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In solving the one dimensional Euler equation in fluid dynamics, we have two functions. What should I do If I want to solve it without considering that one of them is constant?

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$$

How to obtain $u$ based on $p$? I don't know how to approximate $p$ based on orthogonal basis function to solve the equation.

The method of characteristic for this problem is as follows: $$\frac{dt}{1}=\frac{dx}{u}=\frac{du}{−\partial u/\partial x},$$ $$\int udt= \int dx,\ \ \int udu= \int − \frac{\partial u}{\partial x}dx.$$ Is it possible to approximate $p$ based on orthogonal basis function? If yes, How to choose basis function. I want to obtain a mathematical formula for $u$ based on $p$ function that has a boundary condition on$ x=0$.

Arctic Char
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S. Sadeghi
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  • Characteristics work here. This equation is already well studied on this website, see any of the following links: https://math.stackexchange.com/questions/843118/method-of-characteristics-for-burgers-equation-with-rectangular-data?noredirect=1 – K.defaoite Jan 05 '22 at 15:14
  • https://math.stackexchange.com/questions/679703/using-implicit-differentiation-verify-that-u-fx-tu-satisfies-frac-partia?noredirect=1 – K.defaoite Jan 05 '22 at 15:15
  • @K.defaoite Thanks for your reply. My problem is $\frac{\partial p}{\partial x}$. If the third expression is zero, I know th solution. But $\frac{\partial p}{\partial x}$ is not constant or zero. – S. Sadeghi Jan 05 '22 at 15:43
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    The method of characteristics still applies. One of your characteristic equations will simply have an inhomogeneous term. In that case the solution may not be expressible in closed form. – K.defaoite Jan 05 '22 at 16:16
  • @K.defaoite. Thanks. The method of characteristic for this problem is as follows: $$ \frac{dt}{1}=\frac{dx}{u}=\frac{du}{\frac{-\partial p}{\partial x}}$$ $$\int u dt=\int dx,\quad \int u du=\int \frac{-\partial p}{\partial x} dx$$ Is it possible to approximate $p$ based on orthogonal basis function? If yes, How to choose basis function. I want to obtain a mathematical formula for $u$ based on $p$ function that has a boundary condition on $x=0$. – S. Sadeghi Jan 05 '22 at 21:05
  • How you should approximate $p$ depends heavily on the geometry of the problem and the initial/boundary conditions. Without more information it is really hard to say. – K.defaoite Jan 06 '22 at 15:23
  • Your question is ambiguous because you don't say if $p$ is a function of $x$ only or if $p$ is a function of $x$ and $t$ which involves very different calculus. – JJacquelin Mar 01 '23 at 11:46

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