Preface

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This tutorial is made solely for the purpose of education and it is designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts. This means that you can copy and paste all comamnds into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts to your needs for learning how to use the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

Boundary Value Problems

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Consider a second order differential equation

subject to the boundary conditions of the first kind (also called the Dirichlet boundary conditions) 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 f_x and f_y satisfy

then the boundary value problem 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

then the linear boundary value problem 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.