In the following series of web pages, we discuss basic partial differential equations (PDEs for short) of hyperbolic type. The wave equation \( \Box_c u \overset{\mathrm def}{=} u_{tt} - c^2 \Delta u \) is one of the most important representative of hyperbolic equations.
The wave equation usually describes water waves, the vibrations of a string or a membrane, the propagation of electromagnetic and sound waves, or the transmission of electric signals in a cable.
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Introduction to Linear Algebra with Mathematica
A hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n − 1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface.
The wave equation is an important representative of a hyperbolic equation.
The wave equation for a function u(x_{1}, … , x_{n}, t) = u(x, t) of n space variables x_{1}, … , x_{n} and the time t is given by
\[
\square u = \square_c u \equiv u_{tt} - c^2 \nabla^2 u = 0 , \qquad \nabla^2 = \Delta = \frac{\partial^2}{\partial x_1^2} + \cdots + \frac{\partial^2}{\partial x_n^2} ,
\]
with a positive constant c (having dimensions of speed). The operator □ defined above is known as the d'Alembertian or the d'Alembert operator.
Let us consider a partial differential equation with two independent variables.
By a linear change of variables, any equation of the form
are Fourier transforms of the the functions u(x,t), d(x), and v(x) with respect to spacial variable x, respectively. Solution of the above initial value problem is
\[
u^F = d^F \cos\left( c\xi t \right) + v^F \,\frac{\sin\left( c\xi t \right)}{c\xi} .
\]
Upon introducing two auxiliary functions
\[
\Phi (\xi ) = \frac{\sin\left( c\xi t \right)}{c\xi} \qquad\mbox{and} \qquad
\Psi (\xi ) = \cos\left( c\xi t \right)
\]
we can represent the Fourier transform of the solution as the sum
because of Euler's formula\( \sin t = \frac{1}{2{\bf j}} \, e^{{\bf j} t} - \frac{1}{2{\bf j}} \, e^{-{\bf j} t} . \)
We know from Example D in section that
where u_{xx} stands for the partial derivative
\( u_{xx} = \partial^2 u/\partial x^2 , \)
and so on. Assuming that the solution is represented by the power series
where u_{xx} stands for the partial derivative
\( u_{xx} = \partial^2 u/\partial x^2 , \)
and so on. Assuming that the solution is represented by the power series
where u_{xx} stands for the partial derivative
\( u_{xx} = \partial^2 u/\partial x^2 , \)
and so on. Assuming that the solution is represented by the power series
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