rhita3yp

2022-02-18

a) Find a weak formulation for the partial differential equation$\frac{\partial u}{\partial t\text{}}+c\frac{\partial u}{\partial x\text{}}=0$ b) Show that $u=f(x-ct)$ is a generalized solution of$\frac{\partial u}{\partial t\text{}}+c\frac{\partial u}{\partial x\text{}}=0$ for 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)${\int}_{\mathrm{\Omega}}(\frac{\partial u}{\partial t\text{}}+c\frac{\partial u}{\partial x\text{}})\ast \varphi (x)dx=$ $={\int}_{\mathrm{\Omega}}\frac{\partial u}{\partial t}\varphi (x)dx-c{\int}_{\mathrm{\Omega}}u(x,t){\varphi}^{\prime}(x)dx$ 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?

Tate Puckett

Beginner2022-02-19Added 6 answers

a.) The idea of integral solutions is a little more complicated than just integration with a test function. Its

koffiejkl

Beginner2022-02-20Added 7 answers

b.) Use the weak formulation to integrate $u=f(x-ct)$ . Youre

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