Consider the one-dimensional wave equation $u_{tt}-u_{xx} = 0$ with boundary conditions $u(x,0) = g(x) = x^2$ and $u_t(x,0) = h(x) = 1- x^2$. I am trying to apply d'Alembert's formula and et $u(x,t) = \frac{1}{2}[(x+t)^2+(x-t)^2] + \frac{1}{2} \int^{x+t}_{x-t} (1-y^2) dy = ... = x^2 + t^2 + t - x^2 - \frac{1}{3} t^3$. But this neither solves the equation nor the satisfies the boundary conditions. What went wrong?
Solving a one-dimensional wave equation with d'Alembert's formula
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pde
wave-equation
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1In fact, d'Alemblert's formula produces $u(x,t)=x^2+t^2+t-t^3/3-tx^2.$ – 2017-02-21
1 Answers
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The following commands can be used to solve the problem with the use of Maple:
g := x-> x^2; h := x-> 1-x^2;
simplify( 1/2*(g(x-t)+g(x+t)) + 1/2*int( h(xi), xi=x-t .. x+t) );
Maple will respond with
$\displaystyle x^2+t^2+t-\frac{t^3}3-tx^2$
(I'd recommend my book on PDEs in which numerous examples are given how to handle PDEs with the use of Maple and Wolfram Alpha.)