# 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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### Question about characteristics and classification of second-order PDEs

I am currently reading through the book 'Computational Techniques for Fluid Dynamics', by C.A.J. Fletcher. Chapter 2 discusses classification of PDEs by finding the number and nature of their ...
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### Whilst There Are Three Characteristic Equations, Only Two of Them Are Linearly Independent?

Take the general quasi-linear equation $$a(x, y, u)u_x + b(x, y, u)u_y - c(x, y, u) = 0. \tag{1}$$ We assume that there exists a solution of the form $u = u(x, y)$. We can define a solution ...
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### How can I solve $u_{xt} + uu_{xx} + \frac{1}{2}u_x^2 = 0$ with the method of characteristics.

I am trying to solve the following PDE: $u_{xt} + uu_{xx} = -\frac{1}{2}u_x^2$, with initial condition: $u(x,0) = u_0(x) \in C^{\infty}$ using the method of characteristics. I am a beginner with the ...
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### Method of characteristics for a first-order linear PDE: $D_t u + xD_x u = t^3$

In short terms, I must solve a certain first-order PDE. I applied the method of caracteristics to find the integral curves and found an answer that indeed satisfies the equation. However, Wolfram ...
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### Traffic flow modelling - How to identify fans/shocks?

A highway contains a uniform distribution of cars moving at maximum flux in the $x$-direction, which is unbounded in $x$. Measurements show that the car velocity $v$ obeys the relation: $v = 1 − ρ$, ...
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### Entropy solution to inviscid Burgers with triangular initial data

Find the entropy solution of $$\begin{cases} u_t + \left( \frac{u^2}{2} \right)_x = 0 & \text{ in } \mathbb{R}\times(0,\infty) \\ u = g & \text{ on } \mathbb{R}\times\{0\}, \end{cases}$$ ...
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### One-way wave equation IBVP

Plese help me to find the solution of te following equation. For values of $x$ in the interval $[-2,3]$ and $t>0$ we consider the one way wave equation $$u_t+u_x=0$$ with initial data \begin{...
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### $u_t+[u(1-u)]_x=0$ with initial conditions - Need help with rarefaction wave portion

I need help solving $$u_t+[u(1-u)]_x=0$$ with initial conditions $$u(x,0) = \begin{cases} 0.75 & |x|<0.5\\ 0 & else \end{cases}$$ I am trying to use the method of characteristics but I am ...
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### Finding when Cauchy data is characteristic

I have been stuck on the following problem: Consider the Cauchy problem \begin{equation} \frac{\partial^2 u}{\partial x_1 \partial x_2} - 4\frac{\partial^2 u}{\partial x_3^2} + \frac{\partial u}{...
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### Solving Burgers' equation $u_y+uu_x=0$ with arbitrary initial values

Consider the PDE $$u_y+uu_x=0$$ with the condition $u(x,0)=f(x)$. I want to solve this using the method of characteristics. The first thing I've done was to solve the characteristic system. In that ...
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### How to determine strip condition of nonlinear PDE?

I started learning PDE on my Own. I was doing Example 0.14 in the book (1) p. 32 but I stuck at one step. I do not understand how the Author come at the conclusion about strip condition. Example 0....
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### Solve this Semi-Linear PDE (Partial Differential Equation) with the Characteristic Method

I need to solve this linear PDE: $3u_x - 4u_y = y^2$ The initial condition provided is: $u (0,y)= sin(y)$ I need to use the Characteristic Method. I learned the method from this video. I have ...