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.

Filter by
Sorted by
Tagged with
43
votes
1answer
1k views

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 ...
9
votes
2answers
240 views

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 ...
9
votes
2answers
498 views

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 ...
7
votes
1answer
51 views

Are characteristics of $u_t+f(u)_x=0$ always straight lines?

I am studying conservation laws and reviewing the papers I get a doubt. Consider $$u_t+f(u)_x=0$$ with $f$ smooth a conservation law and take the characteristics $$x(t)\,\, ; \,\, x'(t)=f'(u(x(t),t))...
6
votes
2answers
157 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-...
6
votes
1answer
181 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)= ...
6
votes
1answer
133 views

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 ...
6
votes
1answer
756 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: $v = 1 − ρ$, ...
6
votes
1answer
197 views

Method of Characteristics for the Equation $\;\left(x+\alpha y\right)u_{xx} + u_{yy} = 0\:$

Below is Exercise $3.6$ (p.$91$) of "Applied Partial Differential Equations" by Ockendon et al., $2^\mathrm{nd}$ ed.: Show that, if $$\big(x+\alpha y\big)\dfrac{\partial^2 u}{\partial x^2} + \...
5
votes
2answers
109 views

The PDE $u_t = u_{xx}$ follows the path defined by $\dfrac{dx}{dt} = \pm \infty$

I have the PDE $u_t = u_{xx}$ (heat equation). I am then told that, by writing the equation as $(\partial_x + (0)\partial_t)^2 u = u_t$, we see that its characteristics would follow the path defined ...
5
votes
3answers
427 views

Use the method of characteristics to solve $u_t+uu_x+\frac{1}{2}u=0$.

Use the method of characteristics to solve $$u_t+uu_x+\frac{1}{2}u=0, \quad t>0, \quad {-\infty}<x<\infty$$ $$u(x,0)=\sin(x)\quad {-\infty}<x<\infty$$ (solution may be expressed ...
5
votes
1answer
132 views

Solving the Quasilinear PDE $u_{t} - u^2 u_{x} = 0$ with piecewise initial condition.

Solving the quaslinear PDE by the method of characteristics is a bit tricky for me. I was trying to obtain the solution $u$ for the PDE $$u_{t} - u^2 u_{x} = 0$$ The initial condition is given by: ...
5
votes
3answers
245 views

How to solve a system of PDE $u_t+u_x=v, v_t+v_x=-u$

Solve the following initial value problem: $$u_t+u_x=v, \\v_t+v_x=-u, \\u(0,x)=u_0(x), \\ v(0,x)=v_0(x).$$ I did not learn any method to solve a system of PDE so I guess there is a "trick". So ...
5
votes
3answers
86 views

Another attempt at solving a PDE with the method of characteristics

I want to use the method of characteristics to obtain the solution to this PDE, $$\frac{\partial F}{\partial t}=\left(z-t\right)\left(\beta z-\gamma\right)\frac{\partial F}{\partial z}$$ which I've ...
5
votes
2answers
2k views

Find complete integral of $(y-x)(qy-px) = (p-q)^{2}$

Find complete integral for partial differential equation $(y-x)(qy-px) = (p-q)^{2}$ where $p={ \partial z \over \partial x},q={ \partial z \over \partial y}$. My attempt: The given equation is f(...
4
votes
2answers
82 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 $,...
4
votes
3answers
99 views

Problem with solving $u_x+xu_y=1$ using method of characteristics

I got an exercise in my PDE class which I'm struggling to solve. Solve following eq using the method of characteristics $$u_x(x,y)+xu_y(x,y) = 1 \qquad (x,y) \in \mathbb{R}^2$$ $$u(3,y) = y^2 ...
4
votes
2answers
233 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(...
4
votes
3answers
75 views

Solve $\frac{1}{x}z_x+\frac{1}{y}z_y=4$

$$\begin{cases} \frac{1}{x}z_x+\frac{1}{y}z_y=4\\ z(1,y)=y^2-1.\\ \end{cases}$$ So we started with: $$\frac{dx}{dt}=\frac{1}{x}\rightarrow x^2(t,s)=2t+f_1(s)$$ $$\frac{dy}{dt}=\frac{1}{y}\...
4
votes
1answer
118 views

Entropy solution to inviscid Burgers with triangular initial data

Find the entropy solution of $$\begin{cases} u_t + \left( \frac{u^2}{2} \right)_x = 0 & \text{ in } \mathbb{R}\times(0,\infty) \\ u = g & \text{ on } \mathbb{R}\times\{0\}, \end{cases}$$ ...
4
votes
4answers
89 views

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{...
4
votes
1answer
53 views

$\frac{dx}{dt} = p, \frac{dy}{dt} = q$: Solution of these ODE imply the solution is constant along characteristics of the form $qx − py = constant$.

My lecture notes state the following: When we were dealing with first order equations we saw that a differential operator of the form, $$p\frac{\partial}{\partial{x}} + q\frac{\partial}{\...
4
votes
1answer
292 views

Existence/uniqueness and solution of quasilinear PDE

So I have this question in a textbook that I have been trying to solve for review but I can't $$u_t+uu_x=1$$ $$u(x,t)=t\text{ when } 2x-t^2=0$$ So the book says that the solution does not exist at all ...
4
votes
1answer
61 views

Determine the shock regions of quasi-linear equation

Assume the IVP: \begin{cases} z^2 z_x + z_y = 0 \\ z(x,0) = f(x) \\ \end{cases} The condition of existence of (locally) unique solution is: $$ P(t_0) \frac{dy(t_0)}{dt} - Q(t_0) \frac{dx(t_0)}{dt} \...
4
votes
1answer
66 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 ...
4
votes
1answer
140 views

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}{...
4
votes
1answer
307 views

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 ...
4
votes
1answer
77 views

Lagrangian formulation of conservation laws

In some articles that I read, I often encountered the same formulation for one-dimensional conservation laws of the form $$u_t+(F(u)u)_x=0$$ where $F(u)$ may also depend on $u_x$, etc., and $u(t=0,x)=...
4
votes
0answers
89 views

Solve wave equation with non-constant wave speed using method of characterstics?

I am trying to get a better understanding of wave pulses in a domain with a non-constant wave speed. I am trying to solve either one of the two equations: $$\frac{\partial^2u}{\partial t^2}-c(x)^2\...
4
votes
1answer
92 views

Solve transport equation $\frac{\partial\phi}{\partial t}+\phi\frac{\partial\phi}{\partial x}=0$ using method of characteristics

I'm trying to solve the following transport equation $$\frac{\partial\phi}{\partial t}+\phi\frac{\partial\phi}{\partial x}=0$$ subject to the initial condition $$\phi(x,0)=f(x)=\left\{ \...
4
votes
1answer
248 views

Method of characteristic for PDE $xu_x + yu_y = u$

I'm studying method of characteristic. This is a homework. I'm very new to this topic so hope anyone can give a direction. Given a PDE : $xu_x + yu_y = u$. a) Solve it, where the initial curve ...
3
votes
2answers
56 views

Where is my error in solving this first order PDE using method of characteristics?

Update Thanks to the comment below. I fixed a silly error I had. But now after fixing the error, I found I am not able to solve this PDE. So the question really becomes, can this pde be solved ...
3
votes
2answers
344 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{...
3
votes
1answer
207 views

Method of characteristics PDE $u_t +uu_x = x$

In this question you are asked to solve the initial value problem \begin{aligned} u_t +uu_x &= x,\\ u(x, 0) &= f(x) \end{aligned} using the method of characteristics. Using the parameters $s$ ...
3
votes
1answer
140 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,$...
3
votes
1answer
147 views

Extension of the method of characteristics

Solve the pde (advection equation): $$ u_t + c u_x = 0, \hspace{0.5cm} t>0, \hspace{0.5cm} x\in \mathbb{R} $$ with the condition given on an arbitrary curve $t=\tau (x)$, that is: $$ u(x,\tau (x))= ...
3
votes
1answer
86 views

Behavior of the solution to the inviscid Burgers' equation

Consider the inviscid Burgers' equation $u_t+uu_x=0$ with the initial condition $$u_0=\begin{cases} 0, & x<0\\ x, & 0\leq x \leq 1\\ 1, & x>1 \end{cases}$$ I tried to implement ...
3
votes
2answers
68 views

Method of Characteristics: Initial Conditions and the Unique Solution in This Linear PDE Example

Find the general solution of the PDE $pu_x + u_y = u$, where $p$ is constant, in the domain $\{ (x, y) : x \in \mathbb{R}, 0 \le y \le Y \}$, subject to the initial condition $u(x, 0) = g(x), x \in \...
3
votes
1answer
89 views

PDE IVP : $zz_x + z_y = 0, \; \; z(x,0) = -3x$

Exercise : Given the PDE IVP : $$\begin{cases} zz_x + z_y = \quad0 \\ z(x,0) \; \; \; =-3x \end{cases}$$ a) Find the solution of it. b) Determine the lines of the $(x,y)$ plane on which the ...
3
votes
1answer
65 views

Existence and uniqueness of PDE IVP : $z z_x + y z_y = x$, $C : x=t, y=t \; ; \; t >0$

Exercise : Consider the equation $$z z_x + y z_y = x$$ and the initial curve $$C : x=t, y=t \; ; \; t >0$$ Decide whether there is a unique solution, no solution or infinitely many ...
3
votes
1answer
57 views

Form of a solution of an equation similiar to the Burger's Equation

I'm fairly new to the whole field of PDE and I tried to work on some examples in order to become more familiar with the method of characteristics. I've been working on this task: You have the PDE $$\...
3
votes
1answer
53 views

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....
3
votes
1answer
153 views

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 ...
3
votes
1answer
100 views

Solving Quasi-Linear Transport Equation with two shockwaves.

I want to solve the following Partial Differential Equation by using the method of characteristics. This is the transport equation. \begin{align}u_t - (1-2u)u_{x}&=0, &-\infty < x < \...
3
votes
0answers
41 views

Solving the one-dimensional incompressible Navier-Stokes Equations

I am interested in solving the PDE system $$\frac{\partial\rho}{\partial t}(x(t),t)+u(t)\frac{\partial \rho}{\partial x}(x(t),t)=0, \qquad (\text{EQ} \ 1)$$ $$\rho(x(t),t) u'(t)=-\frac{\partial p}{\...
3
votes
2answers
69 views

A Charpit-type problem

Solve in parametric form for $u(x,y)$: $$u + u_x^2 + u_y^2 - 2 = 0$$ with the data $u(0,y) = y$ for $0\leq y \leq 1$ and the restriction $u_x \geq 0$. Determine (and show on a sketch) the domain in ...
3
votes
1answer
57 views

Uniqueness of solution based on characteristic curves

I have a pde $$\begin{cases} u_t − xu_x = 2u & x\in\mathbb{R}, t>0\\ u(x, 0) = \frac{1}{1+x^2} \end{cases}$$ I've solved it using method of characteristics ($u=\frac{1}{1+x^2e^{2t}}e^{2t})$ ...
3
votes
0answers
40 views

Solve the pde $u_xu_y=u$

I want to solve the following pde $$\left\{\begin{array}{cc} u_xu_y=u \mbox{ on $\Omega:=\{(x,y)|x>0\}$} \\ u(0,y)=y^2 \end{array}\right.$$ I supposed that $u$ was a polynomial of two variables ...
3
votes
0answers
140 views

Why is the domain of dependence for a system of hyperbolic PDE’s an interval on the $x$ -axis?

I’m looking at the hyperbolic system ${{\mathbf{u}}_t} + {\mathbf{A}}({\mathbf{u}},x,t){{\mathbf{u}}_x} = {\mathbf{h}}({\mathbf{u}},x,t)$ $\quad$ (1) where ${\mathbf{u}}(x,t) \in {\mathbb{R}^n},\;\;...
3
votes
0answers
57 views

ODE IVP : $u_t + uu_x = 0, \; \; u(x,0) = \cos \pi x$ [duplicate]

Exercise : Show that a smooth solution of the initial value problem $$\begin{cases} u_t + uu_x = 0 \\ u(x,0) = \cos \pi x \end{cases}$$ must satisfy $u = \cos[\pi(x-ut)]$. Also, show that when $...