Mechanization for solving SPP by reducing order method.

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Mechanization for solving SPP by
reducing order methodq
Zhenqing Lia, Weiming Wangb,*
aLaboratory of Quantitative Vegetation Ecology, Institute of Botany,
The Chinese Academy of Sciences, Beijing 100093, China
bSchool of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang
325035, China
Abstract
In this paper, by using the theories and methods of mathematical analysis and com-
puter algebra, a reliable algorithm of reduction of order approximation method for
solving singular perturbation problems was established, a new Maple procedure red-
ordproc was established, too. The procedure redordproc give not only the approx-
imate analytic solutions but also the numerical solutions of the kinetics problems. Some
examples are presented to illustrate the implementation of the program.
? 2004 Published by Elsevier Inc.
Keywords: Singular perturbation problems (SPP); Reduction of order method; Algorithm; Mech-
anization; Maple
0096-3003/$ - see front matter ? 2004 Published by Elsevier Inc.
doi:10.1016/j.amc.2004.11.006
qThe project is supported by the Natural Science Foundation of China (Granted No. 30270249
& No. 10371090) and Natural Science Foundation of Zhejiang (Granted No. M103043).
*Corresponding author.
E-mail addresses: lizhq@ibcas.ac.cn (Z. Li), weimingwang2003@163.com (W. Wang).
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1. Introduction
Many physical processes connected with nonuniform transitions are de-
scribed by differential equations with large or small parameters. If, in problems
arising in this manner, the role of the perturbation is played by leading terms of
the differential operator (or part of them), then the problem is called a singular
perturbation problem (SPP) [1]. Singular perturbation problems occur very fre-
quently in fluid mechanics and other branches of applied mathematics [2].
These problems depend on a small positive parameter in such a way that the
solution varies rapidly in some parts and varies slowly in some other parts
[3]. So, SPP have been received a significant amount of attention in past and
recent years. And some valid methods for solving SPP have been developed
in resent years, such as Taylor series, spline in compression, least-square spec-
tral collection, Pad´e approximation method, etc. [1–11].
But it is a difficult point in the field of solving such problems because of huge
size of calculations and the high complexity in practice. It is well known that
the rapid development of computer science and computer algebra system has
a profound effect on the concept and the methods of mathematical researches
[12–18]. The objective of this paper is to establish a promising algorithm that
can be easily solved singular perturbation problems in Maple with
mechanization.
2. Basic methods
Let?s first recall the basic principles of the reducing order method for solving
singular perturbation problems.
Consider a singular perturbed two-point boundary value problem of the
form:
ey00ðxÞ þ fðxÞyðxÞ0þ gðxÞyðxÞ ¼ hðxÞ;
with conditions
x 2 ½a;b?;
ð2:1Þ
yðaÞ ¼ a;
yðbÞ ¼ b;
where e is a small positive parameter (0 < e ? 1), a and b are given constants,
f(x) and g(x) are assumed to be sufficiently continuously differentiable func-
tions on [a,b], and f(x) ? M > 0 throughout the interval [a,b], where M is some
positive constant. Under these assumptions, (2.1) has a unique solution y(x)
which in general, displays a boundary layer of width O(e) at x = a for some
small values of e.
Theorem [2] 1. The singular perturbation problem (2.1) with conditions y(a) = a,
y(b) = b could be convert into the following form:
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fðxÞp0ðxÞ þ gðxÞpðxÞ ¼ hðxÞ;
d
dx½fðxÞqðxÞ? ¼ gðxÞqðxÞ;
r0ðxÞ ¼ fðxÞ;
Solving the above equations (2.2) to find p(x), q(x) and r(x), then we can get the
solution of (2.1) as follows:
pðbÞ ¼ b;
qðaÞ ¼ a ? pðaÞ;
rðaÞ ¼ 0:
8
>
>
<
:
ð2:2Þ
yðxÞ ¼ pðxÞ þ qðxÞe?rðxÞ=e:
Based on this, when we input the values of x and e into (2.3), we can get the
numerical solutions of (2.1) rapidly.
ð2:3Þ
3. Algorithm for solving SPP by reducing order method with mechanization
It is a new development orientation in the field of mathematics and com-
puter to conduct science calculation by computer [12,13].
Maple, an international mathematical software, is an exchanged computer
algebra system with great ability of symbolic operation, numerical calculation,
coping with graphics, etc. Its powerful functions library and unique interior
programming language provide scientific calculation and programming with
friendly platform [14–18].
In Maple, the process of solving equations (2.3) can be well established a
procedure which can calculate the approximate analytic solutions and numer-
ical solutions of problem (2.1) easily. Now if we want to solve SPP by using the
method of reduction of order, everything we have to do is just to input infor-
mation about the equations, then the program will give out the approximate
analytic solution of the problem. Based on this, when we input the values of
x and e, we can get the numerical solutions of the equations. The main algo-
rithm of redordproc is as follows:
redordproc:=proc(expr,init)
local
f,g,h,p,q,r,px,qx,rx,eqn1,eqn2,eqn3,ini1,ini2,ini3;
f:=coefproc(expr,1);
g:=coefproc(expr,0);
h:=-coefproc(expr,-1);
f:=unapply(f,x);
g:=unapply(g,x);
h:=unapply(h,x);
eqn1:=f(x)*diff(p(x),x)+g(x)*p(x)=h(x);
ini1:=p(op(2,op(1,init)))=op(2,op(2,init));
px:=dsolve(eqn1,ini1,p(x));
p:=unapply(rhs(px),x);
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eqn2:=diff(f(x),x)*q(x)+f(x)*diff(q(x),x)=g(x)*q(x);
ini2:=q(op(1,op(1,init)))=op(1,op(2,init))-
p(op(1,op(1,init)));
qx:=dsolve(eqn2,ini2,q(x));
q:=unapply(rhs(qx),x);
eqn3:=diff(r(x),x)=f(x);
ini3:=r(op(1,op(1,init)))=0;
rx:=dsolve(eqn3,ini3,r(x));
lprint(?The singular perturbation problem can be converted
into:?);
print(eqn1,ini1);
print(eqn2,ini2);
print(eqn3,ini3);
lprint(?Solving above equations, we can get:?);
print(px, qx, rx);
lprint(?Then we can get the approximate analytic solution of
the SPP is
as follows:?);
print(y(x)=rhs(px)+rhs(qx)*exp(-rhs(rx)/epsilon));
end proc:
The use of sub-procedure coefproc we established is extracting the coef-
ficients of a differential-function polynomial.
Easy to read, in this paper, we established the form of readable outputting
for the solving results by use of the commands lprint and print. So, we can
get the whole processes of solving the singular perturbation problems just like
we do them with our hands. For a detailed discussion on readable outputting
we may refer the readers to the article [15] of ours.
For convenience, in the program redordproc, we always let x represent
the independent variable, and set the parameters as following:
expr: the equation which to be solved
init: list of initial conditions of (2.1) in the form [[a,b],[a,b]]
For example, if we consider the singular perturbation problems:
ey00ðxÞ þ yðxÞy0ðxÞ ? eyðxÞ¼ x þ 3;
One just sets:
yð0Þ ¼ e2;
yð1Þ ¼ ln3:
eqns:=epsilon*diff(y(x),x$2)+y(x)*diff(y(x),x)-
exp(y(x))=x+3;
init:=y(0)=exp(2), y(1)=ln3;
redordproc(eqns,[[0,1],[exp(2),ln3]]);
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4. Examples
In this section, there are some examples to illustrate the mechanized process.
Example 1. Consider the following singular perturbation problem:
ey00ðxÞ þ y0ðxÞ ? yðxÞ ¼ 0;
yð0Þ ¼ 1;
The exact solution is given by
yðxÞ ¼ðem2? 1Þem1xþ ð1 ? em1Þem2x
where m1¼?1þffiffiffiffiffiffiffi
To solve this problem by using procedure redordproc, everything we
have to do is just inputting the commands as follows:
x 2 ½0;1?;
yð1Þ ¼ 1:
ð4:1Þ
em2? em1
and m2¼?1?ffiffiffiffiffiffiffi
;
1þ4e
2e
p
1þ4e
2e
p
.
eqns:=epsilon*diff(y(x),x$2)+diff(y(x),x)-y(x)=0;
ini:=y(0)=1, y(1)=1;
redordproc(eqns,[[0,1],[1,1]]);
Then we can get the following results:
The singular perturbation problem can be converted into:
d
dxpðxÞ ? pðxÞ ¼ 0;
pð1Þ ¼ 1;
d
dxqðxÞ ¼ ?qðxÞ;
qð0Þ ¼ 1 ?1
e;
d
dxrðxÞ1;
rð0Þ ¼ 0:
Solving above equations, we can get:
pðxÞ ¼ex
e;
qðxÞ ¼ðe ? 1Þe?x
e
;
rðxÞ ¼ x:
Then we can get the approximate analytic solution of the SPP is as follows:
eþðe ? 1Þe?x?x
e
yðxÞ ¼ex
e
:
If we want to obtain the numerical solutions of the equations, we need to input
the value of x and e. The numerical results of Example 1 are given in Table 1.
Apparently, our results in Table 1 are more approximate with the exact
solutions.
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In Maple, it is a simple thing for inputting the values of x and e to calculate
y(x) as Table 1. So, we do not want to repeat this process in this paper.
Example 2. Consider the equations:
ey00ðxÞ þ 1 ?x
yð0Þ ¼ 0;
One just sets:
2
??
y0ðxÞ ?1
2yðxÞ ¼ 0;
x 2 ½0;1?;
yð1Þ ¼ 1:
ð4:2Þ
eqns:=epsilon*diff(y(x),x$2)+(1-x/2)*diff(y(x),x)-
y(x)/2=0;
ini:=y(0)=0,y(1)=1;
redordproc(eqns,[[0,1],[0,1]]);
We can get:
The singular perturbation problem can be converted into:
1 ?x
2
??d
dxpðxÞ ?1
2pðxÞ ¼ 0;
pð1Þ ¼ 1;
?1
2qðxÞ þ
1 ?1
2x
??d
dxqðxÞ ¼ ?1
2qðxÞ;
qð0Þ ¼ ?1
2;
d
dxrðxÞ ¼ 1 ?1
2x;
rð0Þ ¼ 0:
Solving above equations, we can get:
pðxÞ ¼ ?
1
?2 þ x;
qðxÞ ¼ ?1
2;
rðxÞ ¼ ?1
4x2þ x:
Table 1
Approximate solutions y(x), e = 10?3
xy(x) Result in [2]Exact solution [2]
0.00
0.01
0.02
0.03
0.04
0.05
0.10
0.30
0.50
0.70
0.90
1.00
1.0000000
0.3716050
0.3753109
0.3790832
0.3828929
0.3867409
0.4065696
0.4965853
0.6065305
0.7408182
0.9048373
1.0000000
1.0000000
0.3712379
0.3749439
0.3787160
0.3825260
0.3863742
0.4062043
0.4962382
0.6062278
0.7405963
0.9047471
1.0000000
1.0000000
0.3719724
0.3756784
0.3794502
0.3832599
0.3871079
0.4069350
0.4969324
0.6068334
0.7410401
0.9049277
1.0000000
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Then we can get the approximate analytic solution of the SPP is as follows:
yðxÞ ¼ ?
1
?2 þ x?1
2e
?ð1=4Þx2þx
e
:
Example 3. Solving the following singular perturbation problems:
ey00ðxÞ ? y0ðxÞ ? ð1 þ eÞyðxÞ ¼ 0;
yð0Þ ¼ 1 þ expð?ð1 þ eÞ=eÞ;
x 2 ½0;1?;
yð1Þ ¼ 1 þ1
e:
ð4:3Þ
Input the following commands:
eqns:=epsilon*diff(y(x),x$2)-diff(y(x),x)-(1+epsilon)
*y(x)=0;
ini:=y(0)=1+exp(-(1+epsilon)/epsilon),y(1)=1+1/exp(1);
redordproc(eqns,[[0,1],[1+exp(-(1+epsilon)/epsilon),
1+1/exp(1)]]);
We can get:
The singular perturbation problem can be converted into:
?d
dxpðxÞ þ ð?1 ? eÞpðxÞ ¼ 0; pð1Þ ¼ 1 þ1
e;
?d
dxqðxÞ ¼ ð?1 ? eÞqðxÞ;
qð0Þ ¼ 1 þ e?1þe
e ?e þ 1
e?e;
d
dxrðxÞ ¼ ?1;
rð0Þ ¼ 0:
Solving above equations, we can get:
pðxÞ ¼ðe þ 1Þe?ð1þeÞx
e?e
;
qðxÞ ¼
e?eþ e?1þe
e ? e ? 1
e?e
??
eð1þeÞx
;
rðxÞ ¼ ?x:
Then we can get the approximate analytic solution of the SPP is as follows:
yðxÞ ¼ðe þ 1Þe?ð1þeÞx
e?e
þðe?eþ e?1þe
e ? e ? 1Þeð1þeÞx
e?e
e?x=e:
Example 4
ey00ðxÞ þ x2y0ðxÞ þ 2x3yðxÞ ¼ 0;
yð1Þ ¼ e;
x 2 ½1;2?;
yð2Þ ¼ ln2:
ð4:4Þ
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Input the following commands:
eqns:=epsilon*diff(y(x),x$2)+x ˆ2*diff(y(x),x)+2*x ˆ3*
y(x)=0;
ini:=y(1)=exp(1),y(2)=ln(2);
redordproc(eqns,[[1,2],[exp(1),ln(2)]]);
The approximate solution is as follows:
The singular perturbation problem can be converted into:
x2d
dxpðxÞ þ 2x3pðxÞ ¼ 0;
pð2Þ ¼ ln2;
2xqðxÞ þ x2d
dxqðxÞ ¼ 2x3qðxÞ;
qð1Þ ¼ e1?ln2e?1
e?4
;
d
dxrðxÞ ¼ x2;
rð1Þ ¼ 0:
Solving above equations, we can get:
pðxÞ ¼lnð2Þe?x2
e?4
;
qðxÞ ¼ ?ð?ee?4þ lnð2Þe?1Þex2
ee?4x2
;
rðxÞ ¼1
3x3?1
3:
Then we can get the approximate analytic solution of the SPP is as
follows:
yðxÞ ¼lnð2Þe?x2
e?4
?ð?ee?4þ lnð2Þe?1Þex2
ee?4x2
eðð1=3Þx3?ð1=3ÞÞ=e:
Example 5
ey00ðxÞ þ xy0ðxÞ ? yðxÞ ¼ 0;
yð?1Þ ¼ 1;
x 2 ½?1;1?;
yð1Þ ¼ ln2:
ð4:5Þ
One just sets:
eqns:=epsilon*diff(y(x),x$2)+x*diff(y(x),x)-y(x)=0;
ini:=y(-1)=1, y(1)=2;
redordproc(eqns,[[-1,1],[1,2]]);
The singular perturbation problem can be converted into:
xd
dxpðxÞ ¼ 0;
pð1Þ ¼ 2;
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qðxÞ þ xd
dxqðxÞ ¼ ?qðxÞ;
qð?1Þ ¼ 3;
d
dxrðxÞ ¼ x;
rð?1Þ ¼ 0:
Solving above equations, we can get:
pðxÞ ¼ 2x;
qðxÞ ¼3
x2;
rðxÞ ¼1
2x2?1
2:
Then we can get the approximate analytic solution of the SPP is as follows:
yðxÞ ¼ 2x þ3
x2eðð1=2Þx2?ð1=2ÞÞ=e:
5. Conclusion and remarks
Mechanization of reduction of order method for solving singular perturba-
tion problems has proposed in this study. The procedure redordproc give
not only the approximate analytic solutions of SPP, but also the numerical
solutions. The method is very simple and effective for most of singular pertur-
bation problems.
Of course, in this paper, we give the solution of left-end boundary layer
problems of singular perturbation problems. If we want to obtain the solution
of right-end boundary layer problems, we must convert Eq. (2.1) into the fol-
lowing form:
fðxÞp0ðxÞ þ gðxÞpðxÞ ¼ hðxÞ;
d
dx½fðxÞqðxÞ? ¼ gðxÞqðxÞ;
r0ðxÞ ¼ fðxÞ;
Solving above equations, we can get the solution of (2.1) as follows:
pðaÞ ¼ b;
qðbÞ ¼ b ? pðbÞ;
rðbÞ ¼ 0:
8
>
>
>
>
<
:
yðxÞ ¼ pðxÞ þ qðxÞe?rðxÞ=e:
In Maple, there are three methods to establish functions: arrow operator (!),
command unapply and program. But we find an interesting thing which we
cannot use arrow operator in programming when we want to establish a func-
tion. The only way to do it is using the command unapply to convert an
expression into function.
The results of the examples indicated that the procedure redordproc of
reducing order approximation method had advantages of simple idea, excellent
property for operation and powerful competence. This would be useful for
solving differential equations.
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