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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Prove the distribution of the point $\rho(x,t)$ at time $t$ satisfies an equality

Suppose we want to study the behavior of the gradient descent algorithm on function $F$ . But the discrete case may be a bit hard to analysis. We try to study a continuous version of it as follows. ...
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D'alembert solution with Neumann and Dirichlet boundary conditions

How can I solve the wave equation with the D'alambert solution in a finite domain with one end of the string clamped and the other end free to move vertically? i.e. $u_{tt}=c^2u_{xx}, \quad 0<x<...
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How to prove that two PDE's are related?

Say that I have PDE a) $U_x+U_y=\alpha U$ then I have PDE b) $U_{xx}+U_{yy}=\beta U$ It is obvious that the first and the second are related by that they are composed of two operators which differ by ...
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Min-max principle for eigenvalues in 1d elliptic problem

I have the following eigenvalue problem: \begin{equation} \begin{cases} -u''=\lambda u\\ u(0)=u(\pi)=0 \end{cases} \end{equation} and I have to prove the following min-max priciple for eigenvalues: \...
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Caratheodory equation and Mayer field

Let $L:J \times \Bbb{R} \times \Bbb{R}\to \Bbb{R}$ with $L(t,u,p)$ smooth Lagrangian. Let $(\Omega ,\psi)$ be the fields of extremals, and $\psi$ is the directional fields on it. It can be shown the ...
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Solutions for time-dependent sphere differential equation?

Suppose I have three suitably-smooth real-valued function $x(t), y(t), z(t)$ where $t \in \mathbb{R}$. Let's consider the following differential equation: $$\left( \frac{\partial x(t)}{\partial t} \...
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Fluid in a rotating cylinder

I am so stuck over a partial differential equation. I have the following problem A liquid in a spinning cylinder has $u_r=u_h=0$ and $u_\phi = u$, a time $t=0$ the cylinder stops and this his the ...
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How to prove this estimate for difference quotient operator $\left\|\nabla_{h} u\right\|_{H^{m-1, p}} \leqslant\|u\|_{H^{m, p}}$

I want to prove this estimate of difference quotient operator. $$\left\|\nabla_{h} u\right\|_{H^{m-1, p}\left(\mathbb{R}_{+}^{n}\right)} \leqslant\|u\|_{H^{m, p}\left(\mathbb{R}_{+}^{n}\right)}$$ Here ...
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partial differential equation involving the jacobian

Lets consider a vector function $f:\mathbb{R}^2\to\mathbb{R}^2$ with $x,y = f(z,w)$ and a second function $g:\mathbb{R}^2\to\mathbb{R}$. I want to solve for $f$ the differential equation $$ g(x,y)\...
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Split step Fourier method

I am trying to use the split-step Fourier (SSF) method for solving partial differential equation $$\frac{\partial u(x,t)}{\partial t} =u^*f(x,t)+\hat{H}\, u(x,t) \tag{1}$$ The first term causes ...
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A characterization for the kernel of an elliptic operator

Consider $\Omega \subset \mathbb{R}^{N}$ a smooth domain and Let $\lambda$ an eigenvalue of $-\Delta$. Define the operator $$ Lu = -\Delta u - \lambda u. $$ Now I will use the Theorem 3, page 319 from ...
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For which values of $q$ is $\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^q}{|x-y|^{N+sq}}dxdy$ is finite?

Let $u=u(r)$ be radially symmetric, nonnegative and decreasing function. Let $s\in (0, 1)$ and $p, q\in\mathbb{R}$ such that $1<q<p$ and $ps<N$ with $N\in\mathbb{N}, N\ge 2$. Assume that $u\...
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Prove comparison principle for $u_t + f(u)_x = k u_{xx} + g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Let us consider $$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f \in W^{1,\infty}(\mathbb R)$ and $f \not \equiv 0$ (possibly, we can also add the ...
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PDE $\partial_t f(x,y,t)=cy (\partial_xf)+3x\partial_y(yf)+\frac{1}{2}c^2y^2(\partial_{xx}f) \\$

I have the following PDE $\partial_t f(x,y,t)=cy (\partial_xf)+3x\partial_y(yf)+\frac{1}{2}c^2y^2(\partial_{xx}f) \\$ where $c \in \mathbb{R}$ I would like to know what type of PDE is, any information ...
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Is there any theorem for the solutions $u(t)$, $v(t)$ of the following differential equation? [closed]

Suppose I have the following equation (where $a(t)$, $b(t)$ and $c(t)$ are continuous functions) $a(t)(u'(t))^2 + b(t)u'(t)v'(t) + c(t)(v'(t))^2 = 0$ Is there any theorem that could possibly tell me ...
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CFL condition on 1 dimensional non-homogeneous hyperbolic PDEs

I want to find the CFL condition on the upwind scheme for the following type of 1-dimensional hyperbolic PDE or a system of PDEs \begin{equation} w_t+Aw_x = f(w) \end{equation} Where $A$ is a ...
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2 votes
1 answer
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How can you derive the spacetime Fourier transform of the free Schrodinger evolution rigorously?

I'm trying to compute the spacetime Fourier transform of the free Schrodinger evolution. Consider $f\in L^2(\mathbb{R}^d)$ and $e^{it\Delta}f=:\mathcal{F}^{-1}(e^{-it|\xi|^2}\hat{f}(\xi))$ its free ...
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Prove that $\Delta u=F$ with these conditions has at most one solution

Let $\alpha >0$, and let $\Omega\subset \mathbb{R}^N$ be and open domain. I want to prove that the following problem has at most one solution. $$\Delta u=F \quad \text{in } \Omega$$ $$u=f \quad \...
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how do i solve non linear equation? TOPIC IS PDE [closed]

Find a separated solution of the following nonlinear wave equation: ∂u/∂t=cu ∂y/∂x and What is a separated solution of the 2 -dimensional wave equation (∂^2 u)/(∂t^2 )=a (∂^2 u)/(∂x^2 )+b (∂^2 u)/(∂y^...
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Brezis book, Functional analysis, Sobolev spaces and PDE, problem 8.30

Let $k \in \mathbb{R}$, $k \neq 1$, consider the space $$V= \{ v \in H^1(0,1): v(0) = k v(1)\}$$ and the bilinear form $$B(u,v) = \int_0^1 \left(u'v' + uv\right) ~dx - \left(\int_0^1 u\right) \left(\...
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Finding a solution of partial derivative of a standard derivative

Considering the formula mentioned below, I arrived to the expansion as stated after performing a partial derivative with respect to the x co-ordinate: $$\nabla \left({\frac{dB}{dt} B}\right) = \frac{...
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A question about the meaning of the notation $(\frac{\partial}{\partial \cosh r})^2 F(r,x).$ [closed]

I am studying a book and I found the expression $$(\frac{\partial}{\partial \cosh r})^2 F(r,x).$$ Any help about the meaning of the notation that uses the author? About the exponent, I am sure that he ...
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-1 votes
1 answer
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Complete integral of PDE $x^2p^2+y^2q^2-4=0$

I have the following non-linear first order PDE before me : $x^2p^2+y^2q^2-4=0$ I have to find two complete integrals for this PDE. I wrote the Charpit equations as below: $\dfrac{dp}{2p^2x}=\dfrac{...
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Apply Arzela-Ascoli theorem to unifomly bounded sequence in $H^1$

Let $\{u_{i}\}$ be sequence of smooth function defined on $\Bbb{R}^n$ such that $\|u_i\|_{L^2(\Bbb{R}^n)}$ is uniformly bounded in $i$ and $\|\nabla u_{i} \|_{L^{^2}(\Bbb{R}^n)}$ is also uniformly ...
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2 votes
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Solving $ A\frac{\partial z}{\partial x} + B \frac{\partial^2 z}{ \partial x \partial y} + C \frac{\partial^3 z}{\partial x \partial^2 y} = 0 $? [closed]

Non-mathematician here trying to find a hopefully analytic solution or any constructive directions for solving differential equations of this particular form: Take a function $z(x,y)$, is there any ...
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Using Fourier transform to solve a heat equation on an infinite bar with two different boundary values

I have the given PDE problem \begin{equation} u_t=\alpha u_{xx} \ \ \ \ 0<x<L, t>0 \\ u_x(0,t)=0 \\ u(x,0)= \begin{cases} 0 \ \ \ \ 0<x<L \\ Q \ \ \ \ L<x<\infty \end{cases} \end{...
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Trivial pde formulation

I have an arbitrary function $f(u,v)=0$ where $u,v$ are known and are functions of $x,y,z$. While formulating a first order pde by elimination of the arbitrary function $f$ I end up with a system of ...
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Numerical solution of $2D$ wave equation using Fourier transform and finite differences

This is the $2$-dimensional wave equation $$ u_{tt} = u_{xx} + u_{yy} $$ with initial condition $u(x,y,0)=f(x,y)$ and $u_{t}(x,y,0) = 0$. The inverse Fourier transform used is $$ u(x,y,t) = \iint \hat{...
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4 votes
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Using PDE to represent a markov process

Consider a population of constant size $N + 1$ that is suffering from an infectious disease. We can model that spread of the disease as Markov process. Let $X(t)$ be the number of healthy individuals ...
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Partial Integration in solving a PDE [duplicate]

Background: I've been self-studying a book on Partial Differential Equations by Walter Strauss and ran across a particularly challenging problem in a section on First-Order Linear Equations. The ...
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2D Poisson equation using finite difference method in python

enter image description here# 2D Poisson equation example using finite difference method in python (version 3.10). I want to print the matrix form after solving the equation as given in the image ...
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Reference for time dependent traces

Consider the spaces $H^{1/2}(0,T; L^2(\partial\Omega))$, or $L^2(0,T;H^{3/2}(\Omega))$ and what not. I'm interested in a reference book illustrating the meaning, properties of these spaces (so, ...
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Flood Hydrograph Characteristics with Delta Function

I am trying to understand part of my lecture notes on solving a simple model for a river in the case of a flash flood. We have the equation, $$\frac{\partial A}{\partial t} + cA^m \frac{\partial A}{\...
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Problem solving PDE

Dear all: This question as being solved. Thanks! I have the following PDE: $$\frac{\sin(x)}{u(x,y)}\frac{\partial Q}{\partial x}+\frac{1}{v(y)}\frac{\partial Q}{\partial y}=2k\sin(x) \tag{1}$$ where $...
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Weak convergence and duals for $L_p$ involving time and probability space

Questions are from the theory of PDEs\SPDEs Question 1. Suppose $(V, H, V^\star)$ is a Gelfand triple (embeddings are continuous and dense, so $\|\|_H \le C \|\|_V$ for some $C>0$ etc) of ...
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1 vote
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How to the estimate as application of Strichartz Estimates?

RecallA pair $(q,r)$ is admissible if $q\geq 2, r\geq 2$ and $\frac{2}{q} = N \left( \frac{1}{2} - \frac{1}{r} \right),(q,r,N)\neq(\infty,2,2).$ Strichartz estimates Let $\phi \in L^2(\mathbb R^N),$...
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Effect of boundary conditions on general solution

I am having problems integrating given boundary conditions on a wave-equation. The problem is as stated below. I am no expert in solving PDE's, so please forgive if I oversee something obvious or &...
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What conditions are required to guarantee that my matrix is skew-Hermitian?

Consider the equation $$ \frac{\partial \boldsymbol{T}}{\partial t}=\kappa \space \frac{\partial^2 \boldsymbol{T}}{\partial x^2} $$ with $$ \boldsymbol{T}=(T_1,T_2,...T_N)^T $$ Let $$ T_i=\sum_{j=1}^...
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Estimates on Derivates for One Dimensional Heat Equation

$\textbf{The problem}$: Let $[-r_0,r_0]$ be a segment in $\mathbb{R}$, let $T>0$ and $u$ be a smooth function satisfying: \begin{align} \begin{cases} u_t-\Delta u = 0 & \qquad \text{on $[-r_0,...
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$u$ be the sol of $\Delta u+ a_i(x)\frac{\partial u}{\partial x_i}= f(x)$ if $f\geq 0$ then u is constant and $f=0$

Let $\Omega$ be a bounded domain with smooth boundary . Let $u$ be a solution of the problem $\Delta u+ a_i(x)\frac{\partial u}{\partial x_i}=f(x)$ and $\frac{\partial u}{\partial n}=0$ . Assume that $...
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1 vote
1 answer
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Solving an IVBP for a metal rod dipped from cold to warm water

I have to solve the diffusion equation for a solid rod, which is in a bath of 100 degrees. At $t=0$ it is moved to a second bath, where the water temperature is 0 degrees. I prepare the diffusion ...
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summer reading suggestions

I have just finished 2nd year of my maths degree and I was wondering if anyone had any good recommendations for summer reading, I want to make sure I keep practising so that I don't feel underprepared ...
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The method of separation of variables in solving a PDE

Suppose we have a PDE and we can solve it with the separation of variables and let's say the variables are $x,y,z$. Is the factorized function $f(x,y,z)=g(x)h(y)l(z)$ the only possible solution? Are ...
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Weak convergence in $H^1(\mathbb{R}^N)$ implies weak convergente in $H^1(B)$

Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite ...
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Does Homogenity implies first order PDE

I have a pde say $$f(x,y,z, a,b)=0$$ with $x,y$ being independent variable and $z$ being the dependent variable. If i can write $z=f(x,y,a,b)$ and if $f$ is a homogeneous function then can i say that ...
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1 vote
1 answer
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Proof of the embedding of time dependent Sobolev spaces

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
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2 votes
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Overdetermination of the Maxwell-Equations

I‘ve heard people talk about the overdetermination of the Maxwell equations, which are of course: $$ \nabla \cdot E =\frac{\rho}{\epsilon}\\ $$ $$ \nabla \cdot B=0 \\ $$ $$ \nabla \times E=-\...
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3 votes
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Solve the PDE $(xz-y)p+(yz-x)q=xy-z$ using lagrange method

I have the following PDE, $$(xz-y)p+(yz-x)q=xy-z$$where $p=z_x,\quad q=z_y$ Now having a hard time to get two solution from, $$\frac{dx}{xz-y}=\frac{dy}{yz-x}=\frac{dz}{xy-z}$$ I can't think of any ...
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1 vote
1 answer
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Why $w(0,t)=w(L,t)=0\Longrightarrow w_t(0,t)=w_t(L,t)=0$

Let $w$ be a $C^2$ function in two variables, $x$ and $t$. The domain of $x$ is $[0,L]$ whilst the domain of $t$ is $t\geq 0$. Suppose that $w(0,t)=w(L,t)=0$. The apparently $w_t(0,t)=w_t(L,t)=0$. I ...
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0 votes
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Numerical viscosity from the Crank-Nicolson method

Consider the following nonlinear diffusion PDE \begin{equation}\frac{\partial U}{\partial t} = \frac{\partial^2}{\partial x^2}[\Phi(U)], \end{equation} where $\Phi$ is a smooth nonmonotone function of ...
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