Questions tagged [partial-differential-equations]

Questions on partial (as opposed to ordinary) differential equations - equations involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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PDE solution using convolution and (inverse) Fourier transform

I'm studying PDEs from old course material which includes answers to most of the exercises. Given a PDE $-u''(x)+au(x)=f(x)$ and a function $g(x)=e^{-|x|}$, we should find $u(x)$ by calculating the ...
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A reference for equivalent ways of solving the continuity equation

Consider a vector field $$ X:\mathbb R^d\to \mathbb R^d $$ we can define the continuity equation in a distributional sense for a curve of measures $\mu:[0,T]\to\mathcal M_b(\mathbb R^d)$ as $$ \frac{d}...
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Can we extend this result?

If $X$ Hilbert space and $A:D(A)\subset X \rightarrow X$ nonlinear operator. The Cauchy problem is given by: \begin{eqnarray} f(t)&\in&U_t(t)+AU(t) \;t \in [0,T]\\ U(0)&=&U_0 \end{...
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Spatial derivatives and weak convergence in Bochner spaces

Consider $V=H^1(\Omega)$ and a bounded sequence $f_n$ in $B = L^2((0,1),V)$, which is Hilbert with $(f,g)_B = \int_0^1(f,g)+(\nabla f, \nabla g)$. Then, there exists $f \in B$ with, for all $g \in B$, ...
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Exististence result for semilinear parabolic PDE

I am considering the following semilinear parabolic PDE: $$ \partial_tu+\theta(x-y)\partial_yu+\frac12x^2\partial_{xx}u+\frac12y^2\partial_{yy}u + y^{-3/2}u^2+a=0$$ with terminal condition $$u(T,x,y) =...
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2 answers
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When are the coefficients of Laplace equation based on Fourier series?

in various posts I have made recently on the Laplace equation, I have questioned the methods for finding the coefficients of the harmonic/hyperbolic functions (Y(y) and X(x) of U(x,y). One examples is ...
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4 votes
1 answer
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Reproducing the delta function in physics by using the definition adopted in the theory of distributions

I used to study physics, where the delta function on the real line is defined as a function $\delta$ that satisfies $$\delta(x)=\begin{cases}\infty&,x=0\\ 0&,x\neq0 \end{cases}\tag{$*$}$$ and $...
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Problem with deriving a dimensionless differential equation

I am trying to understand how to convert the following differential equation to its dimensionless form: $${\partial{V(t, r, z)}\over{\partial{t}}} = - {1\over{q}} {dP(t,z)\over{}dz}+ {{m} \over{q} } ...
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A tough Laplace PDE problem

here is a PDE I am having problems with, on the domain $[0,a]\times[0,b]$: \begin{equation} u_{xx}+u_{yy}=0 \\ u(x,0)=0 , u(x,b)=\sin \frac{n\pi}{a}x \\ u(0,y)=u(a,y)=0 \end{equation} From this we see ...
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Do zonoids arise as solutions to partial differential equations?

Consider a vector field $X$ in $M=(0,1)^3$ with source at $a=(0,1,1)$ and sink at $b=(1,0,0).$ I'm interested in understanding the collection of vector fields. I mainly want to describe the mappings ...
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Global version of Cauchy-Kowalevskaya theorem

Does there exist a global version of the Cauchy-Kowalevskaya theorem for linear PDEs ? Global in the sense that the solution exist and is analytic on $\mathbb{R}$ if so are the coefficients of the PDE....
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What can I say about the different classifications of differential equations?

I just want to double check my understanding of differential equations and what questions they are trying to solve so please tell me if this is accurate. Ordinary Differential Equation an equation ...
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Imposing only normal or tangential direction Dirichlet boundary conditions in the weak form of a Poisson equation

Consider the vector Poisson equation \begin{align} -\Delta u &= f \text{ in } \Omega \subset \mathbb{R}^d \tag{1}\\ u\cdot v &= 0 \text{ on } \partial \Omega \tag{2} \end{align} where $v = n$ ...
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What are the maximum and minimum values of the function y=sin(2x)−x , x∈[−π2,π2] ? Can u please write in details it will be covered in exam [closed]

please help me to find out answer and know solution.it will be covered in exam. Can u write it till end? I'd be gratefull to you. Thanks in advance!!
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Prove that for every linear $C^1$ flow $\varphi$ there is a linear operator $T$ on $\mathbb{R}^n$ with $ \varphi _t (x) = e^{tT}x $

A continuous function $\varphi : \mathbb{R} \times \mathbb{R}^n \rightarrow \mathbb{R}^n$ is called a flow if for every $t, s \in \mathbb{R}$ and $x \in \mathbb{R}^n$ $\varphi (0,x)=x, \forall x \in \...
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Question regarding a problem derived from Evans PDE's book:

I came across this problems i few months and go and thought i would give it a try. The problem goes as following: Let $\Omega$ be a bounded subset of $R^n$, $\alpha>0, f \in L^2(\Omega)$ and we ...
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What is the maximal domain where the solution to this quasi linear PDE is defined?

I've solved the following PDE using the method of characteristic curves: $$xu_x + xy^2u_y =u; \\ u(1,y) =y,\quad y \in \mathbb{R}$$ The solution I got is: $$u(x,y) = \frac{xy}{1-y+xy}$$ The question ...
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1 answer
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What requirements must a particular solution of a second order DE meet

Probably a very simple question but when i'm solving simple DE's I am not sure whether to use $te^t$ or simply $e^t$ for a particular solution which involves that kind of solution of course. For ...
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Green's Function for the Heat Equation with Boundary Condition Depending on the Spatial Derivative

I am trying to solve the following problem: Let $\Omega$ = (0, $\infty$), $f$ $\in$ $C$([0, $\infty$) $\times$ $\bar{\Omega}$), $h$ $\in$ $C$([0, $\infty$)). Consider the PDE Problem $$ \left\{ \begin{...
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1 vote
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What are the eigenfunctions of $-\frac{d^2}{dt^2} + \Delta^2$?

Let $U = [0,1]^d$ be the $d$-dimensional hypercube and consider the integral operator $S: L^2(U\times[0,1]) \to L^2(U\times[0,1])$ such that $Sf = g$ if and only if $$ \begin{cases} \frac{d}{dt}g - \...
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Weak Solutions for non-divergence form equation

Given a linear 2nd order equation in divergence form the concept of weak solution can easily be defined taking into account Integration by parts formulae. How should one go about formulating weak ...
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Show that there exists a unique continous function $f$ [closed]

Show that there exists a unique continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$ f(t)=\frac{t^2}{2}+ \int_0^t sf(s)ds, \, \forall t \in \mathbb{R} $$ Show this function is $C^{\...
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0 votes
1 answer
62 views

Unique solution of first differential system [closed]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continous locally Lipschitz function, and let $g:\mathbb{R} \rightarrow \mathbb{R}$ be a continous fuction. Justify that the first order differential ...
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Does the Laplace's equation $\nabla^2 u=0$ could have smooth compact-supported solutions $\in C_c^\infty$? Any example?

Does the Laplace's equation $\nabla^2 u=0$ could have smooth compact-supported solutions $\in C_c^\infty$? Any example? In this answer an user named @NinadMunshi explain that the following function $...
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why is $u(x,y)$ derivable only between two specific curves?

I've solved a PDE and got the resulting function: $$u(x,y) = 2f(y-x) -f(y-2x) +\int_{y-2x}^{y-x} g(s)ds$$ The question then required me to specify the continuity of the function and when does it has a ...
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1 vote
1 answer
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How to interpret the mass conservation equation?

I want to know that the difference is between the following equations: $$ \frac{\delta h_d}{\delta t} = -\nabla.(\vec{v_s} h_d) = -u_s(\frac{\delta h_d}{\delta x}) -v_s(\frac{\delta h_d}{\delta y}) - ...
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Are there possible solutions that solves both Laplace $\nabla^2 u=0$ and the Wave equations $\nabla^2 u=\ddot{u}/c^2$?

Are there possible solutions that solves both Laplace $\nabla^2 u=0$ and the Wave equations $\nabla^2 u=\frac{\ddot{u}}{c^2}$? I would like to know which consequences have for a function if its solves ...
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A proof of the sub-super solution method using Schauder's fixed point theorem

Firstly, I would like to define the notion of a sub solution and a supersolution and enunciate the Schauder's fixed point theorem. Consider the non-linear elliptic problem $$(P) \begin{cases} \begin{...
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Definition of subsolution in sobolev spaces

Consider $\Omega \subset \mathbb{R}^{N}$ a smooth domain and $$Lu = -\text{div}(A(x)\nabla u) + b(x) \cdot \nabla u + c(x)u,$$ a uniformly elliptic operator with $A(x) = (a_{ij}), a_{ij} \in C^{1}(\...
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3 votes
2 answers
67 views

Solution of Klein-Gordon equation in momentum space

I'm solving Klein-Gordon equation in order to get scalar field expression $$(\partial^2 + m^2)\phi=0$$ I expand the solution $\phi$ into Fourier integral in momentum space $$\phi=\int\frac{d^4p}{(2\pi)...
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1 answer
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Solve functional equation $F(s \cdot t)=\int_a^b G(x,t) G(x,s) dx$

I need to describe all functions $G(x,y)$ that have following property: for some $a,b$ the integral $$ \int_a^b G(x,t) G(x,s) dx $$ is known function $F$ of one variable $s \cdot t$, i.e. \begin{...
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Dispersion relation - Fourier transform of spatial operator

In my course I was told that the dispersion relation is defined as the Fourier transform of the spatial operator. For example, consider the linear wave PDE $$\partial_{tt}u(x,t) - \partial_{xx}u(x,t) =...
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2 votes
1 answer
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Cauchy problem and unique solution

Let $f: \Omega \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ be a continuous function on the open subset $\Omega$ which is non-increasing on the second variable, i.e., $f(t,x) \geq f(t,y)$, whenever $...
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4 votes
1 answer
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How does pointwise Hölder continuous on compact subsets not imply locally Hölder continuous?

When Gilbarg and Trudinger introduced the Hölder spaces, they mentioned on page 52 that Furthermore note that local Hölder continuity is a stronger property than pointwise Hölder continuity in ...
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1 vote
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Setting z to 0 in constrained (on x & y) differential?

Let $w = zxe^y + xe^z + ye^z$ with constraint $x^2y + y^2x = 1$. Assuming x is the independent variable, find $\frac{\partial w}{\partial x}$. - MIT's multivariable calculus course We can take the ...
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1 vote
1 answer
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Cauchy problem and right-saturated solution

Consider the Cauchy problem \begin{equation} x''+ax+f(x')=0, x(t_0)=x_0, x'(t_0)=x_1, (1) \end{equation} where $a$ is a positive constant, and $f:\mathbb{R} \rightarrow \mathbb{R}$ is a locally ...
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0 answers
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Duhamel's principle: show that this integral equation solves the inhomogeneous heat equation

Let $\Delta$ be the Laplacian on a bounded domain $\Omega \subseteq \mathbb{R}^{d}$. Let $s \geq 0$. Assume that $(x, t, s) \mapsto$ $v_{F}(x, t, s)$ solves the IVP of the heat equation $$ \begin{...
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2 votes
0 answers
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Existence of symplectic vortices

Consider for a closed Riemann surface $(\Sigma,j_{\Sigma})$, a compact Lie group $G$ and $P$ a principal $G$ bundle over $\Sigma$ and a connection 1-form $A$ on the principal bundle $P$ and $u:P \to M$...
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0 answers
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Solving 3D-Laplace Equations with mixed boundary conditions at the end [closed]

I faced this question in my research. I tried to solve it using the separation of variables and Green's functions but I did not reach any answer. I would appreciate your help. Let $\Omega = [0, x_{1}]\...
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0 answers
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Are there any $L^\infty$ bounds on the eigenfunctions of a first order elliptic dfferential operator on $\mathbb{R}^n$ in terms of eigenvalues?

The question is as in the title. All references I searched for only seem to deal with compact manifolds. So I ask for noncompact cases. For a first-order elliptic differential operator with "...
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1 vote
1 answer
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Existence of a unique solution to Cauchy problem

Let $X$ Hilbert space. Let $\mathcal{A}:D(\mathcal A)\subset X\rightarrow X$ nonlinear and $f:[0,+\infty)\rightarrow X$ $$\begin{cases}U_{t}(t)=\mathcal{A} U(t)+f(t),& \forall t>0, \\ U(0)=U_{0}...
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1 vote
1 answer
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Partial Derivatives involving Change of Variables

I am currently reading Elementary Fluid Dynamics by Acheson and following along $3.9$. In this section, the author uses a change of variables: $z=3c-2c_0$ (where $c_0$ is a constant and $z,c$ are ...
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0 votes
2 answers
71 views

Trigonometric problem in solving a PDE

I'm self-studying partial differential equations with course material from 2018 and I have example solutions to the exercises. I have tried to arrive at the example solution for this PDE already a few ...
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0 votes
1 answer
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Solve $u_t=2u_{xx}+t\sin(x)$

Solve $u_t=2u_{xx}+t\sin(x)$ $t>0,0<x<\pi$ boundary conditions are: $u(0,t)=u(\pi,t)=0$ and $u(x,0)=\sin(x)\cos(x)$ my attempt: solving for the non homog. first and I got that $u(x,t)=\frac{1}...
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1 vote
0 answers
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trying to find asymptotic or approximate solutions to coupled diffusion-advection-decay equations

I am trying to solve a set of diffusion-advection equations with coupled decay (see unanswered coupled diffusion-advection-decay equations). The equations are as follows: \begin{align} u_t - \alpha ...
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5 votes
1 answer
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Laplace-Beltrami operator is essentially self-adjoint on a bounded domain

I came across this lecture note The Poincaré inequality on domains on a webpage. In the first section, it claims that $L$ is essentially self-adjoint on $\mathcal{D}^\infty$. It is known that the ...
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  • 1,083
3 votes
2 answers
117 views

Unsolvable partial differential equation

I was given the following differential equation: $$\left\{\begin{matrix}u_t+2u_x=0, \ \ x>0, \ \ t>0, \\ u(x,0)=\arctan(x), \ \ x>0, \hspace{0.15cm} { }\\ u(0,t)=\dfrac{t}{1+t^2}, \ \ t>0. ...
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1 vote
1 answer
31 views

Developing a correct Ansatz and solving the Laplace equation on a rectangle

I am trying to find a way to efficiently solve the Laplace equation, by generating a rule for making Ansatz. However, once this ansatz is found, I struggle with finding the coefficients and the ...
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-1 votes
1 answer
33 views

Help in understanding inequality involving gradient [closed]

Let $f \in C_0^{1}(\mathbb{R}^N)$, i'm reading a text and the author stated without proof that if we fix $x \in \mathbb{R}^N$ then for every $y \in \mathbb{R}^N$ we have $$|f(x-y)-f(x)| \leq |\nabla f|...
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2 votes
0 answers
22 views

Numerical solution of a nonlinear PDE

I am trying to solve the following one-dimensional PDE $$\partial_tv(t,x)+b(x)\partial_xv(t,x)+\frac{1}{2}\sigma^2(x)\partial_x^2(t,x)+(v-2v^{-1})(t,x)=0 \quad \text{ for } t \in (0,T)$$ with the ...
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