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Oct
16
comment shock waves characteristics
Also, if I understand your work, the method you are using is for solving fully nonlinear first order PDE's, and you are using it wrong. Your problem is quasilinear, and there is no need to introduce $p$ and $q$. This are only introduced in the case the derivatives of $u$ are involved nonlinearly in the equation. I strongly suggest you to study the first chapter of John's Partial Differential Equations, as I believe you are very confused. Any doubts, we can try to help.
Oct
16
comment shock waves characteristics
As timur comments, the characteristic equations are wrong. They should be stated against a parameter, not the involved variables. You are mistaking $t$ with $y$. The construction of the characteristics is based on the supposition that if $x = x(\eta)$ and $t = t(\eta)$, then $$\frac{d}{d\eta}u\big(x(\eta),t(\eta)\big) = u_x x'(\eta) + u_t t'(\eta) = u^2 u_x + u_t = 0$$ and then one says $x'(\eta) = u^2$, $t'(\eta) = 1$, $u'(\eta) = 0$. See my answer for a full analysis.
Oct
16
answered shock waves characteristics
Oct
16
comment The derivative of $f(t, y(t))$ with respect to $t$?
@F'OlaYinka Really? Any multivariable calculus textbook will have the construction. A good introductory textbook is Marsden and Tromba's Vector Calculus.
Oct
16
revised The derivative of $f(t, y(t))$ with respect to $t$?
added tag multivariable-calculus
Oct
16
suggested suggested edit on The derivative of $f(t, y(t))$ with respect to $t$?
Oct
16
answered The derivative of $f(t, y(t))$ with respect to $t$?
Oct
16
revised Showing a certain operator is trace class.
formatted link
Oct
16
comment Showing a certain operator is trace class.
You can make nice links by using the code [nice link description](http://nicelink.direction.here/nicelink)
Oct
16
suggested suggested edit on Showing a certain operator is trace class.
Oct
16
comment Elementary Question about limits
But if you are really too lazy to do it, a quick example would be $\sin x \sim x$ for small $x$. What does that say of $\sin^3 x$? And so on.
Oct
16
revised Elementary Question about limits
corrected tex
Oct
16
comment Elementary Question about limits
Such a mess? Is not a mess. The most you have to use the chain rule once!
Oct
16
suggested suggested edit on Elementary Question about limits
Oct
16
comment shock waves characteristics
Why don't you take a look at Witham's Linear and Nonlinear Waves?
Oct
16
revised A differential equation of Buckling Rod.
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Oct
16
suggested suggested edit on A differential equation of Buckling Rod.
Oct
16
revised A differential equation of Buckling Rod.
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Oct
16
answered A differential equation of Buckling Rod.
Oct
15
revised Special Differential Equation
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