Second order quasi-linear singular perturbedproblem with Neumann boundary conditions and discontinuous term Текст научной статьи по специальности «Математика»

Section 5. Mathematics

Stepanov Vasily Innokentievich, East China Normal University, Ph.D., student, School of Mathematical Sciences

E-mail: [email protected] Ni Mingkang, doctor, of Mathematics Prof., School of Mathematical Sciences East China Normal University,

SECOND ORDER QUASI-LINEAR SINGULAR PERTURBED PROBLEM WITH NEUMANN BOUNDARY CONDITIONS AND DISCONTINUOUS TERM

Abstract: Using boundary functions method in combination with the method of sewing connection proved the existence of a solution and constructed its asymptotic expansion for 2nd order quasi-linear system with Neumann boundary conditions and discontinuous term.

Keywords: singular perturbed problem, boundary functions method, Neumann conditions, asymptotic solution.

1. Introduction

This paper devoted to the illustration of boundary functions method [1, 4] application to the solution of singularly perturbed problems on the example of 2nd order quasi-linear system with Neumann boundary conditions and discontinuous term using sewing connection as it is described in [2]. Using approach proposed by Vasil'eva A.B. in [3, 4] proved the existence of a solution and constructed its asymptotic representation and formulated theorem for solving the singularly perturbed problem given in the paper.

2. Formulation of Problem

Considered following Neumann boundary value problem

jvy" = A(y,t)y' + f(y,t), 0 < t <1, y'(0, ») = y°, y'(l,M) = y1,

HiS0 s '< t0

10 < t < 1.

System (1.1) modified into the following form.

¡z ' = A (y,t )z + f (y,t ),

(3)

z =

dy

(1)

where

0 < t < tn

A (yt )-iA ("(y ,t ),

A(y,tH lW(y,t), to < t < 1,

A

(2)

dt ' (4)

z (0, )) = y0, z (1, )) = y1, where H is a small parameter, t0 is known point, y is a scalar function, f (y ,t) sufficiently smooth enough function. In this paper, considered the particular case when so called contrast structure occurred [5, 6]. Solution of such problem is a step-like contrast structure with a spike at the point t0 and due to discontinuity of A(y,t) and f (y,t) functions.

The solution of described above problem constructed in the following form of function, which satisfies the initial system (1).

3. Main Assumptions

In order to solve stated above singularly perturbed problem, it is required to hold following assumptions.

(5)

(6)

Assumption 1. Functions A(y,t) and f (y,t) are discontinuous at point t = t0. Thus, it is required to satisfy following inequalities at the discontinuity point.

A(-) (y>to) A(+)(y,to), y e Iy,

f (-)(y,to) f (+)(y,to), y e Iy.

Assumption 2. Degenerate equations \aW(y,t)y' + f W(y,t), 0 < t < to,

[ y'(o,= y0, y'(1,= y1.

Have isolated roots (t) on the interval [o,to ] ,and (p2 (t) on the interval [to ,1]. In order to be more precise, here we assumed that p1 (to ) < p2 (to).

Assumption 3. The condition of internal transitional layer existence at the transitional point t = to.

[A(-)(y,t)> o, o < t < to, <A(+) (y,t)< o, to < t < 1.

Here, constructed a solution of the problem (1), with a discontinuity at point t = to.

t o (v)= to + M +-.. + V"tn +... .

Introduced following substitution of variable and used it for the initial equation in the following form:

nz' = A(%,t)z + f^(y,t), = z, (7)

y '(o, n) = y0, y '(1, n) = y \

A new variable introduced:

z (to, n) = — + Zo +... + ^Zn + ■••, y (t o ) = Уo, (8) H

where yo, z{ are unknown yet. Its values will be determined and calculated while the process of asymptotic expansion construction.

Asymptotic expansion of the problem (1.7) constructed in the following form.

y (t, ») = y (t, + Q)y n) z (t, n) = z (t, n) + Q (T)y ,

where £ = -—t° and

y (t, = To (t) + Wi () + --- + vnyn + .-o z(t,n) = z0 (t) + pzi(t) + ... +vnzn + .••, is a regular part of asymptotic expansion.

QV (e,M) = Q(t )y (D + rfV («) + ...+ +MnQ(: v (£)+...,

Q(T )z S )z (|) + z o(0 + ...+

+^nQnT ^ ($) + ...,

(9)

(10)

ai)

(12)

(13)

Series describe boundary layer in the neighborhood of to point.

Assumption 4. On the phase plane (y, z). Vertical

line y = y0,1, intersect separatrix Q. This means, that

y y

I f (s,0,0)ds > 0 , | f (s,i,0)ds > 0 for any

V(0) VI(i)

y e(<Pi (t(-)) .

4. Construction of Formal Asymptotic

For simplifying construction asymptotic solution of the initial problem, two auxiliary problems introduced: Left boundary value problem on the interval [0,t0 ]

Ly" = A(-) (y,t)y'+ f(-) (y,t), 0 < t < -0,

1 y'(0m) = y^ y(t0,M) = p(m).

Right boundary value problem on the interval [t0, i]

\py" = A(+)(y,t)y'+ f (+)(y,t), 10 < t < i

y (t 0, n) = p (p), y^= yi. Function pdefined while the process of asymptotic expansion coefficients calculation. It defined in the form Taylor series with the power of expansion /d.

p (m) = p0+m+---+vnpn +•.

5. Calculation of Regular and Boundary Layer's Asymptotic Coefficients

New variable substitution introduced z = —, as the result equivalent system to the system (12) ob^ined.

^ = A (-)(y ,t )z + f (-)(y ,t ), dt

y '(0, n) = y0, y (t0, n) = p (»), (14)

z(-) ( )

z (t o, V) = — + z 0-) =•••.

Formal solution of the problem (12) on the interval where t e [0,t0 ] constructed in the following form.

y(-)(t, n) = y(-)(t, + Q(-)y(^, ^, (15) z(-)(t, M) = z(-)(t, n) + Qz (£, n), (16)

where

y (")(t = y0-)(t ) + ^7i(-)(t ) + yH(t ) + ^ + + ) + ...,

Q (i)y (t, n) = Q^y W + MQ^y + ^ + + MkQ(-)y (%) + ...,

z(-) (t, n) = z0(-) (t ) + /dz(~) (t ) + _ + i/z{~] (t ) + ...,

Q (-)z (t ,p) =

Q-1)z (5)

' Q0-)z (5) + pQl-)z (5)

+ pkQi-)z (5) + —

1 =

(t - 10 )

dQ(T)z _ y(% + + -— A

d^

Q(T )y,t 0

z (% + ,u)-

+Q(t)Z

+f{T){y(to + Uu) + QWz) - (19)

-a (((T)(+^,u),t> o +u u)-

-f ^((t+UuK+U

^^ = vQ ^z, Q (T)z (+«) = 0, Q Wy (W) = 0,

where t = t0 +

Let us equate coefficients with the highest power (order n~ in (19) and order ¡U), following results, obtained.

In particular, for the main term of the regular part y(+) (t) a solution of the degenerate equation,

[ A *>(' Kt)) > + f(< )((0»,t ) = 0, ^ ^)'

y\ (0) = y0, y\ (1) = y^

(20)

Then in case of y(+) (t), where k > 1 following system of linear equations obtained.

Functions y( )(t,n) and \t,j) considered as regular parts of the asymptotic expansion of the problem's solution. Functions Qk-ty(tand Q()z(t,n) series represent a transitional layer in the neighborhood of transitional point t = 10. It is necessary to mention that series represented transitional layer must satisfy the following condition (-<») = 0.

In a similar way constructed a formal solution of the right-side boundary value problem (13).

y(+)(t, n) = y{+)(t, n) + Q(+)y . (17) z(+\t,^) = z(+)(t, n) + Q (+)z M). (18)

Using a standard approach introduced in the boundary functions method lets insert equations (15-18) into the problems (12-13). Variables with the same power of t, £ are separated and coefficients with the same power of M will be equal to each other.

dd-) = A )(y0 \t )) + F \t ),

=z

(t )

(21)

dt

y f(0 ) = 0,

where p(+)(t) is a function from (y0+),y(+\...,ygj) , k > 1.

From the theory of differential equations, it is known that linear differential equations are uniquely solvable. Thus, all of the coefficients of the regular part defined.

Internal transitional layer constructed using same standard approach. As the result, following system obtained.

= (0)+Q0V0),

dQ0TV_ n(T

dÇ

= Q-r z,

(22)

Q-1 (+<») = 0 , Q0T)y (±») = 0 , Q-?z(0) = z^-z^), Q0)T)y(0) = P0-V(t0).

A new variable introduced and substitution operation result would be y0+)=^; (t0) + Q0+V(%) and ¿W = y°+^, then as the result obtained modified system.

dzSh _ AM(yM t )

jyM- A (y° ,t0h

(23)

zg«) = 0, yW(+«>)=& (t0),

z^z (0 ) = z?, yjr )= P0.

Equation (23) contains differential equations with a separable variable. Using assumptions 2-4 obtained.

dy\

(+)

/

= zg)= j A)(s,t0 )ds,

p0

zg)= j a(T)(s,t0 )ds.

n ( 10 )

From there it is not difficult to see, that

Q0" )y (0 = y 0" ](ï)-vt(t 0 ).

Let us define other coefficients of asymptotic expansion (y(|),Qk-)z(I)) ,where k > 1. As the result, following system of equations obtained.

+

_ A(t) (ß)Qkiz + A( ) (ß)Q?z^y + Gk) (t),

dt "Qk-1Z'

Q&(+<x>) = 0, k(±o) = 0, Q(-lz(0) = zg, QkT)y(0) = pk -y(t0. where G^l^i;) is a known function, A(+)(£), A(:) culated at the point ( (t0 ) + Q0:f)y ,0).

Notice that function contains unknown pk. In order to define and calculate unknown coefficients, the condition of asymptotic expansion derivative equality for the problem (12) at the point t = 10 used.

y '(-)(t 0, n) = y '(+)(t 0, n) Takin into account that,

z '(-)(t 0, )) = z ,(+){t 0, ) (25)

At first, defined and calculated p0. In approximation of jf1, condition (25) rewritten into the next form.

Q^z (0 ) = Q -i z (0 ). (26)

From assumption 2, following inequality obtained.

H(<Pi (t 0 ))H(<Pi (t 0 ))< 0. (29)

Which means existence of p0 e(<Pi (10 (0 )) (24) that H(p0 ) = 0.

Thus, sequentially all of the pk are defined and calculated, where k > i. Following condition obtained as the result.

zk- (t0 ) + Qk-z(0) = z« (t0 ) + Qk+Jz(0). (30) Taking into account, that

d (A A: +

d^

+4KZ)

d(v (t0 ) + Q<f))

d£,

Q^y =

(31)

= AÏ^Q^z + ^zQkt V

Obtained,

dQ^z _ d(A+ G(T) dÇ dÇ

+ G(-)(Ç). (32)

that

By applying integration to the last equation and taking into consideration boundary conditions, obtained In order to solve introduced above problem, assume next equation

s

Qk—z(s) = AQfV(S)+jGjS(s)ds.

By assuming Ç = 0, obtained following expression.

Qk-jz(0) = A(:) (p0,t0)(pk -y((t0)) + {G(-) (s)ds .

H(p)= J A(-)(s,to )ds - { A(+)(s,to )ds . (27)

Pi(fo) 92(to)

Next step is to verify the uniqueness of the described above equation's solution po e ((to),^2 (to)). Thus,

H (nit o ))H ( (t o )) =

A new substitution of the variable introduced

Q(-|z (0) in (30). As the result, obtained equation for

^(t0) (t0) - (28) calculation pk.

= j A(-)(s,t0)dsx J A(+)(s,t0)ds'

9i(to)

n(f0 )

Pk =

!->(<0)-10)+a'-»(P0,t0w(t0)-A<-(P0,t0)y(-\t0)] C'^.10d-CGH(s0d

' 9i ( 10 )

(a (-)(p0,t 0 )-a (%,t 0 ))

( A (_)(p0>t 0 )-A (+)(p0,t 0 )) (33)

Thus, in case of sufficient smoothness, it is possible Theorem. If the assumptions 1-4 are holds and val-

to define and calculate all of the asymptotic expansion's ues of small parameter ¡j,> o sufficiently small enough, coefficients for the function Q^y(£), Q^z(£) for any. there exist smooth solution y(t,p) for the boundary 6. The Existence of Asymptotic Solution value problem (1), which has following asymptotic rep-

resentation

y (t > /) =

Y/k [y(-)+Q(]y(%)]+0(/+2 ), 0 < t < 10,

k=0

n+1 r -1

Y/[y(}+Qt]y(ï)]+o(/n+2), 10 <t < 1.

(34)

Proof. It is known that partial sum of a k -order approximate solution of the initial problem (1) to the order up to O(/dk,and its derivatives up to O(ßk) order. Substitute derivative solutions at the point t0 for proving existence solution of the initial problem and its asymptotic expansion following auxiliary problems considered.

Let us consider first auxiliary problem on the interval

0 < t < tn

0 .

W,,(-) = A(-)(y,t)y'(-) + f(-)(y(-),t), 0 < t < to,

y '(-)(0, w) = y 0, y (-)(to, W) = p (w),

wherep(v) = p0 + Vp1 + ... + ^n+1 (pn+1 + 5), 8 is a parameter.

Next step is to consider the similar problem on the interval 0 < t < 10.

10 < t < 1

(36)

Wy = A (+)(y ,t )y,(+) + f (+)(y (+),t)

y(+)(to,n) = pC4 y'(+)(i,n) = y1.

As mentioned previously every auxiliary problem given above has a solution y( ^ (t,/ and y(+)(t,/ that represented in the following forms.

n+1 r -1

y(-) (t,„) = [t] (t) + Q^y(Z)] + O(+2),(37)

k=0

where Ç =

t -10

n +1

yw (t,ß) = [yk+) (t) + Qi+V(a)] + O(+>) .(38)

k=0

If the derivatives of these functions have the same value at the point t = t0, then these functions are solutions for the initial problem (1).

Here given a demonstration that p can be chose and satisfy following conditions.

du ( ) / \ du() / \ \t 0> = 0' V).

dt

Let us assume that,

dy(-)

dt

(39)

dy

(+)

I (p ' ^-j- (t 0' (t 0' »). (40)

From already described previously asymptotic expansion and derivatives equality condition concluded result.

I(p= {(0) + q{-)-0)-Qi+z(0))

Q (

v"+1 ) = v"+1

.(41)

"5 (A (~\p 0,t o)- A (+)(p0,i o }) From the equation above it is known, that there exists S=S* such that I(5*,^) = 0 . Thus, obtained smooth solution y(t,of initial problem (1).

References:

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3. Vasil'eva A. B. On a system of two singularly perturbed second order quasilinear equations. J. Vychisl. Mat. Mat. Fiz., - Vol. 44. - No 4. - 2004. - P. 650-661.

4. Vasil'eva A. B., Butuzov V. F. Asymptotic expansions of solutions of singularly perturbed equations. Nauka, Moscow - 1973. - 272 p.

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