Ktl-icon-tai-lieu

The shooting method for solving BVPs

Được đăng lên bởi Nguyen Ngoc
Số trang: 11 trang   |   Lượt xem: 580 lần   |   Lượt tải: 0 lần
MATH 337, by T. Lakoba, University of Vermont

7
7.1

69

The shooting method for solving BVPs
The idea of the shooting method

In the first four subsections of this lecture we will only consider BVPs that satisfy the conditions
of Theorems 6.1 or 6.2 and thus are guaranteed to have a unique solution.
Suppose we want to solve a BVP with Dirichlet boundary conditions:
y = f (x, y, y ),

y(a) = α,

y(b) = β .

(7.1)

We can rewrite this BVP in the form:
y =z
z = f (x, y, z)
y(a) = α
y(b) = β .

(7.2)

The BVP (7.2) will turn into an IVP if we replace the boundary condition at x = b with the
condition
z(a) = θ ,
(7.3)
where θ is some number. Then we can solve the resulting IVP by any method that we have
studied in Lecture 5, and obtain the value of its solution y(b) at x = b. If y(b) = β, then we
have solved the BVP. Mostly likely, however, we will find that after the first try, y(b) = β.
Then we should choose another value for θ and try again. There is actually a strategy of how
the values of θ need to be chosen. This strategy is simpler for linear BVPs, so this is the case
we consider next.

7.2

Shooting method for the Dirichlet problem of linear BVPs

Thus, our immediate goal is to solve the linear BVP
y + P (x)y + Q(x)y = R(x) with Q(x) ≤ 0,

y(a) = α,

y(b) = β .

(7.4)

To this end, consider two auxiliary IVPs:

and

u + P u + Qu = R,
u(a) = α, u (a) = 0

(7.5)

v + P v + Qv = 0,
v(a) = 0, v (a) = 1 ,

(7.6)

where we omit the arguments of P (x) etc. as this should cause no confusion. Next, consider
the function
w = u + θv,
θ = const .
(7.7)
Using Eqs. (7.5) and (7.6), it is easy to see that
(u + θv) + P (u + θv) + Q (u + θv) = R,
(u + θv)(a) = α, (u + θv) (a) = θ ,

(7.8)

MATH 337, by T. Lakoba, University of Vermont

i.e. w satisfies the IVP

w + P w + Qw = R,
w(a) = α, w (a) = θ .

70

(7.9)

Note that the only difference between (7.9) and (7.4) is that in (7.9), we know the value of w
at x = a but do not know whether w(b) = β. If we can choose θ in such a way that w(b) does
equal b, this will mean that we have solved the BVP (7.4).
To determine such a value of θ, we first solve the IVPs (7.5) and (7.6) by an appropriate
method of Lecture 5 and find the coresponding values u(b) and v(b). We then choose the value
θ = θ0 by requiring that the corresponding w(b) = β, i.e.
w(b) = u(b) + θ0 v(b) = β .

(7.10)

This w(x) is the solution of the BVP (7.4), because it satisfies the same ODE and the same
boundary conditions at x = a and x = b; see Eqs. (7.9...
MATH 337, by T. Lakoba, University of Vermont 69
7 The shooting method for solving BVPs
7.1 The idea of the shooting method
In the first four subsections of this lecture we will only consider BVPs that satisfy the conditions
of Theorems 6.1 or 6.2 and thus are guaranteed to have a unique solution.
Suppose we want to solve a BVP with Dirichlet boundary conditions:
y

= f(x, y, y
), y(a) = α, y(b) = β . (7.1)
We can rewrite this BVP in the form:
y
= z
z
= f(x, y, z)
y(a) = α
y(b) = β .
(7.2)
The BVP (7.2) will turn into an IVP if we replace the boundary condition at x = b with the
condition
z(a) = θ , (7.3)
where θ is some number. Then we can solve the resulting IVP by any method that we have
studied in Lecture 5, and obtain the value of its solution y(b) at x = b. If y(b) = β, then we
have solved the BVP. Mostly likely, however, we will find that after the first try, y(b) = β.
Then we should choose another value for θ and try again. There is actually a strategy of how
the values of θ need to be chosen. This strategy is simpler for linear BVPs, so this is the case
we consider next.
7.2 Shooting method for the Dirichlet problem of linear BVPs
Thus, our immediate goal is to solve the linear BVP
y

+ P(x)y
+ Q(x)y = R(x) with Q(x) 0, y(a) = α, y(b) = β . (7.4)
To this end, consider two auxiliary IVPs:
u

+ Pu
+ Qu = R,
u(a) = α, u
(a) = 0
(7.5)
and
v

+ Pv
+ Qv = 0,
v(a) = 0, v
(a) = 1 ,
(7.6)
where we omit the arguments of P (x) etc. as this should cause no confusion. Next, consider
the function
w = u + θv, θ = const . (7.7)
Using Eqs. (7.5) and (7.6), it is easy to see that
(u + θv)

+ P (u + θv)
+ Q (u + θv) = R,
(u + θv)(a) = α, (u + θv)
(a) = θ ,
(7.8)
The shooting method for solving BVPs - Trang 2
Để xem tài liệu đầy đủ. Xin vui lòng
The shooting method for solving BVPs - Người đăng: Nguyen Ngoc
5 Tài liệu rất hay! Được đăng lên bởi - 1 giờ trước Đúng là cái mình đang tìm. Rất hay và bổ ích. Cảm ơn bạn!
11 Vietnamese
The shooting method for solving BVPs 9 10 213