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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24 views

General solution of $u_t + u_x = -u^2$

Can someone help me to write down a formula for the general solution to the nonlinear partial differential equation $$u_t + u_x + u^2 = 0$$ and how do I show that if the initial data is positive and ...
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Method of Characteristics for nonlinear PDE

I just started working on classical methods for nonlinear PDE's and I'm kind stucked in some questions. I would appreciate some help with the following Qing Hang question, so I could use it for trying ...
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A necessary condition for the solvablity of two functional equations

This appears in the proof of Charpit's method for solving nonlinear first order PDEs: Consider the two equaions: $$f(x,y,z,p,q)=0,\qquad \qquad (1)$$ $$g(x,y,z,p,q)=0,\qquad \qquad (2)$$ where $f$ ...
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Partial differential equation : $z_x+z_y+z=e^{x+2y},z(x,0)=0$

How to solve this equation: $$z_x+z_y+z=e^{x+2y}$$ with boundary condition as $z(x,0)=0$ I tried $$\frac{dx}{1}=\frac{dy}{1}=\frac{dz}{e^{x+2y}-z}$$ I got one condition as $y-x=a$ how to obtain ...
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A particular PDE of the form $u_y+a(x,y)u_x=0$

Exercise from Qing Han: Exercise 2.4. Find a smooth function $a=a(x,y)$ in $\Bbb R^2$ such that, for the equation of the form $$ u_y + a(x,y) u_x = 0, $$ there does not exist any solution in the ...
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Help solving a linear partial differential equation through the method of characteristics

I've been struggling to solve a partial differential equation of the form $$yu_x-xu_y=u, \qquad u(x,1) = f(x). $$ So far I've been able to use $dx/y=dy/-x=du/u$ to fairly easily find the ...
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How to find domain of Solution of PDE?

I wanted to solve following PDE $$u_x+2xu_y=2xu,$$ $u(x,0)=x^2$ for $x\geq 0$ and $u(0,y)=y^2$ for $y\geq 0$ I had find its solution as follows for $x\geq 0$ $u(x,y)=(x^2-y)e^y$ for $y\geq 0$ $u(...
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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}{\...
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Is there any example of PDE whose projection of 2 distinct characteristics curve intersect?

I know that for any PDE if we consider its any 2 distinct characteristics they never intersect. I also know that for semilinear pde projection of 2 distinct characteristics they never intersect.(I do ...
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IVP of $u_t=(2x+y)u_x+(x+2y)u_y$ and the method of characteristics

Solve the initial value problem: $$\begin{cases}\frac{\partial u}{\partial t}&=(2x+y)\frac{\partial u}{\partial x}+(x+2y)\frac{\partial u}{\partial y},\\u(x,y,0)&=e^{x}\end{cases}$$ for $u=u(x,...
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How to find the entropy solutions and breaking times for the following Riemann problem of the inviscid Burgers' equation?

For the inviscid Burgers' equation $$u_t + uu_x = 0,$$ with initial conditions (correct me if I am wrong, these are piecewise functions) \begin{equation} u_a(x,0)=\left\{ \begin{array}{@{}ll@{}} ...
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How to obtain the characteristics equation for the inviscid Burgers equation?

I am trying to plot the characteristic lines for the inviscid Burgers equation which is $$u_t +uu_x=0.$$ From what I understand, with the initial condition $u(x,0)=f(x)$ and using the method of ...
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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 $\frac{1}{x}z_x+\frac{1}{y}z_y=4$

$$\begin{cases} \frac{1}{x}z_x+\frac{1}{y}z_y=4\\ z(1,y)=y^2-1.\\ \end{cases}$$ 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}\...
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Solving PDE $xu_x+(x+t)u_t=1$ with $u(1,t)=t$ [duplicate]

Solve $xu_x+(x+t)u_t=1$ such that $u(1,t)=t$. Is the solution defined everywhere? I had known that this specific problem is related to a Heat Equation problem. I tried solving for its ...
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Finding explicit solution of PDE $xu_x+tu_t=pu$

Let $p\in \mathbb{R}$ and consider $xu_x+tu_t=pu$. Find a) Characteristic curves for the equation. b) Find an explicit solution for $p=4$, where $u=1$ on the unit circle. What I tried: a) ...
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Method of Characteristics - Lagrange-Charpit Equations

I need to solve the following PDE with initial condition $U(x,0)=U_0(x)$. Once this is one of my first times, I'd like to get second opinions. Many Thanks. \begin{equation} \partial_t U + (\text{...
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Geometric interpretation and solution of Lagrange-Charpit equations

What is the geometric meaning of the Lagrange-Charpit equation, $$P\dfrac{\partial z}{\partial x} + Q \dfrac {\partial z}{\partial y}= R \\Pp+ Qq= R$$ where $P$,$Q$,and $R$ are functions of $x\,$,$\,...
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Transport equation on the whole space $\mathbb{R}^3.$

If one has the equation: $$v\cdot \nabla_x f = g(x),$$ where the domain is $D =\{x:x = (x_1,x_2,x_3)\in\mathbb{R}^3\},$ how does one write down a closed formula for $f$ in terms of $g$ ? I looked at ...
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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 ...
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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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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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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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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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Finding solution of semi-linear PDE using Method of Characteristics

I am given the PDE: $$u_x+u_y+u=e^{x+2y} \quad u(x,0)=0$$ I tried to do this using the method of the characteristics in the following way. First I find the particular solution of this inhomogenous ...
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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
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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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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
66 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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1answer
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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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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
25 views

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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67 views

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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18 views

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
38 views

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
61 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
76 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{...