchy problem
ω′(t) = σ(t, ωt) + γ(h), ω(t) = ϑ(h) for t ∈ [−b0, 0], (16)
is defined on [−b0, a].
Under these assumptions we have∣∣z̃(m)(t)− z(m)
h (t)
∣∣ ≤ ω(t, γ, ϑ) on Eh. (17)
Proof. For ε > 0 we denote by ω( · , γ, ϑ, ε) the maximal solution of the Cauchy problem
ω′(t) = σ(t, ωt) + γ(h) + ε, ω(t) = ϑ(h) + ε for t ∈ [−r, 0].
There exists ε̃ > 0 such that for every 0 < ε < ε̃ the solution ω( · , γ, ϑ, ε) is defined on [−b0, a] and
lim
ε→0
ω(t, γ, ϑ, ε) = ω(t, γ, ϑ) uniformly on [−b0, a].
Set
ω̂h(t) = max
{∣∣z(m)
h (t)− z̃(m)
h (t)
∣∣ : m ∈ K
}
, t ∈ [−b0, a].
We prove that
ω̂h(t) < ω(t, γ, ϑ, ε) for t ∈ [−b0, a], (18)
where 0 < ε < ε̃. It is clear that ω̂h(t) < ω(t, γ, ϑ, ε) for t ∈ [−r, 0]. Suppose by contradiction that
assertion (18) fails to be true. Then the set
I+ =
{
t ∈ (0, a] : ω̂h(t) ≥ ω(t, γ, ϑ, ε)
}
is not empty. If we put t = min I+, it is clear that t > 0 and
D−ω̂h(t) ≥ ω′(t, γ, ϑ, ε). (19)
There is m ∈ K such that ω̂h(t) = z
(m)
h (t)−z̃(m)
h (t) or ω̂h(t) = −
[
z
(m)
h (t)−z̃(m)
h (t)
]
. Let us consider
the first case. We conclude from (13) that m ∈ IntK. It follows from Assumption H[Fh, Gh, ψh] that
D−ω̂h(t) ≤ d
dt
[
z
(m)
h (t)− z̃(m)
h (t)
]
≤
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1371
≤ σ(t, ωt( · , γ, ϑ), ε) + γ(h) + ω̂h(t)
∑
λ∈Λ
Fh.λ
(
t, x(m), (zh)[t,m]
)
.
This gives
D−ω̂(t) < σ(t, ω(t, γ, ϑ, ε)) + ε = ω′(t, γ, ϑ, ε)
which contradicts (19). The case ω̂h(t) = −
[
z
(m)
h (t)− z̃(m)
h (t)
]
can be treated in a similar way. Then
I+ = ∅ and (18) is proved. From (18) we obtain in the limit, letting ε tend to 0, inequality (17).
Theorem 2.1 is proved.
Remark 2.2. Let us consider the following condition:
3A) there exists σ : [0, a] × C([−r, 0],R+) → R+ such that Assumption H[σ] is satisfied and
for w, w̃ ∈ Fc(Bh,R) and for ζ ∈ F (Ah,R) we have∣∣Fh(t, x, w, ζ)− Fh(t, x, w̃, ζ)
∣∣ ≤ σ(t,Wh[w − w̃]),
where (t, x) ∈ Eh, It is clear that Theorem 2.1 remains true if condition 3 is replaced by 3A. We
show in Sections 3 and 4 that assumption 3, is important in our considerations. The operators Fh
generated by (1) or (3) satisfy condition 3 and they do not satisfy 3A.
Remark 2.3. Suppose that the assumptions 1, 2 of Theorem 2.1 are satisfied and there exists
L ∈ R+ such that and for w, w̃ ∈ Xh, ζ ∈ Yh we have∣∣Fh(t, x, w, ζ)− Fh(t, x, w̃, ζ)
∣∣ ≤ L‖w − w̃‖,
where (t, x) ∈ Eh. Then ∣∣z̃(m)(t)− z(m)
h (t)
∣∣ ≤ α̃(h) on Eh
where
α̃(h) = ϑ(h)eLa + γ(h)
eLa − 1
L
if L > 0, (20)
and
α̃(h) = ϑ(h) + aγ(h) if L = 0. (21)
The above estimates are obtained by solving problem (16).
3. First order partial functional differential equations. We construct a numerical method of
lines for (1), (2). The following assumptions of given functions are needed in our considerations.
Assumption H0[F,G]. The functions F : Ω→ Rn, G : Ω→ R are continuous and they satisfy
condition (V ) and there is x̃ ∈ (−b, b), x̃ = (x̃1, . . . , x̃n), such that
(xi − x̃i)Fi(t, x, w) ≥ 0 on Ω for 1 ≤ i ≤ n. (22)
Remark 3.1. Two types of assumptions are needed in theorems on the existence and uniqueness
of classical or generalized solutions to (1), (2). The first type conditions deal with regularity of given
functions. It is assumed in theorems on uniqueness of solutions that F and G are continuous and they
satisfy nonlinear estimates of the Perron-type with respect to functional variables. More restrictive
conditions are needed in theorems on the existence of solutions.
The assumptions of the second type concern the bicharacteristics and they have the following
form. Suppose that z ∈ C(E0 ∪ E,R). Let us denote by g[z]( · , t, x) = (g1( · , t, x), . . . , gn( · , t, x))
the solution of the Cauchy problem
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1372 W. CZERNOUS, Z. KAMONT
y′(τ) = F (τ, y(τ), z(τ,y(τ))), y(t) = x,
where (t, x) ∈ E. Condition (22) asserts that the function gi( · , t, x) is non increasing if x̃i ≤ xi ≤ bi
and it is non decreasing if −bi ≤ xi < x̃i. This property of bicharacteristics and assumptions on
regularity of given functions ensure the existence and uniqueness of classical or generalized solutions
to (1), (2) (see [6] and [11] (Chapter 5)).
Let us denote by Ĥ the set of all h ∈ H satisfying the condition ‖h‖ < min {b?, b?}, where
b? = min {bi − x̃i : 1 ≤ i ≤ n} and b? = min {bi + x̃i : 1 ≤ i ≤ n}.
Solutions of differential difference equations corresponding to (1), (2) are defined on E0.h ∪ Eh.
Equation (1) contains the functional variable z(t,x) which is an element of the space C(D[t, x],R).
Then we need an interpolating operator Th : Fc(Bh,R)→ C(B,R). We assume that Th satisfies the
following condition (V): if w, w̃ ∈ Fc(Bh,R), (t, x(m)) ∈ Eh and w(τ, y) = w̃(τ, y) for (τ, y) ∈
∈ Dh[t,m] then (Thw)(τ, y) = (Thw̃)(τ, y) for (τ, y) ∈ D[t, x(m)]. In the next part of the paper we
adopt additional assumptions on Th.
Write ei = (0, . . . , 0, 1, 0, . . . , 0) ∈ Rn with 1 standing on the i th place. For z : E0.h ∪ Eh → R
and (t, x(m)) ∈ Eh, m ∈ IntK, we write
δ
(m)
i (t) =
1
hi
[
z(m+ei)(t)− z(m)(t)
]
if x
(mi
i ≥ x̃i,
δ
(m)
i (t) =
1
hi
[
z(m)(t)− z(m−ei)(t)
]
if x
(mi
i < x̃i,
and we put i = 1, . . . , n in the above definitions. Set
Fh[z](m)(t) =
n∑
i=1
Fi(t, x
(m), Thz[t,m])δiz
(m)(t) +G(t, x(m), Thz[t,m])
and suppose that ψh : E0.h ∪ ∂0Eh → R is a given function. We approximate classical solutions to
(1), (2) with solutions of differential difference equations
d
dt
z(m)(t) = Fh[z](m)(t), m ∈ IntK, (23)
with the initial boundary conditions
z(m)(t) = ψ
(m)
h (t) on E0.h ∪ ∂0Eh. (24)
We claim that we have obtained a functional differential problem which is a particular case of (4),
(5). For m ∈ IntK we put
I+[m] =
{
i ∈ {1, . . . , n} : x̃i ≤ x(mi)
i < bi
}
,
I−[m] =
{
i ∈ {1, . . . , n} : −bi < x
(mi)
i < x̃i
}
and
Fh(t, x(m), w, ζ) = G(t, x(m), Thw) +
n∑
i=1
Fi(t, x
(m), Thw)δiζ
(θ), (25)
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1373
where (t, x(m), w, ζ) ∈ Ωh × F (Ah,R). The expressions δiζ(θ), 1 ≤ i ≤ n, are defined in the
following way:
δiζ
(θ) =
1
hi
[
ζ(ei) − ζ(θ)
]
for i ∈ I+[m],
δiζ
(θ) =
1
hi
[
ζ(θ) − ζ(−ei)
]
for i ∈ I−[m],
and we put i = 1, . . . , n in the above formulas. It is clear that problem (23), (24) is equivalent to (4),
(5) with the above defined Fh.
We formulate assumptions on F, G, ψ and Th. Let us denote by W : C(B,R)→ C([−r, 0],R+)
the operator given by
W [w](t) = max
{
|w(t, x)| : x ∈ [−2b, 2b]
}
, t ∈ [−r, 0].
AssumptionH?[F,G, ψ]. The functions F : Ω→ Rn, G : Ω→ R satisfy AssumptionH0[F,G]
and
1) there is Υ: [0, a]× C([−r, 0],R+)→ R+ such that Assumption H[Υ] is satisfied and∣∣G(t, x, w)
∣∣ ≤ Υ(t,W [w]) on Ω,
2) ψ ∈ C(E0 ∪ ∂0E,R), ψh ∈ Fc(E0.h ∪ ∂0E,R) and there is α0 : Ĥ → R+ such that∣∣ψ(m)(t)− ψ(m)
h (t)
∣∣ ≤ α0(h) on E0.h ∪ ∂0Eh and lim
h→0
α0(h) = 0,
3) η ∈ C([−b0, 0],R+) and∣∣ψ(m)(t)
∣∣ ≤ η(t),
∣∣ψ(m)
h (t)
∣∣ ≤ η(t) on E0.h
and the maximal solution ω( · , η) to (6) with µ = η satisfies the conditions∣∣ψ(m)(t)
∣∣ ≤ ω(t, η),
∣∣ψ(m)
h (t)
∣∣ ≤ ω(t, η) on ∂0Eh.
Assumption H[Th]. The operator Th : Fc(Bh,R)→ C(B,R) satisfies the conditions:
1) if w, w̃ ∈ Fc(Bh,R), (t, x(m)) ∈ Eh and w(τ, y) = w̃(τ, y) for (τ, y) ∈ Dh[t,m], then
(Thw)(τ, y) = (Thw̃)(τ, y) for (τ, y) ∈ D[t, x(m)],
2) for w, w̃ ∈ Fc(Bh,R) we have
‖Thw − Thw̃‖B ≤ ‖w − w̃‖Bh
,
3) if θh : Bh → R is given by θh(τ, y) = 0 for (τ, y) ∈ Bh, then Th[θh](τ, y) = 0 for (τ, y) ∈ B,
4) for each w : B → R which is of class C1 there is c̃ ∈ R+ such that
‖w − Thwh‖B ≤ c̃‖h‖,
where wh is the restriction of w to Bh.
Remark 3.2. An example of the operator Th satisfying Assumption H[Th] can be found in [11],
Chapter 5.
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1374 W. CZERNOUS, Z. KAMONT
Lemma 3.1. If Assumptions H?[F,G, ψ], H[Th] are satisfied then for each h ∈ Ĥ there exists
a solution ẑh : E0.h ∪ Eh → R to (23), (24) and∣∣ẑ(m)
h (t)
∣∣ ≤ ω(t, η) on E.
Proof. We apply Lemma 2.1. Let us define Gh : Ωh → R, Fh : Ωh → Rκ, Fh = {Fh.λ }λ∈Λ, in
the following way:
Gh(t, x(m), w) = G(t, x(m), Thw),
Fh,θ(t, x
(m), w) =
∑
i∈I−[m]
1
hi
Fi(t, x
(m), Thw)−
∑
i∈I+[m]
1
hi
Fi(t, x
(m), Thw)
and
Fh.ei(t, x
(m), w) =
1
hi
Fi(t, x
(m), Thw) for i ∈ I+[m],
Fh.−ei(t, x
(m), w) = − 1
hi
Fi(t, x
(m), Thw) for i ∈ Ii[m]
and we put i = 1, . . . , n in the above definitions. Set
Fh.λ(t, x(m), w) = 0 for λ ∈ Λ \
{
I+[m] ∪ I−[m] ∪ { θ }
}
,
and Fh(t, x(m), w, ζ) = Gh(t, x(m), w) + Fh(t, x(m), w) ◦ ζ. Then Assumption H[Fh, Gh, ψh] is
satisfied. Our lemma follows from Lemma 2.1.
Now we construct estimates of solutions to (1), (2).
Lemma 3.2. Suppose that Assumption H?[F,G, ψ] is satisfied and v : E0∪E → R is a solution
to (1), (2) and v is of class C1 on E. Then
|v(t, x)| ≤ ω(t, η) on E. (26)
Proof. For ε > 0 we denote by ω( · , η, ε) the maximal solution of the Cauchy problem
ω′(t) = Υ(t, ωt) + ε, ω(t) = η(t) + ε for t ∈ [−b0, 0]. (27)
There is ε0 > 0 such that for 0 < ε < ε0 the solution ω( · , η, ε) is defined on [−b0, a] and
lim
ε→0
ω(t, η, ε) = ω(t, η) uniformly on [−b0, a].
Write
ω̃(t) = max
{
|v(t, x)| : x ∈ [−b, b]
}
, t ∈ [−b0, a].
We prove that
ω̃(t) < ω(t, η, ε) for t ∈ [−b0, a]. (28)
It is clear that ω̃(τ) < ω(τ, η, ε) for τ ∈ [−b0, 0]. Suppose by contradiction that (28) fails to be true.
Then there is t ∈ (0, a] such that
ω̃(τ) < ω(τ, η, ε) for τ ∈ [0, t) and ω̃(t) = ω(t, η, ε).
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1375
Then we have
D−ω̃(t) ≥ ω′(t, η, ε). (29)
There is x ∈ [−b, b] such that ω̃(t) = |v(t, x)|. It follows from condition 3 of AssumptionH?[F,G, ψ]
that (t, x) 6∈ ∂0E and consequently ∂xv(t, x) = θ. Let us consider the case when ω̃(t) = v(t, x).
Then we have
D−ω̃(t) ≤ ∂tv(t, x) ≤ Υ(t, ω̃t) < υ(t, ωt( · , η, ε) + ε = ω′(t, η, ε),
which contradicts (29). The case ω̃(t) = −v(t, x) can be treated in a similar way. This completes the
proof of (28). From (28) we obtain in the limit, letting ε tend to 0, estimate (26).
Lemma 3.2 is proved.
Suppose that Assumptions H?[F,G, ψ] and H[Th] are satisfied. Write ĉ = ω(a, η) and
X[ĉ] =
{
w ∈ C(B,R) : ‖w‖B ≤ ĉ
}
.
Assumption H[F,G, ψ]. The functions F : Ω → Rn, G : Ω → R, ψ : E0 ∪ ∂0E → R satisfy
Assumption H?[F,G, ψ] and there is σ : [0, a] × C([−r, 0],R+) → R+ such that Assumption G[σ]
is satisfied and ∥∥F (t, x, w)− F (t, x, w̃)
∥∥ ≤ σ(t,W [w − w̃]),∣∣G(t, x, w)−G(t, x, w̃)
∣∣ ≤ σ(t,W [w − w̃]),
for (t, x) ∈ E and w, w̃ ∈ X[ĉ].
Remark 3.3. It is important that we have assumed nonlinear estimates of the Perron-type for
‖w‖B, ‖w̃‖B ≤ ĉ. There are differential equations with deviated variables and differential integral
equations such that Assumption H[F,G, ψ holds and global estimates are not satisfied. We give
suitable examples.
Suppose that F̃ : E×R→ Rn, F̃ = (F̃1, . . . , F̃n), and G̃ : E×R→ R are given functions of the
variables (t, x, p). Suppose that φ0 ∈ C(E,R), φ ∈ C(E,Rn) and 0 ≤ φ0(t, x) ≤ t, φ(t, x) ∈ [−b, b]
for (t, x) ∈ E. Write ϕ(t, x) = (φ0(t, x), φ(t, x)) on E. Let F : Ω→ Rn, G : Ω→ R be defined by
F (t, x, w) = F̃ (t, x, w(ϕ(t, x)− (t, x)), G(t, x, w) = G̃(t, x, w(ϕ(t, x)− (t, x)). (30)
Then (1) reduces to the differential equation with deviated variables
∂tz(t, x) =
n∑
i=1
F̃i(t, x, z(ϕ(t, x)))∂xiz(t, x) + G̃(t, x, z(ϕ(t, x))).
For the above F̃ and G̃ we put
F (t, x, w) = F̃
t, x, ∫
D[t,x]
w(τ, y)dy dτ
, G(t, x, w) = G̃
t, x, ∫
D[t,x]
w(τ, y)dy dτ
. (31)
Then (1) is equivalent to the differential integral equation
∂tz(t, x) =
n∑
i=1
F̃i
t, x, t∫
−b0
x∫
−x
z(τ, y)dy dτ
∂xiz(t, x) + G̃
t, x, t∫
−b0
x∫
−x
z(τ, y)dy dτ
.
Suppose that
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1376 W. CZERNOUS, Z. KAMONT
1) F̃ ∈ C(E × R,Rn), G̃ ∈ C(E × R,R) and there are α, β ∈ R+ such that∣∣G̃(t, x, p)
∣∣ ≤ α+ β|p| on E × R,
2) the derivatives ∂pF̃ , ∂pG̃ exist on E × R and ∂pF̃ ∈ C(E × R,Rn), ∂pG̃ ∈ C(E × R,R),
3) the functions ∂pF̃ and ∂pG̃ are unbounded on E × R.
Then the functions F and G given by (30) and by (31) satisfy Assumption H[F,G, ψ] and they
do not satisfy global estimates with respect to functional variables.
Theorem 3.1. Suppose that Assumption H[F,G, ψ] and H[Th] are satisfied and
1) v : E0 ∪ E → R is a solution to (1), (2) and v is of class C1,
2) h ∈ Ĥ and zh : E0.h ∪ Eh → R is a solution to (23), (24).
Then there is α : Ĥ → R+ such that∣∣v(m)
h (t)− z(m)
h (t)
∣∣ ≤ α(h) on Eh and lim
h→0
α(h) = 0, (32)
where vh is the restriction of v to E0.h ∪ Eh.
Proof. We apply Theorem 2.1 to prove (32). Let Γh be defined by the relations
d
dt
v
(m)
h (t) = Fh[vh](m)(t) + Γ
(m)
h (t), t ∈ [0, a], m ∈ IntK.
It follows from Assumption H[Th] and from the definition of δvh = (δ1vh, . . . , δnvh) that there is
γ : Ĥ → R+ such that∣∣Γ(m)
h (t)
∣∣ ≤ γ(h) for t ∈ [0, a], m ∈ IntK and lim
h→0
γ(h) = 0.
There is c̄ ∈ R+ such that ‖∂xv(t, x)‖ ≤ c̄ on E. Let us denote by Yh that class of all ζ ∈ F (Ah,R)
such that ∣∣∣∣ 1
hi
[
ζ(ei) − ζ(θ)
]∣∣∣∣ ≤ c̄, ∣∣∣∣ 1
hi
[
ζ(θ) − ζ(−ei)
]∣∣∣∣ ≤ c̄, 1 ≤ i ≤ n.
Then (vh)〈t,m〉 ∈ Xh for (t, x(m) ∈ Eh. It follows from Lemmas 3.1 and 3.2 that
(vh)[t,m], (zh)[t,m] ∈ Xh and (vh)〈t,m〉 ∈ Yh for (t, x(m)) ∈ Eh.
The operator Fh given by (25) satisfies the condition:∣∣Fh(t, x, w, ζ)− Fh(t, x, w̃, ζ)
∣∣ ≤ (1 + c̄)σ(t,Wh[w − w̃]) (33)
for (t, x) ∈ Eh, w, w̃ ∈ Xh, ζ ∈ Yh. Then all the assumptions of Theorem 2.1 are satisfied and
assertion (32) follows.
Remark 3.4. Note that estimate (33) is not satisfied for all ζ ∈ F (Ah,R).
Let us suppose that all the assumptions of Theorem 3.1 are satisfied and there is L̃ ∈ R+ such
that
‖F (t, x, w)− F (t, x, w̃)‖, |G(t, x, w)−G(t, x, w̃)| ≤ L̃‖w − w̃‖B,
for (t, x) ∈ E, w, w̃ ∈ X[ĉ]. Then there is L ∈ R+ such that∣∣v(m)
h (t)− z(m)
h (t)
∣∣ ≤ α̃(h) on Eh,
where α̃ : Ĥ → R+ is given by (20), (21).
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1377
We apply the results on the numerical method of lines to differential equations with deviated
variables and to differential integral equations. We have transformed initial boundary-value problems
into systems of ordinary functional differential equations. The system such obtained is solved by
using the explicit Euler method. Let us denote by εh the maximal error of the difference method. In
the tables we give experimental values for εh.
Example 3.1. Put n = 2 and E = [0, 0.25]× [−0.5, 0.5]× [−0.5, 0.5]. Consider the differential
equation
∂tz(t, x, y) = x
1 + cosπy
0.5∫
−0.5
z(t, x, s)ds− 2
π
z(t, x, y)
2
∂xz(t, x, y)+
+y
1− cosπx
0.5∫
−0.5
z(t, s, y)ds+
2
π
z(t, x, y)
2
∂yz(t, x, y)+
+xπ2
x∫
0
z(t, s, y)ds+ yπ2
y∫
0
z(t, x, s)ds+ z(t, x, y) + cosπx cosπy (34)
with the initial boundary conditions
z(0, x, y) = 0, (x, y) ∈ [−0.5, 0.5]× [−0.5, 0.5], (35)
z(t,−0, 5, y) = z(t, 0.5, y) = 0, t ∈ [0, 0.25], y ∈ [−0.5, 0.5], (36)
z(t, x,−0.5) = z(t, x, 0.5) = 0, t ∈ [0, a], x ∈ [−0.5, 0.5]. (37)
The solution to (34) – (37) is known. It is v(t, x, y) = (et − 1) cosπx cosπy. Table 3.1 gives the
maximal errors for several step sizes h = (h0, h1, h2).
Table 3.1
(h0, h1, h2) εh Time [s]
(2−8, 2−6, 2−6) 1.47690152 · 10−3 0.037
(2−9, 2−7, 2−7) 7.56321595 · 10−4 0.196
(2−10, 2−8, 2−8) 3.83651338 · 10−4 4.731
(2−11, 2−9, 2−9) 1.93483845 · 10−4 41.351
(2−12, 2−10, 2−10) 9.72385359 · 10−5 497.0004
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1378 W. CZERNOUS, Z. KAMONT
Example 3.2. Put n = 2, E = [0, 0.25]× [−1, 1]× [−1, 1]. Consider the equation
∂tz(t, x, y) =
x
4
[
1 + z(t, sin(x+ y), cos(x+ y))
]2
∂xz(t, x, y)+
+
y
4
[
1 + z(t, sin(x− y), cos(x− y))
]2
∂yz(t, x, y)+
+z(t, 0.5(x+ y), 0.5(x− y)) sin z(t, x, y) + f(t, x, y)z(t, x, y),
(38)
f(t, x, y) = (x2 + y2)(1− 2t)− 1− exp
{
−t
(
x2
2
+
x2
2
)}
sin exp
{
t(x2 + y2 − 1)
}
with the initial boundary conditions
z(0, x, y) = 1, (x, y) ∈ [−1, 1], (39)
z(t,−1, y) = z(t, 1, y) = exp{ty2}, (t, y) ∈ [0, a]× [−1, 1], (40)
z(t, x,−1) = z(t, x, 1) = exp{tx2}, (t, x) ∈ [0, a]× [−1, 1]]. (41)
The solution to (38) – (41) is known. It is z̃(t, x, y) = exp{t(x2 + y2 − 1)}. Table 3.2 gives the
maximal errors for several step sizes h = (h0, h1, h2).
Table 3.2
(h0, h1, h2) εh Time [s]
(2−6, 2−5, 2−5) 1.47049139 · 10−3 0.048
(2−7, 2−6, 2−6) 7.53238436 · 10−4 0.319
(2−8, 2−7, 2−7) 3.81321493 · 10−4 2.924
(2−9, 2−8, 2−8) 1.91826460 · 10−4 26.837
(2−10, 2−9, 2−9) 9.62069745 · 10−5 255.386
Note that the right-hand sides of equations (34) and (38) satisfy the assumptions of Theorem 3.1.
The local Lipschitz condition with respect to unknown function holds and the global Lipschitz con-
dition is not satisfied.
4. Parabolic functional differential equations. We formulate a differential difference problem
corresponding to (3), (2). Write
J =
{
(i, j) : i, j = 1, . . . , n, i 6= j
}
and suppose that we have defined the sets J+, J− ⊂ J such that J+ ∪ J− = J, J+ ∩ J− = ∅.
We assume that (i, j) ∈ J+ if (j, i) ∈ J+. In particular, it may happen that J+ = ∅ or J− = ∅.
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1379
Relations between the sets J+, J− and equation (3) are given in Remark 4.1. Let us denote by H̃ the
set of all h ∈ H satisfying the condition: there is d̃ > 0 such that hi ≤ d̃hj for (i, j) ∈ J.
For z : E0.h ∪ Eh → R, (t, x(m)) ∈ Eh, m ∈ IntK we write
δ+
i z
(m)(t) =
1
hi
[
z(m+ei) − z(m)(t)
]
, δ−i z
(m)(t) =
1
hi
[
z(m)) − z(m−ei)(t)
]
,
where i = 1, . . . , n. The difference operators δ = (δ1, . . . , δn) and δ(2) =
[
δij
]
i,j=1,...,n
are defined
in the following way. Set
δiz
(m)(t) =
1
2
[
δ+
i z
(m)(t) + δ−i z
(m)(t)
]
, δiiz
(m)(t) = δ+
i δ
−
i z
(m)(t), 1 ≤ i ≤ n,
and
δijz
(m)(t) =
1
2
[
δ+
i δ
−
j z
(m)(t) + δ−i δ
+
j z
(m)(t)
]
for (i, j) ∈ J−,
δijz
(m)(t) =
1
2
[
δ+
i δ
+
j z
(m)(t) + δ−i δ
−
j z
(m)(t)
]
for (i, j) ∈ J+.
Let Th : Fc(Bh,R)→ C(B,R) be an interpolating operator. Write
Fh[z](m)(t) =
n∑
i,j=1
Fij(t, x
(m), Thz[t,m])δijz
(m)(t)+
+
n∑
i=1
Fi(t, x
(m), Thz[t,m])δiz
(m)(t) +G(t, x(m), Thz[t,m])δijz
(m)(t)
and suppose that ψh : E0.h ∪ Eh → R is a given function. We consider the functional differential
equations
d
dt
= Fh[z](m)(t), m ∈ IntK, (42)
with the initial boundary condition
z(m)(t) = ψ
(m)
h (t) on E0.h ∪ E∂0Eh. (43)
We will approximate classical solutions of (3), (2) with solutions to (42), (43).
We claim that (42), (43) is a particular case of (4), (5). Write
Fh(t, x, w, ζ) = G(t, x, Thw) +
n∑
i=1
Fi(t, x, Thw)δiζ
(θ) +
n∑
i,j=1
Fij(t, x, Thw)δijζ
(θ), (44)
where (t, x, w) ∈ Ωh, ζ ∈ F (Ah,R). The expressions
δζ(θ) =
(
δ1ζ
(θ), . . . , δnζ
(θ)
)
, δ(2)ζ(θ) =
[
δijζ
(θ)
]
i,j=1,...,n
are defined in the following way. Put
δ+
i ζ
(θ) =
1
hi
[
ζ(ei) − ζ(θ)
]
, δ−i ζ
(θ) =
1
hi
[
ζ(θ) − ζ(−ei)
]
, 1 ≤ i ≤ n.
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1380 W. CZERNOUS, Z. KAMONT
Write
δiζ
(θ) =
1
2
[
δ+
i ζ
(θ) + δ−i ζ
(θ)
]
, δiiζ
(θ) = δ+
i δ
−
i ζ
(θ), 1 ≤ i ≤ n,
and
δijζ
(θ) =
1
2
[
δ+
i δ
−
j ζ
(θ) + δ−i δ
+
j ζ
(θ)
]
for (i, j) ∈ J−,
δijζ
(θ) =
1
2
[
δ+
i δ
+
j ζ
(θ) + δ−i δ
−
j ζ
(θ)
]
for (i, j) ∈ J+.
It is easy to see that problem (42), (43) with the above given Fh is equivalent to (4), (5).
Assumption H0 [F, F,G, ψ]. The functions F : Ω → Mn×n, F : Ω → Rn, G : Ω → R and
ψ : E0 ∪ ∂0E → R satisfy the conditions
1) F, F, G are continuous and they satisfy condition (V ),
2) there is Υ: [0, a]× C([−r, 0],R+)→ R+ such that Assumption H[Υ] is satisfied and∣∣G(t, x, w)
∣∣ ≤ Υ(t,Ww) on Ω,
3) the matrix F is symmetric and for P = (t, x, w) ∈ Ω we have
Fij(t, x, w) ≤ 0 for (i, j) ∈ J−, Fij(t, x, w) ≥ 0 for (i, j) ∈ J+, (45)
and
n∑
i,j=1
Fij(t, x, w)yiyj ≥ 0 for y = (y1, . . . , yn) ∈ Rn, (46)
4) h ∈ H̃ and for P = (t, x, w) ∈ Ω we have
Fii(P )−
n∑
j=1
j 6=i
hi
hj
∣∣FijF (P )
∣∣− hi
2
∣∣Fi(P )
∣∣ ≥ 0, 1 = 1, . . . , n,
5) ψ ∈ C(E0 ∪ ∂0E,R), ψh ∈ Fc(E0.h ∪ ∂0Eh,R) and there is α0 : H̃ → R+ such that∣∣ψ(m)(t)− ψ(m)
h (t)
∣∣ ≤ α0(t) on E0.h ∪ ∂0Eh and lim
h→0
α0(h0 = 0,
6) η ∈ C([−r, 0],R+) and
|ψ(m)(t)| ≤ η(t), |ψ(m)
h (t)| ≤ η(t) on E0.h
and the maximal solution ω( · , η) to (6) with µ = η satisfies the conditions:
|ψ(m)(t)| ≤ ω(t, η), |ψ(m)
h (t)| ≤ ω(t, η) on ∂0E0.h.
Remark 4.1. We have assumed that F satisfies the condition: for each (i, j) ∈ J the function
F̂ij(t, x, w) = sign Fij(t, x, w), (t, x, w) ∈ Ω,
is constant. Inequalities (45) can be considered as definitions of J− and J+.
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1381
Suppose that there is â > 0 such that
Fii(t, x, w)−
n∑
j=1
j 6=i
∣∣Fij(P )
∣∣ ≥ â on Ω for 1 ≤ i ≤ n.
Then condition (46) is satisfied and there is ε0 > 0 such that for ‖h‖ < ε0 and for h1 = h2 = . . .
. . . = hn condition 4 of Assumption H0[F, F,G, ψ] is satisfied.
Lemma 4.1. If Assumptions H0[F, F,G, ψ], H[Th] are satisfied then for each h ∈ H̃ there
exists a solution z̃h : E0.h ∪ Eh → R to (42), (43) and
|z̃(m)
h (t)| ≤ ω(t, η) on Eh.
Proof. We apply Lemma 2.1. Let us define Fh : Ωh → Rκ, Fh = {Fh.λ}λ∈Λ and Gh : Ωh → R
in the following way. Write
Λ0 =
{
λ ∈ Λ: there is i, 1 ≤ i ≤ n, such that λ = ei or λ = −ei
}
,
ΛI =
{
λ ∈ Λ: there is (i, j) ∈ J+ such that λ = ei + ej or λ = −ei − ej
}
,
ΛII =
{
λ ∈ Λ: there is (i, j) ∈ J− such that λ = ei − ej or λ = −ei + ej
}
and
Λ̃ = Λ \
[
Λ0 ∪ ΛI ∪ ΛII ∪ { θ }
]
.
Write
Gh(t, x, w) = G(t, x, Thw),
Fh.θ(t, x, w) = −2
n∑
i=1
1
h2
i
Fii(t, x, Thw) +
∑
(i,j)∈J
1
hihj
∣∣Fij(t, x, Thw)
∣∣
and
Fh.ei(t, x, w) =
1
h2
i
Fii(t, x, Thw)−
n∑
j=1
j 6=i
1
hihj
∣∣Fij(t, x, Thw)
∣∣+
1
2hi
Fi(t, x, Thw),
Fh.−ei(t, x, w) =
1
h2
i
Fii(t, x, Thw)−
n∑
j=1
j 6=i
1
hihj
∣∣Fij(t, x, Thw)
∣∣− 1
2hi
Fi(t, x, Thw),
and we put i = 1, . . . , n in the above formulas. Moreover we put
Fh.ei+ej (t, x, w) = Fh.−ei−ej (t, x, w) =
1
2hihj
Fij(t, x, Thw) for (i, j) ∈ J+,
Fh.ei−ej (t, x, w) = Fh.−ei+ej (t, x, w) = − 1
2hihj
Fij(t, x, Thw) for (i, j) ∈ J−,
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1382 W. CZERNOUS, Z. KAMONT
Fh,λ(t, x, w) = 0 for λ ∈ Λ̃,
and Fh(t, x(m), w, ζ) = Gh(t, x(m), w) + Fh(t, x(m)w) ◦ ζ. It follows that all the assumption of
Lemma 2.1 are satisfied and the assertion follows.
Now we construct estimates of solutions to (3), (2). We say that z ∈ C(E0 ∪ E,R) is of class
C1.2 if z( · , x) : [−b0, a] → R is of class C1 for x ∈ [−b, b] and z(t, · ) : [−b, b] → R is of class C2
for t ∈ [−b0, a].
Lemma 4.2. If Assumption H0[F, F,G, ψ] is satisfied and v : E0 ∪ E → R is a solution to (3),
(2) and v is of class C1.2 then
|v(t, x)| ≤ ω(t, η) on E. (47)
Proof. For ε > 0 we denote by ω( · , η, ε) the maximal solution to (27). There is ε0 > 0 such
that for 0 < ε < ε0 the solution ω( · , η, ε) is defined on [−b0, a] and
lim
ε→0
ω(t, η, ε) = ω(t, η) uniformly on [−b0, a].
Write
ω̂(t) = max
{
|v(t, x)| : x ∈ [−b, b]
}
, t ∈ [−b0, a].
We prove that
ω̂(t) < ω(t, η, ε) for t ∈ [−b0, a], (48)
where 0 < ε < ε0. It is clear that ω̂(t) < ω(t, η, ε) for t ∈ [−b0, 0]. Suppose by contradiction that
(48) fails to be true. Then there is t ∈ (0, a] such that
ω̂(τ) < ω(τ, η, ε) for τ ∈ [−b0, t) and ω̂(t) = ω(t, η, ε).
This gives
D−ω̂(t) ≥ ω′(t, η, ε). (49)
There is x ∈ [−b, b] such that ω̂(t) = |v(t, x)|. It follows from condition 6 of Assumption H0[F, F,
G, ψ] that (t, x) 6∈ ∂0E. Let us consider the case when ω̂(t) = v(t, x). Then we have
∂xv(t, x) = θ and
n∑
i,j=1
∂xixjv(t, x)yiyj ≤ 0 for y = (y1, . . . , yn) ∈ Rn.
The above relations and (46) imply
n∑
i,j=1
Fij(t, x, v(t,x))∂xixjv(t, x) ≤ 0
and consequently
D−ω̂(t) ≤ ∂tv(t, x) ≤ Υ(t, ω̂t) < ω′(t, η, ε),
which contradicts (49). The case ω̂(t) = −v(t, x) can be treated in a similar way. This completes the
proof of (48). From this we obtain in the limit, letting ε tend to 0, estimate (47).
Lemma 4.2 is proved.
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1383
Now we prove that the method of lines (42), (43) is convergent. Suppose that Assumption
H0[F, F,G, ψ] and H[Th] are satisfied. Write ĉ = ω(a, η) and
X[ĉ] =
{
w ∈ C(B,R) : ‖w‖B ≤ ĉ
}
.
Assumption H[F, F,G, ψ]. The functions F : Ω→Mn×n, F : ω → Rn, G : Ω→ R, ψ : E0 ∪
∪ ∂0E satisfy Assumption H0[F, F,G, ψ] and there is σ : [0, a] × C([−r, 0],R+) → R+ such that
Assumption H[σ] is satisfied and the expressions
‖F(t, x, w)− F(t, x, w̃)‖n×n, ‖F (t, x, w)− F (t, x, w̃)‖,
∣∣G(t, x, w)−G(t, x, w̃)
∣∣
for are bounded from above by σ(t,W [w − w̃]), where (t, x) ∈ E, w, w̃ ∈ X[ĉ].
Remark 4.2. It is important that we have assumed nonlinear estimates of the Perron-type for
‖w‖B, ‖w̃‖B ≤ ĉ. There are differential equations with deviated variables and differential integral
equations such that Assumption H[F, F,G, ψ] holds and global estimates are not satisfied. Example
given in Section 3 for first order partial functional differential equations can be extended on parabolic
problems.
Theorem 4.1. Suppose that Assumption H[F, F,G, ψ] and H[Th] are satisfied and
1) v : E0 ∪ E → R is a solution to (3), (2) and v is of class C1.2,
2) h ∈ H̃ and zh : E0.h ∪ Eh → R is a solution to (42), (43).
Then there is α : H̃ : → R+ such that∣∣v(m)
h (t)− z(m)
h (t)
∣∣ ≤ α(h) on Eh and lim
h→0
α(h) = 0. (50)
Proof. We apply Theorem 2.1 to prove (50). We start with the observation that
δijv
(m)
h (t) =
1
2
1∫
0
1∫
0
∂xjxjv(t, x(m) + τhiei + νhjej) dτ dν×
×1
2
1∫
0
1∫
0
∂xjxjv(t, x(m) − τhiei − νhjej) dτ dν,
where (i, j) ∈ J+ and
δijv
(m)
h (t) =
1
2
1∫
0
1∫
0
∂xjxjv(t, x(m) + τhiei − νhjej) dτ dν×
×1
2
1∫
0
1∫
0
∂xjxjv(t, x(m) − τhiei + νhjej) dτ dν,
where (i, j) ∈ J+. Let Γh be defined by the relations
d
dt
v
(m)
h (t) = Fh[vh](m)(t) + Γ
(m)
h (t), t ∈ [0, a], m ∈ IntK.
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1384 W. CZERNOUS, Z. KAMONT
It follows from the above relation and from Assumption H[Th] that there is γ : H̃ → R+ such that∣∣Γ(m)
h (t)
∣∣ ≤ γ(h) for t ∈ [0, a], m ∈ IntK and lim
h→0
γ(h) = 0.
We conclude that vh satisfies (12) – (14). There is c̄ ∈ R+ such that
‖∂xv(t, x)‖ ≤ c̄, ‖∂xxv(t, x)‖n×n ≤ c̄ on E.
Let us denote by Yh the class of all ζ ∈ F (Ah,R) satisfying the conditions:
‖δζ(θ)‖ ≤ c̄, ‖δ(2)ζ(θ)‖n×n ≤ c̄.
Set Xh = X[ĉ]. It follows from Lemmas 4.1 and 4.2 that(
(vh)[t,m], (vh)〈t,m〉
)
∈ Xh × Yh, (zh)[t,m] ∈ Xh.
We conclude from Assumption H[F, F,G, ψ] and H[Th] that there is c? > 0 such that the operator
Fh given by (44) satisfies the condition∣∣Fh(t, x, w, ζ)− Fh(t, x, w̃, ζ)
∣∣ ≤ (1 + c?)σ(t,Wh[w − w̃]), (51)
where (t, x) ∈ Eh, w, w̃ ∈ Xh, ζ ∈ Yh.
Then all the assumptions of Theorem 2.1 are satisfied and condition (50) follows.
Remark 4.3. Note that estimate (51) is not satisfied for all ζ ∈ F (Ah,R).
If all the assumptions of Theorem 4.1 are satisfied and there is L̃ ∈ R+ such that the expressions
‖F(t, x, w)− F(t, x, w̃)‖n×n, ‖F (t, x, w)− F (t, x, w̃)‖, |G(t, x, w)−G(t, x, w̃)|
are bounded from above by L̃‖w − w̃‖B where (t, x) ∈ E and w, w̃ ∈ C[ĉ] then there is L ∈ R+
such that ∣∣v(m)
h (t)− z(m)
h (t)
∣∣ ≤ α̃(h) on Eh,
where α̃ : Ĥ → R+ is given by (20), (21).
We apply the results on the numerical method of lines to differential equations with deviated
variables and to differential integral equations. We have transformed initial boundary-value problems
into systems of ordinary functional differential equations. The system such obtained is solved by
using the explicit Euler method. Let us denote by εh the maximal error of the difference method. In
the tables we give experimental values for εh.
Example 4.1. Put n = 2 and E = [0, 0.25]× [−0.5, 0.5]× [−0.5, 0.5]. Consider the differential
equation
∂tz(t, x, y) = ∂xxz(t, x, y) + ∂yyz(t, x, y) +
1
π2
∂xyz(t, x, y)−
−π2
x∫
0
y∫
0
z(t, µ, ν) dν dµ+
t∫
0
z(τ, x, y)dτ + 2π2z(t, x, y) + (t+ 1) cosπx cosπy
with the initial boundary conditions (35) – (37).
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1385
The solution of the above problem is known. It is v(t, x, y) =
(
et − 1
)
cosπx cosπy (see
Table 3.3).
Table 3.3
(h0, h1, h2) εh Time [s]
(2−10, 2−4, 2−4) 2.38495068 · 10−3 0.099
(2−12, 2−5, 2−5) 6.05392401 · 10−4 0.458
(2−14, 2−6, 2−6) 1.55723669 · 10−4 2.851
(2−16, 2−7, 2−7) 5.51208597 · 10−5 27.469
(2−18, 2−8, 2−8) 2.39516870 · 10−5 732.380
Example 4.2. Put n = 2 and E = [0, 0.25]× [−1, 1]× [−1, 1]. Consider the differential equation
∂tz(t, x, y) = 2∂xxz(t, x, y) + 2∂yyz(t, x, y) +
[
1− z(t, sinxy, cosxy)
1 + z2(t, sinxy, cosxy)
]
∂xyz(t, x, y)+
+z(t, 0.5(x+ y), 0.5(x− y)) + f(t, x, y)z(t, x, y),
f(t, x, y) = x2 + y2 − 1− 8t− 2t2(xy + 4x2 + 4y2)− exp
{
−t
(
x2
2
+
y2
2
)}
with the initial boundary conditions (38) – (41).
The solution of the above problem is known. It is v(t, x, y) = exp
{
t
(
x2+y2−1
)}
(see Table 3.4).
Table 3.4
(h0, h1, h2) εh Time [s]
(2−9, 2−3, 2−3) 2.51412371 · 10−4 0.086
(2−11, 2−4, 2−4) 6.33848401 · 10−5 0.449
(2−13, 2−5, 2−5) 1.58817990 · 10−5 3.877
(2−17, 2−7, 2−7) 3.97273960 · 10−6 55.038
(2−18, 2−8, 2−8) 2.39516870 · 10−5 1120.800
Difference methods described in Section 4 have the following property: a large number of pre-
vious values z(i,m) must be preserved, because they are needed to compute an approximate solution
with t = t(r+1).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1386 W. CZERNOUS, Z. KAMONT
Remark 4.4. Suppose that we apply a difference method to (23), (24) or (42), (43). The superpo-
sition of the numerical method of lines and the difference method for ordinary functional differential
equations leads to difference schemes for original problems. The above examples show that there
are explicit difference schemes which are convergent. It is not our aim to show theoretical results on
such difference schemes.
Remark 4.5. All the theorems on the numerical method of lines presented in the paper can be
extended on weakly coupled functional differential systems.
5. Conclusions. A new theorem, useful for proving convergence of difference schemes for first
order or second order parabolic PDEs, is given. The theory embraces initial boundary problems with
functional dependence, namely integro-differential and deviating variable ones.
A wider class of these dependences has been made treatable, thanks to removing the requirement
of globality on the estimate of growth of coefficients in functional variable. This is also illustrated
by numerical examples of Section 3.
This is the last published work, written jointly by the authors. Professor Zdzisław Kamont passed
away on the 3rd of September 2012, in Gdańsk, Poland. The first author would like to express his deep
gratitude for the years of mentoring and cooperation in mathematics. Requiescat in pace.
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METHOD OF LINES FOR QUASILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS 1387
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Received 28.10.11,
after revision — 05.05.13
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
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| id | umjimathkievua-article-2515 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:24:56Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a9/8aaff8ef2286a1b843ac7edc11627ca9.pdf |
| spelling | umjimathkievua-article-25152020-03-18T19:17:19Z Method of Lines for Quasilinear Functional Differential Equations Метод ліній для квазілінійних функцюнально-диференціальних рівнянь Kamont, Z. Czernous, W. Камонт, З. Черноус, В. We give a theorem on the estimation of error for approximate solutions to ordinary functional differential equations. The error is estimated by a solution of an initial problem for a nonlinear functional differential equation. We apply this general result to the investigation of convergence of the numerical method of lines for evolution functional differential equations. The initial boundary-value problems for quasilinear equations are transformed (by means of discretization in spatial variables) into systems of ordinary functional differential equations. Nonlinear estimates of the Perron-type with respect to functional variables for given operators are assumed. Numerical examples are given. Наведено теорему про оцінку похибки наближених розв'язків звичайних диференціальних рівнянь. Похибка оцінюється за допомогою розв'язку початкової задачі для нелінійного функціонально-диференціального рівняння. Цей загальний результат застосовується при дослідженні збіжності числового методу ліній для еволюції функціонально-диференціальних рівнянь. За допомогою дискретизації по просторових змінних початково-крайові задачі для квазілі-нійних рівнянь зводяться до систем звичайних диференціальних рівнянь. Припускається справедливість нелінійних оцінок перронівського типу відносно функціональних змінних для заданих операторів. Наведено також чисельні приклади. Institute of Mathematics, NAS of Ukraine 2013-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2515 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 10 (2013); 1363–1387 Український математичний журнал; Том 65 № 10 (2013); 1363–1387 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2515/1795 https://umj.imath.kiev.ua/index.php/umj/article/view/2515/1796 Copyright (c) 2013 Kamont Z.; Czernous W. |
| spellingShingle | Kamont, Z. Czernous, W. Камонт, З. Черноус, В. Method of Lines for Quasilinear Functional Differential Equations |
| title | Method of Lines for Quasilinear Functional Differential Equations |
| title_alt | Метод ліній для квазілінійних функцюнально-диференціальних рівнянь |
| title_full | Method of Lines for Quasilinear Functional Differential Equations |
| title_fullStr | Method of Lines for Quasilinear Functional Differential Equations |
| title_full_unstemmed | Method of Lines for Quasilinear Functional Differential Equations |
| title_short | Method of Lines for Quasilinear Functional Differential Equations |
| title_sort | method of lines for quasilinear functional differential equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2515 |
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