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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Solving a PDE with Method of Characteristics

I am currently trying to solve a PDE but am having difficulties. The PDE is: $$(y+u)u_x+yu_y=x-y, \>\>\>\>u(x,1)=1+x$$ Now I found the Characteristic Equations: $$\dot x(s) = y+z, \>\&...
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Having trouble solving a PDE with method of characteristics

I am currently trying to solve a problem I have already solved, but am trying to solve it the way our professor solved it. The PDE is given by: $$yu_x-2xyu_y=2xu, \>\>\>\>\>\>\>...
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Method of Characteristics(Advection equation with initial and boundary condition)

Solve, using the Method of Characteristics, the equation $\frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x}=-\mu\rho$ for $x,t>0$ with the conditions $\rho(x,0)=f(x)$ and $\rho(0,...
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Using method of characteristics to find general solution of PDE $x^3 u_x + y u_x = 4 + 2x^2u$

Use the method of characteristics to find the general solution $u(x; y)$ of the partial differential equation $$ x^3 \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial x} = 4 + 2 x^2 u $$
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Solving the partial differential equation using the method of characteristic to find the general solution

The question is solve the following partial differential equation using the method of characteristic $\frac{\partial{u}}{\partial{x}}-\frac{\partial{u}}{\partial{t}}=2t$, with the condition $u=u(x,t)...
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271 views

Solving 1st order PDE with three variables using method of characteristics

I am trying to use the method of characteristics to solve the following first order PDE in three variables: $$u_x + x \,y \,u_y + 2 x^2 \,z\,\ln z\,u_z = 0 $$ I have begun with the following: $ \...
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PDE by method of characteristics

How do I solve this PDE by method of characteristics? $$2y{\partial z \over \partial x} - {\partial z \over \partial y} = x$$
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1st Order Nonlinear PDE: Understanding Envelopes and Monge Cones

I have a question about envelopes of surfaces. In a book I am reading the following: Suppose $S_a$ is a one parameter family of surfaces in $R^3$ given by $z=w(x,y;a)$ where $w$ depends smoothly on ...
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Problem with finding shock location via the separable ODE $\dot s=(s+1)/2t$.

I want to solve the following separable ODE to find an expression for the shock location $x=s(t)$. The initial condition I prescribe to this problem will be that $s(2)=3$. $$\frac{ds}{dt}=\frac{s+1}{...
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Why is the family of characteristics for a second order linear/semilinear parabolic PDE given by this equation?

Consider a PDE of the form $$ a(x,y) \frac{\partial^2 u}{\partial x^2} + 2b(x,y) \frac{\partial^2 u}{\partial x \partial y} + c(x,y) \frac{\partial^2 u}{\partial y} + d(x,y) \frac{\partial u}{\partial ...
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Finding the Riemann Invariants of a system of 2 PDEs

I've been asked to find the Riemann Invariants for the system: $$ \begin{pmatrix} \cos(v) & 0 \\ 0 & \cos(v) \end{pmatrix} \begin{pmatrix} u_x \\ v_y \end{pmatrix} + \begin{pmatrix} \sin(...
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DG and concept of characteristics

I want to solve the Euler equations in 1D numerically. I am currently using DG-Lax Friedrich flux splitting technique and RK-4 explicit time stepping scheme. Now,using the idea of characteristics ...
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Characteristic method for quasi-linear PDE

I'm trying to solve the Cauchy problem: $u_{x}+xu_{y}=0, x \in\mathbb{R}$ $u(x,-1)=exp(x)$ Parametrization of the surface: $(x,y,z)=(s,-1, e^{s})$ I got to these projected characteristics: $x=\tau+...
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Question about Different Types of First-Order PDEs

I am currently working my way through a question, and wanted some verification for my work. The question reads: In each of the following, $u(x,t)$ is a smooth solution to the PDE for $x\in\mathbb R$...
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238 views

cauchy problem pde

$$(6u+2y)U_x +(3x-6u)U_y+3x+2y=0 , x>0 , y>0 $$ $$U(x,0)=x$$ question: Find the solution for the initial value problem. $\Gamma=(r,0,r)$ I parameterized with r $$\frac{dx(r,s)}{ds}=6u+2y\...
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115 views

Solve the First Order PDE equation

Problem: For equation: $$u=xu_x+yu_y+\frac{1}{2}(u_x^2+u_y^2) $$ find the solution with $$u(x,0)=\frac{1}{2}(1-x^2)$$ Here is what I have so far: Let $$F(x,y,z,p,q)=z-xp-yq-\frac{1}{2}(p^2+...
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Solution of the PDE which has Cauchy Data.

In the case of the quasilinear PDE $$ (y-z) \frac{\partial z}{\partial x} + (z-x) \frac{\partial z}{\partial y} = x-y \, , $$ the method of characteristics yields the following Lagrange-Charpit ...
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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 ...
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Conditions added to a first order PDE

Given this PDE $$xu_x + yu_y + zu_z=0$$ It is asked, in some old paper-and-pencil notes I am reading, to solve it. 1) Firstly, finding the general solution 2) Then, finding the particular solution ...
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Finding Characteristics of second order Partial differential Equation

$y^2r-x^2t=0$ where $r=d^2z/dx^2$ $t=d^2z/dy^2$ I know a way to find characteristics in single order equation. But not in second order. How do we proceed here ?
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Solve partial differential equation using method of characteristics.

Given: $zp + yq = x$ $x_{0} = s$ (x at time t = 0) $y_{0} = 1$ (y at time t = 0) $z_{0} = 2s$ (z at time t = 0) Approach : $ f=zp+yq-x$ Using Characteristic Equations : $x'(t) = f_{p}$ $...
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560 views

Sketching Characteristics of a PDE

Hello I have a question about sketching characteristics, I have 2 Burgers equations that are as follows: $$u_t+u\cdot u_x=0, \>\> u(x,0)=x^3$$ $$u_t+uu_x=0\>\> u(x,0)= \begin{cases} 1 &...
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PDE with 3 variables, solved. Am I right?

I solved an exercise and I need to confront some other point of view. Given this equation $$u_x + u_y + u_z=0$$ I solved by using characteristics, obtaining $$\frac{dx}{1}=\frac{dy}{1}=\frac{dz}{1}$...
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Method of Characteristics: Relationship of constants

Let us look at this simple $(1+1)$-dimensional transport equation: $$\partial_t u+a\partial_x u = 0.$$ When solving this by the method of characteristics we obtain the following equation: $$\frac{...
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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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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 ...
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method of characteristics: finding ode; evans

i have a question regarding the method of characteristics for first order nonlinear pde, as done by evans (§3.2.3.b) a fact that i cannot illuminate myself turns up discussing compatibility ...
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How to solve the following Cauchy problem.

Solve the following Cauchy problem for a first order PDE: $$(2x_1 + x_2)u_{x_1} + (x_2 + 1)u_{x_2} = u^2, \ \ u(x_1, 1) = x_1^2 + 1, \ \ x_1 \ge 0, x_2 \ge 1$$ and find an implicit conldition over $...
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How to solve method of characteristics problem?

Solve the following Cauchy problems for a first order PDE: $$(x_2 + x_2^5)v_{x_1} + x_1 v_{x_2} = 0, v(x_1, 0)$$ $$v_0(x_1, 0) = v_0(s)$$ Attempt at the solution: The characteristic equations are: $...
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Find general solution of the problem [closed]

Find general solution to partial differential equation $$ yz_x+xz_y = (x + y) z ^2 $$ dx/dt = y. (1) dy/dt = x. (2) dz/dt = (x + y) z^2
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Solving a system of semilinear coupled partial differential equations

I would like to know if it is possible to find the solution to the following system of partial differential equations: \begin{equation} \partial_x \mathbf{u}+B(x,y) \cdot \partial_y \mathbf{u}-\mathbf{...
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Characteristics and solution of the differential equation $u_t+x^2u_x=u$ by direct substitution

The exercise is as follows: Find and show in the plane (x, y) characteristics of the following equation: $$u_t+x^2u_x=u$$ Get the general solution and check for direct replacement. I was able to ...
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Charpit's method: Check answer

I have the following question: A complete integral of $f=xpq+yp^2-1=0$ obtained by Charpit's method is: $(a) \ (z+b)^2=4(ax-y)$ $(b) \ (z+b)^2=4(ax+y)$ $(c) \ (z-b)^2=4(ax+y)$ $...
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Method of Characteristics Boundary Condition

So this is a pretty simple question that I just need a bit of clarification on where I am messing up: $$\frac{∂w}{∂t} + 4 \frac{∂w}{∂x} = 0,~~~ w(0,t) = \sin(3t)$$ My method so far: $$\frac{dx}{...
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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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Show that no new maxima or minima can develop in smooth solutions to the conservation equation

I would like to show that no new maxima or minima can develop in smooth solutions to $ \frac{\partial \phi(x, t)}{\partial t} + \frac{\partial }{\partial x}(f(\phi(x, t)) = 0 $ My solution: Since ...
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How do you know when there is a shock

I'm not understanding when a shock exists. I know it has something to do with characteristics intersecting but other than that I'm not sure. For example, for the PDE, $$\frac{\partial\phi}{\partial t}...
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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\{ \...
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Traffic Flow problem space time diagram question

Traffic with constant density, $\rho_0$, is stopped by a red light at $x=0$. If $v(\rho)=1-\rho$, calculate what happends behind the light. Unfortunately the main focus of this example in my notes ...
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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}{...
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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 $$\...
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Solving a first order non linear PDE with the method of characteristics

We have to find the function $u(x,y)$ for the following system: $u_xu_y = xy$ $u(x,y) = y+1$ for $x=y$ Using the method of characterstics I get: $F(x,y,u,u_x,u_y) = u_xu_y-xy = 0$ Defining $p = ...
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127 views

Shockwave trajectory

A traffic flow governed by the equations below is observed at $t = 0$ to have a dip in the density distribution at $x = 0$ given by: $$ \rho (x,t=0) = \left\{\begin{aligned} &a &&: x \lt ...
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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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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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Solving $\frac{\partial \rho}{\partial t}-x u \frac{\partial \rho}{\partial x}=0$ using the method of characteristics

I have some questions about solving a partial differential equation using the method of characteristics. The PDE is: $\frac{\partial \rho}{\partial t}-x u \frac{\partial \rho}{\partial x}=0$ where $...
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Solve inhomogenous Wave Equation with Method of Characteristics

Solve, using the methods of characteristics: $$\begin{cases}u_{tt}(x,t)=u_{xx}(x,t)\\ IC: u(x,0)=u_t(x,0)=0\\ BC: u(0,t)=h(t),\ u(L,t)=0\end{cases}$$ I tried to use the substitution $u(x,t)=v(x,t)+...
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72 views

Are there any use of imaginary characteristics?

For hyperbolic equations, characteristics tell how a signal propagates. For example, the equation $u_{tt} - c^2 u_{xx} = 0$ has two characteristics $x\pm c t$. This means a signal propagates at speeds ...
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Basic Traffic-Flow problem (How to physically interpret the results)

Let $u = u(x,t)$ be the density of cars at $x$ (ie. cars per unit length at $x$). Let also, $f(u) = u^2$ be the flux of cars at any point $x$, with some initial conditions (described below). In ...
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How to solve $uu_{x_1}+u_{x_2}=1$ with characteristic method?

$$uu_{x_1}+u_{x_2}=1\text{ with initial condition }u(x_1,x_1)=\frac{1}{2}x_1$$ I have problem to use following characteristic method to solve it. Let $$x(s) = (x_1(s), x_2(s))$$$$z(s) = u(x(s))$$$$p(...