Mathematics Stack Exchange
Mathematics Stack Exchange

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.

The method of characteristics is a way of solving certain partial differential equations (PDEs) by reducing them to ordinary differential equations (ODEs). Also, the method can be seen as a change of variables to simplify the PDE. It is most often used for 1st order linear or nonlinear PDEs. Use with the tag.

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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 characteristics. However, there is a section about…
Time4Tea
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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 $$x(t)\,\, ; \,\, x'(t)=f'(u(x(t),t))\,\, ; x(0)=x_0$$ Are they always straight lines…
Quiet_waters
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How can I solve $u_{xt} + uu_{xx} + \frac{1}{2}u_x^2 = 0$ with the method of characteristics.

I am trying to solve the following PDE: $u_{xt} + uu_{xx} = -\frac{1}{2}u_x^2$, with initial condition: $u(x,0) = u_0(x) \in C^{\infty}$ using the method of characteristics. I am a beginner with the method of characteristics and PDE in general. Here…
Richard
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How to solve a system of PDE $u_t+u_x=v, v_t+v_x=-u$

Solve the following initial value problem: $$u_t+u_x=v, \\v_t+v_x=-u, \\u(0,x)=u_0(x), \\ v(0,x)=v_0(x).$$ I did not learn any method to solve a system of PDE so I guess there is a "trick". So far we've learned how to solve PDE's of the…
catch22
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Whilst There Are Three Characteristic Equations, Only Two of Them Are Linearly Independent?

Take the general quasi-linear equation $$a(x, y, u)u_x + b(x, y, u)u_y - c(x, y, u) = 0. \tag{1}$$ We assume that there exists a solution of the form $u = u(x, y)$. We can define a solution surface in $(x, y, u)$ space via the implicit form of the…
user423167
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Crossing characteristics?

I am slightly confused about the crossing/noncrossing properties of characteristics for first-order PDEs. Let's start with the case of linear homogeneous first-order PDE, of the general form $$ F(Du,u,x) = {\bf b}(x) \cdot Du(x) + c(x)u(x) =…
user91126
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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}{\partial x}(x(t),t)+\rho(x(t),t)g(x(t),t) \qquad…
Andrew Murdza
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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)= \left\lbrace \begin{aligned} &e^{x}-1 &…
Infinite_28
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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 ${\mathbf{u}}(x,t) \in {\mathbb{R}^n},\;\;x \in \mathbb{R},\;\;t \in [0,\infty ),\;$ and…
displayname
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Why is the solution single-valued?

I have shown that a smooth solution of the problem $u_t+uu_x=0$ with $u(x,0)=\cos{(\pi x)}$ must satisfy the equation $u=\cos{[\pi (x-ut)]}$. Now I want to show that $u$ ceases to exist (as a single-valued continuous function) when…
Evinda
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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 Mathematica gives a different solution. Comments on…
Danilo Gregorin Afonso
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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: $v = 1 − ρ$, where ρ is the number of cars per unit…
user429679
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Method of Characteristics for the Equation $\;\left(x+\alpha y\right)u_{xx} + u_{yy} = 0\:$

Below is Exercise $3.6$ (p.$91$) of "Applied Partial Differential Equations" by Ockendon et al., $2^\mathrm{nd}$ ed.: Show that, if $$\big(x+\alpha y\big)\dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial^2 u}{\partial y^2} =…
Vlad
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PDE: Solving using the Method of characteristics

I am trying to solve this PDE using Method of characteristics: $$(u+e^x)u_x+(u+e^y)u_y=u^2-e^{x+y}$$ I don't know how the next equation is called in English, but it is used to solve the…
PonchoALV98
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How do we solve Burgers' equation using the method of characteristics?

I want to solve \begin{align}&\forall(t,x)\in(0,\infty)\times\mathbb R:\left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}\right)(t,x)=0;\tag1\\&\forall x\in\mathbb R:u(0,x)=u_0(x)\tag2,\end{align} where…
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