0
votes
0answers
20 views

Solving an integral equation in general

I have an integral equation such that $$\int_t^Tf(s)g(s,t)~ds=h(t)$$ where $g$ and $h$ is given. we want to know function $f$ explicitly. As I know, this type of question is about the integral ...
0
votes
1answer
8 views

Change of variables from intinite to bounded support.

I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie $x \in ...
1
vote
0answers
36 views

problem of computing limit

The problem is to prove the following for $n \geq 3$ $$u(0)=\frac{1}{n\alpha (n) r^{n-1}}\int_{\partial B(0,r)} g dS +\frac{1}{n(n-2)\alpha (n)} \int_{B(0,r)} (\frac{1}{|x|^{n-2}} - ...
0
votes
1answer
51 views

Help needed in solving a differential equation

Please help me in solving: $$a^2z+\frac{\partial^2z}{\partial x^2}-\frac{\partial^2 z}{\partial y^2}=0$$ ($a$ is a constant) I plugged this in Wolfram Alpha and it outputs that this is a second ...
0
votes
0answers
61 views

Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
5
votes
0answers
85 views

Solve PDE by getting two ODEs

My goal is to solve this PDE for $f:[-1,1] \times \mathbb{R}_{\ge 0}\rightarrow \mathbb{C}$ $$ \partial_t f(x,t) = -\partial_x^2 f(x,t) + g(t)V(x)f(x,t).$$ I would consider this PDE to be solved if ...
1
vote
1answer
45 views

proof of coarea formula for n dimensional hypersurface in $R^n$

$f:R^n \rightarrow R$ be continuous and summable. please give the proof for these formulas $\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr$ $\frac{d}{dr}\int_{ ...
0
votes
4answers
47 views

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$ Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$

$f(x,y) = g(\sqrt{x^{2}+y^{2}})$. Prove that f is differentiable at $(0,0)$ iff $g'(0)=0$ This was a question on my midterm a few days ago. I've been thinking about it for a while and still cannot ...
1
vote
0answers
20 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
1
vote
1answer
26 views

derivation of transport equation

The amount of pollutant in the interval at time $t$ is $M=\int_0^bu(x,t)dx$ .At later time t+h,the same molecules of pollutant have moved to the right by $ch$ centimetres. ...
0
votes
1answer
69 views

An equality from Fritz John's paper

Prove: $\frac{\rho}{(4\pi)^2}\int_{|\xi|=1}d\omega_\xi\int_{|\eta|=1}f(x^0+r\xi+\rho\eta)d\omega_\eta=\int_{|r-\rho|}^{r+\rho}\frac{\lambda}{8\pi ...
1
vote
1answer
61 views

Partial derivatives-Why does this stand?

In my notes there is the following: $$u_{\xi \eta}=0 \Rightarrow \left\{\begin{matrix} u_{\xi}=0 \Rightarrow u=g(\eta)\\ u_{\eta}=0 \Rightarrow u=f(\xi) \end{matrix}\right.$$ I haven't understood ...
1
vote
0answers
54 views

prove this step in poisson equation

$\phi(x)$ is harmonic in $C^2$ in proof for solution of poisson's equation($- \Delta u=f$) ,this step came it maybe very fundamental but I am not getting it $u(x)= \int_{R^n}\phi(x-y)f(y)\mathrm dy ...
1
vote
1answer
27 views

smoothness of harmonic functions confusion in proof

can you explain that last two steps? how that $$na(n)r^{n-1}$$ disappeared in next integral?I noticed the transformation of variables but still not able to figure out properly.
2
votes
2answers
41 views

An object is travelling in a straight line. Its distance, s meters, from a fixed point at time t seconds is given by the expression

$$s=t^3−t^2−6t$$ a) Find ds/dt when t=3 and interpret this result. b) Find d^2s/dt^2 when t=3 and interpret this result. c) Find the time in seconds when the velocity is 2m/s (d) Using the ...
0
votes
1answer
15 views

extension of Legendre transform of a function to a larger domain

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$ and $u_0$ be a uniformly convex function define on $\Omega$. Suppose the gradient of $u_0$ maps $\Omega$ into a subdomain of $\Omega^*$, i.e. ...
0
votes
1answer
23 views

Calculating Legendre Transform

Let $\Omega,\Omega^*$ be bounded domain in $\mathbb{R}^n$, $B_r(y_0)$ is a small ball of radius $r$ center at $y_0\in\Omega^*$, in $\Omega^*$ define a function $\psi(y)=-\sqrt{(r^2-|y-y_0|^2)}$ on ...
3
votes
0answers
57 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
0
votes
1answer
28 views

Finding solution to a unidirectional nonlinear wave equation

I can do the parts a), b) and c) and find that in part c) that the condition in which the solution will break down is when $1+tf'(x-tu)=0$ However I am unable to part d) I tried ...
0
votes
0answers
38 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
0
votes
0answers
27 views

Laplace's equation boundary conditions

I am supposed to determine a solution from the following boundary conditions in a rectangular area $V$. $f(0,y)=f(x,a) = f(x,0)=0$ and $f(b,y) = C\ sin(\frac{\pi}{a}y)$, for $a,b \in \mathbb{R}$. I ...
2
votes
2answers
104 views

Greens function of 1-d forced wave equation

[ORIGINAL PROBLEM] You are given hat the Green's function $g(x,t,\xi, \phi)$ is $\frac{\partial^2g}{\partial t^2} - \frac{\partial^2g}{\partial x^2}=\delta(t-\tau)\delta(x-\xi)$ with ...
-1
votes
1answer
39 views

Explain the methodology behind the answer to this first-order PDE question

QUESTION: Solve PDE $$ u_t + t^3 u_x = u$$ Conditions: $u(t,0) = t$, $u(0,x) = 1 - e^{-x}$ ANSWER: For $x \le\frac{t^4}{4}$ we have $$\frac{dx}{dt} = t^3; x(0) = x_0 \to x(t) = \frac{t^4}{4} + ...
0
votes
0answers
27 views

How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
0
votes
1answer
34 views

Integrating the Gaussian kernel with absolute value

How do I integrate: $$\dfrac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}|y|e^{\frac{-(\xi-y)^2}{4t}}dy,$$ in terms of of the error function, erf$(x)=\dfrac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$. I have ...
1
vote
1answer
36 views

Implicit Euler Scheme and stability

Find the fixed points of the implicit Euler scheme \begin{equation} y_{n+1}-y_{n}= hf(t_{n+1},y_{n+1}) \end{equation} when applied to the differential equation $y'=y(1-y)$ and investigate their ...
1
vote
0answers
27 views

Establishing a certain bound

If I know the bound of the certain derivatives: $$|\dfrac{\partial^k}{\partial x^{k_1}_1\cdots \partial x^{k_n}_n}V(x)|\leq M\epsilon^{-k} \exp(-m \epsilon^{-1}g(x)),$$ where $k=k_1+\cdots+k_n$ and ...
1
vote
1answer
39 views

Finding the uniquely determined region of a PDE

(a) Solve the equation $yu_x+xu_y=0$ with the condition $u(0,y) = e^{-y^2}$. (b) In which region of the xy plane is the solution uniquely determined? I did the first part but I don't understand ...
0
votes
0answers
38 views

Heat equation form $u_t = u_{xx} + f(x)$.

What is the Heat equation form $u_t = u_{xx} + f(x)$.What is this $f(x)$ means. In the question here it is used. (That is it hasn't asked to change $u_t = \alpha^2 u_{xx}$ as $ u_t = u_{xx} + f(x)$) ...
0
votes
0answers
20 views

Solving PDE using line integral

This question is in regard to the question here In the answer provided there I understand upto the point $ u = \frac 1{a^2+b^2} \int f dt + g(p) \\ $ After this wha tI did in order to compute the ...
1
vote
0answers
52 views

variational derivative

Let $\Omega \subset \mathbb{R}^n,\ n=1,2 \mbox{ or } 3$. Define the following energy $$E=\int_{\Omega} \frac{1}{\varepsilon}\left[f(u)+\frac{\varepsilon^2}{2}|\gamma(n)\nabla u|^2\right]\,dx$$ ...
5
votes
1answer
97 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
3
votes
2answers
185 views

How find this $\frac{yf_{y}-z}{f_{x}}+\frac{xf_{x}-z}{f_{y}}-xf_{x}-yf_{y}+x+y+z=C$ solution

In plane $R^3$,Find $z=f(x,y)$, such the length of the portion of any tangent line to the astroid $$z=f(x,y)$$ cut off by the coordinate axes is constant $C$, This problem is from this post (when I ...
3
votes
1answer
76 views

Solving Wave Equation with Initial Values

I am trying to solve the wave equation: $u_{tt}$ = $u_{xx}$ With initial values: $u(x,0) =\begin{cases} x^3 - x, &\text{for }|x|\le 1,\ \\0, &\text{for }|x|\ge1\end{cases}$ $u_t(x,0) ...
0
votes
2answers
48 views

Solution of the wave equation $\frac{\partial^2 U}{\partial t^2} = c^2\Delta U$?

how to show that $$U(X,t) = r^{−1}\ g(ct − r)\ ,$$ where $r = |X|$, and $X = (x,y,z)$, is a solution of the wave equation in dimension 3, i.e., $U$ satisfies $$ \frac{\partial^2 U}{\partial t^2} = ...
1
vote
1answer
26 views

Could someone walk me through this pde?

$$\frac{\partial^2 y} {\partial x^2} = \lim_{\Delta x \to 0} \frac {1} {\Delta x} \left(\frac{\partial y} {\partial x}_x - \frac{\partial y} {\partial x}_{x+ \Delta ...
1
vote
2answers
68 views

How to prove that $\frac{d}{dt} \left( \int_{a}^{s(t)}f(x,t)dx +\int_{s(t)}^{b}f(x,t)dx \right) = \frac{ds}{dt}(f_{a} - f_{b}) $…

let $s(t) \in (a,b)$ and the function $f(x,t)$ is continuous except at $x=s(t)$, how to prove $$\frac{d}{dt} \left( \int_{a}^{s(t)}f(x,t)dx +\int_{s(t)}^{b}f(x,t)dx \right) = \frac{ds}{dt}(f_{a} - ...
1
vote
1answer
62 views

what is the solution of this question on partial differential equation

what is the solution of this $$\frac{1}{D^2_x-D^2_y}\sin(x-y)$$ please solve, should I move using $\exp(x-y)$ or any other method is there to solve this for particular integral
0
votes
1answer
42 views

is this intengral bounded?

Im just stuck beacause I dont know if this integral in bounded, I was trying to make a change of variable but I cant get to anything: (edited what need is that f is bounded for a fixed x) ...
0
votes
1answer
45 views

Sign of Laplacian at critical points of $\mathbb R^n$

Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative ...
0
votes
1answer
35 views

How to separate $x^2X''v+xX'v+Xv_{yy}+x^2Xv_{zz} = 0$

I have the following partial differential equation: $$x^2X''v+xX'v+Xv_{yy}+x^2Xv_{zz} = 0,$$ where $X = X(x)$ and $v = v(y,z)$. How to separate it? Thank you for any help! From the book: ...
0
votes
1answer
49 views

Separation of variables with three independent variables

I have the following differential equations problem: Derive sets of three ordinary differential equations from the following partial differential equation by separation of variables: ...
1
vote
1answer
48 views

Understanding partial differential equation requirement

I'm reading about separation of variables in my Fourier series book and there is one requirement in a problem I don't understand. Here it is: I don't understand, why can't the $\sqrt{-A}$ take a ...
19
votes
1answer
594 views

Was Euler right?

We have a differential equation $$ y + y' = f(x) $$ and assume $f$ is infinitely differentiable. And we want to find particular solution. Then,I set $$ y_p = f(x)-f'(x)+f''(x)...., $$ i.e., ...
0
votes
1answer
79 views

upper bound of some weird function with an exponencial

well im trying to find the proof of the following statement, but I cant go foward anymore: Let $\Omega= \mathbb{R}^{n} \times (0,\infty) $ $$w(x,t)= \sum^{\infty}_{k=0} ...
0
votes
0answers
35 views

Integral with a jump (Leibniz rule?)

I'm reading a book on pde's and don't understand this: Consider ajump in $\rho$ at $x=x_s$ then: $\frac{d}{dt} [\int_a^{x_s} \rho(x,t)dx +\int_{x_s}^b\rho(x,t)dx]$ $=$ $ [\int_a^{x_s} ...
2
votes
1answer
33 views

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$

Discuss the existence and uniqueness of solutions of the equation $X' = X^{a}$ where $1>a > 0$ and $x(0) = 0.$ First, I notice that this equation is not differentiable at x = 0. Therefore, the ...
1
vote
1answer
42 views

$C^\omega(\Omega)\cap C^\infty_0(\Omega)$.

Let $\Omega$ denote an open connected set in $\mathbf{R}$ (AKA open interval). Is it true and how can we prove it that $C^\omega(\Omega)\cap C^\infty_0(\Omega)$ consists of the zero function alone, ...
0
votes
1answer
38 views

non-homogeneous boundary value prblem - help!

I am studying boundary value problems and I have worked out the partial derivatives and worked out my $V*$ and its constants. Now I have a heat equation to solve using $X(x)T(t)$. Using the ...
0
votes
1answer
53 views

Heat Equation - please help

I've been studying the heat equation and separation of variables and came accross this practise question - I wanted to know if I use the same method I use usually is correct i.e. choose a constant ...