# Questions tagged [characteristics]

The method of characteristics is a way of solving certain partial differential equations by reducing them to ordinary differential equations. It is most often used for 1st order equations. Use with the (pde) tag.

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### Partial Differential Equations - Transport Equation & Characteristics

So I am doing an introductory course to PDEs and I have been introduced to the method of characteristics. Now I have sort-of learnt the method used to solve the transport equation, where the ...
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### Solving a system of semilinear coupled partial differential equations

I would like to know if it is possible to find the solution to the following system of partial differential equations: \begin{equation} \partial_x \mathbf{u}+B(x,y) \cdot \partial_y \mathbf{u}-\mathbf{...
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### Traffic Flow problem space time diagram question

Traffic with constant density, $\rho_0$, is stopped by a red light at $x=0$. If $v(\rho)=1-\rho$, calculate what happends behind the light. Unfortunately the main focus of this example in my notes ...
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### Solve the quasilinear initial value problem by the method of characteristics.

I am trying to solve $u_{t}+uu_{x}=0, \quad x\in\mathbb{R},\quad t>0$ with the initial values $u(x,0)= \begin{cases} 1-x^2, & |x| \leq 1 \\ 0, & |x| > 1 \end{cases}$ I also need ...
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### Method of Characteristics for parametric problems

I am considering the equation $u_t(t,x,\eta)+V(t,x,\eta)\cdot u_x(t,x,\eta)=0$ where $V:[0,T]\times\mathbb{R}^n\times[0,1]^D\to \mathbb{R}^n$. For the parameter independent case, one would solve ...
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### Solving definite integral in two variables.

Solving a PDE with the following boundary problem with arbitrary constant $b$: $$u(0,t)=F(t)=b\int_0^\infty u(a,t)\mathrm{d}a$$ Hint given in the question is as follows: Split this integral in two ...
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### Method of Characteristics - Can we have multiple answers

Determine the characteristics of the equation $z=p^2-q^2$, and find the integral surface which passes through the parabola $4z+x^2=0$, $y=0$. I solved the question and my answer was: The ...
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Consider a PDE of the form $$a(x,y) \frac{\partial^2 u}{\partial x^2} + 2b(x,y) \frac{\partial^2 u}{\partial x \partial y} + c(x,y) \frac{\partial^2 u}{\partial y} + d(x,y) \frac{\partial u}{\partial ... 1answer 220 views ### How to solve the following Cauchy problem. Solve the following Cauchy problem for a first order PDE:$$(2x_1 + x_2)u_{x_1} + (x_2 + 1)u_{x_2} = u^2, \ \ u(x_1, 1) = x_1^2 + 1, \ \ x_1 \ge 0, x_2 \ge 1$$and find an implicit conldition over ... 1answer 118 views ### Equation with non constant coefficients with the method of characteristics?$$\dfrac{\partial u}{\partial t} +c\dfrac{\partial u}{\partial x} = 0$$where c is a constant, subject to u(x,0) =\cosh(x) With the normal method I wrote down (s is the parameter I'm using): ... 0answers 29 views ### Solving the G - equation with method of characteristics Trying to solve$$u_t = |\nabla u| \quad u(0,x) = e^{-|x|^2/2} using method of characteristics. I can solve it without the absolute value, but not sure how to handle them since these solutions ...
Working on the following problem. $y u_x -u u_y = x \\ u(x,x)=-2x$ I've went after this with the method of characteristics. I'm using $(t,s$) as my parametrization variables. In parameterize space I ...