Talk:Boundary value problem

A more informative title would be "Boundary Value Problems for Ordinary Differential Equations".

Introduction

The use of "explicit" would be more informative if an example of an implicit problem were presented.

"An explicit ordinary differential two-point boundary value problem (BVP)" is quite a mouthful. Less cumbersome would be "two-point boundary value problem for ordinary differential equations (BVODES)".

In the fourth last line of the first section, it is not clear what it means for the vector functions $g_a$ and $g_b$ to be independent.

Existence and Uniqueness

In the first line, the comma should be a period.

In the fourth line, it is not clear what "the corresponding IVPs" are.

In the line following equation (4) and in the remainder of the section, "projectile" would be more informative than "shot".

In the sixth last line of the section, \phi should be $\phi$.

In the caption of Figure 1, $\%$ should be deleted.

Shooting or Marching Techniques

Mention is made of "the stability of the initial value problem" and "a stable boundary value problem" but no definition or reference to a discussion of stability is given. Then in the third paragraph of the next section, the terminology "stable (well-conditioned) problem" is used. And in the fifth line after equation (7), mention is made of a well-posed problem and then four lines later to "a BVP that is not well-posed". Some explanation of stability and well-posedness should be provided.

Infinite Intervals

BVODES is preferable to ODE BVPs.

In the second line, there should be a comma after "For example".

In the third line of the second paragraph, "at" should be inserted before "a finite point $L$".

In the fourth line after equation (6), "positive" could be eliminated.

In the fifth line after equation (7), there appears to be a word or words missing in "the requirements exponential dichotomy".

Numerical Methods

The first sentence should be reworded: "section" is not labeled and "that there are" could be omitted.

In the second paragraph, a comma should be inserted after "a user specified error".

In the third paragraph, the discussion of collocation is confusing. In the first sentence, the relationship between the basis and the approximate solution is not explained. Also in that sentence, instead of "by defining" write "defined in terms of", or, alternatively "by an element of a linear space....". It should be mentioned that, in order to obtain optimal accuracy, the collocation points must be chosen judiciously. Instead of "the number of basis functions", I would prefer "dimension of the chosen linear space".

The parenthetical remark concerning Runge-Kutta formulas seems out of place. If mention of these methods is necessary - and I am not sure that it is - it should come later.

Sturm-Liouville Eigenproblems

ODE BVP could be BVODE.

Five lines after (8), \verb+\cite{z}+ needs to be changed

In the seventh and eighth lines after (8) sentences could be combined: "...zeros in $(a,b)$, and the set of eigenfunctions...

In the second last line of the first paragraph, a comma should be inserted after "observed".

The second paragraph is difficult to follow. Are Pruess methods examples of

finite difference or finite element methods, or another class of methods?  It is
not clear to what "Finally" refers; is it to the last step in Pruess methods, or a
third class of methods? A reference to the Prufer transformation would be helpful.