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93 views

Questions about the hyperbolic system of equations

$$u_t+A(x,t,u)u_x=b(x,t,u) \tag 1$$ $$u=(u_1, \dots, u_n), b=(b_1, \dots, b_n)$$ $$A=[a_{ij}], i,j = 1, \dots, n$$ $$$$ We set the question if there are characteristic directions at the path of which ...
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47 views

Hyperbolic equations-characteristic system

Let the system: $$\alpha(x,t,u)u_t+\beta(x,t,u)u_x=f(x,t,u)$$ To find the characteristic equations: $$\frac{du}{ds}=\frac{\partial{u}}{\partial{t}} \frac{dt}{ds}+\frac{\partial{u}}{\partial{t}} ...
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1answer
54 views

Solving the Laplace equation in terms of exponential of hyperbolic trigonometric functions

I'm solving the Laplace equation $U_{xx}+U_{yy}=0$ subject to BC's: \begin{align} U(0,y) &= 0 \\ U(a,y) &= 0 \\ U(x,0) &= 0 \\ U(x,b) & = \left\{x \text{ for } x \in ...
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0answers
47 views

Integral with $\cosh$ and $\log$ in the integrand

I am trying to find a good way to simplify (or even solve) the following integral: $$ \int_0^{-\log \varepsilon} \cosh^3r \frac{\partial^2}{\partial r^2} \log (\det E) dr $$ where $r > 0$ is a ...
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55 views

How do I write this as a solution for hyperbolic sine and cosine?

Consider the following PDE. Where $u(x,t) = X(x)T(t)$ $u_{tt}+u_t = u_{xx}$ $u(0,t)=u(\pi,t)=0$ $u(x,0)=0$ $u_t(x,0)=10$ I am having trouble solving for my $T(t)$, it comes down ...