0
votes
0answers
21 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
27 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 ...
0
votes
0answers
6 views

Modified Symplectic Euler

Simple harmonic motion: $y'= -z $, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
1
vote
1answer
29 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 ...
0
votes
0answers
20 views

A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses \begin{equation} y_{n+1}-y_{n}=hf_{n} \tag P \end{equation} as predictor and \begin{equation} ...
1
vote
0answers
25 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
33 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
34 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
16 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
44 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
80 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 ...
0
votes
0answers
55 views

What is changed in Navier-Stokes equations after Otelbaev's proof?

What is changed in Navier-Stokes equations by Otelbaev's Strong solution of Navier-Stokes equations? does it make any change to Navier-Stokes formulation?
3
votes
2answers
183 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
63 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
0answers
18 views

Find the value of the origin inside the unit ball.

Here, $\Delta$=0 inside the unit ball in $\mathbb{R}^{3}$ and $u(1,\varphi,\theta)=sin^{2}(\varphi)$. Where is the value of u at the origin? I am not sure if $\Delta=0$ is suppose to be $\Delta u = ...
0
votes
0answers
39 views

Harmonic function in the plane. Show for any point P…

Let f(x,y) be a harmonic function in the plane. Show that for any point P, $$f(P)= \frac{1}{\pi r^{2}} \int\int_{Dr(P)} f$$ Here, Dr(P) is a disc of radius r centered at P. So I know that the ...
0
votes
2answers
43 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
25 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
62 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
60 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
39 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
35 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
29 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
46 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
42 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
573 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
75 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
30 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
31 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
38 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
27 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
48 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 ...
1
vote
1answer
28 views

For which constants does the following converge to a delta function?

let $g_n(x,y)=\frac{c_n}{1+n^2(x^2+y^2)}$ for which constants, $c_n$ , does the function converge to a delta function as n becomes arbitrarily large? My intitial thought, corroborated by plots in ...
0
votes
0answers
28 views

Fourier Sine Transform Identity Relation through Integration by Parts

This is purely for my own recreational interest. I've spent the last few days trying to demonstrate to myself that the Fourier Sine Transform and the inverse Fourier Sine Transform return their ...
0
votes
1answer
50 views

Energy for the 1D Heat Equation

So consider the heat equation on a rod of length $L$, $u_t (x,t) = c^2 u_{xx} (x,t)$, $\forall (x,t) \in [0,L]$ x $\mathbb{R}^+ $, and the energy at time $t$ defined as, ...
3
votes
2answers
87 views

Laplacian identity.

Consider $f$ and $g$ smooth functions. How to prove the following identity: $$\Delta\left(\frac{f}{g}\right)=\frac{1}{g}\Delta f-\frac{2}{g}\nabla\left(\frac{f}{g}\right).\nabla g-\frac{f}{g^2}\Delta ...
10
votes
1answer
95 views

Non-divergence form of a 2nd order PDE

This might be a trivial question but I'm very rusty with regards to calculus and am new to PDEs. How would you write the following second order quasilinear equation in it's non-divergence form: The ...
0
votes
1answer
34 views

Solving an initial value problem (PDE)

$$∂u/∂x + ∂u/∂y = 1,$$ $$u(x,0) = \mathrm{e}^x$$ My prof hasn't explained how to solve these very well. I think it has something to do with the method of characteristics, but I'm not entirely sure ...
3
votes
2answers
101 views

Solution of a partial differential equation.

Find $u$ if $$\dfrac{\partial^2 u}{\partial x^2} = 6xy, \,\,u(0,y) = y, \,\,\dfrac{\partial u}{\partial x}(1,y)=0.$$ I have tried by laplace transformation $$\displaystyle ...
2
votes
1answer
67 views

Symmetry in partial derivatives.

I was wondering how the relationship $$x_j \partial_i f(x) = x_i \partial_j f(x)$$ means, that a function has rotational symmetry? I mean with rotational symmetric, that the value of $f$ at a point ...
0
votes
1answer
29 views

Canonical Separation of variables

Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$? Insert proper grammar here.
1
vote
1answer
48 views

How to see that solution of PDE increases with time

How do I tell if the solution of $u_t=au_{xx}+bu$ on $[0,1]\times(0,T]$ and $a,b>0$ is increasing in time? $u(x,0)=g(x), u(0,t)=u(1,t)=0$.
0
votes
0answers
26 views

Differentiation - Change of region

here's a small lemma, found in a book on Partial Differential Equations, which I need help on.: For $0 \le r \le R$, let $B_r=\{x:|x| \le R - r\}$. Show that if $f$ is a continuous function on ...
0
votes
1answer
18 views

For $\theta_t=\theta_{xx}.$ Find by substitution solutions of the form $\theta(x,t) = f(t)\exp[-(x+a)^2/4(t+b)] $ for arbitrary $a$ and $b$.

Consider the diffusion equation $$\theta_t=\theta_{xx}.$$ Find by substitution solutions of the form $$\theta(x,t) = f(t)\exp[-(x+a)^2/4(t+b)] $$ for arbitrary $a$ and $b$. Now I used the ...
2
votes
0answers
57 views

Change of variables for linear differential operators

I am trying to find an expression for a change of variables (invertible and $C^{\infty}$) of a linear differential operator in $\mathbb R ^d \newcommand{\t}{\tilde} \newcommand{\p}{\partial}$. i) ...
0
votes
2answers
214 views

General solution of $(u.u_{xy}) - (u_x.u_y) = 0$

$(u.u_{xy}) - (u_x.u_y) = 0$ I'm in bad need to answer this question. Please help. I considered $D= (d/dx)$ & $D'= (d/dy)$ so I came to $(U^2)DD'-(U^2)DD'=0$ but its already obvious that these ...
1
vote
1answer
44 views

Show that $k \bigtriangleup T+ \nabla k \cdot \nabla T=0$ leads to $\bigtriangleup (k^{1/2} T)- \frac{\bigtriangleup(k^{1/2})}{k^{1/2}}k^{1/2}T=0$

I am trying to see the following derivation given in the book 'Kernel Functions and Elliptic Differential Equation in Mathematical Physics' by S. Bergman and M. Schiffer. We deduced the differential ...
2
votes
1answer
52 views

Partial differentials vs normal differential (notation question/clarification only)

In physics, it seems like the use of $\dfrac{dy}{dx}$ and $\dfrac{\partial y}{\partial x}$ are used somewhat interchangeably. My understanding is that, technically $\dfrac{dy}{dx}$ is only ...
1
vote
2answers
41 views

line of sources for differential equation

It is well known that for representation of point source with some intensity $q(t)$ in some PDE we use delta-function $\delta({\bf r} - {\bf r}')$. If problem requires usage of line of sources which ...
1
vote
1answer
52 views

Please explain. Really I dont understand and I need to learn. Pde: : example of finding particular integral

When we look at the solution part, there is a statement The PI of the given PDE is obtained as follows After the statement, I dont really understand all of the calculation. Espacially, After the ...