0
votes
0answers
40 views

Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
0
votes
0answers
36 views

checking the solution of PDE

Let $u(x)$ be an entire solution of $\Delta u = 1$ on $R^{n}$, $n>1$. If $u(x)$ is also convex, that is $(D^{2}u(x))$ is non-negative definite for all $x$. Then $u(x)$ is given be a quadratic ...
1
vote
0answers
44 views

Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
0
votes
0answers
33 views

Estimation of solution to $u_t=u_{xx}+x^3u_x$ using integrals

Let $$u_t=u_{xx}+x^3u_x$$ With: $$u(0,x)=u_0(x)$$ $$u(t,0)=u(t,l)=0$$ Find an energy approximation of $u$ on $(0,T) \times(0,l)$. By multiplting by $u$ we get: ...
0
votes
2answers
48 views

Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases} $$ doesn't admit weak solutions. I'm proceeding by ...
1
vote
0answers
39 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}} - ...
1
vote
1answer
22 views

when is : $\int_{R^n\backslash\{0\}}|{f(x) \over |x|^{n-2}}| dx < \infty$ given $f$ is summable and continuous?

While trying to complete the calculations in the question : Value of $u(0)$ of the Dirichlet problem for the Poisson equation I came across a point where I need clarification. $$F(r) := ...
0
votes
0answers
35 views

Solve this heat equation problem.

$$ u_t-f(t)u_{xx}=0 \textrm{, over } \mathbb{R}^N\times]0,\infty[ \\ u(x,0)=u_0(x)\textrm{, over } x\in \mathbb{R}^N, u_0\in S(\mathbb{R}^N) $$ where $u\in C^2(\mathbb{R}^N\times]0,\infty[)\cap ...
0
votes
2answers
62 views

To solve a PDE by separation of variables $\displaystyle 4\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=3u$

To Solve: $\displaystyle 4\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=3u$ My attempt: Let $\displaystyle u=X(x)Y(y)$. So, $\displaystyle 4X'Y +XY'=3XY$ Separating the variables, ...
1
vote
1answer
10 views

To solve a PDE by separation of variables

To Solve: $\displaystyle py^3+qx^2=0$ where $p = \dfrac{\partial z}{\partial x}$, $q = \dfrac{\partial z}{\partial y}$. My attempt: Let $\displaystyle z=X(x)Y(y)$. So, $\displaystyle ...
1
vote
1answer
26 views

To solve a non-homogeneous linear PDE

To solve a non-homogeneous linear PDE $\displaystyle \frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial x \, \partial y}+\frac{\partial z}{\partial y}-z=e^{-x}$ My Attempt: Putting ...
0
votes
0answers
5 views

non-homogeneous linear PDE $\displaystyle (D+D'-1)(D+2D’-3)z=4+3x+6y$

$\displaystyle (D+D'-1)(D+2D’-3)z=4+3x+6y$ My Attempt: C.F.= $\displaystyle e^{x}\phi_1(y-x)+e^{3x}\phi_2(y-2x)$ For P.I., $\displaystyle (1-(D+D'))(3-(D+2D’))z=4+3x+6y$ P.I.= $\displaystyle ...
0
votes
1answer
23 views

To Solve $\displaystyle q+xp=p^2$ using Charpit's Method

p = ∂z/∂x q = ∂z/∂y My Attempt: $\displaystyle \frac{dx}{-(x-2p)}=\frac{dy}{-1}=\frac{dz}{-p(x-2p)-q}=\frac{dp}{2p}=\frac{dq}{0}$ So, do I get.. $\displaystyle ...
0
votes
0answers
5 views

To Solve $\displaystyle p(p^2+1)+(b-z)q=0$ using Charpit's Method

p = ∂z/∂x q = ∂z/∂y My Attempt: $\displaystyle \frac{dx}{-3p^2-1}=\frac{dy}{b-z}=\frac{dz}{-p(3p^2+1)-q(b-z)}=\frac{dp}{-pq}=\frac{dq}{-q^2}$ Now, I can't think ...
0
votes
0answers
13 views

To Solve $\displaystyle pxy+pq+qy=yz$ using Charpit's Method

p = ∂z/∂x q = ∂z/∂y My Attempt: $\displaystyle \frac{dx}{-py-q}=\frac{dy}{-p-y}=\frac{dz}{-p^2y+qy}=\frac{dp}{0}=\frac{dq}{px+q+qy}$ Now, $\displaystyle dp=0 $ ...
0
votes
0answers
14 views

Solving PDEs using Charpit's Method

To Solve : $\displaystyle 1+p^2=qz$ I have solved this equation till auxiliary equation: $\displaystyle \frac{dp}{-pq}=\frac{dq}{-q^2}=\frac{dz}{2p^2-qz}=\frac{dx}{2p}=\frac{dy}{z} $ ...
0
votes
1answer
29 views

Solve initial value problem (C.S.I.R)?

The initial value problem is $$ \frac{\partial u}{\partial t} +x\frac{\partial v}{\partial x} = x, \ \ 0 \leq x \leq 1, \ \ t > 0 \ \ and$$ $$ u(x,0) = 2x \ \ has$$ a unique solution $u(x,t$ ...
1
vote
0answers
33 views

Solving $\displaystyle p(1-q^2)=q(1-z)$

To Solve: $\displaystyle \frac{\partial z}{\partial x} \left \{1-\left(\frac{\partial z}{\partial y}\right)^2\right \}=\frac{\partial z}{\partial y}(1-z)$ My Attempt: Assume $z$ is a function of ...
2
votes
1answer
32 views

Solving $\displaystyle x(z-2y^2)\frac{\partial z}{\partial x}=\left(z-\frac{\partial z}{\partial y}\right)(z-y^2-2x^3)$.

To Solve: $$\displaystyle x(z-2y^2)\frac{\partial z}{\partial x}=\left(z-\frac{\partial z}{\partial y}\right)(z-y^2-2x^3)$$ Forming the subsidiary equations: $\displaystyle ...
0
votes
1answer
18 views

To Solve a linear PDE of first order 2

To Solve: $\displaystyle y^2\frac{\partial z}{\partial x}-xy\frac{\partial z}{\partial y}=x(z-2y)$ My attempt: Forming the subsidiary equations: $\displaystyle ...
0
votes
1answer
20 views

Form a PDE by eliminating the arbitrary function

Form a PDE by eliminating the arbitrary function (i) $\displaystyle F(xy+z^2,x+y+z)=0 $ (ii) $\displaystyle F(x+y+z,x^2+y^2+z^2)=0 $ How do I proceed for any of these?
2
votes
1answer
22 views

Doubt on solution of PDE

To Solve: $\displaystyle (x^2-y^2-z^2)\frac{\partial z}{\partial x}+2xy\frac{\partial z}{\partial y}=2xz$ Subsidiary equation: $\displaystyle \frac{dx}{x^2-y^2-z^2}=\frac{dy}{2xy}=\frac{dz}{2xz}$ ...
1
vote
2answers
37 views

To Solve a linear PDE of first order

To Solve: $\displaystyle \cos(x+y)\frac{\partial z}{\partial x}+\sin(x+y)\frac{\partial z}{\partial y}=z$ My attempt: Forming the subsidiary equations: $\displaystyle ...
1
vote
1answer
46 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_{ ...
1
vote
1answer
32 views

To Solve a PDE by direct integration

$\displaystyle \frac{\partial^2 z}{\partial x^2}=a^2z $, given that when $\displaystyle x=0, \frac{\partial z}{\partial x}=a\sin y$ and $\displaystyle \frac{\partial z}{\partial y}=0 $ The solution ...
1
vote
2answers
69 views

How do i solve this simple PDE? $ \partial_x u + \partial_y u =0$

Can you please give me a hint on how to start this differential equation? $u$ is a two variable real function. $$ \partial_x u + \partial_y u =0$$ Thank you in advance!
2
votes
2answers
34 views

Solution of a PDE

Solve $\displaystyle \frac{\partial^2 z}{\partial x^2}+z=0$, given that when $x=0$, $z=e^y$ and $\displaystyle \frac{\partial z}{\partial x}=1$. My Attempt: Integrating w.r.t x twice (keeping y ...
1
vote
2answers
17 views

Solve this PDE using a change of variable.

Solve the following PDE using a change of variable: $$ \alpha^2 \dfrac{\partial^2 z}{\partial x^2} - \beta^2 \dfrac{\partial^2 z}{\partial y^2} = 0$$ This is my attemp: Let the following change of ...
2
votes
4answers
84 views

Solving Laplace's Equation - weird boundary conditions?

The potential is given by: $$V = \sum_{n=0}^{\infty} \left[a_n r^n +b_nr^{-(n+1)}\right] P_n(cos \theta) $$ I want to find potential for $r \geq a$ using th definition $I_n = \int_0^1 P_n(x) \space ...
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 ...
1
vote
0answers
26 views

Finding the homomorphism that links the linear part of a dynamical system to the nonlinear part.

here is a picture of my problem Basically what i have is that i was told i could find this homomorphism by doing the following ...
0
votes
0answers
55 views

solution to Laplace Equation

I am currently working on a homework problem for my boundary value problem class. Solve the Laplace equation with the following conditions $u(0,y)=2\sin\dfrac{\pi y}{3}$ $\dfrac{du}{dx}(a,y)=0$ ...
1
vote
1answer
50 views

Exists $C$ constant: $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$

Show that there exists $C$ constatant such that $||f||_{L^{\infty}(\mathbb{R})}\leq\{ ||f||_{L^2(\mathbb{R}}+||f'||_{L^2(\mathbb{R})} \}, \forall f\in S(\mathbb{R})$. This is a question in my ...
1
vote
1answer
49 views

Show this inequality in the “heat equation” problem.

Let $(u,t)$ the $C^2$ solution of the equation $$ u_t=u_{xx}+u, \textrm{ over } [0,a]\times[0,T]\subset \mathbb{R}^2 $$ where $T>0$ Show that $$ \max\limits_{[0,a]\times[0,T]} |u| ...
1
vote
1answer
34 views

Electromagnetic fields and Laplace equations along a square

I'd like to solve Laplace equation satisfying the following BCs: $$\phi(x,y=0)=0$$ $$\phi(x=0,y)=0$$ $$\phi(x,y=1)=9\sin(2\pi x)+3x$$ $$\phi(x=1,y)=10\sin(\pi y)+3x$$ where $0\leq x,y\leq 1$. I have ...
0
votes
1answer
41 views

Helmholtz equation in Laplace equation of cylinder

I have a cylinder of height $H$ and radius $a$. I am about to find $u(\rho, \phi, z)$ that solves the equation $$\Delta u = 0 $$ with the given boundary conditions $$u(a, \phi, z) =0,$$ $$u(\rho, ...
0
votes
0answers
24 views

Maximum and Minimum principle of Laplace equation

I want to prove that if $u=u(x, y, t)$ is a solution of the equation: $$\frac{\partial u}{\partial t} = \Delta u,\;\;\;\;\;\;\ where \; \Delta u = \frac{\partial ^2}{\partial x^2} + \frac{\partial ...
1
vote
1answer
24 views

Solve PDE by separation (Fourier-) method

I have to solve the following pde by a separation approach: $$ x^2 u_{xx} + u_{yy} - xu_x - u = 0. $$ So I put $u(x,y) = g(x) f(y)$, substituting yields $$ x^2 g''(x) f(y) + g(x) f''(y) - x g'(x) ...
0
votes
0answers
21 views

Green's first or second or both identities?

(i) Let $F(x) = (-1/2) e^{-\mid x\mid}$. Show that $0=\int_{\mid x \mid >\epsilon} (F''-F) \varphi =\int_{\mid x \mid >\epsilon} (\varphi'' -\varphi)F + \int_{\mid x \mid >\epsilon}F''\varphi ...
1
vote
0answers
60 views

Using Fourier Transforms to Solve the Heat Equation PDE In Infinite Three Dimensions

Problem: Using Fourier transforms, solve for $u(x,y,z,t)$, where $$u_t=D\nabla^2 u$$ $$ -\infty<x,y,z<\infty,t>0$$ $$D>0, u(x,y,z,0)=f(x)f(y)f(z)$$ and $u\rightarrow 0$ as ...
2
votes
1answer
97 views

PDE Evans Chapter 7 problem 16

Problem 16 of chapter 7 states Use problem 15 to prove that if $u$ is the semigroup solution in $X=L^2(U)$ of $$ \left\{ \begin{array}{rl} u_t - \Delta u =0 & \text{in } U_T \\ u=0 & ...
1
vote
1answer
113 views

Linear evolution equation inequality (Evans chapter 7 problem 9)

I'm trying to prove an inequality from Evans' PDE book (Chapter 7 Problem 9). It's inequality (54) in $\S7.1.3$ and (59) in $\S7.2.3$. Problem: Given $u \in H^2(U) \cap H_0^1(U)$ there exists ...
0
votes
1answer
38 views

Using the Spherical mean, show that $V(x)=\frac{|y^2|-|x|^2}{|y-x|^N}$ is hamonic over $R^N\setminus\{y\}$.

Let $y\in R^N$. Show that $V(x)=\frac{|y^2|-|x|^2}{|y-x|^N}$ is hamonic over $R^N\setminus\{y\}$. This is an exercise of my first course in PDE. My doubt I know that the Laplace's equation is: ...
0
votes
0answers
24 views

Show there is a single harmonic function $u_1$ over $\Omega$ wich coincides with $u$ in $\Omega_+$

Let $\Omega$ a open pathwise-connected subset from $R^N$ such that $$ x=(x_1,...,x_{N-1},X_N)\in \Omega \Rightarrow (x_1,...,x_{N-1},-X_N) \in \Omega $$ Let $u$ a harmonic function over ...
0
votes
1answer
36 views

Show the equality holds for any $x \in [0, \pi]$

We are considering a $2\pi$ periodic function defined on $x\in(-\pi,\pi)$ by $$f(x) = \pi - x, 0<x<\pi $$ and 0 otherwise. I already computed the full Fourier series is equal to: $$f(x) = ...
2
votes
0answers
44 views

Characteristic curves of 2nd-order PDEs under invertible coordinate transformations

First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know. Given a linear, scalar, ...
-1
votes
1answer
64 views

Heat Equation Steady state question

Say you have a slab of material occupying the region $0\leq x\leq a$. Heat is supplied at a constant unit rate so the temperature $T(x,t)$ satisfies $$\frac{\partial T}{\partial t}= k ...
0
votes
0answers
34 views

Solving $u_{yy} + (2-x)u_y - 2xu = 1$

I want to solve the pde $$ u_{yy} + (2-x)u_y - 2xu = 1 $$ so if I treat $x$ in the coefficients as arbitrary but fixed it is equivalent to solving the ode $$ y'' + (2-x) y' - 2x y = 1. $$ For the ...
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
0answers
27 views

Finite Difference Scheme for a PDE on non-rectilinear coordinates

Consider poisson's equation on the domain $0 \leq x \leq 1$ $0 \leq y \leq H(x)$. Change the coordinates to $\xi=x$, $\eta=y/H(x)$. Construct a FDS that gives a positive definite symmetric matrix. ...