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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4
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1answer
70 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)=...
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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 ...
4
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1answer
56 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} \...
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PDE $u_t = f - u_x - u$ of a heating system

I found the following PDE in a book, describing a kind of a heating system: $\partial_tu=f(t)-\partial_xu-u$ where $u=u(t,x)$ and $(t,x)\in(0,\infty)\times(0,1)$. After own research i found out ...
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2answers
76 views

Gravity dominated flow in porous medium

I have the following equation modelling flow in a porous medium modeled by $$ (u^2)_x+u_t=0, \hspace{15pt}x>0,\hspace{5pt}t>0 $$ I have the initial condition: $$ u(x,0)=\frac{1}{2}, \hspace{...
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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 ...
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2answers
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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 ...
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1answer
37 views

question on a step used in book solution for PDE using method of characteristics

On page 213, from book nonlinear partial differential equations by Lokenath Debnath, 3rd edition. I do not understand one step in the solution. Book shows how to solve the pde $(y-z)u_x+(z-x)u_y+(x-y)...
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Equivalence of methods for semilinear equations

For i) we directly obtain $x(s,\tau) = x_0(s)e^{a\tau}$, $y(s,\tau) = y_0(s)e^{b\tau}$, $u(s,\tau) = u_0(s)e^{c\tau}$ and for ii) we obtain the general solution by intersecting $y - \frac{b}{a}x = ...
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PDE system and domain of definition

Consider the PDE system, defined for $u>0$: $u_x + vu_y + u^2v_y = 0$, $v_x + u_y + vv_y = 0$. It is easy to show that $\log u \pm v$ are Riemann invariants on $\frac{dy}{dx} = v\pm u$ (I do not ...
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Characteristics for PDE $u_x + 3u_y - (3x-y)u = -2 $

Question : I am willing to solve the PDE :$$u_x(x,y) + 3u_y(x,y) - (3x-y)u(x,y) = -2 $$ Thoughts : The characteristics are : $$\frac{\mathrm{d}x}{1} = \frac{\mathrm{d}y}{3} \implies u_1 = 3x-y$$ ...
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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))...
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Hard Partial Differential Equations - Characteristics

$$ (x-y)\frac{\partial u}{\partial x}+(x+y)\frac{\partial u}{\partial y}=\alpha u $$ where α is a constant, with initial condition $u(x, 0) = x^2$ , $x > 0$. How do I solve this partial ...
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Charpit PDE, rays parallel or perpendicular to boundary

The rays are all possible curves $(x(\tau), y(\tau))$. The derivation of the equations is clear and the condition of defining $p_0, q_0$ is $\frac{dx_0}{ds}\frac{dy_0}{d\tau} - \frac{dy_0}{ds}\frac{...
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1answer
68 views

Solve the characteristic equations numerically to solve a PDE

Do you know any introductory references about integrating numerically the characteristic equations and deduce the general solutions of a pde? More precisely I am interested in the transport equation $...
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1answer
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Triple-valued solution to Riemann problem - Profile of the bulge

I am studying conservation laws and hyperbolic systems, particularly, Burgers' equation and shocks, and have a doubt at pages 48/49 of the book Numerical Methods for Conservation Laws by R.J. LeVeque (...
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2answers
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Finding an Integral Surface

Consider finding the integral surface of $$x^2 p + xy q = xyz-2y^2$$ which passes through the line $x=y e^y$ in the $z=0$ plane. Attempt In Lagrange's subsidiary form $$\frac{dx}{x^2}=\frac{dy}{...
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Characteristic method , boundry conditions

I have problem with solving following task: Using characteristic method find solution $ u(x,y) $ of $$ \begin{aligned} x^{2} u_{x}+y^{2} u_{y} &=(x+y) u \quad \text { for }(x, y) \in \mathbb{R}^{...
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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
63 views

Method of characteristics for $x u_y + u_x=u$

I am trying to solve the PDE $$x\frac{\partial u}{\partial y}+\frac{\partial u}{\partial x}=u,$$ subject to the conditions $u(x,0)=0$ and $u(0,y)=y$. The characteristic equations are: $$\frac{dx}{...
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Method of characteristics, understanding/setting out method

Suppose I have the PDE: $u_x + u_y = 1 - u, u(x, x+x^2) = \sin(x), x > 0$ The Lagrange Charpit equations are $\frac{dx}{dt} = 1, x(0, s) = s$, $\frac{dy}{dt} = 1, y(0, s) = s + s^2$ $\frac{...
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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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1answer
60 views

Method of characteristics $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + 2\frac{\partial^2 u}{\partial x \partial y}=0$

I know how to solve problems with form like this (via method of characteristics): $$a(x,y) u_{x}+b(x,y)u_y=c(x,y).$$ But I got this problem: $$ \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}...
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1answer
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Solve a Nonlinear PDE

I want to solve the problem $$x_1u_{x_1} + 2x_2u_{x_2} + u_{x_3}=3u,~~~~~ u(x_1,x_2,0) = g(x_1,x_2).$$ I believe I am on the right track but would really like feedback. This is what I have so far: ...
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1answer
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Solving the PDE $xu_{x} + 2yu_{y} = 3u, u(x,y,0) = g(x,y)$

I was trying to solve the PDE: $xu_{x} + 2yu_{y} = 3u$ with $u(x,y,0) = g(x,y)$ I thought of using method of characteristics So the initial curve looks like $x = a, y=b , z = g(a,b)$ with $\frac{...
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Linear advection equation with coefficient given at data points

I am working on a problem to solve some particle population balances. In the analysis of some experiments I got an equation of this type: $$\frac{\partial u}{\partial y} - \alpha(x, y) \frac{\partial ...
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1answer
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IVP for linear first-order PDE $3u_x + 4u_y + 5u_z =0$

I solved an old problem (I don't remember if I have already posted this problem: forgive me, if so) $$ \begin{cases} 3u_x + 4u_y + 5u_z =0\\ u(1,y,0)=2y-6 \end{cases} $$ I quite easily obtained the ...
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1answer
58 views

Characteristic curves PDE $uu_x+u_y=u$ with $u(x,0)=-x$

I have the following PDE: $$uu_x+u_y=u\qquad , y>0$$ $$u(x,0)=-x$$ I need to find the ground curves, solve the PDE and find where the solution is valid. First parametrize the initial data: $$x=s, ...
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1answer
72 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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First order PDE $u_t - u^2u_x = 0$ with piecewise initial conditions [duplicate]

I have an initial value problem: $$ u_t - u^2u_x = 0 \quad \text{with} \quad u(x,0) = g(x) = \begin{cases} -\frac{1}{2}, & x\leq 0 \\ 1, & 0 < x <1 \\ \frac{...
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1answer
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Finite Characteristic of a Ring [duplicate]

If $H$ is a ring and it has a finite non-zero characteristic $p$ then ring is finite. I couldn’t any counter example for this claim. Can anyone help me please?
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1answer
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Validity of solution to PDE $xu_x - uu_t = t$ obtained from characteristics

I have a boundary-value problem: $$ xu_x - uu_t = t $$ with boundary conditions: $$ u(1, t)= t, -\infty<t<\infty $$ Finding the characteristic equations is no problem, and I get a general ...
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1answer
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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 ...
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1answer
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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
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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: ...
2
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1answer
53 views

Solving $u_{\alpha} + u_{t} = -\mu t u, t>0$ .

An age structured population with distribution $u(a,t)$ over age $a$ has a death rate increasing linearly with time and constant birth rate $b$, $u(\alpha,0) = u_{0}(\alpha)$. Model is $u_{\alpha} + ...
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1answer
51 views

Solving quasi-linear PDE, $x \frac{\partial{u}}{\partial{x}} - u\frac{\partial{u}}{\partial{t}} = t$ , $u(1,t) = t$, $-\infty < t < \infty$ [duplicate]

I am trying to solve the quasi-linear PDE $x \frac{\partial{u}}{\partial{x}} - u\frac{\partial{u}}{\partial{t}} = t$ , $u(1,t) = t$, $-\infty < t < \infty$ using method of characteristics. $\...
4
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1answer
65 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 ...
2
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1answer
116 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
47 views

Principle of conservation of mass and the shock speed

Consider the Burgers equation $u_t+uu_x=0$ with the initial condition $$u_0(x) = \begin{cases} u_l,x\leq0\\ u_r,x>0 \end{cases}$$ Consider the region in the $xt$−plane given by $[−1, X] × [0, T]$ ...
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3answers
70 views

Cauchy problem for quasi-linear pde $u_t+uu_x=1$

I am solving following Cauchy IVP: $$u_t+uu_x=1,$$ $x$ is real,$t>0$, and initial condition is $$u(t^2/4,t)=t/2$$ and found contradictory results:-Parametrizing the given initial curve as follows $...
2
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1answer
67 views

Method of characteristics non-linear PDE

Consider the following initial-value problem: $$xu_x-uu_t=t$$ $$u(1,t)=t$$ I've come to the follow characteristic equations: $$\frac{\mathrm{d}x}{\mathrm{d}\tau }=x,\,\,\,\frac{\mathrm{d}t}{\...
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0answers
58 views

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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1answer
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PDE characteristics clarification of paper

I am curious how the following circled part is found. I have been trying to figure it out for the past half hour or so and I can't find the computation that gives that number $x(t)=\sqrt{2t}$. It ...
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1answer
50 views

How to solve $u_t + u_x =0$ with $u_0(x) = \sin(\pi x)$ with Characteristics?

Consider the initial-boundary value problem (IBVP) for the convection equation \begin{array} { l } { u _ { t } + u _ { x } = 0 \quad x \in [ a ( t ) , b ( t ) ] , t \in [ 0 , T ] } \\ { u ( x , 0 ) = ...
2
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2answers
98 views

Solve PDE using method of characteristics with non-local boundary conditions.

Given the population model by the following linear first order PDE in $u(a,t)$ with constants $b$ and $\mu$ : $$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0$$ $$u(a,0)=u_0(a)\,\,\,a≥0$$ $$u(0,t)=F(t)=b\...
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0answers
22 views

Proving that a ring with some properties is commutative

A is a ring with the next properties: a) the order of $1$ is p (prime) in the group $(A,+)$ b) there exists $B \subset A$ with $p$ elements such that : for all $x,y \in A$, exists $b \in B$ which ...
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1answer
15 views

$k\in \mathbb Z,\ (k,p)=1$ the element $k\cdot1_A$ is invertible

Let $A$ be a ring. The order of $1_A$ in (A,+) is p (prime). For $k\in \mathbb Z,\ (k,p)=1$ the element $k\cdot1_A$ is invertible. I tried to prove this. $(k,p)=1 \to \exists m,n\in \mathbb Z\ s.t \...
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0answers
29 views

PDE-find general solution and solve initial value problem [closed]

I stuck with method of characteristics here, how to find the general solution and IVP? ${u_{xx} + 4u_{xy}+3u_{yy}}={0}, -\infty <x<+\infty, t >0 $ $ u(x,0)=0, -\infty <x<+\infty, $ ...
2
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3answers
58 views

Method of Characteristics for $u_t + uu_x = -2u$

Consider the following quasi-linear PDE : $u_t + uu_x = -2u$, with the boundary condition $u(0,t) = e^{-t}$. Show, using the method of characteristics, that the solution to this boundary value ...