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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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Inhomogenous nonlinear transport equation $u_t+uu_x = -Du$

We have the following setup: $$u_t+uu_x = -Du \\ u(x,0)=\sin x.$$ The question is to find the time $T_s$ of a first shock formation. So basically, I need to solve the equation using method of ...
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1answer
174 views

Cauchy Problem for inviscid Burgers' equation

Consider the Cauchy Problem of finding $u(x,t)$ such that $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0,x\in\mathbb{R},t>0$$ $$u(x,0) = u_0(x), x\in\mathbb{R}$$ Which choices ...
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147 views

Why is the solution single-valued?

I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-...
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1answer
1k views

Rarefaction and shock waves colliding in Burgers' equation

I'm confident I've solved all but the last segment of this problem, to which I have an answer that just doesn't seem right. The problem is to solve the inviscid Burgers' equation $$u_{t}+uu_{x}=0$$ ...
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1answer
130 views

How to take this exterior derivative of the expression $du - \sum_i p_i dx_i$?

I am reading the wikipedia page about applying the method of characteristics in the fully nonlinear case. We have the fully nonlinear equation $$ \tag{1} F(x_1, \cdots, x_n , u, p_1, \cdots, p_n) = 0,$...
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1answer
112 views

Shock of Burgers equation $u_t+uu_x=0$ at $t=0$

Consider the Burgers equation $u_t+uu_x=0$ with the initial condition $$u_0(x) = \begin{cases} u_l,x\leq0\\ u_r,x>0 \end{cases}$$ My attempt to solve this: Using the method of characteristics we ...
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1answer
113 views

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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1answer
170 views

Treating shocks with conservation laws

I'm trying to solve Burgers equation $(\rho_{t} + \rho \rho_{x} = 0$), interpreted as a conservation law for $\int \rho dx$, with $$\rho (x_{0} ,0) = \begin{cases} 0 \quad & x_{0} \leq 0 \\ x \...
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1answer
308 views

Finding shock curves, Burgers equation

I'm working on this problem: $u_{t}+uu_{x}=0\\ u(0,x)=-x \mathbb{1}_{[a,b]}$. For the case when $a<b=0$ I want to find a shock curve starting from point $(t,x)=(1,0)$. On the field without ...
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1answer
75 views

Entropy solution of the Burgers' equation 2

I'm trying to construct a solution to the following problem: $u_{t}+uu_{x}=0\\ u(0,x)=-x \mathbb{1}_{[a,b]}$. For the case when $0<a<b$ I try to find a shock curve starting from point $(t,x)=(...
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1answer
973 views

Singular solution of partial differential equation

If complete integral of differential equation $$ x (p^2 +q^2) = zp $$ ( p is partial derivative of z with respect to x and q is partial derivative of z with respect to y ) Passing through $x=0$ ...
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2answers
165 views

How to solve $u_t + uu_x =\delta(x)$

I am studying from old exams and there is a problem which is traffic flow with a ramp. I have never seen this type of problem in class, so for the simple case, how would I solve $$u_t + uu_x =\delta(...
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1answer
726 views

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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1answer
154 views

IVP for nonlinear PDE $u_t + \frac{1}{3}{u_x}^3 = -cu$

I'm trying to solve the following partial differential equations: $$ u_t + \frac{1}{3}{u_x}^3 = 0 \tag{a} $$ $$ u_t + \frac{1}{3}{u_x}^3 = -cu \tag{b} $$ with the initial value problem $$ u(x,0)=h(x)= ...
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2answers
76 views

Quasi-linear pde $u_t + x u u_x = 0$, find shock time

Using the characteristics method, show that the Cauchy Problem for the quasi-linear equation $$u_t + x u u_x = 0 \qquad u(0, x) = \phi(x) = \frac \pi 2 - \arctan(x)$$ has two shock times, $t^*_\pm $,...
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1answer
64 views

$u_t+[u(1-u)]_x=0$ 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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1answer
126 views

Analyze : $u_t-u^2u_x +cu =0, u(x,0)=g(x)$

Analyze : $u_t-u^2u_x +cu =0 $, $ u(x,0)=g(x)$. From This we have following $$\begin{align} \frac{dt}{ds} &=1 \\ \frac{du}{ds} &=c \\ \frac{dx}{ds} &=-u^2 \end{align}$$ then how to ...
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2answers
188 views

Method of characteristics - eliminating variables

I am trying to follow a guide for the method of characteristics; quoting the first example: We use the method of characteristics to solve the problem $ 2u_x - u_y = 0, \;\; u(x, 0) = f(x) $ ...
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0answers
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Solve partial differential equation by Charpit's equations

Solve partial differential equation, $$F=xp^2-ypq+y^2q-y^2z=0,\;\;\;\; p = \frac{\partial z}{\partial x},\;\;\;\;q = \frac{\partial z}{\partial y}$$ My attempt: $F_x = p^2, \\F_y = -pq+2yq-2yz, \\...
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1answer
121 views

How to Solve This PDE and Deal with Boundary Value Using Method of Characteristics. Work Shown.

The function $u(x,y)$ satisfies $u_y + u_x = 0$ in $x > 0$, $y > 0$ together with the initial condition $u(x, 0) = \sin(x)$, $x > 0$ and the boundary condition $u(0, y) = \sin(y)$, $y > 0$....
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326 views

Burgers' equation with triangular initial data

The problem is: Consider Burgers' equation, $$u_t +uu_x = 0 $$ $$ u(x,0) = f(x) $$ where $$f(x) = \begin{cases} 1 - |x-2| &\mbox{if}\,\, 1\leq x \leq3, \\ 0 &\mbox{otherwise}.\end{...
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1answer
75 views

Why is $u(x, y) = A(x - y)$ the general solution for this PDE?

My Professor's notes had this problem: Solve the PDE $u_x + u_y = 0$ in the domain $y > \phi(x)$, $x \in \mathbb{R}$ given that $u = g(x)$ on the curve $y = \phi(x)$, where $\phi(x) = \frac{x}{1 ...
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2answers
823 views

Solution of Burgers' equation

Can you help me to solve this problem and explain the method used? $u\equiv u(x,t)$ with $x\in\mathbb{R}$ $$ u_t+\left( \frac{1}{2}u^2\right)_x=0 $$ with initial data $$u_0(x)=u(x,0)= \begin{cases} ...
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1answer
269 views

Shockwaves in traffic flow

I have been struggling on this problem for a while now so here it is: I am looking to work out when the shockwaves occur in the traffic flow model given below: A traffic flow governed by the ...
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1answer
77 views

Entropy solution - Burgers' equation

Why doesn't the following problem have a solution for $t\ge1$? $u_{t}+uu_{x}=0\\ u(0,x)=-x$. The characteristics don't intersect and they cover the whole space above t=1.
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1answer
43 views

Shock in the Inviscid Burgers Equation

I'm trying to implement the Finite-Volume method using local Lax-Friedrichs flux function. I found an example to the Riemann-problem, here is the solution plot: So from characteristics $$u=\begin{...
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1answer
71 views

Solve one-dimensional form of Euler’s equations

This is a home work problem. Please find the problem in the image attachment. Problem : Consider the one-dimensional form of Euler's equations for isentropic flow and assume that pressure $p$ is ...
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2answers
78 views

Method of characteristics for $f f_x + f_y = 1$. Where is the solution valid?

Suppose we have a PDE that can be solved with the method of characteristics \begin{align} F(\nabla u, u , x) = 0 \text{ in $U$}\\ u|_\Gamma = g \text{ on $\Gamma$ } \end{align} Where $\Gamma \...
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1answer
51 views

Solving PDE $yu_x - xu_y + x^2 - y^2 = 0$ with method of characteristics

I am struggling to understand one 'trick' that is used in the solution of the ODE's from method of characteristics, which currently doesn't make any sense to me: 13. Solve the 1st order PDE $$ ...
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1answer
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Method of Characteristics - Second order derivatives - Need help finding

given that $\zeta _{(x,y)}= \ln y+\frac{1}{x}$ and $\phi_{(x,y)}= 4\ln y-\frac{1}{x}$ , I've been asked to find $\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}, \frac{\partial^2 u}{\...
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3answers
118 views

PDE IVP - Characteristics, why is my method wrong?

Solve the IVP $$ \begin{cases} u_t + cu_x = 1 & c \in \mathbb{R} \\ u(x,0) = \sin x \end{cases}$$ To solve this, I have used characteristics as follows: Note that $$\frac{\...
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3answers
196 views

$\left\{\begin{array}{lll} f_{t}+xf_{y}=0\\ f|_{t=0}=f_{0}(x,y) \end{array}\right.$

Given $f\in C^{1}({\bf R}^{3})$ , then find the solution of the following pde : $$\left\{\begin{array}{lll} f_{t}+xf_{y}=0\\ f|_{t=0}=f_{0}(x,y) \end{array}\right.$$ I tried to use the ...
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1answer
161 views

Domain solutions on partial differential equations

$$(1+x^2)\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}=0$$ integrating to give: $\arctan x=\ln y+A$ I managed to find the general solution as $u(x,y)=F(A)=F(\ln y -\arctan x)$ I ...
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4answers
855 views

What is the geometric interpretation of the solution to PDE $xu_x+yu_y=0$

I have the following PDE $$xu_x+yu_y=0$$ for which I get the characteristic function $$y=cx$$ along which the u(x,y) is constant. The general solution is $$u(x,y)=f(\frac{y}{x})$$. I understand ...
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0answers
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Characteristics of First-Order PDEs Example: How Was This Integration Done?

From Essential Partial Differential Equations, by Griffiths, Dold, and Silvester: Example 4.2 (Half-plane problem) Solve the PDE $pux + quy = f(x, y)$ (with $p$ and $q$ constant) in the domain $\...