HW 8 Mathematics 503, Mathematical Modeling, CSUF , July 12, 2007

Nasser M. Abbasi

June 15, 2014

problem:

Write down the equations that determine the solution of the isoperimetric problem

b ∫ p(x)y′2+ q(x)y2 dx→ min a

Subject to

∫b 2 r(x)ydx= 1 a

where $p,q,r$ are given functions and $y(a)= y(b)= 0$ .

Answer

Since $y(x)$ is ﬁxed at each end, this is not a natural boundary problem. Therefore one can use the auxiliary lagrangian approach, where we write the auxiliary Lagrangian $L∗$ as

|------------| | ∗ | --L-=-L+λG---|

Where $L(x,y,y′)= p(x)y′2+ q(x)y2$ , and $G= r(x)y2$ and $λ$ is the Lagrangian multiplier. Hence

|--------------------------| | ∗ ′2 2 2 | | L = p(x)y +q (x)y+ λr(x)y | ---------------------------

Hence now we write the solution as the Euler-Lagrange equation, but we use $L∗$ instead of $L$

Therefore the diﬀerential equation is

|-------------------------| | (p(x)y′)′− y(q(x)+ λr(x))= 0| ---------------------------

This is a sturm-Liouville eigenvalue problem. The solution $y(x)$ from the above will contain 3 constants. 2 will be found from boundary conditions, and the third, which is $λ$ is found from plugging in the solution $y(x)$ into the constraint given: $∫b r(x)y2dx= 1 a$

HW 8 Mathematics 503, Mathematical Modeling, CSUF , July 12, 2007

Contents

1 Problem 6 page 204 section 3.6