# 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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### 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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### 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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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: $$\frac{\partial u}{\partial y} - \alpha(x, y) \frac{\partial ... 1answer 38 views ### IVP for linear first-order PDE 3u_x + 4u_y + 5u_z =0 I solved an old problem (I don't remember if I have already posted this problem: forgive me, if so)$$ \begin{cases} 3u_x + 4u_y + 5u_z =0\\ u(1,y,0)=2y-6 \end{cases} $$I quite easily obtained the ... 1answer 61 views ### Characteristic curves PDE uu_x+u_y=u with u(x,0)=-x I have the following PDE:$$uu_x+u_y=u\qquad , y>0u(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, ...
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 ...
### 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{...