Generalized Euler method for nonlinear first order partial differential equations
Classical solutions of nonlinear first order partial differential equations are approximated in the paper by solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on...
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| Zitieren: | Generalized Euler method for nonlinear first order partial differential equations / Z. Kamont, J. Newlin-Łukowicz // Нелінійні коливання. — 2003. — Т 6, № 4. — С. 456-474. — Бібліогр.: 12 назв. — англ. |
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irk-123456789-1769892021-02-10T01:26:32Z Generalized Euler method for nonlinear first order partial differential equations Kamont, Z. Newlin-Łukowicz, J. Classical solutions of nonlinear first order partial differential equations are approximated in the paper by solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the method are given. The proof of the stability of the difference problem is based on a comparison method. Nonlinear estimates of the Perron type are assumed for increment functions. This new approach to a numerical solving of nonlinear equations is generated by a method of quasilinearization for mixed problems. Numerical examples are given Класичнi розв’язки нелiнiйних диференцiальних рiвнянь з частинними похiдними першого порядку наближаються розв’язками квазiлiнiйних систем рiзницевих рiвнянь. Наведено достатню умову для збiжностi запропонованого методу. Доведення стiйкостi рiзницевої задачi базується на методi порiвняння. Вважається, що функцiя приросту задовольняє нелiнiйнi оцiнки перронiвського типу. Цей новий пiдхiд до числового розв’язання нелiнiйних рiвнянь базується на методi квазiлiнеаризацiї для мiшаних задач. Наведено числовi приклади. 2003 Article Generalized Euler method for nonlinear first order partial differential equations / Z. Kamont, J. Newlin-Łukowicz // Нелінійні коливання. — 2003. — Т 6, № 4. — С. 456-474. — Бібліогр.: 12 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/176989 517.944 en Нелінійні коливання Інститут математики НАН України |
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Classical solutions of nonlinear first order partial differential equations are approximated in the paper by
solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the
method are given. The proof of the stability of the difference problem is based on a comparison method.
Nonlinear estimates of the Perron type are assumed for increment functions.
This new approach to a numerical solving of nonlinear equations is generated by a method of quasilinearization for mixed problems. Numerical examples are given |
| format |
Article |
| author |
Kamont, Z. Newlin-Łukowicz, J. |
| spellingShingle |
Kamont, Z. Newlin-Łukowicz, J. Generalized Euler method for nonlinear first order partial differential equations Нелінійні коливання |
| author_facet |
Kamont, Z. Newlin-Łukowicz, J. |
| author_sort |
Kamont, Z. |
| title |
Generalized Euler method for nonlinear first order partial differential equations |
| title_short |
Generalized Euler method for nonlinear first order partial differential equations |
| title_full |
Generalized Euler method for nonlinear first order partial differential equations |
| title_fullStr |
Generalized Euler method for nonlinear first order partial differential equations |
| title_full_unstemmed |
Generalized Euler method for nonlinear first order partial differential equations |
| title_sort |
generalized euler method for nonlinear first order partial differential equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176989 |
| citation_txt |
Generalized Euler method for nonlinear first order partial differential equations / Z. Kamont, J. Newlin-Łukowicz // Нелінійні коливання. — 2003. — Т 6, № 4. — С. 456-474. — Бібліогр.: 12 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT kamontz generalizedeulermethodfornonlinearfirstorderpartialdifferentialequations AT newlinłukowiczj generalizedeulermethodfornonlinearfirstorderpartialdifferentialequations |
| first_indexed |
2025-07-15T14:56:48Z |
| last_indexed |
2025-07-15T14:56:48Z |
| _version_ |
1837725289155657728 |
| fulltext |
UDC 517.944
GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER
PARTIAL DIFFERENTIAL EQUATIONS
УЗАГАЛЬНЕНИЙ МЕТОД ЕЙЛЕРА
ДЛЯ НЕЛIНIЙНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
З ЧАСТИННИМИ ПОХIДНИМИ ПЕРШОГО ПОРЯДКУ
Z. Kamont, J. Newlin-Łukowicz
Techn. Univ. Gdańsk
11/12 Gabriel Narutowicz Street,
80-952 Gdańsk, Poland
Classical solutions of nonlinear first order partial differential equations are approximated in the paper by
solutions of quasilinear systems of difference equations. Sufficient conditions for the convergence of the
method are given. The proof of the stability of the difference problem is based on a comparison method.
Nonlinear estimates of the Perron type are assumed for increment functions.
This new approach to a numerical solving of nonlinear equations is generated by a method of quasili-
nearization for mixed problems. Numerical examples are given.
Класичнi розв’язки нелiнiйних диференцiальних рiвнянь з частинними похiдними першого по-
рядку наближаються розв’язками квазiлiнiйних систем рiзницевих рiвнянь. Наведено достатню
умову для збiжностi запропонованого методу. Доведення стiйкостi рiзницевої задачi базується
на методi порiвняння. Вважається, що функцiя приросту задовольняє нелiнiйнi оцiнки перро-
нiвського типу.
Цей новий пiдхiд до числового розв’язання нелiнiйних рiвнянь базується на методi квазiлi-
неаризацiї для мiшаних задач. Наведено числовi приклади.
1. Discretization of mixed problems. For any metric spaces X and Y we denote by C(X,Y )
the set of all continuous functions defined on X and taking values in Y . We will use vectorial
inequalities with the understanding that the same inequalities hold between their correspon-
ding components. Write
E = [0, a]× [−b, b] and E0 = {0} × [−b, b],
where a > 0, b = (b1, . . . , bn) ∈ Rn with bi > 0 for 1 ≤ i ≤ n. Set Ω = E × R × Rn, and
suppose that f : Ω → R is a given function of the variables (t, x, p, q) where x = (x1, ..., xn)
and q = (q1, ..., qn). Let us assume that f ∈ C(Ω, R) and that the first order partial derivatives
(∂q1f(P ), . . . , ∂qnf(P )) = ∂qf(P )
exist for P = (t, x, p, q) ∈ Ω. Write
∂
(i)
+ E = {(t, x) ∈ E : xi = bi}, ∂
(i)
− E = {(t, x) ∈ E : xi = −bi}, 1 ≤ i ≤ n,
c© Z. Kamont, J. Newlin-Łukowicz, 2003
456 ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 457
and
∂0E =
n⋃
i=1
(
∂
(i)
+ E ∪ ∂(i)
− E
)
.
In the following we will assume that
∂qif(t, x, p, q) > 0 for (t, x, p, q) ∈ ∂(i)
+ E ×R×Rn, 1 ≤ i ≤ n, (1)
and
∂qif(t, x, p, q) < 0 for (t, x, p, q) ∈ ∂(i)
− E ×R×Rn, 1 ≤ i ≤ n. (2)
Suppose that ϕ : E0 ∪ ∂0E → R is a given function. We consider the problem consisting of the
nonlinear differential equation
∂tz(t, x) = f(t, x, z(t, x), ∂xz(t, x)) (3)
and the initial boundary condition
z(t, x) = ϕ(t, x) for (t, x) ∈ E0 ∪ ∂0E, (4)
where ∂xz = (∂x1z, . . . , ∂xnz). A function v : E → R is called a classical solution of the above
problem if
(i) v ∈ C(E,R) and v is of class C1 on E\(E0 ∪ ∂0E),
(ii) v satisfies (3) on E\(E0 ∪ ∂0E) and initial boundary condition (4) holds.
We formulate now a finite difference problem corresponding to (3), (4). Let N and Z be
the sets of natural numbers and integers, respectively. For x, y ∈ Rn, x = (x1, . . . , xn), y =
= (y1, . . . , yn), we write
x � y = (x1y1, . . . , xnyn) and ‖x‖ =
n∑
i=1
|xi|.
We define a mesh on the set E in the following way. Let (h0, h
′), h′ = (h1, ..., hn), stand for
steps of the mesh. For h = (h0, h
′) and (r,m) ∈ Z1+n wherem = (m1, ...,mn), we define nodal
points as follows:
t(r) = rh0, x(m) = m � h′, x(m) = (x(m1)
1 , . . . , x(mn)
n ).
Let us denote by ∆ the set of all h = (h0, h
′) such that there is N = (N1, ..., Nn) ∈ Nn with the
propertyN �h′ = b. We assume that ∆ 6= ∅. There isN0 ∈ N such thatN0h0 ≤ a < (N0 +1)h0.
Write
R1+n
h {(t(r), x(m)) : (r,m) ∈ Z1+n},
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
458 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
Ih = {t(r) : 0 ≤ r ≤ N0}
and
Eh = E ∩R1+n
h , ∂0Eh = ∂0E ∩R1+n
h ,
Er,h = Eh ∩
(
[0, t(r)]×Rn
)
where 0 ≤ r ≤ N0.
For functions z : Eh → R, u : Eh → Rn, u = (u1, ..., un), and ξ : Ih → R, we write
z(r,m) = z(t(r), x(m)), u(r,m) = u(t(r), x(m)), ξ(r) = ξ(t(r)) and
‖z‖r,h = max
{
|z(i,m)| : (t(i), x(m)) ∈ Er,h
}
,
‖u‖r,h = max
{
|u(i,m)| : (t(i), x(m)) ∈ Er,h
}
,
where 0 ≤ r ≤ N0. Let ej = (0, ..., 0, 1, 0, ..., 0) ∈ Rn, 1 standing in the j-th place. We denote
by δ0 the difference operator with respect to variable t and by δ = (δ1, ..., δn) the difference
operator for spatial variables (x1, ..., xn) = x. Write
δ0z
(r,m) =
1
h0
z(r+1,m) − 1
2n
n∑
j=1
(
z(r,m+ej) + z(r,m−ej)
) (5)
and
δjz
(r,m) =
1
2hj
(
z(r,m+ej) − z(r,m−ej)
)
for 1 ≤ j ≤ n. (6)
A classical difference method for the mixed problem consists in replacing partial deriva-
tives in (3) by the above difference operators. This leads to the difference equation
δ0z
(r,m) = f
(
t(r), x(m), z(r,m), δz(r,m)
)
(7)
with the initial boundary condition
z(r,m) = ϕ
(r,m)
h on E0,h ∪ ∂0Eh (8)
where δz(r,m) = (δ1z
(r,m), . . . , δnz
(r,m)) and ϕh : E0,h∪∂0Eh → R is a given function. Sufficient
conditions for the convergence of method (7), (8) to a classical solution of (3), (4) are given in
the following theorem.
Theorem 1. Suppose that
1) f ∈ C(Ω, R), the derivatives (∂q1f, . . . , ∂qnf) = ∂qf exist on Ω and ∂qf ∈ C(Ω, Rn);
2) conditions (1), (2) are satisfied and there is A ∈ R+ such that
|f(t, x, p, q)− f(t, x, p̄, q)| ≤ A|p− p̄| on Ω; (9)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 459
3) h ∈ ∆ and there is M = (M1, . . . ,Mn) ∈ Rn such that Mi > 0 for 1 ≤ i ≤ n and
h′ ≤ Mh0;
4) the estimates
1− nh0
hj
∣∣∂qjf(t, x, p, q)
∣∣ ≥ 0, 1 ≤ j ≤ n,
are satisfied on Ω;
5) v : E → R is a solution of (3), (4) and v is of class C1 on E and there is a function
α0 : ∆ → R+ such that∣∣ϕ(r,m) − ϕ(r,m)
h
∣∣ ≤ α0(h) on E0,h ∪ ∂0Eh and lim
h→0
α0(h) = 0;
6) zh : Eh → R is a solution of (7), (8).
Then there is a function α : ∆ → R+ such that
‖vh − zh‖r,h ≤ α(h) for 0 ≤ r ≤ N0 and lim
h→0
α(h) = 0
where vh is the restriction of v to the set Eh.
The above theorem may be proved by a method used in [1 – 3]. Note that the Lipschitz
condition (9) may be replaced in Theorem 1 by a nonlinear estimate of the Perron type.
Remark 1. Suppose that all the assumptions of Theorem 1 are satisfied and the solution
v : E → R of (3), (4) is of class C2.
Then there are C0, C ∈ R+ such that we have the following error estimate for method
(7), (8):
‖vh − zh‖r,h ≤ C0α0(h) + Ch0, 0 ≤ r ≤ N0.
The above result may be proved by methods used in [1 – 3].
Note that the classical Euler method [1, 2] is not applicable to problem (3), (4). Now we
formulate a new class of difference problems corresponding to (3), (4). We need next assumpti-
ons on f .
AssumptionH0[f ]. Suppose that f ∈ C(Ω, R) and
1) the partial derivatives(
∂x1f(P ), . . . , ∂xnf(P )
)
= ∂xf(P ), ∂pf(P ),
(
∂q1f(P ), . . . , ∂qnf(P )
)
= ∂qf(P )
exist for P = (t, x, p, q) ∈ Ω and ∂xf, ∂qf ∈ C(Ω, Rn), ∂pf ∈ C(Ω, R);
2) conditions (1), (2) are satisfied.
Write
E′h =
{
(t(r), x(m)) : 0 ≤ r ≤ N0 − 1,−N < m < N
}
.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
460 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
Let us denote by (z, u), u = (u1, . . . , un), the unknown functions of the variables (t(r), x(m)).
Write
P (r,m)[z, u] =
(
t(r), x(m), z(r,m), u(r,m)
)
,
Q(r,m)[z, u] =
(
t(r), x(m), z(r,m), u(r+1,m)
)
.
We consider the quasilinear system of difference equations
δ0z
(r,m) = f
(
Q(r,m)[z, u]
)
+
n∑
j=1
∂qjf
(
Q(r,m)[z, u]
)(
δjz
(r,m) − u(r,m)
j
)
(10)
and
δ0u
(r,m)
i = ∂xif
(
P (r,m)[z, u]
)
+ ∂pf
(
P (r,m)[z, u]
)
u
(r,m)
i +
+
n∑
j=1
∂qjf
(
P (r,m)[z, u]
)
δju
(r,m)
i , 1 ≤ i ≤ n, (11)
with the initial boundary condition
z(r,m) = ϕ
(r,m)
h and u(r,m) = ψ
(r,m)
h on E0,h ∪ ∂0Eh, (12)
where ϕh : E0,h ∪ ∂0Eh → R and ψh : E0,h ∪ ∂0Eh → Rn are given functions. The difference
operators δ0 and δ = (δ1, . . . , δn) are defined in the following way. Suppose that the functions
(z, u) are given on the sets Er,h and Er+1,h, respectively, where 0 ≤ r < N0. Then we put
δ0z
(r,m) =
1
h0
(
z(r+1,m) − z(r,m)
)
, δ0u
(r,m)
i =
1
h0
(
u
(r+1,m)
i − u(r,m)
i
)
, 1 ≤ i ≤ n. (13)
The difference operators with respect to spatial variables are defined in the following way:
if ∂qjf
(
Q(r,m)[z, u]
)
≥ 0 then δjz
(r,m) =
1
hj
(
z(r,m+ej) − z(r,m)
)
, (14)
if ∂qjf
(
Q(r,m)[z, u]
)
< 0 then δjz
(r,m) =
1
hj
(
z(r,m) − z(r,m−ej)
)
, (15)
if ∂qjf
(
P (r,m)[z, u]
)
≥ 0 then δju
(r,m)
i =
1
hj
(
u
(r,m+ej)
i − u(r,m)
i
)
, 1 ≤ i ≤ n, (16)
if ∂qjf
(
P (r,m)[z, u]
)
< 0 then δju
(r,m)
i =
1
hj
(
u
(r,m)
i − u(r,m−ej)
i
)
, 1 ≤ i ≤ n. (17)
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 461
The difference problem consisting of system of difference equations (10), (11) with δ0 and δ
given by (13) – (17) and initial boundary condition (12) is called a generalized Euler method for
problem (3), (4). There exists exactly one solution (zh, uh) : Eh → R1+n, uh = (uh,1, . . . , uh,n),
of the above difference problem. For this purpose note that if (zh, uh) are defined on Er,h,
0 ≤ r < N0, and (t(r+1), x(m)) ∈ E′h then(
t(r), x(m+ej)
)
, and
(
t(r), x(m−ej)
)
∈ Eh for 1 ≤ j ≤ n.
It follows that u(r+1,m)
h may be calculated from (11), (12) and then z(r+1,m) may be calculated
from (10), (12). Then by induction the solution exists and it is unique on Eh.
There are two main differences between the classical result presented in Theorem 1 and our
methods.
1. If we apply difference problem (7), (8) then we approximate the spatial derivatives of z
in (3) with the use of difference expressions (6). In our method we approximate the derivatives
with respect to x by using solutions of difference quations (11) which are generated by the
original problem.
2. Suppose that we calculate the number z(r+1,m)
h by using method (7), (8). Then we apply
the vector δz considered at the point (t(r), x(m)). In our method we calculate z(r+1,m)
h by means
of (10) and we need the vector uh at the point (t(r), x(m)) and also at the point (t(r+1), x(m)).
Difference problem (10) – (12) is obtained in the following way. Suppose that Assumption
H0[f ] is satisfied and that the function ϕ : E0 ∪ ∂0E → R is of class C1. Existence theory for
classical or generalized solutions to mixed problem (3), (4) is based on a method of quasili-
nearization. The method consists in replacing problem (3), (4) with the following quasilinear
differential system for unknown functions (z, u), u = (u1, . . . , un), of the variables (t, x):
∂tz(t, x) = f (U [z, u; t, x]) +
+
n∑
j=1
∂qjf (U [z, u; t, x])
(
∂xjz(t, x)− uj(t, x)
)
(18)
and
∂tui(t, x) = ∂xif (U [z, u; t, x]) + ∂pf (U [z, u; t, x])ui(t, x) +
+
n∑
j=1
∂qjf (U [z, u; t, x]) ∂xjui(t, x), 1 ≤ i ≤ n, (19)
with the initial boundary condition
z(t, x) = ϕ(t, x) and u(t, x) = ∂xϕ(t, x) for (t, x) ∈ E0 ∪ ∂0E (20)
where U [z, u; t, x] = (t, x, z(t, x), u(t, x)). Note that each equation of system (18), (19) depends
on the unknown functions (z, u) and it contains partial derivatives of only one scalar function.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
462 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
Under natural assumptions on given functions, the following properties of problem (18) –
(20) may be proved ([4, 5]):
(A) If (z̃, ũ) : E → R1+n, ũ = (ũ1, . . . , ũn), is a classical solution of (18) – (20) then ∂xz̃ = ũ
on E.
(B) If ṽ : E → R is a solution of (3), (4) and v is of class C2 onE then the functions (ṽ, ∂xṽ)
satisfy (18) – (20).
(C) If (z̃, ∂xz̃) : E → R1+n is a solution of (18) – (20) then z̃ satisfies (3), (4).
Difference problem (10) – (12) is a discretization of (18) – (20). The principal significance of
system (10), (11) is that the method of discretization of (18), (19) depends on the properties
of the functions (∂q1f, . . . , ∂qnf) and on the previous values of the unknown functions. System
(18), (19) has the following property: the differential equations of bicharacteristics for (18) and
for (19) are the same and they have the form
η′(t) = −∂qf
(
t, η(t), z(t, η(t)), u(t, η(t))
)
.
This property of system (18), (19) is important in the investigation of the stability of difference
problem (10) – (17).
Remark 2. Suppose that f ∈ C(Ω, R) and
1) the derivatives (∂q1f, . . . , ∂qnf) = ∂qf exist on Ω and ∂qf ∈ C(Ω, Rn);
2) the Lipschitz condition (9) is satisfied and conditions (1), (2) hold.
Then the classical solution of mixed problem (3), (4) is unique. This result may be proved
by a method of differential inequalities.
Note that condition (9) may be replaced in the above statement by a nonlinear estimate of
the Perron type.
Existence results for mixed problem (3), (4) may be deduced from [4, 5], see also [6].
The papers [1 – 3] initiated the investigations of difference methods for nonlinear partial
differential equations and weakly coupled systems. Initial problems on the Haar pyramid and
initial boundary-value problems were considered. The main question in the theory of numerical
methods for nonlinear differential equations is to find a difference equation generated by the
original problem, which is stable. The method of difference inequalities or theorems on linear
recurrent inequalities are used in the investigation of the stability. The monograph [7] conta-
ins an exposition of recent developments on difference methods for hyperbolic differential or
functional differential equations.
The paper is organized as follows. In Section 2 we give sufficient conditions for the conver-
gence of the generalized Euler method. In the next section we consider some modification
of method (10) – (12) and we prove a convergence result under assumptions that the deri-
vatives ∂xf, ∂pf, ∂qf satisfy nonlinear estimates of the Perron type with respect to variables
(p, q). Numerical examples are given in the last part of the paper. Our results are a continuation
of paper [8], where the generalized Euler method was considered for initial problems on the
Haar pyramid. We use in the paper general ideas concerning difference equations which were
introduced in [2, 9, 10].
2. Convergence of the generalized Euler method. We formulate next assumptions on f .
AssumptionH[f ]. Suppose that Assumption H0[f ] is satisfied and
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GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 463
1) there is A ∈ R+ such that
‖∂xf(P )‖, |∂pf(P )|, ‖∂qf(P )‖ ≤ A
for P = (t, x, p, q) ∈ Ω;
2) there is L ∈ R+ such that the terms
‖∂xf(t, x, p, q)− ∂xf(t, x, p̄, q̄‖, |∂pf(t, x, p, q)− ∂pf(t, x, p̄, q̄|,
‖∂qf(t, x, p, q)− ∂qf(t, x, p̄, q̄‖
are bounded from above by L(|p− p̄|+ ‖q − q̄‖).
We prove the main result on the generalized Euler method.
Theorem 2. Suppose that Assumption H[f ] is satisfied and
1) h ∈ ∆ and
1− h0
n∑
j=1
1
hj
∣∣∂qjf(t, x, p, q)
∣∣ ≥ 0 on Ω; (21)
2) the function ϕ : E0 ∪ ∂0E → R is of class C2 and v : E → R is a solution of problem (3),
(4) and v is of class C2 on E;
3) the functions (zh, uh) : Eh → R1+n, where uh = (uh,1, . . . , uh,n), satisfy (10) – (12) with
δ0 and δ given by (13) – (17) and there is a function α0 : ∆ → R+ such that
|ϕ(r,m) − ϕ(r,m)
h |+ ‖∂xϕ(r,m) − ψ(r,m)
h ‖ ≤ α0(h) on E0,h ∪ ∂0Eh (22)
and lim
h→0
α0(h) = 0.
Then there is a function α : ∆ → R+ such that
‖vh − zh‖r,h + ‖∂xvh − uh‖r,h ≤ α(h), 0 ≤ r ≤ N0, (23)
and lim
h→0
α(h) = 0, where vh and ∂xvh are the restrictions of v and ∂xv respectively to the set Eh.
Proof. Write w = ∂xv and w = (w1, . . . , wn). Let us denote by wh = (wh,1, ..., wh,n) the
restriction of w to the set Eh and put
ξh = vh − zh, λh = wh − uh (24)
with λh = (λh,1, ..., λh,n). Let the functions ωh,0, ωh,1 : Ih → R be defined by
ω
(r)
h,0 = ‖ξh‖r,h, ω
(r)
h,1 = ‖λh‖r,h, (25)
where 0 ≤ r ≤ N0. Set ωh = ωh,0 + ωh,1. We will write a difference inequality for the function
ωh. We first prove some properties of ωh,0. Let the functions Γh,0,Λh,0 : E′h → R be defined by
Γ(r,m)
h,0 = δ0v
(r,m)
h − ∂tv(r,m) +
n∑
j=1
∂qjf
(
P (r,m)[v, w]
)(
∂xjv
(r,m) − δjv(r,m)
h
)
(26)
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464 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
and
Λ(r,m)
h,0 = f
(
P (r,m)[v, w]
)
− f
(
Q(r,m)[zh, uh]
)
+
+
n∑
j=1
∂qjf
(
Q(r,m)[zh, uh]
)
u
(r,m)
h,j −
−
n∑
j=1
∂qjf
(
P (r,m)[v, w]
)
w
(r,m)
j +
+
n∑
j=1
{
∂qjf
(
P (r,m)[v, w]
)
− ∂qjf
(
Q(r,m)[zh, uh]
)}
δjv
(r,m)
h . (27)
Put
Wh[ξh](r,m) = ξ
(r,m)
h + h0
n∑
j=1
∂qjf
(
Q(r,m)[zh, uh]
)
δjξ
(r,m)
h .
It follows from (10) and (18) that ξh satisfies the difference equation
ξ
(r+1,m)
h = Wh[ξh](r,m) + h0
[
Γ(r,m)
h,0 + Λ(r,m)
h,0
]
(28)
where (t(r), x(m)) ∈ E′h. Write
J+[r,m] =
{
j ∈ {1, . . . , n} : ∂qjf
(
Q(r,m)[zh, uh]
)
≥ 0
}
,
J−[r,m] = {1, . . . , n}\J+[r,m].
It follows from the definition of the operator δ that the function ξh satisfies the relation
Wh[ξh](r,m) = ξ
(r,m)
j
1− h0
n∑
j=1
1
hj
∣∣∣∂qjf (Q(r,m)[zh, uh]
)+
+ h0
∑
j∈J+[r,m]
1
hj
∂qjf
(
Q(r,m)[zh, uh]
)
ξ
(r,m+ej)
h −
− h0
∑
j∈J−[r,m]
1
hj
∂qjf
(
Q(r,m)[zh, uh]
)
ξ
(r,m−ej)
h .
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GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 465
It follows from assumptions (1), (2), (21) and from (6) that∣∣∣Wh[ξh](r,m)
∣∣∣ ≤ ω
(r)
h,0 on E′h. (29)
Since the functions (v, w) satisfy equation (18), it follows that there is γ0 : ∆ → R+ such that∣∣∣Γ(r,m)
h,0
∣∣∣ ≤ γ0(h) on E′h and lim
h→0
γ0(h) = 0. (30)
Let c̃ ∈ R+ be such a constant that
|∂tv(t, x)|, ‖∂xv(t, x)‖,
(31)
‖∂txv(t, x)‖, ‖∂xxv(t, x)‖ ≤ c̃ on E
where
∂xxv(t, x) =
[
∂xixjv(t, x)
]
i,j=1,...,n
and
‖∂xxv(t, x)‖ = max
{
n∑
i=1
|∂xixjv(t, x)| : 1 ≤ j ≤ n
}
.
According to Assumption H[f ] we have∣∣∣Λ(r,m)
h,0
∣∣∣ ≤ Aω
(r)
h,1 + (A+ 2c̃L)
[
ω
(r)
h,0 + ω
(r+1)
h,1 + c̃h0
]
(32)
where (t(r), x(m)) ∈ E′h. Applying (28) – (32) we conclude that the recurrent inequality
ω
(r+1)
h,0 ≤ ω(r)
h,0 +Ah0ω
(r)
h,1 + h0(A+ 2c̃L)
[
ω
(r)
h,0 + ω
(r+1)
h,1
]
+
+ h0[γ0(h) + h0c̃(A+ 2c̃L)] (33)
is satisfied for 0 ≤ r ≤ N0 − 1.
Now we write a difference inequality for ωh,1. Let the functions
Γh = (Γh,1, . . . ,Γh,n) : E′h → Rn,
Λh = (Λh,1, . . . ,Λh,n) : E′h → Rn
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466 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
be defined by
Γ(r,m)
h,i = δ0w
(r,m)
h,i − ∂tw(r,m)
i +
+
n∑
j=1
∂qjf
(
P (r,m)[v, w]
)(
∂xjw
(r,m)
i − δjw(r,m)
h,i
)
, 1 ≤ i ≤ n, (34)
and
Λh,i = ∂xif
(
P (r,m)[v, w]
)
− ∂xif
(
P (r,m)[zh, uh]
)
+
+ ∂pf
(
P (r,m)[v, w]
)
w
(r,m)
i − ∂pf
(
P (r,m)[zh, uh]
)
u
(r,m)
h,i +
+
n∑
j=1
[
∂qjf
(
P (r,m)[v, w]
)
− ∂qjf
(
P (r,m)[zh, uh]
)]
δjw
(r,m)
h,i , 1 ≤ i ≤ n. (35)
Write
Ũh[λh,i](r,m) = λ
(r,m)
h,i +
+ h0
n∑
j=1
∂qjf
(
P (r,m)[zh, uh]
)
δjλ
(r,m)
h,i , 1 ≤ i ≤ n,
and
Uh[λh](r,m) =
(
Ũh[λh,1](r,m), . . . , Ũh[λh,n](r,m)
)
.
It follows from (11) and (19) that the function λh satisfies the difference equation
λ
(r,m)
h = Uh[λh](r,m) + h0
[
Γ(r,m)
h + Λ(r,m)
h
]
(36)
where (t(r), x(m)) ∈ E′h. Write
I+[r,m] =
{
j ∈ {1, . . . , n} : ∂qjf
(
P (r,m)[zh, uh]
)
≥ 0
}
,
I−[r,m] = {1, . . . , n}\I+[r,m].
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GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 467
Then we have
Ũh[λh,i](r,m) = λ
(r,m)
h,i
1− h0
n∑
j=1
1
hj
∣∣∣∂qjf (P (r,m)[zh, uh]
)∣∣∣
+
+ h0
∑
j∈I+[r,m]
1
hj
∂qjf
(
P (r,m)[zh, uh]
)
λ
(r,m+ej)
h,i −
− h0
∑
j∈I−[r,m]
1
hj
∂qjf
(
P (r,m)[zh, uh]
)
λ
(r,m−ej)
h,i , 1 ≤ i ≤ n.
According to (1), (2), (21) and (6), we have∥∥∥Uh[λh](r,m)
∥∥∥ ≤ ω
(r)
h,1 on E′h. (37)
Since the functions (v, w) satisfy equations (19), it follows that there is γ : ∆ → R+ such that
‖Γ(r,m)
h ‖ ≤ γ(h) on E′h and lim
h→0
γ(h) = 0. (38)
It follows from Assumption H[f ] that
‖Λ(r,m)
h ‖ ≤ Lω
(r)
h (1 + 2c̃) +Aω
(r)
h,1 on E′h,
and consequently
ω
(r+1)
h,1 ≤ ω
(r)
h,1 + h0γ(h) + Lh0(1 + 2c̃)ω(r)
h +Ah0ω
(r)
h,1 (39)
where 0 ≤ r ≤ N0− 1. Combining (33) with (39) we see that there are L̃ > 0 and γ̃ : ∆ → R+
such that
ω
(r+1)
h ≤ ω
(r)
h (1 + L̃h0) + h0γ̃(h), 0 ≤ r ≤ N0 − 1,
and lim
h→0
γ̃(h) = 0 which implies that
ω
(r)
h ≤ α0(h)eL̃a + γ̃(h)
eL̃a − 1
L̃
, 0 ≤ r ≤ N0. (40)
This completes the proof.
Remark 3. Suppose that all the assumptions of Theorem 2 are satisfied and there isM ∈ Rn+
such that h′ ≤ Mh0. Then there are C0, C ∈ R+ such that we have the following error estimate
for method (10) – (12):
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468 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
‖vh − zh‖r,h + ‖∂xvh − uh‖r,h ≤ C0α0(h) + Ch0
where 0 ≤ r ≤ N0.
The above result is a consequence of (30), (38) and (40).
Remark 4. In the results on error estimates we need estimates for the derivatives of the
solution v of problem (3), (4). One may obtain them by the method of differential inequalities.
The results given in [11] (Chapter VII ) and [12] for initial problems on the Haar pyramid can
be easily extended to mixed problems.
3. Convergence of the difference method with nonlinear estimates. We prove that the genera-
lized Euler method is convergent if the Lipschitz condition for the derivatives ∂xf, ∂pf, ∂qf is
replaced by a nonlinear estimate of the Perron type. We change equation (10) in this case. We
consider the quasilinear system of difference equations consisting of equations (11) and
δ0z
(r,m) = f
(
P (r,m)[z, u]
)
+
n∑
j=1
∂qjf
(
P (r,m)[z, u]
)(
δjz
(r,m) − u(r,m)
j
)
(41)
with initial boundary condition (12). Difference expressions
δ0z
(r,m) and δ0u
(r,m)
i , δu
(r,m)
i =
(
δ1u
(r,m)
i , . . . , δnu
(r,m)
i
)
, 1 ≤ i ≤ n,
are given by (13) and (16), (17), respectively. Difference operator δz(r,m) is given by (14), (15)
with P (r,m)[z, u] instead of Q(r,m)[z, u]. Numerical results obtained by the above difference,
method are better than those obtained by (7), (8). We show a numerical example.
Our basic assumptions are the following.
AssumptionH[σ]. The function σ : [0, a]×R+ → R+ is continuous and
1) σ is nondecreasing with respect to both variables and σ(t, 0) = 0 for t ∈ [0, a];
2) for each c ∈ R+ and d ≥ 1 the maximal solution of the Cauchy problem
η′(t) = cη(t) + dσ(t, η(t)), η(0) = 0,
is η(t) = 0, t ∈ [0, a].
Assumption H̃[f ]. Suppose that Assumption H0[f ] and condition 1 of Assumption H[f ]
are satisfied and there is a function σ : [0, a]×R+ → R+ satisfying Assumption H[σ] and such
that the terms
‖∂xf(t, x, p, q)− ∂xf(t, x, p̄, q̄)‖, |∂pf(t, x, p, q)− ∂pf(t, x, p̄, q̄)|,
‖∂qf(t, x, p, q)− ∂qf(t, x, p̄, q̄)‖
are bounded from above by σ(t, |p− p̄|+ ‖q − q̄‖).
Theorem 3. Suppose that Assumption H̃[f ] is satisfied and
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GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 469
1) h ∈ ∆ and condition (21) holds;
2) the function ϕ : E0 ∪ ∂0E → R is of class C2 and v : E → R is a solution of problem (3),
(4) and v is of class C2 on E;
3) the function (zh, uh) : Eh → R1+n, uh = (uh,1, . . . , uh,n), satisfies (11), (41) and (12);
4) there is a function α0 : ∆ → R+ such that initial boundary estimate (22) is satisfied and
lim
h→0
α0(h) = 0.
Then there is a number ε0 > 0 and a function α : ∆ → R+ such that condition (23) is
satisfied for ‖h‖ < ε0 and lim
h→0
α(h) = 0.
Proof. Write w = ∂xv. Let ξ : Eh → R and λh : Eh → Rn be the functions defined
by (24) where vh and wh are the restrictions of v and w to the mesh Eh. We can now proceed
analogously to the proof of Theorem 2. It follows easily that the functions ωh,0 and ωh,1 given
by (25) satisfy the recurrent inequalities
ω
(r+1)
h,0 ≤ ω
(r)
h,0 + 2Ah0ω
(r)
h + 2c̃h0σ
(
t(r), ω
(r)
h
)
+ h0γ0(h) (42)
and
ω
(r+1)
h,1 ≤ ω
(r)
h,1 +Ah0ω
(r)
h + (1 + 2c̃)σ
(
t(r), ω
(r)
h
)
+ h0γ(h) (43)
where 0 ≤ r ≤ N0 − 1 and ωh = ωh,0 + ωh,1. The functions γ0 and γ are given by conditions
(30) and (38) respectively, the constant c̃ is such that estimates (31) hold. The functions Γh,0,
Γh and Λh are defined by (26), (34) and (35) respectively, whereas the function Λh,0 is given by
(27) with P (r,m)[zh, uh] instead of Q(r,m)[zh, uh]. Adding inequalities (42) and (43) we conclude
that the function ωh satisfies the recurrent inequality
ω
(r+1)
h ≤ ω
(r)
h + ch0ω
(r)
h + dh0σ
(
t(r), ω
(r)
h
)
+ h0(γ0(h) + γ(h)), 0 ≤ r ≤ N0 − 1, (44)
where c = 3A, d = 1 + 4c̃. Consider the Cauchy problem
η′(t) = cη(t) + dσ(t, η(t)) + γ0(h) + γ(h), η(0) = α0(h).
It follows from Assumption H[σ] that there is ε0 > 0 such that for ‖h‖ < ε0 there exists the
maximum solution ηh of the above problem and ηh is defined on [0, a]. Moreover we have
lim
h→0
ηh(t) = 0 uniformly on [0, a].
The function ηh satisfies the recurrent inequality
η
(r+1)
h ≥ η
(r)
h + ch0η
(r)
h + dh0σ
(
t(r), η
(r)
h
)
+ h0 (γ0(h) + γ(h)) , 0 ≤ r ≤ N0 − 1. (45)
Since ω(0)
h ≤ η
(0)
h , (44) and (45) show that ω(r)
h ≤ η
(r)
h for 0 ≤ r ≤ N0. Then we get (23)
with α(h) = ηh(a) for ‖h‖ < ε0. This proves the theorem.
Remark 5. The results of the paper can be extended to weakly coupled nonlinear systems
with initial boundary conditions.
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470 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
4. Numerical experiments. Now we give an example of a classical difference method and the
generalized Euler method for a nonlinear equation.
Let n = 2 and E = [0, 1] × [−1, 1] × [1,−1]. Denote by z an unknown function of the
variables (t, x, y). Consider the differential equation
∂tz(t, x, y) = x∂xz(t, x, y) + y∂yz(t, x, y) +
+
1
2
sin
[
x∂xz(t, x, y)− y∂yz(t, x, y)
]
− 4z(t, x, y) + f(t, x, y), (46)
with the initial boundary condition
z(t, x, y) = 0 for (t, x, y) ∈ E0 ∪ ∂0E. (47)
Here
f(t, x, y) =
1
2
xy(x2 − 1)(1− y2)− txy(1− x2y2)− 1
2
sin
[
txy(x2 − y2)
]
.
The exact solution of this problem is known. It is
w(t, x, y) =
t
2
xy(x2 − 1)(1− y2).
Consider the classical difference equation for (46), (47)
z(r+1,m) =
1
4
z(r,m1+1,m2)
[
1 + 2x(m1)h0
h1
]
+
+
1
4
z(r,m1−1,m2)
[
1− 2x(m1)h0
h1
]
+
+
1
4
z(r,m1,m2+1)
[
1 + 2y(m2)h0
h2
]
+
+
1
4
z(r,m1,m2−1)
[
1− 2y(m2)h0
h2
]
+
+
h0
2
sin
[
A(r,m)
]
− 4h0z
(r,m) + h0f
(r,m), (48)
where 0 ≤ r ≤ N0, −N ≤ m ≤ N , N = (N1, N2), m = (m1,m2), and the initial boundary
condition
z(r,m) = 0 for
(
t(r), x(m1), y(m2)
)
∈ E0,h ∪ ∂0Eh (49)
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GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 471
where
A(r,m) = x(m1)
(
z(r,m1+1,m2) − z(r,m1−1,m2)
)
(2h1)−1 −
− y(m2)
(
z(r,m1,m2+1) − z(r,m1,m2−1)
)
(2h2)−1.
Denote by z̃h : Eh → R the solution of (48), (49).
Now we construct the generalized Euler method for problem (46), (47). Let us denote by
(z, u, v) unknown functions of the variables
(
t(r), x(m1), y(m2)
)
. Write
F (t, x, y, p, q1, q2) = xq1 + yq2 +
1
2
sin(xq1 − yq2)− 4p+ f(t, x, y)
and
P (r,m) =
(
t(r), x(m1), y(m2), z(r,m), u(r,m), v(r,m)
)
.
Consider the system of difference equations
z(r+1,m) = z(r,m) + h0F
(
P (r,m)
)
+
+ h0∂q1F
(
P (r,m)
)(
δ1z
(r,m) − u(r,m)
)
+
+ h0∂q1F
(
P (r,m)
)(
δ2z
(r,m) − v(r,m)
)
,
and
u(r+1,m) = u(r,m) + h0∂xF
(
P (r,m)
)
+ h0∂pF
(
P (r,m)
)
u(r,m) +
+ h0∂q1F
(
P (r,m)
)
δ1u
(r,m) + h0∂q2F
(
P (r,m)
)
δ2u
(r,m),
v(r+1,m) = v(r,m) + h0∂yF
(
P (r,m)
)
+ h0∂pF
(
P (r,m)
)
v(r,m) +
+ h0∂q1F
(
P (r,m)
)
δ1v
(r,m) + h0∂q2F
(
P (r,m)
)
δ2v
(r,m),
with the initial boundary conditions
z(r,m) = 0 on E0,h ∪ ∂0Eh,
u(0,m) = v(0,m) = 0 on E0,h
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472 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
and
u(r,m1,N2) = u(r,m1,−N2) = 0,
v(r,N1,m2) = v(r,−N1,m2) = 0,
u(r,N1,m2) = u(r,−N1,m2) = t(r)y(m2)
[
1−
(
y(m2)
)2
]
,
v(r,m1,N2) = v(r,m1,−N2) = −t(r)x(m1)
[(
x(m1)
)2
− 1
]
,
where 0 ≤ r ≤ N0, −N ≤ m ≤ N . The difference operators
(
δ1z
(r,m), δ2z
(r,m)
)
,
(
δ1u
(r,m), δ2u
(r,m)
)
,
(
δ1v
(r,m), δ2v
(r,m)
)
are defined according to our theory.
Let us denote by (zh, uh, vh) : Eh → R3 the solution of the above difference problem. We
give the following information on the errors of the methods. Write
η̃
(r)
h = max
{
|w(r,m) − z̃(r,m)
h | : −N ≤ m ≤ N
}
,
η
(r)
h = max
{
|w(r,m) − z(r,m)
h | : −N ≤ m ≤ N
}
, 0 ≤ r ≤ N0,
and
ε̃
(r)
h =
1
(2N1 + 1)(2N2 + 1)
N∑
m=−N
∣∣∣w(r,m) − z̃(r,m)
h
∣∣∣ ,
ε
(r)
h =
1
(2N1 + 1)(2N2 + 1)
N∑
m=−N
∣∣∣w(r,m) − z(r,m)
h
∣∣∣ , 0 ≤ r ≤ N0.
The numbers η̃(r)
h and η(r)
h are the maximal errors with fixed t(r). The numbers ε̃(r)
h and ε(r)
h are
the arithmetical mean of the errors with fixed t(r). The values of the functions η̃h, ηh, ε̃h, εh are
listed in the tables.
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GENERALIZED EULER METHOD FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS 473
Table of errors for h0 = 10−3, h1 = h2 = 10−2
t(r)
(
η̃(r); η(r)
) (
ε̃(r); ε(r)
)
0,4 1 · 10−3; 2 · 10−4 1 · 10−3; 1 · 10−4
0,5 2 · 10−3; 3 · 10−4 1 · 10−3; 1 · 10−4
0,6 3 · 10−3; 4 · 10−4 1 · 10−3; 2 · 10−4
0,7 3 · 10−3; 5 · 10−4 1 · 10−3; 2 · 10−4
0,8 4 · 10−3; 6 · 10−4 2 · 10−3; 2 · 10−4
0,9 5 · 10−3; 7 · 10−4 2 · 10−3; 2 · 10−4
1,0 5 · 10−3; 8 · 10−4 2 · 10−3; 3 · 10−4
Table of errors for h0 = 10−4, h1 = h2 = 5 · 10−3
t(r)
(
η̃(r); η(r)
) (
ε̃(r); ε(r)
)
0,4 3 · 10−3; 1 · 10−4 1 · 10−3; 5 · 10−5
0,5 5 · 10−3; 1 · 10−4 2 · 10−3; 7 · 10−5
0,6 6 · 10−3; 2 · 10−4 2 · 10−3; 9 · 10−5
0,7 8 · 10−3; 2 · 10−4 3 · 10−3; 9 · 10−5
0,8 9 · 10−3; 3 · 10−4 4 · 10−3; 1 · 10−4
0,9 1 · 10−2; 3 · 10−4 4 · 10−3; 1 · 10−4
1,0 1 · 10−2; 4 · 10−4 5 · 10−3; 1 · 10−4
Note that η(r)
h < η̃
(r)
h and ε(r)
h < ε̃
(r)
h for all the values of t(r).
We give also the following information on the errors of the methods. Write
η̃h = max
{
η̃
(r)
h : 1 ≤ r ≤ N0
}
,
ηh = max
{
η
(r)
h : 1 ≤ r ≤ N0
}
and
ε̃h =
1
N0
N0∑
r=1
ε̃
(r)
h , εh =
1
N0
N0∑
r=1
ε
(r)
h .
The values of the errors (η̃h, ηh) and (ε̃h, εh) are listed in the following table.
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
474 Z. KAMONT, J. NEWLIN-ŁUKOWICZ
Table of global errors
(h0, h1 = h2) (η̃h; ηh) (ε̃h; εh)
10−3; 5 · 10−2 0,050628; 0,000425 0,010604; 0,000679
10−3; 10−2 0,005727; 0,000890 0,001063; 0,000147
5 · 10−4; 10−2 0,010666 ; 0,000890 0,001979 ; 0,000147
10−4; 10−2 0,034160 ; 0,000810 0,006635 ; 0,000147
10−4; 5 · 10−3 0,012884; 0,000447 0,002387; 0,000074
Note that ηh < η̃h and (εh < ε̃h) for all the values h.
Thus we see that the errors of the classical difference method are larger than the errors
of the generalized Euler method. This is due to the fact that the approximation of the spatial
derivatives of z in the new method is better than the respective approximation of ∂xz in the
classical case. Methods described in Theorems 2 and 3 have the potential for applications in the
numerical solving of first order nonlinear differential equations.
1. Kowalski Z. A difference method for certain hyperbolic systems of nonlinear partial differential equations
of first order // Ann pol. math. — 1967. — 19. — P. 313 – 322.
2. Kowalski Z. On the difference method for certain hyperbolic systems of nonlinear partial differential equati-
ons of the first order // Bull. Acad. pol. sci. Sér. sci. math., astron. et. phys. — 1968. — 16. — P. 297 – 302.
3. Pliś A. On difference inequalities corresponding to partial differential inequalities of first order // Ann. pol.
math. — 1968. — 28. — P. 179 – 181.
4. Cinquini S. Sopra i sistemi iperbolici di equazioni a derivate parziali (nonlineari) in piu variabili indipendenti
// Ann. mat. pura ed appl. — 1979. — 120. — P. 201 – 214.
5. Cinquini Cibrario M. Sopra una classe di sistemi di equazioni non lineari a derivate parziali in piu variabili
indipendenti // Ibid. — 1985. — 140. — P. 223 – 253.
6. Tran Duc Van, Miko Tsuji, and Nguyen Duy Thai Son. The characteristic method and its generalizations for
first-order nonlinear partial differential equations. — Boca Raton: Chapman & Hall/CRC, 2000.
7. Kamont Z. Hyperbolic functional differential inequalities and applications. — Dordrecht: Kluwer Acad.
Publ., 1999.
8. Baranowska A., Kamont Z. Finite difference approximations for nonlinear first order partial differential
equations // Univ. Iagell. Acta Math. — 2002. — 40. — P. 15 – 30.
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York: Springer, 1996.
10. Magomedov K. M., Kholodov A. S. Mesh-characteristics numerical methods. — Moscow: Nauka, 1988 (in
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12. Lakshmikantham V., Leela S. Differential and integral inequalities. — New York; London: Acad. Press, 1969.
Received 11.06.2003
ISSN 1562-3076. Нелiнiйнi коливання, 2003, т . 6, N◦ 4
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