Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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What rule of integration by substitution am I breaking?

Context - Method of characteristics to integrate PDE given initial conditions: $$ u_t + a u_x + b u = f(t, x) \tag{1a.}$$ $$ u(0, x) = u_0(x) \tag{1b.}$$ I am quoting some intermediate results ...
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1answer
106 views

Isolated singularity of harmonic function

I am working with a book by Axler, Bourdon and Ramey and find the following problem: Suppose $u$ is a harmonic function on $B \setminus \{0 \}$ such that $$ |x|^{n-2} u(x) \to 0, \qquad x \to 0 $$ ...
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284 views

Provide a bond pricing differential equation and invoke Feynman-Kac

Grateful for any assistance. Consider the process: $dZ=r(t)Z\,dt$ , where $r(t)$ is stochastic and $Z=Z(r,t;T)$ is a zero coupon bond. Provide a bond pricing differential equation and invoke ...
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1answer
151 views

Solution to the heat equation with mixed boundary conditions and step function.

PDE with the given intial and boundary conditions $\gamma \frac{\partial^{2}p}{\partial x^{2}}=\frac{\partial p}{\partial t}$ Initial condition: $p(x,t=0)=0$ Outer Boundary Condition: ...
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0answers
119 views

Mean Value Property for Harmonic Functions (clarifying Axler's proof)

I'm going through Axler's proof of the mean value property, and I'm a little puzzled. Since this proof is in the first few pages of his book, I thought I'd ask for clarification before going further ...
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47 views

Evans PDE mappings into better spaces

Evans PDE chapter 5.9 theorem 4 (mappings into better space), Evans wrote in the proof: In addition, $\bar{u}^{'}\in L^2(0,T;L^2(V))$ , with the estimate: \begin{equation} ...
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1answer
271 views

Weak derivative as an $L^2$ limit of the difference quotient

Let $u \in H^1(\mathbb{R})$. Show that $$\left\| \frac{u(x+h)-u(x)}{h} - u' \right\|_2 \to 0\quad \text{ as } h \to 0, $$ where $u' \in L^2(\mathbb{R})$ is the weak derivative of $u$. In other words, ...
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1answer
29 views

Derivative of a function in the context of Sobolev spaces

Consider $B_1$ the unitary ball in $R^n$ centered in the origin and $2 \leq p < \infty$. Let $ \psi \in W^{1,p}(B_1)$. Let $h \in W^{1,p}(B_1) $ with $h - \psi \in W^{1,p}_{0}(B_1)$ with ...
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1answer
155 views

Black Scholes PDE

How to show that $V_1(S,t)=S\frac{\partial V(S,t)}{\partial S} $ satisfies Black-Scholes PDE given as $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + ...
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1answer
63 views

shifting points near a $C^1$ boundary (argument from Evans' PDE book)

Let $U \subset \mathbb{R}^n$ be open, bounded, and $C^1$. $U$ being $C^1$ means that, for any point $x^0 \in \partial U$, there exists $r > 0$ and a $C^1$ function $\gamma : \mathbb{R}^{n-1} \to ...
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41 views

Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
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1answer
181 views

Trace operator counterexample

This is homework so no answers please Let $U$ be bounded with a $C^1$ boundary. Show that a ''typical'' function $u \in L^p(U) \ (1 \leq p < \infty)$ does not have a trace on $\partial U$. ...
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1answer
30 views

Asymptotic Estimate

Consider the following Sturm–Liouville problem $$u''+\lambda u=0, \ 0<x<1$$ $$u(0)-u'(0)=0, \ u(1)+u'(1)=0.$$ Obtain an asymptotic estimate for large eigenvalues. I solved the problem and ...
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1answer
78 views

Show that f solves the so called wave equation

Task $\text{Let } \; c \in \mathbb{R} \; \text{ be a given parameter, with } \; c > 0$ $\text{ Show that } \; f: (\mathbb{R}^3 \setminus \{ \vec{0} \}) \times \mathbb{R} \to \mathbb{R} \; ...
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1answer
38 views

Local Estimates for higher order homogeneous elliptic operators

For $u\in W^{2k}_2(\mathbb{R^n})$, $k\geq 1$, it is well known (see, for example, Exercise 12.9.4 in Krylov, N. "Lectures on Elliptic and Parabolic Equations in Sobolev Spaces") that the following ...
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2answers
44 views

Estimate in Sobolev Spaces

Let $u\in H^0(U)\cap H^1_0(U)$ and $v_k\in C^\infty_c$(U) such that $v_k\rightarrow u$ in $H^1_0(U)$ and $w_k\in C^\infty (U)$ such that $w_k \rightarrow u $ in $H^2(U)$. I want to show that $ \int_U ...
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1answer
42 views

Equivalence of lens shaped domain and the existence of a smooth time function

A lens shape doimain is defined here as: Defn A lens-shaped domain $D\subset M$ based on $\Sigma$ is the image of a smooth map $\Phi: \Sigma\times (-1,1) \to M$ where $\Sigma\subset M$ is a compact, ...
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1answer
71 views

Legendre Polynomials Recursion Problem

Using the recurrence equation for Legendre Polynomials: $$(k+1)P_{k+1}(x)=(2k+1)xP_k(x)-k P_{k-1}(x) \text{ , } k \in \mathbb{N}$$ Compute the Integral: $$ \int_{-1}^1xP_k(x)P_{k+1}(x)dx $$ I am ...
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2answers
107 views

Solving a PDE via method of characteristics

I'm interested in solving the following PDE via the method of characteristics: $$\frac{\partial f}{\partial t} - ax\frac{\partial f}{\partial p}+ bp \frac{\partial f}{\partial x} = 0,$$ with ...
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1answer
40 views

A quick question regarding Green's Functions

I've been given the following conditions for some 3D function, $\phi$; $$\nabla^2 \phi(x,y,x) = 0$$ $$\phi(x,y,0) = f(x,y)$$ My question is, would the equivalent Green's function problem be; ...
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1answer
94 views

Solving second order partial differential equation

I' trying to solve this differential equation: $$y^2 \frac{\partial ^2 u}{\partial y^2} - 2xy \frac{\partial ^2 u}{\partial x \partial y} + x^2 \frac{\partial ^2 u}{\partial x^2} + 2y \frac{\partial ...
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27 views

Removable singularities for Dirichlet problems of Laplace equaions?

It is already known that if $u$ is harmonic in $\Omega\backslash\{x_0\}$ where $\Omega$ is a pre-compact domain in $\mathbb{R}^n$, $n\geq2$ and $u=o(|x-x_0|^{2-n})$ when $x\to x_0$, then the singular ...
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1answer
190 views

Solving the non-homogeneous heat equation with homogeneous Dirichlet boundary conditions

The problem: Solve $$ \frac{dT}{dt} + k \frac{d^{2}T}{dx^2} = \exp \Bigl[-\alpha t\Bigr] \sin\left( \frac{2\pi x}{L}\right)\text{ with }T(0,t) = T(L,t) = 0,\ T(x,0) = f(x) $$ with $k$, $\alpha > ...
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1answer
69 views

Extension theorems in Sobolev spaces: Solving for constants

I saw this problem in PDE book and tried searching for the idea behind solving it which I have not been able to find yet. If we have $n\ge2$, $B=\{x\in\mathbb R^n:|x|<1\}$ and ...
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48 views

How can I solve this PDE?

$\dfrac{\partial \hat{Q}}{\partial t} - \dfrac{Am}{\rho} \dfrac{\partial ^3Q}{\partial t \partial z^2} = 0$ I really do not know which method could I use to solve it!
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41 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
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49 views

Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde?

I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which ...
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1answer
67 views

Unique solution for PDE?

How can one tell if a solution is existent or unique? For example: $yu_y+uu_x=u-y$ $u(x,1)=x$ I've found the solution to be $u=x+1-y$, but have been told there are infinitely many solutions. Is ...
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1answer
183 views

Cauchy problem for the inhomogeneous wave equation written as $u_{xy}=1$

I have a question on a PDE assignment that's giving me problems interpreting. Solve the following Cauchy problem for the inhomogeneous wave equation: $u_{xy} = 1$ $u(x,-x)=6$ ...
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65 views

Can you give a nonconstant function to show difference between The Weak Maximum Principle and The Strong Maximum Principle

We know that the Weak Maximum Principle and Strong Maximum Principle in every PDE book,such as Theorem 3.1 and Theorem 3.5 in David Dilbarg's book. But I never see a author give a nonconstant ...
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3answers
178 views

PDE - Energy - Wave Equation

I dont know how solve a) and b), I'm read the book of Walter Strauss, but I have a lot of doubts, for the c) first, I tried estimate $(k(t)e^{-2at})'$ and integrate the inequal, but not unwind... :( ...
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91 views

Validity of approximating a difference equation with a differential equation

Consider the following two equations: $$ \begin{cases} A_k(n+1)-A_k(n)=\beta \displaystyle \frac{ A_{k-1}(n)- A_k(n)}{\alpha+ n} + \delta_{k, \beta} \\ \\ \displaystyle ...
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1answer
106 views

Partial differential equation (heat equation with other terms)?

Can some one help me solve the following PDE with the given intial and boundary conditions? $\gamma t\frac{\partial^{2}f}{\partial x^{2}}=t\frac{\partial f}{\partial t}-\alpha f$ Initial condition: ...
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1answer
72 views

Dirichlet Problem

I have to solve the following Dirichlet Problem $$\Delta u=0\quad\text{in}\,\,\, D,$$ $$u(\mathrm{e}^{it})=\frac{1}{2}(\mathrm{e}^{it}+\mathrm{e}^{-it}),$$ for $$u \in C^2(D)\cap C(\overline{D}).$$ ...
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55 views

Is it possible to solve this set of equations?

Let's have system of equations: $$ \tag 1 [\nabla \times \mathbf E ] = -\frac{\partial \mathbf B}{\partial t} , $$ $$ \tag 2 [\nabla \times \mathbf B] = \sigma \mathbf E + A(\mu \mathbf K + C \mathbf ...
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1answer
42 views

What are difference among natural boundary, exit boundary, regular boundary and killing boundary??

In the paper i'm reading, they used the terminologies, natural boundary, exit boundary, regular boundary and killing boundary. I can't find the difference of them and definition of them. Tell me ...
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21 views

Numerical Overflow in Dirichlet Boundary Value Problem On High Dimensional State

I am using multigrid methods to solve a quasilinear parabolic pde with Dirichlet boundaries. The problem is too long to reproduce here, but my question is more practical than theoretical: The state ...
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52 views

A particular case of Gelfand triple

I am currently working on hyperbolic equations on bounded domains. For this reason, I am considering in particular functions $u$ such that: $u \in L^2(0;T;H^1_0(\Omega)), \quad u' \in ...
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72 views

A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
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2answers
375 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \frac{1}{2} ...
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1answer
74 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
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24 views

Estimates for the wave equation

Spose $ u $ solves the wave equation on $ U \subset \mathbb{R}^3 $ with initial conditions $ u (x, 0) = g(x)$ and $ u_t(x,0) = h(x)$, where lower script indicates partial differentiation. Then we have ...
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1answer
34 views

Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
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1answer
103 views

How to solve parabolic equation via implicit Euler in 2 dimensions?

I have the following parabolic equation: $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} $$ over domain $(x,y)\in [0,10] \times [0,10]$ ...
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1answer
42 views

Laplace equation - PDE

For the PDE:$$u_{xx}+u_{yy}=0$$ $$u(0,y)=\sin\pi y, \ u(1,y)=0$$ $$u(x,0)=u(x,1)=0.$$ I have that $u(x,y)=X(x)Y(y)$, then $$-Y(y)=\mu Y(y), \ Y(0)=Y(1)=0$$ $$X''(x)=\mu X(x), \ X(1)=0.$$ Thus, ...
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42 views

How to deal with $u_{ttt}$ in derivatives estimates of $u_{tt}$ if $u_{ttt}$ is not defined?

Suppose we have proved the equation $$u_{tt}-u_{xx}=0\quad\text{in}\quad(0,T)\times(0,\ell)$$ with some boundary and initial conditions has an unique solution $u\in C^2(0,T,H^2(0,\ell))$ and we need a ...
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73 views

How to interpret dual space convergence in norm $L^2(0,T;V^*)$ (sobolev--bochner spaces)

Let $V \subset H \subset V^*$ be a Gelfand triple. Define $W = \{ u \in L^2(0,T;V) \mid u' \in L^2(0,T;V^*)$. A result is that $C^\infty([0,T];V)$ is dense in $W$. So given $w \in W$, there exists ...
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1answer
63 views

Analytic solution for a type of PDE systems

Peace be upon you, I have the following system of partial differential equations \begin{align*} \begin{cases} \frac{\partial}{\partial a}S(a,b,c,d)=f_1(a)\\ \frac{\partial}{\partial ...
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53 views

How can two PDE's have the same classical solution, but different weak solutions?

For example: The implicit solution for the inviscid Burgers' Equation, in two forms: $u_y+(\frac{1}{2} u^2)_x = 0$ and $(u^2)_y+(\frac{2}{3} u^3)_x = 0$ share the same smooth solutions, but they ...
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75 views

Derivatives of Differential operators

Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that: \begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y ...