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Questions tagged [linear-pde]

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3
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0answers
23 views

Regularity of Solution for the Kdv equation

Let $u_{t}+u_{xxx}=f,\,\, u(x,0)=0,\,x\in(0,1), \, t\in[0,T]$ $u(0,t)=0,u(1,t)=0, u_{x}(1,t)=0$. Prove that \begin{equation} \boxed{\lVert u \rVert_{L^{2}(0,1;H^{2}(0,1))}\leq C\lVert f\rVert_{ L^{...
0
votes
0answers
18 views

Wave equation with source

I've been trying to solve the following equation: $$u_{tt} - u_{xx} =f(x, t) $$ With the following contour and initial conditions: $u(0,t)=\phi_1(t)$ $u(L,t)=\phi_2(t)$ $u(x,0)=\psi_1(x)$ $u_t(x,0)=\...
1
vote
0answers
6 views

Greens function representation of nonlinear Poisson equation

Let $L$ be an operator and suppose the Green's function exists. That is there exist a function $G$ such that $LG=\delta$ where $\delta$ is the Dirac delta function. If $L$ is linear, one can represent ...
0
votes
0answers
7 views

I can't find the particular solution for PDE with Undetermined Coefficient

How do i solve this linear inhomogeneous PDE? $\left(D_x^3-3D_x^2D_y-4D_xD_y^2+12D_y^3\right)z=\sin(2x+y)$ When i used Undetermined Coefficient, i got $0$. And I'm sure it's true $0$ cz i do this ...
0
votes
0answers
14 views

Linear advection equation with coefficient given at data points

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 ...
0
votes
0answers
16 views

Estimates of Hessian of Heat Equation

I am studying the heat equation \begin{align*} u_t - \Delta u = f \end{align*} where $u \in C^\infty(\bar{\Omega} \times (0,1])$ has compact support on $\Omega$ for all $t > 0$. My objective is to ...
1
vote
0answers
29 views

PDE Book recommendations for Undergraduate

What is the best book of partial differential equation that contains homogeneous and inhomogeneous high order linear PDE such as undetermined coefficient methods to find particular solutions from PDE (...
-1
votes
1answer
31 views

Find a particular solution for linear PDE [closed]

Please help me to find the particular solution for this PDE $(D_x^2+D_y^2)z=x^2y^2$ I know how to find the homogeneous solution. But i don't know how to find the particular solution.
0
votes
1answer
35 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 ...
1
vote
1answer
28 views

Two questions about PDE (sum of solutions)

In an old exercise, I tried to solve a the following problem $$ \begin{cases} u_t + xu_x = u\\ u(0,x)=x^3 \end{cases} $$ I solved the equation with the equalities $$\frac{1}{dt}=\frac{x}{dx}=\frac{...
1
vote
1answer
25 views

Integrating factor in canonical form of second-order linear equations

In the hyperbolic PDE, I have ticked the part I do not understand. How do they get it to $v_s(r,s)= r-1 + C(s)e^{-r}$ in the canonical form process? In the textbook, it's said that they're using some ...
1
vote
0answers
25 views

Exact solution of advection PDE with numerical scheme

Consider the advection pde $v_t + a v_x = 0$ and let $R = \frac{a \Delta t }{\Delta x}$. Then, the FTFS scheme is given by $$ u_k^{n+1} = u_k^n - R (u_{k+1}^n - u_k^n ) $$ and $u_k^n = u(k \Delta x,...
2
votes
1answer
32 views

Infinitely many solutions for a first order Cauchy problem.

Is this correct that the following Cauchy problem has infinitely many solutions? ‎\begin{cases}‎ ‎xu_t+u_x=0 \\‎ ‎u(x,0)=\cos x‎ ‎\end{cases}‎ Using the method of characteristics it is obvious ...
0
votes
0answers
23 views

Uniqueness of solution on Advection Diffusion equation

Let $\Omega\in\mathbb{R}^{n}$ be a bounded connected open set. I have the following Advection-diffusion Equation given by \begin{align} \nabla\cdot\left(\mathbf{V}\psi-D\nabla \psi\right)&=F\quad \...
1
vote
0answers
37 views

Prove the comparison principle for viscosity solution of the Laplace equation [closed]

This is a basic question, but I cannot find where it is addressed. How does one pove a comparison principle for the Laplace equation $\Delta u = 0$ with Dirichlet boundary condition $u = 0$ on a ...
1
vote
0answers
25 views

Norm of a multivalued function

If i have a function defined as follows $F:[0,T]\to H^{1}_{0}(\Omega)\times L^{2}(\Omega)$;$F=(F_1,F_2)$. My question: how does the norm $\|F\|_{H^{1}_{0}(\Omega)\times L^{2}(\Omega)}$ defined?? Is ...
5
votes
0answers
98 views

6 linear PDE for only 3 unknowns?

Let $x \in (0,L)$, $t \in (0,T)$, and let $u_0 = u_0(x) \in \mathbb{R}^3$, $g=g(t) \in \mathbb{R}^3$, $P = P(x,t) \in \mathbb{R}^3$ and $Q = Q(x,t) \in \mathbb{R}^3$ be continuously differentiable ...
0
votes
0answers
30 views

How do I obtain a solution of this Helmholtz equation?

I want to get a solution of the following Helmholtz equation. $$ \Delta u+ k^2 u = f, \qquad x \in \mathbb{R}^{n}, \; k>0$$ Using the Fourier transform, I have \begin{align*}u&=\int_{\mathbb{...
1
vote
0answers
46 views

A differential operator associated with a vector field on the torus

Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$. We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows: $T(f)=...
2
votes
0answers
45 views

Two PDE for one matrix-valued unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $P = P(x,t) \in \mathbb{R}^{3\times 3}$, $Q = Q(x,t) \in \mathbb{R}^{3\times 3}$, $R_0 = R_0(x) \in \mathbb{R}^{3\times 3}$ and $G= G(t) \in \mathbb{R}^{3 \...
0
votes
1answer
11 views

Local smoothing for Airy equation

This question is from Tao's Nonlinear Dispersive Equation, Exercise 2.55. Show that smooth solutions $u\in C_{t,\text{loc}}^\infty \mathcal{S}_x(\mathbb{R}\times\mathbb{R}\to\mathbb{R})$ to the Airy ...
0
votes
1answer
35 views

$u_{xx} + u_{yy} = 1$ in disc with radius $1$

Consider the inhomogeneous elliptic equation $u_{xx} + u_{yy} = 1$ (this is often called a Poisson equation) in the disc $x^2 + y^2 < 1$, with the boundary condition $u = a$ on the boundary $x^...
4
votes
1answer
78 views

Two PDE for one unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions. My ...
1
vote
0answers
28 views

PDE with Method of Characteristics and domain of solution

I wanted to solve the following PDE with initial condition $$ \left\{\begin{array}{c} xu_t+u_x=0\\ u(0,x)=f(x) \end{array}\right.$$ Proving that: (i) if $f(x) = \sin(x),$ then it is impossible to ...
2
votes
2answers
88 views

Solve PDE using method of characteristics with non-local boundary conditions.

Given the population model by the following linear first order PDE in $u(a,t)$ with constants $b$ and $\mu$ : $$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0$$ $$u(a,0)=u_0(a)\,\,\,a≥0$$ $$u(0,t)=F(t)=b\...
1
vote
0answers
26 views

Existence of the solution of the 3D Micropolar equations [closed]

Please how to show the local existence for the solution of the 3D micropolar equations in a Besov space setting ? $\left\{ \begin{array}{l} \partial_tu-(\nu+k)\Delta u-2k\nabla\times w+u\nabla u+\...
1
vote
1answer
30 views

Non-Linear Differential Equation Change of Variable

The function $v(x,t)$ satisfies: $$\frac{\partial v}{\partial t} = \frac{\partial^2v}{\partial x^2} + \left(\frac{\partial v}{\partial x}\right)^2$$ for $0<x<1$, the initial condition $v(x,0)...
1
vote
1answer
57 views

Use the Laplace transform to solve $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$

Use the Laplace transform to solve the following initial boundary value problem for the wave equation $ u_{tt}(x, t) − c^2u_{xx}(x, t) = 0$ $u(x, 0) = 0$, $u_{t}(x, 0) = 0 ∀x > 0$ , and $u(0, t) ...
0
votes
1answer
30 views

Problem with IVP (PDE)

I'm not very familiar with the Method of Characteristics for PDEs but have been reading up with several *.pdfs and YouTubes in the past few days. Despite that I got stuck on a relatively simple semi-...
0
votes
1answer
58 views

Solving system of PDEs with gradient

I have the following system of PDEs: $\partial_x q = -c(b_1 \partial_x u + b_2 \partial_y u)$ $\partial_y q = -c(\partial_y u + b_2 \partial_x u)$ where $b_1,b_2,c$ are constants, with $c$ ...
0
votes
0answers
73 views

Continuous dependence of first Dirichlet Eigen value on the domain.

Let $\Omega$ be a open, bounded domain in $R^n$. Let $Lu=\sum (a_{ij}(x)u_{x_i})_{x_j} +c(x) u$ be the operator where $a_{ij}=a_{ji}$ are in $C^1$ and $c(x)$ is continuous. Let $\Omega_0$ be a bounded ...
0
votes
1answer
15 views

Regularity of linear pde with smooth coefficients

Consider $au_x+bu_y+cu_z=f$ on $\mathbb{T}^3$ where $a,b,c,f$ are in $C^\infty$ and $\forall (x,y,z)\in\mathbb{T}^3,|a|,|b|,|c|>1$. If there exists $C^1$ solution to this pde, can we say that it ...
0
votes
0answers
42 views

Monge cone and nonlinear first-order PDE

I am trying to understand the idea behind the Monge cone for nonlinear first-order PDEs. Can anyone please give me the intuition behind the Monge cone and the way it is used to obtain ODEs from the ...
2
votes
2answers
59 views

Solution for $u_t+u_x=0$ using characteristics

P. Dravek and G. Holubova, Elements of Partial Differential Equations, Section 3.4 Exercise 22: Show that the initial value problem $$u_t + u_x = 0,\; u(x,t) = x \;\text{ on }\; x^2+t^2=1.$$ ...
2
votes
0answers
45 views

Can we extend the Riesz potential convolution operator for the Laplacian to a continuous operator from $L^p$ to $\mathcal{S}'$ if $p\ge\frac{n}{2}$?

If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define: $$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$ Then $K_n$ is locally ...
2
votes
1answer
54 views

Can we say anything about the first distributional derivatives of $g$, where $g$ is the solution to $-\Delta g =f\in L^p$ given by Riesz potential?

If $n\ge3$ and $\omega_n$ is the $n-1$-dimensional Hausdorff measure of the unit sphere in $\mathbb{R}^n$, define: $$K_n(x):=\frac{1}{(n-2)\omega_n} \frac{1}{|x|^{n-2}}.$$ Then $K_n$ is locally ...
2
votes
1answer
38 views

What does constant along characteristic mean

When we have linear or quasilinear first order pde $$ a(x,y,u) u_x + b(x,y,u) u_y = c(x,y,u) $$ And suppose we have found characteristics with prescribed Cauchy data $u |_{\text{curve}} = \phi$ I ...
2
votes
1answer
53 views

Showing scheme is consistent with PDE $v_t+v_x = 0$

For the advection equation $v_t + v_x = 0$, For practice, i want to show the scheme FTCS $$ u_i^{n+1} = u_i^n - \frac{\Delta t }{2 \Delta x } (u_{i+1}^n - u_{i-1}^n ) $$ is consistent and ...
4
votes
4answers
79 views

One-way wave equation IBVP

Plese help me to find the solution of te following equation. For values of $x$ in the interval $[-2,3]$ and $t>0$ we consider the one way wave equation $$u_t+u_x=0$$ with initial data \begin{...
0
votes
1answer
24 views

Solution to simple linear 2nd order partial differential equation with boundary conditions

I am trying to find out the form for solutions to this equation: $\partial_x^2u(x,M) = 0$, where x and M are not independent (M is the running maximum of process x, but I don't think that's necessary ...
0
votes
0answers
34 views

Finding solution at some specific points for wave equation.

Given the differential equation $$\frac{\partial u}{\partial t}+v\frac{\partial u}{\partial x}=0$$ where $u=u(x,t)$, $x$ has units of feet, and $t$ has units of seconds, determine the following ...
1
vote
0answers
48 views

Reducing Second order PDE System to First-Order

I am confused as to how to choose the variables for $a, b, c, d, e$ and $f$ when reducing the second-order PDE systems to first-order. This is the question I am referring to for help:
0
votes
1answer
43 views

Uniqueness: linear first order pde with constant coefficients

Let us say I find the characteristic lines of some easy PDE $a U_x + b U_y = 0$ to be $bx-ay=c$, where $b, a, c$ are constants. Now, we say the solution must be constant along those lines, so it HAS ...
3
votes
3answers
55 views

Is it useful to convert a higher order PDE into a 1st order system?

I just learned how a higher order PDE can be converted into a system of PDEs.I am just wondering whether this is a standard way to solve some higher order PDEs which are immune to other methods or is ...
1
vote
1answer
67 views

Classifying linear first-order PDE system (elliptic, hyperbolic, or parabolic)

Consider the constants $$\begin{aligned} & (\text i.)\; a_1 = b_1 = a_2 = b_2 = 1 \\ & (\text {ii}.)\; a_1 = b_2 = 1, \quad b_1 = 0, \quad a_2 = -1 \\ & (\text {iii}.)\; a_1 = b_1 = b_2 =...
0
votes
1answer
36 views

PDE with complex roots of characterictic solution

I have the following equation $$\frac{\partial^2 V}{\partial x^2 }+\frac{\partial^2 V}{\partial y^2}=-4\pi(x^2+y^2)$$ Such that $V(x,y)$ is real and $V(x,y)=0$ at $y=0$. As it is clear form the ...
1
vote
0answers
34 views

Characteristics for nonhomogeneous wave equation $y_{tt}=y_{xx} + f$

Consider the initial- and boundary-value problem $$\eqalign{ & {y_{tt}} = {y_{xx}} + f(t,x){\text{ }}{\text{, (t}}{\text{,x)}} \in {\text{(0}}{\text{,}}\infty {\text{)}} \times {\text{(0}}{\text{...
1
vote
1answer
35 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 ...
0
votes
0answers
12 views

Laplace form partial differential equation. [duplicate]

We have to solve for the general solution of $p\cos (x+y) +q \sin (x+y) = z$ Using multipliers $(1,1,0) I get $$\frac{d(x+y)}{\cos (x+y) + \sin (x+y)}=\frac{dz}{z}$$ This gives the first solution $...
1
vote
1answer
24 views

Partial differential equation in Lagrange form.

Solve the pde. $$(my(x+y)-nz^2)p-(lx(x+y)-nz^2)q=(lx-my)z$$ Using method of multipliers$(1,1,(x+y)/z)$ I got one of the solutions $$(x+y)z=c_1$$ I can't figure out what the other solution might be....