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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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Partial Differential Equations Question: Find an explicit expression for the solution of the IVP [on hold]

Find an explicit expression for the solution of the IVP $$ \begin{cases} u_{t}(x,t)+u_{x}(x,t)+u(x,t)=e^{t+2x}\\ \\ u(0,x)=0, \end{cases} $$ by using the method of characteristics
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0answers
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characterestics method for nonhomogenuous wave equation

$$\eqalign{ & {y_{tt}} = {y_{xx}} + f(t,x){\text{ }}{\text{, (t}}{\text{,x)}} \in {\text{(0}}{\text{,}}\infty {\text{)}} \times {\text{(0}}{\text{,1)}} \cr & y(t,0) = y(t,1) = 0 \cr &...
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1answer
29 views

Basic question about a first-order linear equation

I am just learning PDE. My lecture notes say the following: Consider the IVP $$ \begin{cases} u_t + a u_x = 0 \\ u(x,0) = \phi(x) \end{cases} $$ where $a \in \mathbb{R}$. Our goal is to reduce this ...
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1answer
62 views

Finding the time when the speed of discontinuity becomes time-dependent in traffic flow

I am trying to use the following conservation law: $$u_t+f(u)_x=0 \ \ \ \ \text{where} \ \ \ f(u)=u(1-u).$$ IC: $u(x,0)=\frac{1}{4}$ for BC: $u(0,t)=1$ for $t>0$. I found the solution ...
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1answer
21 views

Solve $u_x + 4xu_y = 1 + u^2$ for $u(0,y)=y$

I got a weird result so I'm not sure I did this right Let the initial condition be $u(0, y_0) = y_0 $ for some $y_0$ By the method of characteristics let $$\frac{dx}{ds} = 1 \to x = s + A$$ $$x(s=0)...
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Are both answers for $xu_x + yu_y = 0$ valid?

Solving this problem by the method of characteristic curves we have to solve the ODE $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y}{x}$$ which gives us $$C = \ln(y/x)$$ where $C$ is constant. ...
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1answer
28 views

Characteristic coordinates $ξ(x, y)$ and $η(x, y)$ for $xu_{xx} + u_{yy} = 0$ when $x<0$

How would I determine the characteristic coordinates for $xu_{xx} + u_{yy} = 0$? This PDE reads $au_{xx} + 2b u_{xy} + cu_{yy} = 0$ with $a=x, b=0, c=1$. The polynomial equation $a\lambda^2 -2b\...
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2answers
39 views

Quasilinear PDE using method of characteristics

The equation is: $yu_x+uu_y=-xy$ with initial conditions $u=y$ on $x=0$ I first find that $\frac{dx}{y}=\frac{dy}{u}=-\frac{du}{xy}$ Solving $\frac{dx}{y}=\frac{dy}{u}$ we get, $ux=\frac{1}{2}y^2+...
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0answers
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Exact solution for two coupled non-homogeneous transport equations

I want to solve the following system $$\eqalign{ & {y_t} = -{y_x} + z{\text{ in (0}}{\text{,T)}} \times {\text{(0}}{\text{,1)}} \cr & {z_t} = {z_x} + y{\text{ in (0}}{\text{,T)}} \times {...
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1answer
33 views

Behavior of the solution of the eikonal equation

Consider the nonlinear first-order initial-value problem: $$(u_t )^2 + (u_x )^2 = 1$$ with initial condition $u(x, 0) = {−\sqrt{1+x^2}}$. Find its solution for all $t>0$ using the method of ...
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1answer
54 views

Characteristic curves for second-order Tricomi equation

Consider the Tricomi equation $$yu_{xx} + u_{yy} = 0$$ Find ordinary differential equations describing the real characteristic curves and solve these ODEs to obtain equations for the ...
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1answer
72 views

Solve IVP of $(u_t )^2 + (u_x )^2 − u^2 = 0$ using method of characteristics

Consider the nonlinear first-order initial-value problem: $$(u_t )^2 + (u_x )^2 − u^2 = 0$$ with initial condition $u(x, 0) = Ae^{−\sqrt{1+x^2}}$. (a) Find its solution for all $t > 0$ ...
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2answers
74 views

Shock formation condition in IVP of $u_t + uu_x + \alpha u = 0$

Consider $u_t + uu_x + \alpha u = 0$ for $t > 0$, all $x$ where $\alpha > 0$ is a constant. Find the characteristic equations for the equation with initial data $u(x, 0) = f(x)$ given. Show ...
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2answers
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Method of characteristics for $f f_x + f_y = 1$. Where is the solution valid?

Suppose we have a PDE that can be solved with the method of characteristics \begin{align} F(\nabla u, u , x) = 0 \text{ in $U$}\\ u|_\Gamma = g \text{ on $\Gamma$ } \end{align} Where $\Gamma \...
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2answers
45 views

Characteristic Curve of a PDE

Question from a Exam: Consider the pde $xu_{xx}+2x^2u_{xy}=u_x-1$. Find the characteristic curves of the above. Can someone please tell me how are these types of problems handled? I dont want ...
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0answers
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existence of solutions for Cauchy problems

Consider the equation $$(1-\cos x)u_{tt} - u_{tx} - u_{xx} = 0$$ with Cauchy data $$u(x,0) = f(x), u_t(x,0) = g(x),\text{ for } f,g\in\mathcal{C}^2$$ What compatibility condition do $f$ and $g$ have ...
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1answer
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charpits method to solve $u_x^2 + yu_y = u$

$u_x^2 + yu_y = u$ subject to $u(x,1) = 1 +x^2 /4$ for $-\infty < x < \infty $ Setting $p = du/dx$ and $q = du/dy$, I get $p^2 +yq = u$ and so I am able to write the diffeq as $F(p,q,y,u) = p^2 ...
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2answers
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Domain of definition for $u_x + uu_y = 1$

How do i find the domain of definition for $u_x + uu_y = 1$ with $u = x/2$ on $y=x$ , $0 \leq x \leq 1$ I parametrise by letting $y=s$ , $x=s$ , $u=s/2$ , $0 \leq s \leq 1$ at $t=0$ The ...
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1answer
44 views

Nonhomogeneous Semi-Linear PDE with the Characteristic Method

I need to solve this semi-linear PDE: $u_x - 3u_y = \sin (y) + \cos (x)$ The initial condition provided is: $ u (t,t)= t^2$ I need to use the Characteristic Method. I learned the method from this ...
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1answer
41 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 ($u=\frac{1}{1+x^2e^{2t}}e^{2t})$ ...
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0answers
37 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 ...
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1answer
889 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 ...
3
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1answer
50 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 ...
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1answer
54 views

Inhomogeneous linear transport equation

Let $(a,b)$ be a subintervall of $(0,1)$. We consider the nonhomogeneous transport equation $$\eqalign{ & {y_t}(t,x) + c{y_x}(t,x) = {1_{(a,b)}}(x)f(t){\text{ }}{\text{, }}\left( {{\text{t}}{\...
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1answer
26 views

Solving a PDE for a function of 3 variables

I'm looking to solve the PDE below for a function $f(p,R,t)$. $\frac{\delta f(p,R,t)}{\delta t}-D(R,t)\,p\,\frac{\delta f(p,R,t)}{\delta p}=0 $ I'm aware that the method of characteristics is ...
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1answer
56 views

Long time evolution of Burgers' equation ($t\to\infty$)

Problem Draw the characteristics and describe the evolution for $t \to \infty $ of the solution of the problem $$ \begin{align}\begin{cases}u_{t} + u u_{x} = 0 & t > 0 , x \in \mathbb{R} \...
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1answer
58 views

Inviscid Burgers equation

Burgers Equation Consider the initial value problem for Burger's equation $$ \begin{align}\begin{cases} u_{t} + u u_{x} = 0 \\ u(x,0) = \phi(x) \end{cases} \end{align} \tag{1}$$ our ...
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0answers
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Compute the Characteristics of a PDE

Let $\displaystyle u_x^2+u_y^2=n_0^2$ be given, with the initial condition that $u(x,2x)=1$ and $n_0\in\mathbb{R}$ I want to find a solution using the methods of characteristics. I computed the ...
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1answer
37 views

Solving a PDE using Characteristics

Let $\displaystyle (x,y)\cdot Du = au,\ a>0$. I computed the characteristics to be $\begin{align} \dot{\vec{x}}(s) &= \vec{x}(s)\\ \dot{z}(s)&=\vec{x}(s)\cdot\vec{p}(s)=az(s) \end{align}...
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1answer
27 views

Quasilinear 2nd order PDE, apply initial data to general solution

The question involves finding the solution to a partial differential equation. The general solution that I found was $u(x,t)=F(x^{2}-t^2 e^{u})$ and the initial condition is $u(x,0)=2\ln (x)$. The ...
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1answer
125 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)= ...
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3answers
50 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 ...
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2answers
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How to Find Functions with Given Condition

I am given that $\frac{\partial F}{\partial x}(x) + \frac{\partial F}{\partial y}(-y) = 0$ and that $F(x,-1) = x^2$. I am supposed to find $2$ $F(x,y)$'s that satisfy these conditions. I have found ...
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1answer
27 views

Explanation for two points in the paper on first-order PDE

I have a question about the following paper: https://web.stanford.edu/class/math220a/handouts/firstorder.pdf I have two questions about two points that I thought I understood, but I'm not sure now. ...
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1answer
31 views

Trouble with solution for transport equation.

I have a question about the following paper: https://web.stanford.edu/class/math220a/handouts/firstorder.pdf My question is on the end of the second page, when the author solves the transport ...
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Pde First order characteristic lines Proof about $yu_x+xu_y=u^2$

I need to understand the proof of the following statement: Let $$yu_x+xu_y=u^2$$ if $u(x,y)$ is a function that's $C^1(\Bbb R)$ and the graph of $u(x,y)$ is on $\Bbb R^3 $is a union of the ...
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2answers
53 views

PDE from first Order $yu_x+xu_y=u^2$ Solving with characteristic lines

I need help with the following question: The given equation is: $$yu_x+xu_y=u^2$$ Show that the characteristic lines of the equation are: $$\begin{cases} x(t)=C_1e^t+C_2e^{-t}&\\ y(t)=C_1e^t-...
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1answer
54 views

Method of characteristic: why a characteristic curve cannot intersect the initial curve in two points

Let consider a general linear first order PDE in $xy$ space, ie $$a(x,y)u_x+b(x,y)u_y=c(x,y)u+d(x,y),\qquad (x,y)\in U$$ where $U$ is an open and connected subset of $\mathbb{R}^2$ and $a,b,c,d \in C^...
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0answers
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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},\;\;...
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1answer
83 views

Partial Differential Equation, help with initial conditions

Show that the initial value problem $u_t+u_x=0$, with $u=x$ on $x^2+t^2=1$ has no solution. However, if the initial data are given only over the semicircle that lies in the half-plane $x+t\le0$, the ...
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1answer
156 views

Charpit's Method with a condition for parameter

Solve $$\ p^2-3q^2-u=0 $$ given $\ u(x,0)=x^2 $ ,using Charpit's method to find complete solution. I have tried to do the following, since the given equation is of the form $\ f(p, q, u)=0$ as per ...
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1answer
36 views

Solution to system of first order wave PDEs

I am solving the following system of equations (the Streeter-Phelps equation models pollution of a river) by the method of characteristics. $$b_t + v b_x = -k_1 b,$$ $$D_t + v D_x = k_1 b - k_2 D,$$ $...
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1answer
29 views

How can I find the second constant to solve this PDE?

I'm trying to solve the following first order PDE: $$3u_x+y^2u_y=\frac{x}{y}u$$ Applying the method of characteristics we have the following: $$\frac{dx}{3}=\frac{dy}{y^2}=\frac{ydu}{xu}$$ It can ...
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2answers
72 views

Where is the mistake in solving this PDE?

Given the initial condition $u(x,1)=x+1$ I'm asked to find the solution of the following PDE : $$(y+u)u_x+yu_y=x-y$$ Using the method of characteristics we know that the characteristic equations ...
2
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1answer
53 views

How do I explain my steps in solving this PDE? Characteristics

I'm taking an exams in which I'll be asked to solve a first order PDE. In solving it I will use the method of characteristics, which I know how to use but I can't explain my steps properly. For ...
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1answer
60 views

Solving the PDE $\frac{\partial u}{\partial t}+f(x)\frac{\partial u}{\partial x}+u=0$ subject to $u(x,0)=e^{-b x}$.

I am hoping to obtain a closed-form solution to the PDE $$\frac{\partial u}{\partial t}+f(x)\frac{\partial u}{\partial x}+u=0$$ subject to the initial condition $u(x,0)=e^{-b x}$. If this is not ...
3
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1answer
77 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 $s$ ...
2
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2answers
52 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 $,...
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1answer
35 views

Application of Method of Characteristics

I want to solve the following transport equation with the method of characteristics. $$u_t - yu_x + xu_y = 0$$ $$u(0, x, y) = x + y$$ I read several questions here and found various sources for ...
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0answers
20 views

A first order hyperbolic problem using method of characteristics

Consider the initial value problem for the equation $u_t+au_x=f(t,x)$ with $u(0,x)=1$ and $f(t,x)=1$ if $x\ge 0$ and $0$ otherwise. Assume that $a$ is positive. Show that the solution is given by $u(t,...