# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, $\displaystyle dp=0$ ...
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### Solving PDEs using Charpit's Method

To Solve : $\displaystyle 1+p^2=qz$ I have solved this equation till auxiliary equation: $\displaystyle \frac{dp}{-pq}=\frac{dq}{-q^2}=\frac{dz}{2p^2-qz}=\frac{dx}{2p}=\frac{dy}{z}$ ...
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### 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 ...
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### 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 ...
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### 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 $u(0,y)=2\sin\dfrac{\pi y}{3}$ $\dfrac{du}{dx}(a,y)=0$ ...
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### 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 ...
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(i) Let $F(x) = (-1/2) e^{-\mid x\mid}$. Show that $0=\int_{\mid x \mid >\epsilon} (F''-F) \varphi =\int_{\mid x \mid >\epsilon} (\varphi'' -\varphi)F + \int_{\mid x \mid >\epsilon}F''\varphi ... 0answers 60 views ### Using Fourier Transforms to Solve the Heat Equation PDE In Infinite Three Dimensions Problem: Using Fourier transforms, solve for$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 ... 1answer 98 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 $$\left\{ \begin{array}{rl} u_t - \Delta u =0 & \text{in } U_T \\ u=0 & ... 1answer 116 views ### Linear evolution equation inequality (Evans chapter 7 problem 9) I'm trying to prove an inequality from Evans' PDE book (Chapter 7 Problem 9). It's inequality (54) in \S7.1.3 and (59) in \S7.2.3. Problem: Given u \in H^2(U) \cap H_0^1(U) there exists ... 1answer 38 views ### Using the Spherical mean, show that V(x)=\frac{|y^2|-|x|^2}{|y-x|^N} is hamonic over R^N\setminus\{y\}. Let y\in R^N. Show that V(x)=\frac{|y^2|-|x|^2}{|y-x|^N} is hamonic over R^N\setminus\{y\}. This is an exercise of my first course in PDE. My doubt I know that the Laplace's equation is: ... 0answers 24 views ### Show there is a single harmonic function u_1 over \Omega wich coincides with u in \Omega_+ Let \Omega a open pathwise-connected subset from R^N such that$$ x=(x_1,...,x_{N-1},X_N)\in \Omega \Rightarrow (x_1,...,x_{N-1},-X_N) \in \Omega $$Let u a harmonic function over ... 1answer 36 views ### Show the equality holds for any x \in [0, \pi] We are considering a 2\pi periodic function defined on x\in(-\pi,\pi) by$$f(x) = \pi - x, 0<x<\pi $$and 0 otherwise. I already computed the full Fourier series is equal to:$$f(x) = ... 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, ... 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 $$\frac{\partial T}{\partial t}= k ... 0answers 34 views ### Solving u_{yy} + (2-x)u_y - 2xu = 1 I want to solve the pde$$ 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 ... 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 ...
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. ...