# a) Find a weak formulation for the partial differential equation{\partial u\over\partial t

a) Find a weak formulation for the partial differential equationb) Show that $u=f\left(x-ct\right)$ is a generalized solution offor any distribution $f$What i already haveI know that in order to find a weak form of a pde, we need to multiply it by a test function, then integrate it. Also, to find a generalized solution, we need to find a weak solution and just multiply it by the Heaviside function.Let's take any test function $\varphi$, then we have (integrating by parts second part of the integral)$={\int }_{\mathrm{\Omega }}\frac{\partial u}{\partial t}\varphi \left(x\right)dx-c{\int }_{\mathrm{\Omega }}u\left(x,t\right){\varphi }^{\prime }\left(x\right)dx$where $\varphi$ vanishes at boundaries. So, is it the final form or can we proceed further? And how am I supposed to find a generalized solution?
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Tate Puckett
a.) The idea of integral solutions is a little more complicated than just integration with a test function. Its
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koffiejkl
b.) Use the weak formulation to integrate $u=f\left(x-ct\right)$. Youre