0
votes
0answers
9 views

Wave Characteristic diagram question

Say $F(x)=v(x)=0$ when $|x| \geq a$ whilst $F(x)=h(x)$ and $v(x)=c \ h'(x)$, both when $|x|<a$. As we know, the wave equation in D'Alembert's form is: $$ ...
0
votes
1answer
30 views

Show the equality holds for any $x \in [0, \pi]$

We are considering a $2\pi$ periodic function defined on $x\in(-\pi,\pi)$ by $$f(x) = \pi - x, 0<x<\pi $$ and 0 otherwise. I already computed the full Fourier series is equal to: $$f(x) = ...
1
vote
0answers
28 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
1
vote
2answers
56 views

Question about separation of variables

This is for the heat equation, where $$\frac{\partial U}{\partial t}-k \frac{\partial^2 U}{\partial x^2}=1$$ with the conditions $$U(0,t)=0, \; U(x,0)=0 \text{ and } \frac{\partial U}{\partial t} ...
1
vote
1answer
19 views

Steps in the solution of Korteweg-deVries PDE

In the following solution of the Korteweg-deVries PDE $$ u_t + 6uu_x + u_{xxx} = 0 \qquad (3.1) $$ I do not understand the second integration step and how they arrive at the expression for the ...
0
votes
0answers
30 views

Show $u(x,t)$ is analytic in time

$$u_t + u_x + u u_x - u_{xxt} = 0$$ {know: $u$ can be differentiated $\infty$ times with respect to $t$. this fact may or may not be helpful in the proof} how would one approach such problem? i ...
0
votes
0answers
9 views

what does well posdeness results tells us concerning non linear evolution equations?

Consider a nonlinear Shr\"odinger equation, $$iu_{t}+\bigtriangleup u + f(u)= 0, u(0)= u_{0}$$ where $u(t, x)$ is complex valued function of $(t,x) \in \mathbb R \times \mathbb R^{n}$, $i=\sqrt{-1}, ...
0
votes
0answers
29 views

PDE Heat Equation Question: Finding T(x,t) with limited information.

Say our equation for temperature at position x and time t is shown by: $$ T(x)=T_0(1-x/a) $$ This equation holds for a rod of length a from x=0 to x=a. Initially T(0,t)=$T_0$ and T(a,t)=0. ...
0
votes
1answer
44 views

Heat Equation Steady state question

Say you have a slab of material occupying the region $0\leq x\leq a$. Heat is supplied at a constant unit rate so the temperature $T(x,t)$ satisfies $$\frac{\partial T}{\partial t}= k ...
0
votes
0answers
30 views

Solving $u_{yy} + (2-x)u_y - 2xu = 1$

I want to solve the pde $$ u_{yy} + (2-x)u_y - 2xu = 1 $$ so if I treat $x$ in the coefficients as arbitrary but fixed it is equivalent to solving the ode $$ y'' + (2-x) y' - 2x y = 1. $$ For the ...
0
votes
1answer
12 views

Solving a PDE: basic first-order hyperbolic equation $u_t+cu_x=0$

So I have to solve the first-order hyperbolic equation $u_t+cu_x=0$ and $c$ as a constant. It is a PDE, since there is the time and spatial variable, but I'm overwhelmed by the maths given in books of ...
0
votes
1answer
14 views

Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
3
votes
1answer
44 views

How to solve Cauchy problem?

I'm new to this problem. Here is the question. $$(y+xz)z_x+(x+yz)z_y=z^2-1$$ Find the integral surface which the curves it passes are $y=1$ and $z=x^2$ By Lagrange system i found $u$ and $v$. We ...
0
votes
0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
0
votes
0answers
21 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
0
votes
0answers
10 views

Determining the expressions of the coefficients of the full Fourier series from the Complex Series

Let $l \gt 0$ be a positive real number, and $\phi:[-l,l]\rightarrow\mathbf{R}$ be the function defined by: $$\forall x\epsilon[-l,l], \phi(x)=e^x $$ (1) Calculate the coefficients of the Full ...
1
vote
1answer
42 views

Combining two partial differential equations into one

I have the equation $$ \frac {\partial v}{\partial t}= \gamma \left(1-\frac{v}{v_0}\right)+\alpha \left(1-\frac{v}{v_0}\right)\rho-\beta (\rho-\rho_0) $$ and the mass conservation equation $$ ...
0
votes
1answer
40 views

Non-linear Partial Differential Equations

So I know how to solve linear partial differential equations but I am stuck on this new type of problem that is a nonlinear pde. The question is: Determine the solution of $\frac{\partial ...
1
vote
1answer
27 views

Partial Differential Equations

This is actually a very easy problem but I am brand new to this subject and I just don't the mechanism on how to do it yet. The question is: Determine the solution of $\frac{\partial \rho}{\partial ...
0
votes
0answers
21 views

Advanced numerical solution of differential equations

Show that the explicit Runge-Kutta scheme \begin{equation} \frac {y_{n+1} -y_{n}}{h}= \frac{1}{2} [f(t,y_{n} + f(t+h, y_{n}+hk_{1})] \end{equation} where $k_{1} = f(t,y_{n})$ applied to the equation ...
0
votes
1answer
14 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
1
vote
0answers
23 views

Help with semilinear PDE problem

I need some help: Let $T>1$, $\Omega_T=\{(x,y)\in\mathbb{R}\>|\>x\in(1,T)\}$, $\Gamma=\{(x,y)\in\mathbb{R}\>|\>x=1\}$ and $g\in C^1(\mathbb{R})$. Prove that there exists $T>1$ such ...
1
vote
1answer
25 views

Extension of a function from the edge.

How can I extend the function $f\in W^{1-\frac{1}{p},p}(\mathbb R\times\lbrace0\rbrace)$ up to $g\in W^{1,p}(\mathbb R\times(0,\infty))$?
2
votes
2answers
55 views

Nonlinear PDE from Riemannian Geometry

I am wondering if anyone knows an approach to finding solutions to the following PDE: $-e^{-2u}\Delta u=\alpha$. Here $u=u(x,y)$ is an unknown real-valued function of 2 variables and $\alpha$ is a ...
2
votes
1answer
37 views

:How to find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$?

question : find the general solution of $(y+ux)u_x+(x+yu)u_y=u^2-1$ $\frac{dx}{dt}=y+ux,\frac{dy}{dt}=x+yu,\frac{du}{dt}=u^2-1$ I dont know how to start. is this quasilinear ? edit 1: tried ...
1
vote
0answers
24 views

Easy ode question.

I forget ODE. Please solve this. Thanks. $M=M(u,v))$ $M_{uu}+M=0$ I guess I need to take $D=\frac{d M}{d u}$ Then I need to write $D^2+1=0$ But I cannot remember properly.
1
vote
1answer
60 views

How to find the solution of $u_x-u_y=u^2$

I want to find the solution $u(x,y)$ of the PDE $$ u_x-u_y=u^2 \quad \text{where} \quad u_x = \frac{\partial u}{\partial x}\,,~u_y = \frac{\partial u}{\partial u} \, $$ The boundary condition at ...
1
vote
1answer
27 views

Traffic flow vs Density

This is a pretty simple question but I can't seem to understand it conceptually. The question is: If the traffic flow is increasing as $x$ increases ($\frac{\partial q}{\partial x}>0$), explain ...
0
votes
1answer
45 views

How to find the solution of $4u_x+8u_y-u=1$?

find the solution of the cauchy equation below $4u_x+8u_y-u=1$ $u(x,3x)=cos(x)$ my attempt: char. eq. is $\dfrac{dy}{dx}=\dfrac{8}{4}=2$ $dy=2~dx$, $y=2x+k$ we can take $\xi=x$ and ...
0
votes
1answer
44 views

Another diffusion partial differential equation, or Sturm-Liouville eigenvalue ODE

What is the solution for the following diffusion partial differential equation (initial value problem)? $$\frac{\partial f}{\partial t} = \pm\frac{\partial f}{\partial x}+(ax+b)^2\frac{\partial^2 ...
1
vote
2answers
58 views

Show all the harmonic functions over $\mathbb{R}^N\setminus\{0\}$ such that $u(x)=f(|x|)$.

This is an exercise of my course of PDE: Show all the harmonic functions over $\mathbb{R}^N\setminus\{0\}$ such that $u(x)=f(|x|)$. My Attempt A function $g$ is called harmonic if $\Delta ...
0
votes
1answer
23 views

Determining value of infinite sum after computing full Fourier Series

I have computed the Full Fourier Series of the function $\phi:[-\pi,\pi] \rightarrow \Bbb{R}$ defined by $\forall x \epsilon[-\pi,\pi], \phi(x)=|\sin(x)|$ to be: $$ \phi(x) = {2\over\pi}+{1\over\pi} ...
1
vote
1answer
53 views

Solving Wave Equations with different Boundary Conditions

Right now I'm studying the wave equation and how to solve it with different boundary conditions (i.e. $u(x,0);u(0,t);u_t(x,0);u_x(x,0);u(x,x);u_t(x,x)...$) I know how to solve it when the boundary ...
4
votes
2answers
98 views

Uniqueness for tempered distributional Cauchy problems

Question. Assume that $U\in C^1(\,[0, \infty)\to \mathcal{S}'(\mathbb{R}^n)\,)$ is a solution to the following tempered distributional Cauchy problem $$\tag{CP}\begin{cases} \frac{ d U}{dt} = f ...
2
votes
0answers
19 views

non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
0
votes
0answers
30 views

Solving a simple differential equation

I have an equation which I wish to solve $\dfrac{\partial\rho(x,t)}{\partial t}=-\dfrac{\partial\left(\rho(x,t)\bar{v}(x,t)\right)}{\partial x}$ where I know the $\rho(x,t)$ and want to find ...
1
vote
1answer
23 views

Uniqueness of a solution to a 1st order PDE

I'm asked to discuss the uniqueness of the solution for $$u_y+u_x=u-x-y$$ $$u(x,-2)=x$$ I've found the solution to be $$u(x,y)=x+y+2$$ But I don't know how to prove the uniqueness of the solution.
0
votes
1answer
27 views

Application of maximum principle

If I have a maximum principle of the form: If $\psi\geq 0$ of the boundary. Then $L\psi\geq 0$ on the domain $D$ implies $\psi\geq 0$ on the closure of $D$, for some linear operator $L$ defined in ...
0
votes
1answer
34 views

Elliptic partial differential equations

Consider the following elliptic PDE: $$ \Delta u=f(u), $$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have ...
0
votes
0answers
42 views

Show that a system has at most one solution using the energy method?

We are given the one dimensional heat equation with source $f$: $${\partial u \over \partial t} - k {\partial ^2 u \over \partial ^2 x} = f(t,x)$$ over $x\epsilon(0,l)$ and for $t\gt0$ with Neumann ...
1
vote
1answer
76 views

Solution to a Partial Differential Equation

May I know how to solve the following partial differential equation? $\displaystyle\frac{\partial z}{\partial x}=A\displaystyle\frac{\partial z}{\partial y}+B\displaystyle\frac{\partial^2 z}{\partial ...
1
vote
1answer
44 views

Where to specify boundary conditions

The problem asks me where I need to specify boundary values for the linear PDE problem: $u_t + xu_x + yu_y = 0$ on the domain $\Omega = x^2 + y^2 \le 1$. Using characteristics I get that $u(x,y,t) = ...
2
votes
0answers
21 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
0
votes
1answer
24 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
1
vote
1answer
49 views

solve partial differential equation

$$y^2u_x + xu_y = \sin(u^2) \\ u(x,0)=x$$ I get the projected characteristic curve on xy plane easily. However, cannot get the other one. actually the problem is getting the value of $U_{xx}, ...
1
vote
1answer
54 views

Having trouble integrating for use of energy method to prove uniqueness

We are given $u_{tt} - c^2u_{xx} + ru_t$. To prove only one solution exists, I am taking w = $u_1 - u_2$, assuming they are both solutions to the given wave equation. So: $u_{tt} - c^2u_{xx} + ...
0
votes
1answer
23 views

Weak convergence of the 4-th degree of a weak convergent sequence

Good day! We solve an optimal control problem $$ J(u) = \|y - y_d\|^2 \to \inf $$ where $y$ is a solution of the PDE $$ \frac{dy}{dt} + Ay = Bu. $$ $A$ is a nonlinear operator, $(Bu, v) = ...
2
votes
2answers
68 views

Elliptic PDE, uniqueness of solution

I'm considering a partial differential equation of the form $$\nabla^2 u + \mathbf{a}\cdot\nabla u = 0$$ with Dirichlet boundary conditions, where $\mathbf{a}$ is a (smooth, nonconstant) vector ...
0
votes
0answers
40 views

How to solve this second order PDE

I have this differential equation. $$Dc''+H=0 $$ where the partial of the concentration is with respect to z the distance. H is the rate per unit volume of particles generated and D is the diffusion ...
1
vote
1answer
44 views

Advanced Topic in Numerical solution of Differential Equations?

i) IF $\frac{dy}{dt} = - \frac{∂H}{∂z}, \frac{dz}{dt}= \frac{∂H}{∂y}$ where H is a function of $y$ and $z$, show that $H(y,z)$ is constant in time. ii) Take a $H(y,z) =Ay^2 + 2Hyz + Bz^2$ where ...