Implicit difference methods for first order partial differential functional equations
We present a new class of numerical methods for quasilinear first order partial functional differential equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that th...
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irk-123456789-1778872021-02-18T01:27:19Z Implicit difference methods for first order partial differential functional equations Kepczynska, A. We present a new class of numerical methods for quasilinear first order partial functional differential equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that the new methods are considerably better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators with respect to the functional variable. Розглянуто новий клас чисельних методiв для квiзiлiнiйних функцiонально-диференцiальних рiвнянь першого порядку з частинними похiдними. Розглянутi чисельнi методи є рiзницевими схемами, що задаються неявно вiдносно часової змiнної. Наведено повний аналiз збiжностi методiв i приклад, що показує значну перевагу нових методiв над явними схемами. Доведення стiйкостi базується на технiцi порiвняння з нелiнiйною оцiнкою перронiвського типу для заданого оператора вiдносно функцiональної змiнної. 2005 Article Implicit difference methods for first order partial differential functional equations / A. Kepczynska // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 201-215. — Бібліогр.: 11 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177887 517.9 en Нелінійні коливання Інститут математики НАН України |
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We present a new class of numerical methods for quasilinear first order partial functional differential
equations. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show by an example that the new
methods are considerably better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators with respect to the functional
variable. |
| format |
Article |
| author |
Kepczynska, A. |
| spellingShingle |
Kepczynska, A. Implicit difference methods for first order partial differential functional equations Нелінійні коливання |
| author_facet |
Kepczynska, A. |
| author_sort |
Kepczynska, A. |
| title |
Implicit difference methods for first order partial differential functional equations |
| title_short |
Implicit difference methods for first order partial differential functional equations |
| title_full |
Implicit difference methods for first order partial differential functional equations |
| title_fullStr |
Implicit difference methods for first order partial differential functional equations |
| title_full_unstemmed |
Implicit difference methods for first order partial differential functional equations |
| title_sort |
implicit difference methods for first order partial differential functional equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177887 |
| citation_txt |
Implicit difference methods for first order partial differential functional equations / A. Kepczynska // Нелінійні коливання. — 2005. — Т. 8, № 2. — С. 201-215. — Бібліогр.: 11 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT kepczynskaa implicitdifferencemethodsforfirstorderpartialdifferentialfunctionalequations |
| first_indexed |
2025-07-15T16:07:09Z |
| last_indexed |
2025-07-15T16:07:09Z |
| _version_ |
1837729715329171456 |
| fulltext |
UDC 517 . 9
IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER PARTIAL
DIFFERENTIAL FUNCTIONAL EQUATIONS
НЕЯВНI РIЗНИЦЕВI МЕТОДИ
ДЛЯ ДИФЕРЕНЦIАЛЬНО-ФУНКЦIОНАЛЬНИХ РIВНЯНЬ
ПЕРШОГО ПОРЯДКУ
A. Kȩpczyńska
Gdańsk Univ. Technology
Gabriel Narutowicz Str., 11 – 12 80 – 952 Gdańsk, Poland
We present a new class of numerical methods for quasilinear first order partial functional differential
equations. The numerical methods are difference schemes which are implicit with respect to time vari-
able. We give a complete convergence analysis for the methods and we show by an example that the new
methods are considerably better than the explicit schemes. The proof of the stability is based on a compari-
son technique with nonlinear estimates of the Perron type for given operators with respect to the functional
variable.
Розглянуто новий клас чисельних методiв для квiзiлiнiйних функцiонально-диференцiальних
рiвнянь першого порядку з частинними похiдними. Розглянутi чисельнi методи є рiзницевими
схемами, що задаються неявно вiдносно часової змiнної. Наведено повний аналiз збiжностi ме-
тодiв i приклад, що показує значну перевагу нових методiв над явними схемами. Доведення стiй-
костi базується на технiцi порiвняння з нелiнiйною оцiнкою перронiвського типу для заданого
оператора вiдносно функцiональної змiнної.
1. Introduction. For any metric spaces X and Y we denote by C(X, Y ) the class of all continuous
functions from X to Y. We will use vectorial inequalities with the understanding that the same
inequalities hold between their corresponding components.
We consider the sets
e = [0, a]× (−b, b), D = [−d0, 0]× [−d, d]
where a > 0, b = (b1, . . . , bn), d = (d1, . . . , dn), and di ≥ 0, bi > 0 for 1 ≤ i ≤ n, d ∈ Rn
+,
R+ = (0,+∞). Let c = (c1, . . . , cn) = b + d and
E0 = [−d0, 0]× [−c, c], ∂0E = [0, a]×
(
[−c, c] \ (−b, b)
)
.
Put Ω = E0 ∪ E ∪ ∂0E. For a function z : Ω → R and for a point (t, x) ∈ [0, a] × [−b, b] we
define the function z(t,x) : D → R as follows:
z(t,x)(s, y) = z(t + s, x + y), (s, y) ∈ D.
The function z(t,x) is the restriction of z to the set [t− d0, t]× [x− d, x + d] and this restriction
is shifted to the set D. For a function w ∈ C(D,R) we write
‖w‖D = max{|w(t, x)| : (t, x) ∈ D}.
c© A. Kȩpczyńska, 2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2 201
202 A. KȨPCZYŃSKA
Suppose that
f : E × C(D,R) → Rn, f = (f1, . . . , fn),
g : E × C(D,R) → R, ϕ : E0 ∪ ∂0E → R
are given functions. We will deal with the quasilinear differential functional equation
∂tz(t, x) =
n∑
i=1
fi(t, x, z(t,x))∂xiz(t, x) + g(t, x, z(t,x)) (1)
and the initial boundary condition
z(t, x) = ϕ(t, x) on E0 ∪ ∂0E. (2)
A function υ : Ω → R is a classical solution of (1), (2) provided:
(i) υ ∈ C(Ω, R) and the partial derivatives ∂tυ, ∂xυ = (∂x1υ, . . . , ∂xnυ) exist on E,
(ii) υ satisfies (1) on E and condition (2) holds.
We are interested in establishing a method of approximation of solutions to problem (1),
(2) by means of solutions of associated difference functional equations and in estimating the
difference bet ween the exact and approximate solutions.
In resent years, a number of papers concerning numerical methods for initial or initial
boundary-value problems related to first order partial functional differential equations have
been published.
Difference approximations of nonlinear equations with initial boundary conditions were
considered in [1, 2]. The convergence result for a general class of difference methods related
to initial problems and solutions defined on unbounded domain can be found in [3]. Error esti-
mates implying the convergence results for initial problems on the Haar pyramid were consi-
dered in [4 – 6].
All this considerations have the following property: the main question in the investigations
of numerical methods is to find a difference functional equation which is stable and satisfies a
consistency condition with respect to the original problem. The method of difference inequali-
ties and theorems on recurrence inequalities are used in the investigation of the stability. The
convergence results are based also on a general theorem on error estimates of approximate
solutions to functional difference equations of the Volterra type with initial or initial boundary
condition. The monograph [7] contains an exposition of the theory of difference methods for
nonlinear hyperbolic functional differential problems.
In the paper we start the investigation of implicit difference methods for quasilinear functi-
onal differential equations. We prove that under natural assumptions on given functions and
on the mesh there is a class of implicit difference schemes for (1), (2) which is convergent. The
stability of the methods is investigated by using the comparison technique. It is important in
our considerations that we assume the nonlinear estimates of the Perron type for given functi-
ons with respect to the functional variable. Our results are based on general ideas for finite
difference equations which were introduced in [8, 9].
The paper is organized as follows. In Section 2 we formulate an implicit difference functi-
onal problem corresponding to (1), (2) and we prove that there is exactly one solution of a di-
fference scheme. In Section 3 we prove a theorem on the error estimate for implicit difference
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER . . . 203
functional problems. Section 4 deals with a convergence result and an error estimate for the
method. A numerical example is given in Section 5.
We list below examples of equations which can be obtained from (1) by specializing the
operators f and g.
Example 1. Suppose that
f̃ : E ×R → Rn, f̃ = (f̃1, . . . , f̃n), g̃ : E ×R → R
and
α : E → R1+n, α = (α0, α
′), α′ = (α1, . . . , αn),
are given functions, and
−d0 ≤ α0(t, x)− t ≤ 0, −d ≤ α′(t, x)− x ≤ d, (t, x) ∈ E.
Write
f(t, x, w) = f̃(t, x, w(α0(t, x)− t, α′(t, x)− x)),
g(t, x, w) = g̃(t, x, w(α0(t, x)− t, α′(t, x)− x)),
where (t, x, w) ∈ E × C(D,R). Then equation (1) reduces to the equation with deviated vari-
ables,
∂tz(t, x) =
n∑
i=1
f̃i
(
t, x, z(α(t, x))
)
∂xiz(t, x) + g̃(t, x, z(α(t, x))).
Example 2. For the above f̃ and g̃ we put
f(t, x, w) = f̃
t, x,
∫
D
w(s, y)dyds
,
g(t, x, w) = f̃
t, x,
∫
D
w(s, y)dyds
.
Then (1) is the differential integral equation
∂tz(t, x) =
n∑
i=1
f̃i
∫
D
z(t + s, x + y)dyds
∂xiz(t, x) + g̃
∫
D
z(t + s, x + y)dyds
.
Existence results for mixed problems (1), (2) can be found in [7, 10, 11].
2. Difference functional equations. We will denote by F (X, Y ) the class off all functions
defined on X and taking values in Y where X and Y are arbitrary sets. For x, y ∈ Rn, x =
= (x1, . . . , xn), y = (y1, . . . , yn) we write
‖x‖ =
n∑
i=1
|xi|, x � y = (x1y1, . . . , xnyn).
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
204 A. KȨPCZYŃSKA
We define a mesh on the set Ω in the following way. Let (h0, h
′), h′ = (h1, . . . , hn), stand
for steps of the mesh. Let us denote by H the set of all h = (h0, h
′) such that there are K0 ∈ Z
and K = (K1, . . . ,Kn) ∈ Nn with the properties K0h0 = d0 and K � h′ = d. For h ∈ H and
(r, m) ∈ Z1+n, where m = (m1, . . . ,mn), we define nodal points as follows:
t(r) = rh0, x(m) = m � h′, x(m) =
(
x(m1), . . . , x(mn)
)
.
Let N0 ∈ N be defined by the relations N0h0 ≤ a < (N0 + 1)h0. Write
R1+n
h =
{(
t(r), x(m)
)
: (r, m) ∈ Z1+n
}
and
Eh = E ∩R1+n
h , Eh,0 = E0 ∩R1+n
h , Dh = D ∩R1+n
h ,
∂0Eh = ∂0E ∩R1+n
h , Ωh = Ω ∩R1+n
h .
Moreover we put
Eh,r = Eh ∩
([
−d0, t
(r)
]
×Rn
)
where 0 ≤ r ≤ N0 and
E′
h =
{(
t(r), x(m)
)
∈ Eh : 0 ≤ r ≤ N0 − 1
}
,
Ih =
{
t(r) : 0 ≤ r ≤ N0
}
, I ′h = Ih\
{
t(N0)
}
.
For a function z : Ωh → R we write z(r,m) = z(t(r), x(m)). For the above z and for a point
(t(r), x(m)) ∈ Eh we define the function z[r,m] : Dh → R by the formula
z[r,m](τ, s) = z
(
t(r) + τ, x(m) + s
)
, (τ, s) ∈ Dh.
The function z[r,m] is the restriction of z to the set(
[t(r) − d0]× [x(m) − d, x(m)d]
)
∩R1+n
h
and this restriction is shifted to the set Dh. Let ei = (0, . . . , 0, 1, 0, . . . , 0) ∈ Rn, 1 standing on
the i-th place, 1 ≤ i ≤ n. Suppose that the functions
fh : E′
h × F (Dh, R) → Rn, fh = (fh,1, . . . , fh,n),
gh : E′
h × F (Dh, R) → R, ϕh : Eh,0 ∪ ∂0Eh → R
are given. We consider the difference functional equation
δ0z
(r,m) =
n∑
i=1
fh,i
(
t(r), x(m), z[r,m]
)
δiz
(r+1,m) + gh
(
t(r), x(m), z[r,m]
)
(3)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER . . . 205
with the initial boundary condition
z(r,m) = ϕ
(r,m)
h on Eh,0 ∪ ∂0Eh. (4)
The difference operators δ0 and δ = (δ1, . . . , δn) are defined in the following way. Put
δ0z
(r,m) =
1
h0
(
z(r+1,m) − z(r,m)
)
(5)
and
δiz
(r,m) =
1
hi
(
z(r,m+ei) − z(r,m)
)
if fh,i
(
t(r), x(m), z[r,m]
)
≥ 0, (6)
δiz
(r,m) =
1
hi
(
z(r,m) − z(r,m−ei)
)
if fh,i
(
t(r), x(m), z[r,m]
)
< 0 (7)
where 1 ≤ i ≤ n.
Let us consider the explicit difference equation corresponding to (3)
δ0z
(r,m) =
n∑
i=1
fh,i
(
t(r), x(m), z[r,m]
)
δiz
(r+1,m) + gh
(
t(r), x(m), z[r,m]
)
. (8)
It is clear that the difference problem (4), (8) with δ0 and δ given (5) – (7) has exactly one
solution ũh : Ωh → R.
We prove that under natural assumptions on fh and gh there exists exactly one solution uh :
Ωh → R of the implicit difference functional problem (3), (4). We will approximate classical
solutions of (1), (2) with solutions of (3), (4).
We first prove a maximum principle for difference inequalities generated by (3), (4). Write
B = (−b, b), B? = [−c, c],
and Rn
h = {x(m) : m ∈ Zn}. We consider the sets
Bh = B ∩Rn
h, B?
h = B? ∩Rn
h, ∂0Bh = B?
h\Bh.
Theorem 1. Suppose that h ∈ H, fh : E′
h × F (Dh, R) → Rn and 0 ≤ r ≤ N0 − 1, is fixed.
1. If zh : Eh,r+1 → R satisfies the implicit difference inequality
z
(r+1,m)
h ≤ h0
n∑
i=1
fh,i
(
t(r), x(m), (zh)[r,m]
)
δiz
(r+1,m)
h
for x(m) ∈ Bh and µ ∈ Zn, µ = (µ1, . . . , µn), is such that z
(r+1,µ)
h = M, where
M = max
{
z
(r+1,m)
h : x(m) ∈ B∗
h
}
and M > 0, (9)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
206 A. KȨPCZYŃSKA
then x(µ) ∈ ∂0Bh.
2. If zh : Eh,r+1 → R satisfies the implicit difference inequality
z
(r+1,m)
h ≥ h0
n∑
i=1
fh,i
(
t(r), x(m), (zh)[r,m]
)
δiz
(r+1,m)
h
for x(m) ∈ Bh and µ̃ ∈ Zn is such that z
(r+1,eµ)
h = M̃, where
M̃ = min
{
z
(r+1,m)
h : x(m) ∈ B∗
h
}
and M̃ < 0,
then x(eµ) ∈ ∂0Bh.
Proof. Consider the case 1. Write
J
(r,m)
+ [zh] =
{
i : 1 ≤ i ≤ n and fh,i
(
t(r), x(m), (zh)[r,m]
)
≥ 0
}
, (10)
J
(r,m)
− [zh] = {1, . . . , n} \ J
(r,m)
+ [zh]. (11)
Suppose that x(µ) ∈ Bh. Then
z
(r+1,µ)
h ≤ h0
∑
i∈J
(r,m)
+ [zh]
1
hi
fh,i
(
t(r), x(µ), (zh)[r,m]
) [
z
(r+1,µ+ei)
h − z
(r+1,µ)
h
]
+
+ h0
∑
i∈J
(r,m)
− [zh]
1
hi
fh,i
(
t(r), x(µ), (zh)[r,m]
) [
z
(r+1,µ)
h − z
(r+1,µ−ei)
h
]
.
This gives
z
(r+1,µ)
h
[
1 + h0
n∑
i=1
1
hi
∣∣∣fh,i
(
t(r), x(µ), (zh)[r,µ]
)∣∣∣] ≤
≤ h0
∑
i∈J
(r,m)
+ [zh]
1
hi
fh,i
(
t(r), x(µ), (zh)[r,µ]
)
z
(r+1,µ+ei)
h −
− h0
∑
i∈J
(r,m)
− [zh]
1
hi
fh,i
(
t(r), x(µ), (zh)[r,µ]
)
z
(r+1,µ−ei)
h ≤
≤ h0M
n∑
i=1
1
hi
∣∣∣fh,i
(
t(r), x(µ), (zh)[r,µ]
)∣∣∣ .
We thus get z
(r+1,µ)
h ≤ 0 which contradicts (15). Then x(µ) ∈ ∂0Bh which is our claim. In a
similar way we prove that x(eµ) ∈ ∂0Bh in the case 2. This completes proof.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER . . . 207
Lemma 1. If h ∈ H, fh : E′
h × F (Dh, R) → Rn and gh : Eh × F (Dh, R) → Rn then
difference functional problem (3), (4) with δ0 and δ defined by (5) – (7) has exactly one solution
uh : Ωh → R.
Proof. Suppose that 0 ≤ r ≤ N0 − 1 is fixed and uh : Eh,r → R is known. Then (3), (4) is
a linear system, which allows to calculate u
(r+1,m)
h for x(m) ∈ Bh. The homogeneous problem
corresponding to (3), (4) has the from
z(r+1,m) = h0
n∑
i=1
fh,i
(
t(r), x(m), z[r,m]
)
δiz
(r+1,m), (12)
z(r+1,m) = 0 on Eh,0 ∪ ∂0Eh. (13)
It follows from Theorem 1 that system (12), (13) has exactly one zero solution. Therefore the
problem (3), (4) has exactly one solution for any choice of the function gh : Eh × F (Dh, R) →
→ R. Then the numbers u
(r+1,m)
h , x(m) ∈ Bh, exist and they are unique. Since uh is given on
Eh,0, the proof is completed by induction.
3. Approximate solutions of difference functional equations. Let us denote by Fh the Ni-
emycki operator corresponding to (3), i.e.,
Fh[z](r,m) =
n∑
i=1
fh,i
(
t(r), x(m), z[r,m]
)
δiz
(r+1,m) + gh
(
t(r), x(m), z[r,m]
)
.
Then we consider the difference functional equation
δ0z
(r,m) = Fh[z](r,m) (14)
with initial boundary condition (4). Suppose that vh : Ωh → R and γ, α0 : H → R+ are such
functions that ∣∣∣δ0v
(r,m)
h − Fh[vh](r,m)
∣∣∣ ≤ γ(h) on E′
h, (15)
∣∣∣ϕ(r,m)
h − v
(r,m)
h
∣∣∣ ≤ α0(h) on Eh,0 ∪ ∂0Eh (16)
and
lim
h→0
γ(h) = 0, lim
h→0
α0(h) = 0.
The function vh satisfying the relations (15), (16) is considered as an approximate solution of
problem (4), (14). We give a theorem on the estimate of the difference between the exact and
approximate solutions of (4), (14).
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 2
208 A. KȨPCZYŃSKA
Assumption H[fh, gh]. 1. The function σh : I ′h×R+ → R+ satisfies the following conditions:
(i) σh(t, ·) : R+ → R+ is continuous and nondecreasing for each t ∈ I ′h;
(ii) σh(t, 0) = 0 for t ∈ I ′h, and for each c ≥ 1 the difference problem
η(r+1) = η(r) + h0cσh
(
t(r), η(r)
)
, 0 ≤ r ≤ N0 − 1, (17)
η(0) = 0 (18)
is stable in the following sense: if γ, α0 : H → R+ are such functions that
lim
h→0
γ(h) = 0, lim
h→0
α0(h) = 0,
and ηh : Ih → R+ is a solution of the problem
η(r+1) = η(r) + h0cσh
(
t(r), η(r)
)
+ h0γ(h), 0 ≤ r ≤ N0 − 1,
η(0) = α0(h),
then there is α : H → R+ such that η
(r)
h ≤ α(h) for 0 ≤ r ≤ N0 and lim
h→0
α0(h) = 0.
2. The functions fh : E′
h × F (D,R) → Rn, gh : E′
h × F (Dh, R) → R satisfy the estimates
‖fh(t, x, w)− fh(t, x, w)‖ ≤ σh(t, ‖w − w‖Dh
),
|gh(t, x, w)− gh(t, x, w)| ≤ σh(t, ‖w − w‖Dh
)
on Ωh.
Theorem 2. Suppose that the Assumption H[fh, gh] is satisfied and
1) ϕh : Eh,0 ∪ ∂0Eh →R and the function uh : Ωh →R is a solution of the problem (4), (14);
2) h ∈ H and the functions vh : Ωh → R, are such that the estimates (15), (16) are satisfied;
3) there is c0 ∈ R+ such that the estimate∣∣∣δiv
(r,m)
h
∣∣∣ ≤ c0,
(
t(r), x(m)
)
∈ Eh, (19)
is satisfied for 1 ≤ i ≤ n.
Then there exists a function α : H → R+ such that∣∣∣u(r,m)
h − v
(r,m)
h
∣∣∣ ≤ α(h) on Eh (20)
and
lim
h→0
α(h) = 0.
Proof. Let Γh : E′
h → R be the function defined in the relation
δ0v
(r,m)
h = Fh[vh](r,m) + Γ(r,m)
h . (21)
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IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER . . . 209
It follows from (15) that |Γ(r,m)
h | ≤ γ(h) on E′
h. Since uh satisfies (4), (14), we have
(uh − vh)(r+1,m)
[
1 + h0
n∑
i=1
1
hi
∣∣∣fh,i
(
t(r), x(m), (uh)[r,m]
)∣∣∣] = (uh − vh)(r,m)+
+ h0
∑
i∈J
(r,m)
+ [uh]
1
hi
fh,i
(
t(r), x(m), (uh)[r,m]
)
(uh − vh)(r+1,m+ei)−
− h0
∑
i∈J
(r,m)
− [uh]
1
hi
fh,i
(
t(r), x(m), (uh)[r,m]
)
(uh − vh)(r+1,m−ei)+
+ h0
n∑
i=1
1
hi
(
fh,i
(
t(r), x(m), (uh)[r,m]
)
− fh,i
(
t(r), x(m), (vh)[r,m]
))
δiv
(r+1,m)
h +
+ h0
(
gh
(
t(r), x(m), (uh)[r,m]
)
− gh
(
t(r), x(m), (vh)[r,m]
))
− h0Γ
(r,m)
h ,
where J
(r,m)
+ [uh] and J
(r,m)
− [uh] are given by (10), (11). Write
ε
(r)
h = max
{
|(uh − vh)(r,m) : −K ≤ m ≤ K
}
for r = 0, . . . , N0.
Then we get the following difference inequality:
ε
(r+1)
h ≤ max
{
ε
(r)
h + h0(1 + c0)σh
(
t(r), ε
(r)
h
)
+ h0γ(h), α0(h)
}
.
Let us consider now the difference problem
η(r+1) = η(r) + h0(1 + c0)σh
(
t(r), η
(r)
h
)
+ h0γ(h), 0 ≤ r ≤ N0 − 1,
η(r) = α0(h),
and its solution ηh. It follows from the above considerations that
ε
(r)
h = η
(r)
h , 0 ≤ r ≤ N0.
Now we obtain the assertion of Theorem 2 from the stability of problem (17), (18).
4. Convergence of implicit difference methods. We consider a class of difference problems
(3), (4) where fh and gh are superpositions of f and g with a suitable interpolating operator.
Assumption H[Th]. Suppose that the operator Th : F (Dh, R) → F (D,R) satisfies the
following conditions:
1) if w, w̃ ∈ F (Dh, R) then Th[w], Th[w̃] ∈ C(D,R) and
‖Th[w]− Th[w̃]‖D ≤ ‖w − w̃‖Dh
,
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210 A. KȨPCZYŃSKA
2) if w : D → R is of class C1 then there is γ : H → R+ such that
‖Th[w]− w‖D ≤ γ(h) and lim
h→0
= 0,
where wh is the restriction of w to the set Dh.
We will approximate solutions of (1), (2) with solutions of the difference functional equation
δ0z
(r,m) =
n∑
i=1
fi
(
t(r), x(m), Thz[r,m]
)
δiz
(r+1,m) + g
(
t(r), x(m), Thz[r,m]
)
(22)
with initial boundary condition (4).
Assumption H[f, g]. 1. The functions f : E × F (D,R) → Rn and g : E × F (D,R) → R
are continuous.
2. There is σ : [0, a]×R+ → R+ such that
(i) σ is continuous and nondecreasing with respect to both variables;
(ii) σ(t, 0) = 0 for t ∈ [0, a] and each c ≥ 1, ε ≥ 0, and the maximal solution ω(·, ε) of the
Cauchy problem
w′(t) = cσ(t, w(t)) + ε, w(0) = ε, (23)
is defined on [0, a] and w(t, 0) = 0 for t ∈ (0, a);
(iii) the estimates
‖f(t, x, w)− f(t, x, w)‖ ≤ σ(t, ‖w − w‖D),
|g(t, x, w)− g(t, x, w)| ≤ σ(t, ‖w − w‖D)
are satisfied on E × F (D,R).
Theorem 3. Suppose that Assumptions H[Th] and H[f, g]are satisfied and
1) h ∈ H and the function uh : Ωh → R is a solution of (4), (22) and there is α0H → R+
such that ∣∣∣ϕ(r,m) − ϕ
(r,m)
h
∣∣∣ ≤ α0(h) on Eh,0 ∪ ∂0Eh and lim
h→0
α0(h) = 0; (24)
2) v : Ω → R is a solution of (1), (2) and v is of class C1 on Ω.
Then there is α : H → R+ such that∣∣∣u(r,m)
h − v
(r,m)
h
∣∣∣ ≤ α(h) and lim
h→0
α(h) = 0. (25)
Proof. We prove that the functions
fh(t, x, w) = f(t, x, Th[w]), gh(t, x, w) = g(t, x, Th[w])
satisfy all the assumptions of Theorem 2. We first show that problem (17), (18) is stable in the
sense of Assumption H[fh, gh]. Let the functions α0, γ : H → R+be such that
lim
h→0
α0(h) = 0, lim
h→0
γ(h) = 0.
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IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER . . . 211
Let us consider the difference problem
η(r+1) = η(r) + h0cσh
(
t(r), η(r)
)
+ h0γ(h), 0 ≤ r ≤ K − 1, (26)
η(0) = α0(h), (27)
and its solution ηh : Ih → R+. Denote by ωh : [0, a) → R+ the maximal solution of the
problem
w′(t) = cσ(t, w) + γ(h), (28)
w(t) = α0(h). (29)
It is easily seen that η
(i)
h ≤ ω
(i)
h for 0 ≤ i ≤ N0 and
lim
h→0
ω(t) = 0 uniformly on [0, a].
Then we have η
(i)
h ≤ ωh(a) for 0 ≤ i ≤ N0 and problem (28), (29) with ε = 0 is stable.
Moreover we have
‖fh(t, x, w)− fh(t, x, w)‖ = ‖f(t, x, Th[w])− f(t, x, Th[w])‖ ≤
≤ σ (t, ‖Th[w − w]‖D) ≤ σh (t, x, ‖w − w‖Dh
) .
In the same way we get the inequality
‖gh(t, x, w)− gh(t, x, w)‖ ≤ σh (t, ‖w − w‖Dh
) .
Then the assertion of Theorem 3 follows from Theorem 2.
Remark 1. There are the following consequences of Theorem 3. In classical theorems concer-
ning difference methods for quasilinear functional differential problems it is assumed that
1− h0
n∑
i=1
1
hi
|fi(t, x, w)| ≥ 0 on E × C(D,R), (30)
see [3]. It is important in our considerations that we have omitted the above assumption.
Now we give an example of the operator Th satisfying Assumption H[Th]. Put
S∗ = {(j, s) : j ∈ {0, 1}, s = (s1, . . . , sn), si ∈ {0, 1} for 1 ≤ i ≤ n} .
Let w ∈ F (Dh, R) and (t, x) ∈ D. There exists (t(r), x(m)) ∈ Dh such that
t(r) ≤ t ≤ t(r+1), x(m) ≤ x ≤ x(m+1),
(
t(r+1), x(m+1)
)
∈ Dh.
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212 A. KȨPCZYŃSKA
We define
(Thw)(t, x) =
∑
(j,s)∈S∗
w(r+j,m+s)
(
Y − Y (r,m)
h
)(j,s)(
1− Y − Y (r,m)
h
)1−(j,s)
where (
Y − Y (r,m)
h
)(j,s)
=
(
t− t(r)
h0
)j n∏
k=1
(
xk − x
(mk)
k
hk
)sk
and (
1− Y − Y (r,m)
h
)1−(j,s)
=
(
1− t− t(r)
h0
)1−j n∏
k=1
(
1−
xk − x
(mk)
k
hk
)1−sk
and we take 00 = 1 in the above formulas.
Lemma 2. Suppose that the function w : D → R is of class C2 and denote by wh the restricti-
on of w to the set Dh. Let C̃ be a constant such that
|∂tt(t, x)| ≤ C̃,
∣∣∂txj (t, x)
∣∣ ≤ C̃,
∣∣∂xixj (t, x)
∣∣ ≤ C̃ on D,
where 1 ≤ i, j ≤ n. Then
‖Th[w]− w‖D ≤ C̃
h2
0 + 2h0
n∑
i=1
hi +
n∑
j,i=1
hjhi
.
The proof of the above lemma can found in [7], Chapter 5. We omit details.
Now we give an error estimate for method (4), (22).
Theorem 4. Suppose that
1) all the assumptions of Theorem 3 are satisfied with σ(t, p) = Lp and the solution v : Ω →
→ R of differential problem (1), (2) is of class C2;
2) the constant C̃ ∈ R+ is such that
|∂ttv(t, x)| ≤ C̃,
∣∣∂txjv(t, x)
∣∣ ≤ C̃,
∣∣∂xixjv(t, x)
∣∣ ≤ C̃ on Ω,
where 1 ≤ i, j ≤ n, and there exists d ∈ R+ such that
|fi(t, x, p)| ≤ d, 1 ≤ i ≤ n.
Then ∣∣∣u(r,m)
h − v
(r,m)
h
∣∣∣ ≤ η̃
(r)
h (31)
where
η̃
(r)
h = α0(h)(1 + h0cL)r + γ(h)
(1 + h0cL)r − 1
cL
,
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IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER . . . 213
γ(h) =
1
2
C̃h0 + (1 + c0)Lγ̃(h) +
1
2
dC̃
n∑
i=1
hi,
γ̃(h) = C̃
h2
0 + 2h0
n∑
i=1
hi +
n∑
j,i=1
hjhi
.
Proof. The difference operators δ0 and δ satisfy the conditions∣∣∣δ0v
(r,m) − ∂tv
(r,m)
∣∣∣ ≤ 1
2
C̃h0,
∣∣∣δiv
(r,m) − ∂xiv
(r,m)
∣∣∣ ≤ 1
2
C̃hi, 1 ≤ i ≤ n.
It follows from above estimates and from Assumption H[Th] that∣∣∣Γ(r,m)
h
∣∣∣ ≤ ∣∣∣δ0v
(r,m) − ∂tv
(r,m)
∣∣∣+
+
∣∣∣∣∣
n∑
i=1
fi
(
t(r), x(m), (Th[v])(r,m)
)
δiv
(r+1,m) −
n∑
i=1
fi
(
t(r), x(m), v(r,m)
)
∂xiv
(r+1,m)
∣∣∣∣∣+
+
∣∣∣g (t(r), x(m), (Th[v])(r,m)
)
− g
(
t(r), x(m), v(r,m)
)∣∣∣ ≤
≤ 1
2
C̃h0 + (1 + c0)L γ̃(h) +
1
2
dC̃
n∑
i=1
hi.
The function η̃h is a solution of the problem
η(r+1) = η(r)(1 + h0cL) + h0γ(h), 0 ≤ i ≤ N0 − 1 η(0) = α0(h),
which is equivalent to (26), (27) for σ(t, p) = Lp. Then from Theorem 3 we get the assertion
(31).
5. Numerical examples. For n = 2 we put
E = [0, 1]× [−1, 1]× [−1, 1], E0 = {0} × [−1, 1]× [−1, 1].
Consider the quasilinear differential equation
∂tz(t, x, y) = xy2∂xz(t, x, y) + yx2∂yz(t, x, y) + f(t, x, y)z(t, x, y)+
+ z(t, 0, 5(y − x), 0, 5(x + y)) + z(t, 0, 5(x + y), 0, 5(x− y))− e−tx − ety, (32)
where
f(t, x, y) = (x− y)(1 + txy),
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214 A. KȨPCZYŃSKA
with the initial boundary conditions
z(0, x, y) = 1, (x, y) ∈ [−1, 1]× [−1, 1],
z(t,−1, y) = et(−1−y), t ∈ [0, 1], y ∈ [−1, 1],
z(t, 1, y) = et(1−y), t ∈ [0, 1], y ∈ [−1, 1], (33)
z(t, x,−1) = et(x+1), t ∈ [0, 1], x ∈ [−1, 1],
z(t, x, 1) = et(x+1), t ∈ [0, 1], x ∈ [−1, 1].
The solution of the above problem is given by v(t, x) = et(x−y). Put h = (h0, h1, h2) and assume
that h1 = h2.
Write
Eh = {(tr, xm1 , ym2) : 0 ≤ r ≤ N0, −K ≤ m1,m2 ≤ K}
where N0h0 ≤ 1 ≤ (N0 + 1)h0, Kh1 = Kh2 = 1. Let us denote by zh : Eh → R the solution
of the implicit difference problem corresponding to (32), (33). We consider also the function
z̃h : Eh → R which is a solution of the classical difference equation corresponding to (32),
(33). Write
η
(r)
h =
1
(2K + 1)2
K∑
m1=−K
K∑
m2=−K
∣∣∣z(r,m1,m2)
h − v(r,m1,m2)
∣∣∣ ,
η̃
(r)
h =
1
(2K + 1)2
K∑
m1=−K
K∑
m2=−K
∣∣∣z̃(r,m1,m2)
h − v(r,m1,m2)
∣∣∣ .
The numbers η
(r)
h and η̃
(r)
h are the arithmetical mean of the errors with fixed t(r). The values of
the functions ηh and η̃h are listed in the table.
Table of errors (η̃h, ηh) for h0 = 0, 01, h1 = h2 = 0, 01 :
t = 0, 60 1, 41414e + 003 1, 65029e− 003
t = 0, 70 4, 06659e + 004 2, 03410e− 003
t = 0, 80 1, 10407e + 006 2, 50142e− 003
t = 0, 90 2, 91547e + 007 3, 07663e− 003
t = 1, 00 7, 59963e + 008 3, 78804e− 003
It follows that the results obtained by the implicit difference method are better than those
obtained by the classical scheme. This is due to the fact that we need the relation (30) for steps
of the mesh in the classical case. We do not need this assumption in our implicit difference
method.
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IMPLICIT DIFFERENCE METHODS FOR FIRST ORDER . . . 215
1. Brandi P., Kamont Z., Salvadori A. Approximate solutions of mixed problems for first order partial diffrential
functional equations // Atti Semin. mat. e fis. Univ. Modena. — 1991. — 39. — P. 277 – 302.
2. Kamont Z. Finite difference approximations for first order partial differential functional equations // Ukr.
Math. J. — 1994. — 46, № 7. — P. 985 – 996.
3. Leszczynski H. Convergence results for unbounded solutions of first order nonlinear differential functional
equations // Ann. pol. math. — 1996. — 54. — P. 1 – 16.
4. Jaruszewska-Walczak D., Kamont Z. Difference methods for quasilinear hyperbolic differential functional
systems on the Haar pyramid // Bull. Belg. Math. Soc. — 2003. — 10. — P. 267 – 290.
5. Jaruszewska-Walczak D., Kamont Z. Numerical methods for hyperbolic functional differential problems on
the Haar pyramid //Computing. — 2000. — 65. — P. 45 – 72.
6. Prza̧dka K. Difference methods for nonlinear partial differential functional equations of the first order //
Math. Nachr. — 1988. — 138. — P. 105 – 123.
7. Kamont Z. Hyperbolic functional differential inequalities and applications. — Dordrecht: Kluwer Acad.
Publ., 1999.
8. Godlewski E., Raviart P. A. Numerical approximation of hyperbolic systems of conservation laws. — New
York: Springer, 1996.
9. Magomedov K. M., Kholodov A. S. Mesh-characteristic numerical methods. — Moscow: Nauka, 1988 (in
Russian).
10. Człapiński T. On the mixed problem for quasilinear partial differential functional equations of the first order
// Z. anal. Anwend. — 1997. — 16. — S. 463 – 478.
11. Umanaliev M. I., Vied’ J. A. On integral differential equations with first order partial derivatives // Differents.
Uravneniya. — 1989. — 25. — S. 465 – 477 (in Russian).
Received 13.01.2005
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