About estimations of difference for the partial integro-differential equation with small parameter at the leading derivative

0.1 MB·English

by  I. Kopshaev

About estimations of difference for the partial integro-differential
equation with small parameter at the leading derivative
6 I. Kopshaev ∗
0
0 February 2, 2008
2

n
a
J
Abstract

0
2 Thispaperisdevotedtothestudyofthesingularlyperturbedsecondorderpartialintegro-differential
equations. The estimation of thesolutions of Cauchy problem is obtained.

]
A
1 Introduction
C
.
h
Constructing of asymptotic decompositions of solutions of singularly perturbed differential equations has
t
a great theoretical and practical importance. In this line of investigations, the fundamental results obtained
m
by A.N. Tihonov [1], A.B. Vasyliyeva [2], K.A. Kasymov [3], M.I. Imanaliyev [4], L.A. Lyusternik [5] and
[ others.

However, the results, obtained in [1]-[5], still not generalized for the partial second order integro-
1
v differential equations of Volterra type.
2 The aim of this paper is the study of the Cauchy problem with initial jump for the singularly perturbed
9 partial second order integro-differential equations.
4
1
0 2 Preliminaries
6
0
Consider in G={(t,x):0≤t≤1,λ≤x≤λ+1} following problem
/
h
t L y =εH2[y]+A(t,x)H[y]+B(t,x)y =
a ε
m
: t
v
=F(t,x)+ (K (t,s,x)H[y(s,x)]+K (t,s,x)y(s,x))ds, (1)
i Z 1 0
X 0
r ′
a y(0,x,ε)=π0(x), ε·yt(0,x,ε)=π1(x). (2)
Here ε > 0 - a small parameter, t,x - independent variables, y = y(t,x,ε) - unknown function, A(t,x),
B(t,x), F(t,x), K (t,s,x) and π ,(i=0,1) - functions given in G, operators
i i
H[y]=<e(t,x)·grady >, H2[y]=H[H[y]],
where < · > denotes inner product of vectors e(t,x) = (1,Q(t,x)) and grady = ∂y,∂y , Q(t,x) is also
∂t ∂x
(cid:16) (cid:17)
given in G, the function λ(t) is a solution of characteristic equation
dx
=Q(t,x). (3)
dt
∗Institute of mathematics of NAS of KAZAKHSTAN, 125 Pushkina str., 050010 Almaty, KAZAKHSTAN email: kop-
[email protected]
1
Consider also disturbed problem
L y =A(t,x)H[y ]+B(t,x)y =
0 0 0 0
t
=F(t,x)+ (K (t,s,x)H[y (s,x)]+K (t,s,x)y (s,x))ds, (4)
1 0 0 0
Z
0
y (0,x)=π (x). (5)
0 0
obtained from (1), (2) when ε=0.
Suppose, that
1) A(t,x), B(t,x), F(t,x), K (t,s,x), λ(t) and π ,(i=0,1) - continuous functions in G.
i i
2) conditions
inf A(t,x)≥γ >0, inf Q(t,x)≥σ >0, inf π (x)≥σ >0,
i
(t,x)∈G (t,x)∈G (t,x)∈G
λ(0)=0,λ(1)=1,
is satisfied, where γ and σ - some real numbers.
3 Estimation of difference
In this section, I prove estimations of the difference between perturbed and unperturbed Cauchy problems.
Using theorem2 from[6], is notdifficult to prove,that solutiony(t,x,ε)ofproblem(1), (2) doesn’tgoes
to solution of problem (4), (5) when ε→0.
Consider the following problem
t
L y =F(t,x)+ (K (t,s,x)H[y (s,x)]+K (t,s,x)y (s,x))ds+∆(t,x), (6)
0 0 1 0 0 0
Z
0
y (0,x)=π (x)+∆ (x), (7)
0 0 0
where ∆(t,x), ∆ (x) - not for a while yet unknown functions.
0
Function ∆ (x) is call to be named initial jump of solution of problem (1), (2), function ∆(t,x) - initial
0
jump of integral term of equation (1).
Suppose, that solution y (t,x) of problem (6), (7) when t=t = ε|lnε| satisfies the condition
0 0 γ
y (t ,x)=y(t ,x,ε), x∈G. (8)
0 0 0
Theorem 1. Let conditions 1),2) be satisfied. Then for the difference between solution of problem (1), (2)
and the solution of problem (6), (7) in G ⊂G have place following estimations
1
′
|y(t,x,ε)−y (t,x)|≤K·ε·|lnε|+K·ε·|y (t ,x,ε)|+K· max |K (t,0,x)·∆ (x)−∆(t,x)|,
0 t 0 1 0
(t,x∈G)
′ ′ ′
|y (t,x,ε)−y (t,x)|≤K·ε·|lnε|+K·ε·|y (t ,x,ε)|+
t 0t t 0
+K· max |K1(t,0,x)·∆0(x)−∆(t,x)|+K· 1+|yt′(t0,x,ε)| ·e−γε(t−t0),
(t,x∈G) (cid:16) (cid:17)
′ ′ ′
|y (t,x,ε)−y (t,x)|≤K·ε·|lnε|+K·ε·|y (t ,x,ε)|+
x 0x t 0
+K· max |K1(t,0,x)·∆0(x)−∆(t,x)|+K· 1+|yt′(t0,x,ε)| ·e−γε(t−t0), (9)
(t,x∈G) (cid:16) (cid:17)
where K - some constant independent on t and ε, G ={(t,x):0<t ≤t≤1,λ≤x≤λ+1}.
1 0
2
Proof. Indeed, in (1) assign y(t,x,ε) = y (t,x)+u(t,x,ε), and taking into consideration (7), (8), we
0
obtain for u(t,x,ε) following problem
t
L u=f(t,x,ε)+ (K (t,s,x)H[u(s,x,ε)]+K (t,s,x)u(s,x,ε))ds, (10)
ε 1 0
Z
t0
′ ′
u(t ,x,ε)=0, u (t ,x,ε)=y (t ,x,ε), (11)
0 t 0 t 0
where function f(t,x,ε) has a representation
t ∂K (t,s,ϕ)
f(t,x,ε)=K (t,0,ϕ)·∆ (ψ)+ K (t,s,ϕ)− 1 u(s,ϕ,ε)ds−∆(t,x)−εH2[y (t,x)],
1 0 Z (cid:18) 0 ∂s (cid:19) 0
t0
and estimation
|f(t,x,ε)|≤K·ε·|lnε|+ max |K (t,0,x)·∆ (x)−∆(t,x)|, (t,x)∈G, (12)
1 0
(t,x∈G)
where ϕ=ϕ(t,ψ) - a solution of characteristic equation (3), ψ =ψ(t,x)=x - first integral of equation (1)
0
[6]. Applying to the problem (10), (11) theorem 1 from [6], and taking into consideration (12), we obtain
(9). Theorem is proved.
4 Initial jumps of solutions and integral term
The aimof this sectionis to define the conditions in the presence ofwhichthe solutionof perturbedCauchy
problem goes to the solution of unperturbed problem.
Taking into consideration (9), assign
∆(t,x)=∆ (x)·K (t,0,x). (13)
0 1
Then from theorem1, we obtain, that
limy(t,x,ε)=y (t,x), limy′(t,x,ε)= ∂y0(t,x), (t,x)∈G .
x→0 0 x→0 t ∂t 1
Further, for define ∆ (x), integrate equation (1) along characteristicx=ϕ(t,ψ) on t from 0 to t . Then we
0 0
have
t0 ′
ε·H[y (t ,ϕ,ε)]−π (ψ)+A(t ,ϕ)y (t ,ϕ,ε)−A(0,ϕ)π (ψ)− A (t,ϕ)−B(t,ϕ) ×
0 0 1 0 0 0 0 Z0 (cid:16) t (cid:17)
t0 t
×y(t,ϕ,ε)dt= F(t,ϕ)+ (K (t,s,ϕ)H[y(s,ϕ,ε)]+K (t,s,ϕ)y(s,ϕ,ε))ds dt (14)
1 0
Z (cid:18) Z (cid:19)
0 0
From (14), passage to the limit when ε→0, and taking into consideration (7), ( 9), (13), obtain
π (ψ) π (ψ)
∆ (x)= 1 , ∆(t,x)= 1 ·K (t,0,ϕ). (15)
0 A(0,ψ) A(0,ψ) 1
Thus, in G , if equalities (15) is satisfied, then difference between solution y(t,x,ε) of problem (1), (2) and
1
solution y (t,x) of problem (6), (7) will be enough small with ε.
0
References
[1] A. N. Tihonov. About dependence of the solutions of differential equations on small parameter. J.
Mathematical collection, 22(64): 193–204,1948.
3
[2] A. B. Vasyliyeva, V.F. Butuzov. The asymptotic methods in the theory of singularly pertubations.
Moscow, 1990.
[3] K. A. Kasymov. Linear second order singularly pertubed equations. Almata-Ata, 1981.
[4] M. I. Imanaliev. The asymptotic methods in the theory of singularly. pertubed integro-differential
systems. Frunze, 1972.
[5] M.I.Vishik,L.A.Lyusternik. Theregularexpressionandboundaryfibreforlineardifferentialequations
with small parameter. J. Progress of the mathematical scinces, 12(5): 3–122,1957.
[6] I.T. Tazhimuratov, I.R. Kopshaev. About estimation of solutions of linear singularly pertubed partial
equations of Volterra type. J. Bulletin of Kazakh National University, 31(6): 925–931,1982.
4