MATLAB Mathematics - SERC - Index of

Boundary Value Problems for ODEs
Note The demo mat4bvp contains the complete code for this example. The
demo uses nested functions to place all functions required by bvp4c in a single
M-file. To run this example type mat4bvp at the command line. See “BVP
Solver Basic Syntax” on page 5-65 for more information.
1 Rewrite the problem as a first-order system. To use bvp4c, you must
rewrite the equations as an equivalent system of first-order differential
equations. Using a substitution y 1 = y and y 2 = y′ , the differential
equation is written as a system of two first-order equations
y 1 ′ = y 2
y 2 ′ = –( λ – 2 q cos 2x)y 1
Note that the differential equations depend on the unknown parameter λ .
The boundary conditions become
y 1 ( 0) – 1 = 0
y 2 ( 0) = 0
y 2 ( π) = 0
2 Code the system of first-order ODEs. Once you represent the equation as
a first-order system, you can code it as a function that bvp4c can use.
Because there is an unknown parameter, the function must be of the form
dydx = odefun(x,y,parameters)
The following code represents the system in the function, mat4ode. Variable
q is shared with the outer function:
function dydx = mat4ode(x,y,lambda)
dydx = [ y(2)
-(lambda - 2*q*cos(2*x))*y(1) ];
end % End nested function mat4ode
See “Finding Unknown Parameters” on page 5-72 for more information
about using unknown parameters with bvp4c.
5-69