Preface

This chapter gives an introduction to some techniques for solving boundary value problems. For more deep information we refer the reader to other sourses.

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Glossary

Boundary Value Problems (BVP)

Ordinary differential equations (ODEs) may be divided into two classes: linear equations and nonlinear equations. The latter have a richer mathematical structure than linear equations and generally much more difficult to solve in closed form. Unfortunately, the techniques applicable for solving second order nonlinear ODEs are not available at an undergraduate level. Therefore, we concentrate our attention on linear differential equations. Nevertheless, some techniques for solving nonlinear equations are presented at the end of this chapter.

A broad class of analytical solutions methods and numerical algorithms methods have been used for handling boundary value problems; we mention some of them:

Consider the differential equation of the second order

\[ x'' = f(t,x,x' ) \qquad\mbox{for} \qquad a \le t \le b, \]
with the boundary conditions
\[ x(a) = \alpha \qquad\mbox{and}\qquad x(b) = \beta . \]
This equation together with condition is called a boundary value problem.

The sufficient conditions (necessary conditions are unknown yet) that guarantee that a solution to the formulated above boundary value problem exists should be checked before any numerical scheme is applied; otherwise, a list of meaningless output may be generated. Some general conditions are stated in the following theorem. As usual, derivatives are denoted by primes.

Theorem (Boundary Value Problem): Let f(t,x,y) be continuous on the region \( R = \{ (t,x,y)\,:\, a \le t \le b, \ - \infty < x < \infty , \ - \infty < y < \infty \} \) and the its partial derivatives \( \partial f / \partial x = f_x (t,x,y) \) and \( \partial f / \partial y = f_y (t,x,y) \) be also continuous on R. If there exists a positive constant M for which fx and fy satisfy

\[ \begin{split} f_x (t,x,y) > 0 \qquad &\mbox{for all } (t,x,y) \in R \mbox{ and} \\ \left\vert f_y (t,x,y) \right\vert \le M \qquad &\mbox{for all } (t,x,y) \in R, \end{split} \]
then the boundary value problem
\[ x'' = f(t,x,x' ) \qquad\mbox{with } x(a) = \alpha \quad\mbox{and}\quad x(b) = \beta \]
has a unique solution x = x(t) over \( a \le t \le b . \) ■

Corollary (Linear Boundary Value Problem): Suppose that f in the previous theorem has the form \( f(t,x,y) = p(t)\,y + q(t)\,x + r(t) \) and that f and its derivatives fx = q(t) and fy = p(t) are continuous on R. If there exists a positive constant M for which p(t) and q(t) satisfy

\[ \begin{split} q(t) > 0 \qquad \mbox{for all } t \in [a,b], \\ |p(t)| \le M = \max_{a \le t \le b} \,\{ \,|p(t)| \, \} , \end{split} \]
then the boundary value problem
\[ x'' = p(t)\,x' (t) + q(t)\,x(t) + r(t) \qquad \mbox{with } x(a) = \alpha , \quad x(b) = \beta \]
has a unique solution x = x(t) for \( a \le t \le b . \) ■

 

Linear Differential Operators

The general linear differential equation of the second order is an equation of the form

\[ a_2 (x) \,\frac{{\text d}^2 y}{{\text d}x^2} + a_1 (x) \,\frac{{\text d} y}{{\text d}x} + a_0 y(x) = g(x) , \]
where \( a_2 (x) , \ a_1 (x) , \ a_0 (x) \) are known coefficients and g(x) is a given function, known as driven term, forcing term, or nonhomogeneous term. We will use variables x and t as independent variables, and other (lower case) letters for dependent variables. Derivatives are usually also denoted by primes (y' or y''); however, it is a custom to use Newton's notation and denote derivatives with respect to time (denoted by t) with dots, so instead of y' we will use \( \dot{y} . \) Using the derivative operator, \( \texttt{D} = {\text d}/{\text d}x , \) we can rewrite the above differential equation in the operator form:
\[ L\left[ \texttt{D} \right] y \equiv \left[ a_2 (x) \,\texttt{D}^2 + a_1 (x) \, \texttt{D} + a_0 \,\texttt{I} \right] y(x) = g(x) , \]
so we write it as \( L\left[ \texttt{D} \right] y = g . \) Here \( \texttt{I} \) is the identity operator (which we will usually drop). Then solution of the differential equation \( L\left[ \texttt{D} \right] y = g \) can be written as \(y = L^{-1}\left[ \texttt{D} \right] g . \) Note that the above (right) inverse differential operator L-1 is not unique and contains two arbitrary constants. In order to eliminate these constants, we have to impose auxiliary conditions (either initial or boundary).

 

  2. Keskin, A.U., Boundary Value Problems for Engineers: with MATLAB Solutions, Springer; 1st ed. 2019.

 

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