MATLAB TUTORIAL for the First Course. Part IV: Boundary Value Problems

Prof. Vladimir A. Dobrushkin

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Boundary Value Problems

Consider a second order differential equation

\[ x'' (t) = f(t,x,x' ) \qquad \mbox{for} \quad a \le t \le b, \]
subject to the boundary conditions of the first kind (also called the Dirichlet boundary conditions)
\[ x (a) = \alpha \qquad \mbox{and} \qquad x(b) = \beta . \]
Generally speaking, a boundry value problem may have a unique solutions, may have many solutions, or may have no solution. The conditions that guarantee that a solution to the formulated above Dirichlet boundary value problem exists should be checked before any numerical scheme is applied; otherwise,a list of meaningless output may be generated. The general conditions are stated in the following theorem.

Theorem: Suppose that f(t,x,y) is continuous on the region \( R = \left\{ (t,x,y)\, : \, a \le t \le b, \ -\infty < x < \infty , \ -\infty < y < \infty \right\} \) and that \( \partial f/\partial x = f_x \quad \partial f/\partial y = f_y \) are continuous on R. If there exists a positive constant M for which fx and fy satisfy

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

The notation \( y = x' (t) \) has been used to distinguish the third variable of the function \( f(t,x,x' ) . \) Recall that we also use the dot notation for derivatives with respect to time variable t: \( \dot{x} = x' (t) . \) Finally, the special case of linear differential equations is worthy of mention.

Theorem: Assume that f in the previous theorem has the linear form \( f(t,x,x' ) = p(t)\,x' + q(t)\,x + r(t) , \) and that f and its partial derivatives \( q(t) = \partial f/\partial x \) and p(t) are continuous on R. If there exists a positive constant M for which p(t) and q(t) satisfy

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

Conditions of the above theorem are fulfilled for constant coefficient equations when p(t) = p and q(t) = q. In this case, a closed form formula for solution is possible to obtain. 

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