# CH E374 Lecture Notes - Lecture 7: Stabilisation Force In Bosnia And Herzegovina, Secant Method, Rhode Island Route 2

by OC252573

School

University of AlbertaDepartment

Chemical EngineeringCourse Code

CH E374Professor

Joe MmbagaLecture

7This

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Lecture Notes CHE 374 Computational Methods in Engineering

MODULE 7: Ordinary differential equations – Boundary value problems (BVP)

Basics

Figure 1

Consider a rod whose ends are maintained at constant temperatures, T1 and TL as shown

in Figure 1. Remember the heating rod problem. The rod loses heat to the environment

which is at a temperature Ta. (ambient temperature). The rate of heat loss by convection

from the sides of the rod depends on the heat transfer coefficient, hc, and the ambient

temperature, Ta. The partial differential equation governing temperature distribution as

a function of x and t is

2

2()

TT

hc Ta T

tx

∂∂

=+ −

∂∂ (7.1)

.

Suppose we are only interested in the steady-state solution. At steady-state 0

T

t

∂=

∂. Then

the PDE turns into an ODE:

2

2()0

dT hc Ta T

dx +−=

(7.2)

The two boundary conditions of this second-order ODE are 1

(0)Tx T== and

()

L

Tx L T==.

Note that the conditions are given at two different locations (x=0 and x=L). This is

called a boundary value problem. The solution methods for initial value problem

ODE’s (Module 6) can not directly deal with boundary value problems. In initial value

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problems the state of the system is fully known at one point (e.g. at t=0). From there on

we can march forward (e.g. with a Runge-Kutta method) to determine the solution at

any moment in time.

In boundary value problems, the state of the system is not fully known at one point and

we have no point to start our march from.

There are two solution approaches to boundary value problems, namely: The shooting

Method and Finite difference Method

Shooting Method

Suppose we can guess the boundary conditions required at one of the domains. We can

then solve this problem by using the initial value methods discussed in initial value

problem (module 6). If our guessed value gives the correct value at the end of the

domain, we have arrived, otherwise we have to guess again, and again until

convergence.

Finite difference Methods

Think again of the steady-state heating rod problem: we replace the function

(

)

Tx with

the temperature at a set of discrete points i

T (discretization) and transform the ODE and

the boundary conditions into a set of algebraic equations that we can solve with the

methods from Module 2 (non-linear equations) or Module 3 (linear equations) .

We explain these two methods in the following subsections:

The Shooting method

For (systems of) linear ODE’s the shooting method does not need iteration. As an

example we take a system of two linear ODE’s:

() () ()

() () ()

1112 23

241526

dy gx ygx ygx

dx

dy gx ygx ygx

dx

=+ +

=+ +

(7.3)

with boundary conditions

() ()

11,22,

and

s

f

ya y yb y==.

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