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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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Meaning of the method of characteristics at crossing points of characteristic lines

Given a system of linear PDE's of first order such as $w^1_t + cw^1_x = 0,w^2_t - cw^2_x =0,$ one usually uses the method of characteristics to find the solution $(w^1(x,t),w^2(x,t))$ on each point of ...
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Help in solving PDE with charasteristic equation problem

It's my first time on this sort of problem, so I am having some trouble solving this PDE: $\frac{\partial u}{\partial t}+e^{-x}\frac{\partial u}{\partial x}+\frac1y\frac{\partial u}{\partial y}=0$ and ...
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How to find the characteristic form of coupled PDEs?

I'm reading a textbook named linear and nonlinear waves. In chap.14, a method was used to solve the coupled equations which can reduce PDEs to ODEs. But I don't know how to find the needed ...
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A simple question about $b(x)\cdot \nabla u(x)=0$

I want to find solutions $u$ to the equation $$b(x,y,z)\cdot\nabla u(x,y,z)=0\quad (x,y,z)\in D$$ $$u(1,0,z)=1,\, u(0,y,0)=0$$ where $D$ is the domain (a simplex) defined by $$D=\{(x,y,z)\in[0,1]^3: 0\...
Diplodokus's user avatar
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1 answer
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Solving a PDE that looks like Kinetic Fokker Planck without diffusion

I am interested to solve the following PDE . $$\partial_t f(t,x,y)=-y\partial_xf(t,x,y)+(x+y)\partial_y f(t,x,y)+\alpha f(t,x,y)$$ with $f:\mathbb{R}^+\times\mathbb{R}\times \mathbb{R}\to\mathbb{R}$. ...
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Can we clear out $a,b,c$ in four equations: $ax+by+cz=0, cx+ay+bz=0, a+b+c=1, a^2+b^2+c^2=1$?

Can we clear out $a,b,c$ in four equations: $ax+by+cz=0, cx+ay+bz=0, a+b+c=1, a^2+b^2+c^2=1$? In my search of characteristic cone of $u_{xy}+u_{yz}+u_{zx}=0$, I need to clear out $a,b,c$ in the above ...
xldd's user avatar
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1 answer
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How to solve a method of characteristics problem with only a point boundary condition? [closed]

I am familiar with the method of characteristics. However, I have a problem that appears to only have a specified solution at a single point: Solving for $V(x,y)$, where $V_xy-V_y(ax+bx^2y) = -cx^2y^2-...
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Monge's Equations and Characteristics of $u_t + (x+u)u_x = 0$ with $u(x,0) = x$

Consider the nonlinear wave problem $$ \begin{cases} u_t + (x+u)u_x = 0\\ u(x,0) = x \end{cases} $$ a) Write down Monge's equations in the for $du/d\tau$...,etc, and solve them. b) Plot the ...
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Solution's definition PDE - Characteristics

Solve $$\label{eq_0}\tag{1}\begin{cases} \partial_tu + (x-1) \partial_xu = 0\\ u(0,x) = x^3 \end{cases}$$ in the plane region $Q : = \{(t,x) \in \Bbb R^2 : x<1\}$ Using characteristic I found $$\...
Turquoise Tilt's user avatar
1 vote
2 answers
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Solution to Inviscid Burger's equation given piecewise initial condition

I'm taking Non-Linear PDEs course this semester. I'm stuck in this HW problem. Solve the IVP: $ u_t+uu_x =0, \ \ x \in \mathbb{R}, \ t\ge 0 $ $$ u(x,0) = \left\lbrace \begin{aligned} & 1 &&...
sai saandeep's user avatar
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On the behavior of some first order linear PDEs such as $u_x + x u_t = 0$

I'm currently teaching a PDE course and running through a lot of examples with students to build intuition. I came across an example that has me a bit stumped though. Many first order linear (...
Cameron Williams's user avatar
2 votes
1 answer
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Solve a first order partial differential equation with the boundary condition $u(x,x+x^2)=\sin(x)$ instead of a initial value.

The problem I want to solve is the following one: $$ \begin{cases} u_x + u_t + u = 1,\\ u(x,x+x^2)=\sin(x). \end{cases} $$ I know how to use method of characteristic only if a initial value condition ...
Adam Červenka's user avatar
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PDE solution via characteristics

Solve for $u \equiv u(x,y)$ $$-y u_x+xu_y=0$$ I have to use characteristic method and I have a little problem with the solution proposed. My attempt starts considering auxiliary system $$\frac{dx}{ds}...
Turquoise Tilt's user avatar
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1 answer
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Semilinear equation with characteristics

I have to solve the following equation by the method of characteristics $$\frac{\partial T}{\partial t}+v\frac{\partial T}{\partial z}=\frac{2 U}{RC_F}(T_M-T),$$ where $v,C_F,U,R,T_M$ are known ...
Gonzalo de Ulloa's user avatar
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First-order inhomogeneous PDE in general form

I am solving the initial value problem: $$u_t +u_x = -\sigma(x)u+m(x), \quad u(x,0)=\phi(x), \quad u(0,t)=\gamma(t)$$ I need to get solution like this: \begin{equation*} u(x,t)=\phi(x-t) e^{-\int_{x-t}...
Kyle Crane's user avatar
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Difficulty solving first order linear PDE, characteristics hard to obtain.

The first order linear PDE I am looking at it is. $$(\cos(y) + \cos(x+y))z_x-(\cos(x)+\cos(x+y))z_y=0$$ When I try to solve for the characteristics, I obtain the following ODE's. $$\frac{dx}{dt}=\cos(...
Adam Reddy's user avatar
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1 answer
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Solve this PDE $yu_x-xu_y=2xyu$ (problem with general solution)

Find the general solution of the part differential equation $$yu_x-xu_y=2xyu$$ So $$\frac{dx}{y}=-\frac{dy}{x}=\frac{du}{2xy}$$ Which in turn implies $$\int xdx =- \int ydy \Rightarrow \frac{x^{2}}{2}=...
Gregory99's user avatar
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Fully nonlinear 1st order PDE solved with method of characteristics

I know there are some similar problem posted on here, but I can't find the solution to this particular problem: $$u+\frac{1}{2}(u_x)^2+u_y=0\qquad\text{in }\mathbb{R}\times(-\infty,0)$$ $$u(x,0)=-x^2\...
JackpotWizard 180's user avatar
1 vote
1 answer
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How to show a PDE is locally/globally solvable

I'm having some issues showing a PDE is locally or globally solvable. I'm familiar with the Local Existence Theorem in Evans' book on PDE, but I don't know how to apply it. Here's an example: Use the ...
JackpotWizard 180's user avatar
1 vote
2 answers
160 views

Solve this PDE $xu_x+yu_y+xyu_z=0,u(x,y,0)=x^2+y^2$ (problem with general solution)

Find the particular solution of the partia differential equation $$ xu_x+yu_y+xyu_z=0 $$ that satisfies the Cauchy condition $$ u(x,y,0)=x^{2}+y^{2} $$ So: $$ \frac {dx}{x}=\frac {dy}{y} = \frac{dz}{...
Gregory99's user avatar
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2 answers
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Solve this PDE $u_x+u_y+2u_z=0$,$u(1,y,z)=yz$ (problem with initial conditions/particular solution)

Find the particular solution of the part differential equation $u_x+u_y+2u_z=0$ that satisfies the data $u(1,y,z)=yz$ So: $\frac {dx}{1}=\frac {dy}{1} = \frac{dz}{2}$ $\int dx=\int dy \Rightarrow x=y+...
Gregory99's user avatar
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1 answer
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Finding solution of semilinear PDE using method of characteristics

Using Evans' book, I'm trying to find an explicit solution of the following problem: Consinder the boundary value problem $$x_1^2u_{x_1}(x_1,x_2)-x_2^2u_{x_2}(x_1,x_2)=u^2(x_1,x_2)\qquad (x_1,x_2)\in\...
JackpotWizard 180's user avatar
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What to do when the discriminant of second order linear PDE changes sign?

I am trying to tackle a second order(also of two variables), linear pde with variable coefficients. I don't want to post the problem because I would prefer to solve it myself, but the coefficients are ...
Adam Reddy's user avatar
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Solving the advection equation with an absorbing boundary

Setup: A tool I have used as a tool to learn about stochastic processes has been to solve deterministic problems with stochastic machinery. For example, for a Brownian process with drift $$ \dot{x} = ...
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2 votes
1 answer
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Assigning canonical form to the differential equation using characteristics

The problem statement is to simplify it, i.e. give it a canonical form to this differential equation using the characteristics to achieve that. However, even after many hints and trials on similar ...
Arbatus's user avatar
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Solving this differential equation using method of characteristics

We have the following differential equation: $$u_xx-2u_xy+u_yy+9u_x+9u_y-9u=0$$ The goal is to solve with the method of characteristics, in other words to simplify, to give the canonical form. But I ...
Arbatus's user avatar
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1 answer
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Solve for $\rho$: $\frac{\partial \rho}{\partial t}-u(t,x)\frac{\partial \rho}{\partial x}=\rho\frac{\partial }{\partial x}u(t,x)$.

I'm trying to solve the PDE $$\frac{\partial \rho}{\partial t}-u(t,x)\frac{\partial \rho}{\partial x}=\rho\frac{\partial }{\partial x}u(t,x)$$ for $\rho:[0,1]\times\mathbb{R}\to \mathbb{R}$ for a ...
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PDE $(c_1t+c_2)\frac{\partial u}{\partial t}-\left(c_1x+c_3\right)\frac{\partial u}{\partial x}=c_1u$ by method of characteristics

I'm trying to solve $$(c_1t+c_2)\frac{\partial u}{\partial t}-\left(c_1x+c_3\right)\frac{\partial u}{\partial x}=c_1u$$ for $c_1,c_2,c_3\in\mathbb{R}$. Here is what I am trying (but somthing is off): ...
Gomes93's user avatar
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1 answer
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Method of Characteristics - $X(t;x_0)$

My professor gives us the transport equation with an initial condition and use the method of characteristics to solve. However, when he asks us to find the ODE satisfied by characteristic curves, he ...
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Solving a second-order partial differential equation with non-linear first order terms

I am very new in Stack Exchange. I would like to solve the equation $\phi_{xy}+\phi_y^2-\phi_x^2=0$. I read a summary of the method of characteristics in the book of Zwillinger, and there is ...
Jean's user avatar
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0 answers
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Change of variables from characteristic to initial condition/Derivative of ODE solution with respect to initial condition

I am trying to emulate a change of variables performed in A model of physiologically structured population dynamics with nonlinear individual growth rate (Calsina, Saldaña) in Equation (8). The idea ...
LNass's user avatar
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1 vote
1 answer
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How to get a solution using method of characteristics for a quadratic initial data

I am trying to solve this nonlinear PDE using method of characteristics $$u_t + uu_x = -u$$ $$u(x,0) = u_0(x)$$ We get two ODEs: $du/dt = -u$ and $dx/dt = u$ The solution is $$u(x,t) = u_0(x_0)e^{-t}$$...
Rudinberry's user avatar
1 vote
1 answer
88 views

Solving a linear first-order PDE [closed]

I am looking to solve a linear first-order PDE of the form $$(e^{\beta x} - \alpha x -1 ) f_x(x,t) - f_t(x,t) + \gamma x f(x,t) =0$$ with boundary conditions $f(0,t)=1$ and $f(x,0)=e^{x\lambda}$. I ...
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1 answer
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Quasilinear Transport Equation

Assume $u: \mathbb{R^{2} \times \mathbb{R}_{\geq 0}} \to \mathbb{R}$ satisfy the following PDE $$ \nabla u \cdot \langle 1, y, -x \rangle =u$$ where $\nabla u = \langle \partial_{t} u, \partial_{x} u,...
Matha Mota's user avatar
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1 answer
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Find the solution for given PDE along unit circle

The problem assigned to me is: For the equation $$u_x^2 + u_y^2 = u^2$$ Find the solution when $u|_\Gamma$ = 1, where $\Gamma$ is unit circle at origin. I was taught about method of characteristics ...
user1170874's user avatar
3 votes
0 answers
151 views

Why is this the method of characteristics?

The following question refers to ref. 1, the equation are numbered alike with a slightly different notation. The author claims to solve a renormalization group (RG) equation - so the context is ...
Mr. Feynman's user avatar
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2 answers
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General solution to PDE $xu_x + yu_y = 2xy$, why using characterics in non-parametric form I get an incorrect general solution?

I am trying to solve the PDE \begin{align} xu_x + yu_y = 2xy \end{align} using the method of characteristics. So the characteristic equations are \begin{align} \frac{dx}{x} = \frac{dy}{y} = \frac{du}{...
jaxolotl's user avatar
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1 answer
35 views

How to solve this first order PDE using method of characteristics, cannot find any resources anywhere online. [closed]

The PDE in question is $u_{x}u_{y}=1, u(x,0)=\sqrt{x}$. The process for solving a first order PDE using method of characteristics when the terms are summed is relatively well-established, but I cannot ...
deleted's user avatar
0 votes
1 answer
139 views

Method of characteristic curves

I'm trying to understand the method of characteristic curves, in that purpose I have these two exercises, but i think somwhere i get totaly wrong with this I need to allocate areas of a constant type ...
Maxim Bokov's user avatar
3 votes
0 answers
131 views

Solving for traffic flow density as a function of time and determining when it's no longer valid

So I'm working on the following problem. When given an initial density of: \begin{equation} \rho(x,0)= \begin{cases} \frac{\rho_{max}}{4}, & x<0\\ \rho_{max}, & 0 < x ...
Sprawk48's user avatar
2 votes
1 answer
196 views

Proof of [Lee, Theorem 22.35] The Cauchy Problem for a Hamilton-Jacobi Equation

Relevant Background (for completeness, can be skipped) The statement of Theorem 22.35 in Lee's Introduction to Smooth Manifolds is Suppose $M$ is a smooth manifold, $W \subseteq T^*M$ is an open ...
Tob Ernack's user avatar
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What is the intuition behind the method of characteristics for a second order PDE?

I understand the idea behind the method of characteristics as applied to first-order PDEs: watching how $u(x,y)$ changes along special curves $(x(s),y(s))$ simplifies the problem to a set of coupled ...
Mohit Kumar's user avatar
3 votes
1 answer
96 views

Method of characteristic to solve Sharpe-Lokta model

The conservation law for the population is, $$ \underbrace{\frac{\partial}{\partial t} x(t,a) + \frac{\partial}{\partial a} x(t,a)}_{\text{directional derivative}} = -\mu(a) x(t,a) dt\tag1 $$ where $x(...
N00BMaster's user avatar
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43 views

PDE with initial condition via method of characteristics

Solve $xu_x+(x+2y-2)u_y=x+y, \space x>0, \quad u(1,y)=y$ My attempt: $x'(t)=x(t) \space \Rightarrow \space x(t)=c_1 e^t$ $y'(t)=x(t)+2y(t)-2 \space \Rightarrow \space y'(t)-2y(t)=c_1 e^t-2 \space \...
Li boang's user avatar
2 votes
1 answer
61 views

Characteristics Method PDE - Solution Verification

$$\begin{cases}-x\partial_xu +y\partial_yu =-x^2u \\ u(x,1) = e^{-x} \end{cases}$$ I recently ask a question on PDE and I hope I understand how it works. Let $z(t)=u(x(t),y(t))$ $$\begin{cases}\dot{x}...
Turquoise Tilt's user avatar
0 votes
2 answers
266 views

Charasteristic Method for PDE

Hi i'm struggling a little with solutions of PDE. I have to solve the following $$\begin{cases}\partial_xu +y^2\partial_yu = 2yu+y^2\\ u(0,y)=y \end{cases}$$ I want to use the method of ...
Turquoise Tilt's user avatar
0 votes
1 answer
99 views

Characteristic function on I is uncountable

I studied characteristic function in real analysis in Richard R goldberg book for methods of real analysis and there is an exercise problem asking to prive that characteristic function on I is ...
Lakshmi Priya's user avatar
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32 views

First order differential equation by the method of characteristics

Solve $-yU_x + xU_y= U$, initial conditions $U(x,0)= P(x)$. Characteristic equations: $X_t =-Y$, $Y_t=X$, $U_t=U$. The book says $X(t,s) =F_1(s)\cos t + F_2(s)\sin t$, $Y(t,s)=F_1(s)\sin t -F_2(s)\...
fred's user avatar
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1 vote
0 answers
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On Bishop's 'Tensor Analysis on Manifolds' Problem 4.10.1

Before I can state the actual problem I have to list some preceding definitions from § 4.10 in Bishop's book: Given is the first-order PDE $F(x^1, \dots, x^n, p_1, \dots, p_n, z) = 0$ with $p_i = \...
Alfons Winkel's user avatar
1 vote
0 answers
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Literature on imposing boundary conditions on the method of characteristics solution to PDEs

I am currently studying the method of characteristics. So far I understand what I read in literature. The problem is that I didn't find a good explanation for imposing boundary conditions anywhere. ...
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