1. Computational Methods 2018/2019 Fall Chapter 9-A
CSE 551
Computational Methods
2018/2019 Fall
Chapter 9-A
Ordinary Differential Equations

2. Outline
Taylor Series Methods
Runge-Kutta Methods
Stability and Adaptive Runge-Kutta and Multistep Methods

3. References
W. Cheney, D Kincaid, Numerical Mathematics and Computing, 6ed,
Chapter 10

4. Taylor Series Methods
Initial-Value Problem: Analytical versus Numerical Solution
Solving Differential Equations and Integration
Taylor Series Methods
Taylor Series Method of Higher Order
Types of Errors

5. Initial-Value Problem: Analytical versus Numerical Solution
An ordinary differential equation (ODE)
equation - one or more derivatives of an unknown function
A solution - differential equation
specific function - satisfies the equation

6. c - an arbitrary constant
Examples
t - independent variable
x - dependent variable.
name of the unknown function of the independent variable t:
c - an arbitrary constant

7. not determine a unique solution function
a DE not, in general,
not determine a unique solution function
accompanied by auxiliary conditions
together with the DE
specify the unknown function precisely.

8. Inıtial-Value Problem
initial-value problem for a first-order DE.
x - function of t,
(1): initial-value problem
t - time and t = a - initial instant in time.
determine the value of x at any time t before or after a.

9. Examples
examples of initial-value problems, with solutions:
A numerical solution DE
table;
functional form of the solution remains unknown

10. function f – depend on t and x
If f not involve x, - second example –
DE solved - indefinite integration.
illustrate,
C - x(5) = 17. C = 4 ln(5) − arctan(5) − 108.

11. numerical solution DE:
(a) the closed form solution may be very complicated and difficult to evaluate or
(b) there is no other choice; that is, no closed-form solution can be found
e.g., for the DE
solution - taking the integral of the right-hand side.
can be done in principle but not in practice.
a function x exists
dx/dt - right-hand member (3)
but it is not possible to write x(t) in terms of familiar functions..

12. Solving ODEs on a computer
require a large number of steps with small step size,
significant amount of roundoff error can accumulate.
multiple-precision computations may be necessary on small-word-length computers.

13. Solving Differential Equations and Integration
close connection between
solving DEs and integration
e.g.:
Integrating from t to t + h,

14. Replacing the integral - numerical integration rules formula for solving the differential equation.
Euler’s method - obtained from the left rectangle approximation
The trapezoid rule

15. Since x(t + h) appears on both sides of this equation, it is called an implicit formula. If
Euler’s method
x(t + h) on the right-hand side, then we obtain the Runge-Kutta formula of
order 2—namely, Equation (10) in Section 10.2.

16. Fundamental Theorem of Calculus,
approximate numerical value for the integral
can be computed by solving the following initial-value problem for x(b):

17. Taylor Series Methods
represent the solution of a DE locally by a few terms of its Taylor series.
assume that
solution function x - represented Taylor series:

18. numerical purposes,
Taylor series truncated after m+1 terms
compute x(t + h) rather accurately
h is small and
x(t), x’(t), x’’(t), , x(m)(t) - known.
When terms through hmx(m)(t)/m! included in the Taylor series, the method
Taylor series method of order m.

19. Euler’s Method Pseudocode
Taylor series method of order 1 - Euler’s method
approximate values of the solutions to the initial-value problem:
over the interval [a, b],
first two terms - Taylor series (5) :
the formula:

20. can be used to step from t = a to t = b with n steps of size h = (b−a)/n.
The pseudocode can be written as follows, where some prescribed values for n, a, b, and
xa are used:

21. pseudocode for Euler’s method

22. Example 1
Using Euler’s method, compute an approximate value for x(2)
for the differential equation
x’ = 1 + x2 + t3
with the initial value x(1) = −4
using 100 steps.

23. Solution
Use the pseudocode above with the initial values given and combine with the following function:
The computed value is x(2) ≈

24. computer program - execute Euler’s method
simple problem:
x(2) ≈
The solution, x(t) = et ,
points produced by Euler’s method - dots.
why the dots are always below the curve?

25. Euler’s metho curves 0

26. some questions such as:
How accurate are the answers?
Are higher-order Taylor series methods ever needed?
Euler’s method - not very accurate
only two terms in the Taylor series (5) –used
truncation error is O(h2).

27. Taylor Series Method of Higher Order
Example 1:
Taylor series method of higher order.
initial-value problem:
functions DE differentiated several times with respect to t (chain rule):

28. numerical values of t and x(t) known,
x’(t), x’’(t), x’’’(t), and x(4)(t)
first five terms - Taylor series (5)
x(1) = −4, suitable starting point,
n = 100 – h.
approximation - x(a + h) - from (5) and (8)
same process - repeated
compute: x(a + 2h) form
x’(a + h), x’’(a + h), , x(4)(a + h).

30. The assignment statements that
determine - interval
a = 1 t2 = b, and 100 steps.
In each step,
current value of t
integer multiple of the step size h.
The assignment statements that
define x’, x’’, x’’’, and x(4) carrying out calculations of the derivatives according
to (8)

31. evaluation of the TS in (5) nested multiplication,
five terms. polynomial in h,
nested multiplication,
The computation t ← t + h
may cause a small amount of roundoff error to accumulate in the value of t
avoided by using t ← a + kh.

32. two terms in TS (Euler’s method) not accurate - five terms :
Euler’s Method TS Method (Order 4)
x(2) ≈ x(2) ≈
correct value to more significant figures:
x(2) ≈
computations were done with more precision
show that lack of precision was not a contributin factor.

33. Types of Errors
what sort of accuracy expected?
Are all the digits for the variable x accurate?
not!
not easy to say - how many digits are reliable
a coarse assessment:
Since terms up to (1/24)h4x(4)(t) included,
first term not included in TS: (1/120) h5x(5)(t).
The error may be larger than this,
but the factor h5 = (10−2)5 ≈ 10−10 affecting only the tenth decimal place.

34. Trancation Error
two types of errors
At each step,
if x(t) known and
x(t+h) computed from the first fewterms of the Taylor series,
an error occurs truncated theTaylor series
truncation error - local truncation error.
In the example, roughly (1/120) h5x(5)(ξ ).
local truncation error of order h5 - O(h5).

35. The second type of error
Accumulaed Error
The second type of error
accumulated effects of all local truncation errors
calculated value of x(t +h) - in error
x(t) is already wrong
previous truncation errors
another local truncation error occurs in the computation of x(t + h)

36. Additional sources of errors: roundoff error
not serious in any one step of the solution procedure,
after hundreds or thousands of steps,
accumulate and contaminate the calculated solution seriously.
an error - made at a certain step
carried forward into all succeeding steps

37. Runge-Kutta Methods
Taylor Series for f(x, y)
Runge-Kutta Method of Order 2
Runge-Kutta Method of Order 4

38. Runge-Kutta Methods
imitate the Taylor series method
without analytic differentiation of the original DE.
Taylor series method on the initial-value problem

39. differentiating the function f . a method for solving (1)
RK method of order 2
need to obtain x, x’ , . . .
differentiating the function f .
serious obstacle
a method for solving (1)
writing a code to evaluate f
The Runge-Kutta (RK) methods.
RK method of order 2
low precision
RK method of order 4 - common use.

40. Hearth of an IV Problem
heart of - initial-value problem
advancing the solution function one step at a time;
formula for x(t + h) - known quantities
x(t), x(t − h), x(t − 2h), . . .
At the beginning, only x(a) is known.

41. Taylor Series for f(x, y)
Taylor series in two variables:
f and all partial derivatives evaluated at (x, y)

42. Taylor series - truncated
error term or remainder term - restore the equality:
point (xb, yb) lies on the line segment
joins (x, y) to (x + h, y + k) in the plane
subscripts - denote partial derivatives.

43. functions - order of subscripts immaterial;
e.g. fxt = ftx .
special cases:

44. Runge-Kutta Method of Order 2
Runge-Kutta method of order 2 - a formula:
two function evaluations of the special form
linear combination of these - added to the value of x at t - at t +h:

45. determine constants w1, w2, α, and β
(5) as accurate as possible
reproduce as many terms as possible in the TS

46. One way to force them to agree up
compare (5) with (6).
One way to force them to agree up
through the term in h - set w1 = 1 and w2 = 0
x = f .
Euler’s method - order of precision is only 1.
Agreement up through the h2 term is possible by a more adroit choice of parameters.
apply two-variable form TS - final term in (5).
n = 2 - two-variable TS Formula (3),
with t, αh, x, and βh f
playing the role of x, h, y, and k, respectively:

47. new form for (5):

48. Equation (6) - new form by using DE (1).
x’ = f ,
(6) implies that
Agreement between (7) and (8)

49. solution
The resulting second-order Runge-Kutta method is then, from Equation (5),:
equivalently,

50. (10) - solution function at t + h –
two evaluations of the function f
other solutions - nonlinear System (9) possible.
For example, α can be arbitrary,

51. Error Term
error term for R-K methods of order 2:
none of the secondorder
RK methods - widely used
error O(h3).

52. Runge-Kutta Method of Order 4
algorithm – IV Problem (1)
fourth-order Runge-Kutta method.

53. solution at x(t + h) obtained evaluating the function f four times
About Error
solution at x(t + h) obtained
evaluating the function f four times
The final formula agrees with the Taylor expansion up to and including the term in h4
The error contains h5
no lower powers of h.
Without knowing the coefficient of h5 in the error,
cannot be precise about the local truncation error.

54. Pseudocode

55. illustrate use of pseudocode, IV problem
Illustration
illustrate use of pseudocode,
IV problem
exact solution: x(t) = 1 + t + tan(t − 1).
on the interval [1, ] RK procedure:
step size:
dividing the length of the interval by the number of steps, say, n = 72.

57. The final value of the computed numerical solution:
x(1.5625) =

58. General-purpose routines RK algorithm monitor - truncation error
Software
General-purpose routines RK algorithm
monitor - truncation error
make necessary adjustments
step size as the solution progresses.
In general terms, the step size
can be large when the solution is slowly varying
but should be small when it is rapidly varying.

59. Stability and Adaptive Runge-Kutta and Multistep Methods
An Adaptive Runge-Kutta-Fehlberg Method
An Industrial Example
Adams-Bashforth-Moulton Formulas

60. An Adaptive Runge-Kutta-Fehlberg Method
numerical solution of initial-value problems,
estimate the precision attained in the computation.
error tolerance
the numerical solution must not deviate from the true solution beyond this tolerance.
error tolerance - largest allowable step size
local truncation error,
determining appropriate step size – difficult
small step size - on one portion of the solution curve,
larger one - uffice elsewhere.

61. An Adapteive Procedure
methods for automatically adjusting the step size IV problem.
One simple procedure:
fourth-order RK method.
To advance the solution curve from t to t + h, one step of size h
RK formulas
two steps of size h/2 to arrive at t + h
If no truncation error:
x(t + h) would be the same for both procedures
difference numerical results –
estimate of the local truncation error

62. An Adapteive Procedure
if this difference - within the prescribed tolerance
the current step size h is satisfactory
If this difference exceeds the tolerance
the step size is halved
If the difference is very much less than the tolerance, the step size is doubled.

63. Fehlberg Method of Order 4
The procedure - not recommended.
A more sophisticated method by Fehlberg
Fehlberg method of order 4 – RK type and uses these formulas:

64. Fehlberg Method of Order 4 (cont.)
requires one more function evaluation than the classical RK method of order 4,
questionable value alone.

65. additional function evaluation
fifth-order Runge-Kutta method

66. The difference between the values of x(t + h)
obtained from the fourth- and fifth-order procedures
estimate of the local truncation error
fourth-order procedure
six function evaluations –
fifth-order approximation,
together with an error estimate!

67. pseudode RK45

68. pseudode RK45 (cont.)

69. Non Adaptive Use of RL45

70. print the error estimation at each step.
can use it in an adaptive
procedure,
error estimate ε - when to adjust the step size to control the single-step error.

71. A Simple Adaptive Process
In the RK45 procedure, the fourth- and
fifth-order approximations for x(t + h), say, x4 and x5, are computed from six function evaluations,
and the error estimate ε = |x4 − x5| is known.
From user-specified bounds on the allowable error estimate
(εmin  ε  εmax),
the step size h is doubled or halved as needed
to keep ε within these bounds.
A range for the allowable step size h is also specified
by the user (hmin  |h|  hmax).
the user must set the bounds (εmin, εmax, hmin, hmax) carefully
the adaptive procedure does not get caught in a loop,
trying repeatedly to halve and double the step size from the same point
to meet error bounds that are too restrictive for the given differential equation.

72. adaptive process :

73. Parameter List
The procedure - RK45 Adaptive.
parameter list
f: function f (t, x) for the DE,
t and x : contain the initial values,
h:the initial step size,
tb: the final value for t,
itmax: the maximum number of steps to be taken in going from a = ta to b = tb,
εmin and εmax: lower and upper bounds on the allowable error estimate ε,
hmin and hmax: bounds on the step size h,

74. Parameter List (cont.)
and iflag: an error flag that returns one of the following values:
iflag Meaning
Successful march from ta to tb
Maximum number of iterations reached
On return, t and x are the exit values, and h is the final step size value considered or used:

75. In the pseudocode,
several conditions must be checked
to determine the size of the final step,
floating-point arithmetic is involved and the step size varies.

76. a = 3, b = 5, and c = 0.2 are constants. adaptive RK formulas,
An Industrial Example
A first-order DE arose in the modeling of an industrial chemical process :
a = 3, b = 5, and c = 0.2 are constants.
adaptive RK formulas,
identify (and program) the function f
f (t, x) = sin t + 0.2x.
brief pseudocode for solving the equation on the interval [0, 10]
particular values assigned to the parameters in the routine RK45 Adaptive:

77. obtain approximation: x(10) ≈ 135.917

78. Adams-Bashforth-Moulton Formulas
strategy - numerical quadrature formulas
solve single first-order ODE.
the values of the unknown function computed at several points to the left of t,
t, t − h, t − 2h, , t − (n − 1)h
compute x(t + h).

79. the theorems of calculus:
abbreviation: f j = f (t − ( j − 1)h, x(t − ( j − 1)h))

80. numerical integration formula. simplest case - for the interval [0, 1]
use values of the integrand at points 0,−1,−2, , 1−n
in the case of an Adams-Bashforth formula.
basic rule - change of variable
the rule any other interval with any other uniform spacing.

81. a rule of the form
n coefficients cj at our disposal.
interpolation theory
formula - exact for all polynomials of degree n −1.
integrating each function 1, r, r 2, , r n−1 exactly.
appropriate equation:

82. system Au = b n equations in n unknowns
Ai j = (1 − j )i−1,
bi = 1/i .
output: vector of coefficients
( 55/24 , −59/24 , 37/24 , −3/8)
. higher-order formulas by changing the value of n in the code.
Adams-Moulton formulas, start with a quadrature rule of the form

83. tt+h 01
A program similar to the one above yields the coefficients
9/24 , 19/24 ,− 5/24 , 1/24
The distinction between the two quadrature rules is that one involves the value of the integrand at 1 and the other does not.

84. How do we arrive at formulas for
tt+h g(s) ds from the work already done?
Use the change of variable from s to σ given by s = hσ − t.
In these considerations, think of t as a constant
The new integral: 01 g(hσ + t) dσ,
treated with either of the two formulas already designed for the interval [0, 1]. For example,

85. method of undetermined coefficients
obtain the quadrature formulas does
not, by itself, provide the error terms that we would like to have.
assessment of the error - interpolation theory,
integrating an interpolating polynomial

86. Computational Methods 2018/2019 Fall Chapter 9-B
CSE 551
Computational Methods
2018/2019 Fall
Chapter 9-B
Systems of Ordinary Differentia Equations

87. Outline
Methods for First-Order Systems
Higher-Order Equations and Systems
Adams-Bashforth-Moulton Methods

88. References
W. Cheney, D Kincaid, Numerical Mathematics and Computing, 6ed,
Chapter 11

89. Methods for First-Order Systems
Uncoupled and Coupled Systems
Taylor Series Method
Vector Notation
Systems of ODEs
Taylor Series Method: Vector Notation
Runge-Kutta Method
Autonomous ODE

90. Methods for First-Order Systems
ODEs
single DE, first order
with an accompanying auxiliary condition.
systems of several first-order equations.

91. Uncoupled and Coupled Systems
The sun and the nine planets - system of particles
Newton’s law of gravitation
position vectors of the planets
system of 27 functions,
and the Newtonian laws of motion
system of 54 first-order ODEs
In principle,
past and future positions of the planets can be obtained by solving these equations numerically.

92. Example
two equations with two auxiliary
conditions
x and y be two functions of t:
with initial conditions

93. initial-value problem - system of two first-order DEs.
not possible - solve
either of the two DEs by itself
first equation governing x involves the unknown function y,
and the second equation governing y involves the unknown function x.
two DEs - coupled.
analytic solution:

94. simpler example:
initial conditions:
not coupled
solved separately - two unrelated initialvalue problems

95. Taylor Series Method
Taylor series (TS) method for System (1)
differentiating the equations :

96. few terms of the Taylor series:
program to proceed
from x(t) to x(t + h)
and from y(t) to y(t + h)
few terms of the Taylor series:
together with equations for the various derivatives.
x and y and all their derivatives - functions of t;
x = x(t), y = y(t), x’ = x’(t), y’ = y’(t), and so on.

97. A pseudocode program
generates and prints a numerical solution
from 0 to 1
in 100 steps:
Terms up to h4 have been used in the Taylor series.

99. Vector Notation
(1) vector notation:
initial conditions

100. more general problem:
where
F vector two components right-hand sides in (1).
depends on t and X, - F(t, X).

101. Systems of ODEs
ystem of n first-order differential equations.
(4), ODE in vectornotation.

102. Taylor Series Method: Vector Notation
m-order Taylor series method
X = X(t), X’ = X’(t), X’’ = X’’(t), and so on.

103. pseudocode TS method - order 4
preceding problem
simple change of variables
an array and, inner loop.

105. a two-dimensional array
instead of all the different derivative variables;
di j ↔ xi( j ) .
easy to program
computer language supports vector operations.

106. Runge-Kutta Method
RK extend to systems of DEs.
fourth-order RK method for (4):
where
X = X(t), and all quantities are vectors with n components except variables t and h.

107. A procedure Runge-Kutta procedure
system to be solved - (4) n equations in
The user furnishes:
initial value of t,
the initial value of X,
the step size h,
and the number of steps to be taken, nsteps.
procedure XP System(n, t, (xi ), ( fi )) – needed:
evaluates the right-hand side of (4) for a given value of array (xi )
stores the result in array ( fi ).

108. pseudocode RK4_System1

109. pseudocode RK4_System1

110. pseudocode RK4_System1

111. Illustration
illustrate use of this procedure,
use System (1) for example.
rewritten in the form of (4)

113. A numerical experiment:
compare the results of the TS method and the
RK method with the analytic solution of System (1)
At the point t = 1.0, the results :
TS RK AS
x(1.0) ≈
y(1.0) ≈

114. use mathematical software routines found in Matlab, Maple, or Mathematica to
obtain the numerical solution of the system of ordinary differential equations (1)
For t over
the interval [0, 1], invoke an ODE procedure to march from t = 0 at which x(0) = 1
and y(0) = 0 to t = 1 at which x(1) = and y(1) =
To obtain the numerical solution of the ordinary differential equation defined for t over
the interval [1, 1.5], invoke an ordinary differential equation solving procedure to march
from t = 0 at which x(1) = 2 and y(1) = −2 to t = 1.5 at which x(1.5) ≈ and
y(1.5) ≈

115. Autonomous ODE
system of differential equations in vector form
variable t - explicitly separated from the other variables
new variable x0 is t in disguise
add a new DE x’0 = 1.
a new initial condition x0(a) = a.
increase # of DEs from n to n + 1

116. system in vector form:
system of two equations given by (1).
system with three variables:

117. auxiliary condition for X :
X(0) = [0, 1, 0]T .
a system of n + 1 first-order differential equations

118. in general vector notation:

119. A system of DEs without the t variable explicitly present autonomous.
numerical methods not require
x0 = t or f0 = 1 or s0 = a.
For an autonomous system, the classical fourth-order Runge-Kutta method for System (6) uses these formulas:
X = X(t),

120. procedure RK4 System1 modified:
beginning the arrays with 0 rather than 1
and omitting the variable t.

122. Higher-Order Equations and Systems
Higher-Order Differential Equations
Systems of Higher-Order Differential Equations
Autonomous ODE Systems

123. Higher-Order Equations and Systems
initial-value problem for ODEs of order higher than 1.
A DE of order n - n auxiliary conditions
initial conditions are needed to specify the solution of the differential equation precisely
e.g., a particular second-order initial-value problem:

124. Without the auxiliary conditions, the general analytic solution:
c1 and c2 - arbitrary constants
To select one specific solution,
c1 and c2 fixed,
two initial conditions
x(0) = 0 - c2 = −3/8 ,
and x’(0) = 0 forces c1 = 0.

125. Higher-Order Differential Equations
general form of an ODE with initial conditions
solved numerically - turning it into a system of first-order differential equations.
define new variables x1, x2, , xn:

126. original initial-value problem (2) equivalent to

127. in vector notation,

128. problem - transformed - new variables, dictionary illustration:
relationship between the new and the old variables
illustration:
into a form - solution by the Runge-Kutta procedure.

129. A chart summarizing the transformed problem:
corresponding first-order system:
X(0) = [3, 7, 13]T .

130. Systems of Higher-Order Differential Equations
introducing new variables,
transform a system of DEs of various orders into a larger system of first-order equations
e.g.,
solved by Runge-Kutta procedure

131. transform:
X(1) = [2,−4,−2, 7, 6]T .

132. Autonomous ODE Systems
t present on the right-hand side of (3)
x0 = t and x0 ’ = 1 introduced to form an autonomous system of ODEs in vector notation.

133. a higher-order system of DEs (2)
written in vector notation as

134. Example
the ODE system in (4) in autonomous form:
X(1) = [1, 2,−4,−2, 7, 6]T .

135. Adams-Bashforth-Moulton Methods
A Predictor-Corrector Scheme

136. A Predictor-Corrector Scheme
initial-value problem
single-step numerical methods
if X(t) – known at t,
X(t + h) computed
no knowledge at points earlier than t
RK and TS methods:
compute X(t + h) in terms of X(t) and various values of F.

137. More efficient methods computing X(t + h) multistep methods.
X(t), X(t −h), X(t −2h), . . .
computing X(t + h)
multistep methods.
drawback: at the beginning of the numerical solution,
no prior values of X are available
start - single-step method, such as RK,
transfer - multistep procedure
enough starting values computed.

138. a multistep formula - Adams-Bashforth method

139. Xb (t + h) - predicted value of X(t + h) computed by (2).
solution X computed at four points t, t − h, t − 2h, t − 3h
(2) used to compute X (t + h)
one evaluation of F - required for each step.
considerable savings over the fourth-order RK procedure;
four evaluations of F per step.
consideration of truncation error and stability
permit a larger step size in RK method
and make it much more competitive

140. Corrector
(2) not used by itself
predictor,
another formula – corrector
Adams-Moulton formula:
(2) predicts a tentative value of X(t +h),
(3) computes this X value more accurately
predictor-corrector scheme.

141. initial values of X specified at a,
three steps RK method - determine enough X values
ABM procedure can begin
The fourth-order Adams-Bashforth and Adams-Moulton formulas, started with the fourth-order RK method,
Adams-Moulton method.
Predictor and corrector formulas of the same order
only one application of the corrector formula needed

142. Computational Methods 2018/2019 Fall Chapter 9-C
CSE 551
Computational Methods
2018/2019 Fall
Chapter 9-C
Boundary-Value Problems for Ordinary Differential Equations

143. Outline
Shooting Method

144. References
W. Cheney, D Kincaid, Numerical Mathematics and Computing, 6ed,
Chapter 14.1

145. Shooting Method
A boundary-value problem
exemplified: second-orderODE
solution function prescribed at the endpoints of the interval

146. a DE - general solution involves two arbitrary parameters.
particular solution - two conditions must be given
initial-value problem:
x and x’ specified at some initial point
In this problem:
two points of the form (t, x(t)) - solution curve passes
(0, 1) and (π/2,−3)
general solution of DE:
x(t) = c1 sin(t) + c2 cos(t),
two conditions -boundary values
determine: c1 = −3 and c2 = 1.

147. problem - unable to determine the general solution model the problem:
step-by-step numerical solution (1)
requires two initial conditions
one condition at t = a

148. guess x’(a), then
carry out the solution of IVproblem as far as b, and
hope that the computed solution agrees with β;
that is, x(b) = β.
If not
go back and change guess for x’(a)
Repeating this procedure until hit the target β good method
learn something from the various trials
systematic ways of utilizing this information, shooting.

149. final value x(b) - solution of initial-value problem
depends on guess for x’(a)
Everything else remains fixed in this problem. Thus,
the differential equation x = f (t, x, x’) and the first initial value, x(a) = α, do not change.
If assign a real value z to the missing initial condition,

150. then the initial-value problem can be solved numerically.
x’(a) = z
then the initial-value problem can be solved numerically.
value of x at b - function of z - ϕ(z)
for each choice of z - obtain a new value for x(b),
and ϕ is the name of the function with this behavior.
know very little about ϕ(z),
but compute or evaluate it.
an expensive function to evaluate
each value of ϕ(z) is obtained only after solving an initial-value problem.

151. shooting method combines
any algorithm for the initial-value problem with
any algorithm for finding a zero of a function

152. Shootin metho illustrated

153. Shooting Method Algorithm
a function ϕ(z) is computed:
Solve the initial-value problem
on the interval [a, b]. Let ϕ(z) = x(b)
objective: adjust z until find a value for which
ϕ(z) = β
linear interpolation between ϕ(z1) and and ϕ(z2),

154. given two values of ϕ, we pretend that
where z1 and z2 :
two guesses for the initial condition x’(a).
given two values of ϕ, we pretend that
ϕ is a linear function and determine an appropriate value of z based on this hypothesis.
A sketch of the values of z versus ϕ(z) might look like Figure The strategy just outlined
is the secant method for finding a zero of
ϕ(z) − β.

155. To obtain an estimating formula for the next value z3, we compute ϕ(z1) and ϕ(z2) on
the basis of values z1 and z2, respectively.

156. By similar triangles
repeat this process and generate the sequence
all based on two starting values z1 and z2.

157. This procedure for solving the two-point boundary-value problem
is then as follows: Solve the initial-value problem

158. from t = a to t = b, letting the value of the solution at b be denoted by ϕ(z). Do this twice
with two different values of z, say, z1 and z2, and compute ϕ(z1) and ϕ(z2). Now calculate a
new z, called z3, by Formula (2). Then compute ϕ(z3) by again solving (4). Obtain z4 from
z2 and z3 in the same way, and so on. Monitor

159. ϕ(zn+1) − β
to see whether progress is being made. When it is satisfactorily small, stop. This process
is called a shooting method. Note that the numerically obtained values x(ti ) for a ti b
must be saved until better ones are obtained (that is, one whose terminal value x(b) is closer
to β than the present one) because the objective in solving Problem (3) is to obtain va

160. x(t) for values of t between a and b.
The shooting method may be very time-consuming if each solution of the associated
initial-value problem involves a small value for the step size h. Consequently, we use a
relatively large value of h until |ϕ(zn+1) − β| is sufficiently small and then reduce h to
obtain the required accuracy.

161. Example
What is the function ϕ for this two-point boundary-value problem?

162. Solution
The general solution of the differential equation is x(t) = c1et +c21e-t . The solution of the
initial-value problem is

163. Modifications and Refinements
Many modifications and refinements are possible. For instance, when ϕ(zn+1) is near β, one
can use higher-order interpolation formulas to estimate successive values of zi . Suppose,
for example, that instead of utilizing two values ϕ(z1) and ϕ(z2) to obtain z3, we utilize the
four values
to estimate z5. set up a cubic interpolatin polynomial p3 for

164. f
nd solve
for z5. Since p3 is a cubic, this would entail some additional work. A better way may be to
set up a polynomial p3 to interpolate the data

165. and then use p3(β) as the estimate for z5
and then use p3(β) as the estimate for z5. This procedure is known as inverse interpolation