Differential Equations

Part II Mathematica and partial differential equations

The references provided in the second part of this book may be of assistance; see [94], [93], [95], [65], [99], [82], [63], and [71].

A partial differential equation is a differential equation that contains unknown functions of several independent variables and their partial derivatives. If we restrict to one unknown function u of three independent variables $\mathit{x}\mathrm{,}\mathit{y}$, and z, it is a relation of the form

(II-0)$\begin{array}{c}\mathit{F}\mathrm{(}\mathit{x}\mathrm{,}\mathit{y}\mathrm{,}\mathit{z}\mathrm{,}\mathit{u}\mathrm{,}{\partial }_{\mathit{x}}\mathit{u}\mathrm{,}{\partial }_{\mathit{y}}\mathit{u}\mathrm{,}{\partial }_{\mathit{z}}\mathit{u}\mathrm{,}{\partial }_{\mathit{x}\mathrm{,}\mathit{x}}\mathit{u}\mathrm{,}{\partial }_{\mathit{y}\mathrm{,}\mathit{y}}\mathit{u}\mathrm{,}{\partial }_{\mathit{z}\mathrm{,}\mathit{z}}\mathit{u}\mathrm{,}{\partial }_{\mathit{x}\mathrm{,}\mathit{y}}\mathit{u}\mathrm{,}{\partial }_{\mathit{y}\mathrm{,}\mathit{z}}\mathit{u}\mathrm{,}{\partial }_{\mathit{z}\mathrm{,}\mathit{x}}\mathit{u}\mathrm{,}{\partial }_{\mathit{x}\mathrm{,}\mathit{x}\mathrm{,}\mathit{x}}\mathit{u}\mathrm{,}\dots \mathrm{)}=0.\end{array}$

We suppose that function u is real valued, has at least two independent real variables, and is differentiable as many times as it is required.

Sometimes we take into account cases where the unknown function depends ...