Tagged Questions

2answers
32 views

Is this PDE not solvable using characteristic lines?

My PDE book solved the following equation using the method of characteristic lines: $3u_x - 2u_y + u = x, u = u(x,y).$ I then encountered the following problem in the exercises: $u_x + u_y - u = 0$. ...
2answers
81 views

Solving a wave equation: $a^2\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$

Find product solution $$a^2\frac{\partial^2 u}{\partial x^2}=\frac{\partial^2 u}{\partial t^2}$$ by the method of separation of variables So first off: $$u(x,t)=X(x)T(t)$$ ...
1answer
41 views

Derivation of Lagrange-Charpit Equations

I am working through the derivation of the Lagrange-Charpit equations presented in this Wikipedia article: http://en.wikipedia.org/wiki/Method_of_characteristics#Fully_nonlinear_case I am interested ...
2answers
60 views

Confused over the solution of partial differential equation $xu_x+u_t=0$

Consider, $$\displaystyle x\frac{\partial u}{\partial x}+\frac{\partial u}{\partial t} = 0$$ with initial values $t = 0 : \ u(x, 0) = f(x)$ and calculate the solution $u(x,t)$ of the above ...
0answers
31 views

Dirichlet Eigenvalues of Laplace-Beltrami operator in Hyperbolic space

Consider the hyperbolic half-plane $\mathbb{H}=\lbrace (x,y)\in\mathbb{R}^2: y>0 \rbrace$ with standard Riemannian metric. The Laplace-Beltrami Operator can be written as ...
2answers
30 views

4answers
67 views

Differential equation with the solution of $(1+ax/2)\exp(-ax)$

Is there any linear differential equation which has following solution $$y=(1+ax/2)\exp(-ax)$$ $a$ is constant.
1answer
99 views

My first partial differential equation attempt

have I solved this correctly? My textbook is asking for the relation between $\alpha$ and $\beta$: $$\frac{\partial{u}}{\partial{t}}=\frac{\partial^2{u}}{\partial{x^2}}$$ Textbook's proposed ...
0answers
16 views

Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
0answers
32 views

1answer
38 views

Solution to $u_t+\Delta^2u+\Delta u=0$

Suppose there exists a solution to *$u_t+\Delta^2u+\Delta u=0$ of the form $u(x,y,t)=c(t)e^{i\pi(x/4\pi+y/4\pi)}$. I need to find such a function $c(t)$. Plugging $u(x,y,t)$ into *, I got ...
2answers
29 views

Finding polynomial satistying potential equation and boundary conditions

Can someone help me with this problem? I know that this polynomial is a solution of Poisson's equation.
3answers
52 views

Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
0answers
44 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 ...
0answers
126 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$-u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u$$ where $\lambda$ is a ...
1answer
20 views

How rigorous is multiplying both sides of an eqaution for the differential of a function?

I have to solve this equation: $$-C_0 f + \frac{1}{2}f^2 +\frac{d^2 f}{d X^2}=A$$ where $C_0$ and $A$ are two real nonzero constant; $f:\mathcal{R}\to \mathcal{R}$ I have seen that the person who ...
1answer
25 views

Why are we discarding solutions to this heat equation?

Dirichlet problem on unit disc in polar: $u_{rr} + (1/r) + (1/r^2)u_{\theta\theta} = 0$ $u(1,\theta) = f$ Period in $\theta$ gives $u(r,\theta) = \sum R_n(r) e^{in\theta}$ Inserted into our PDE ...
0answers
25 views

Space-dependent diffusivity and finite-differences

I want to implement a finite difference code of this simple diffusion equation with space-dependent diffusivity: $$\partial_{t}u =D\partial_{x}^{2}u+\partial_{x}D\cdot\partial_{x}u$$ I go for a ...
2answers
76 views

Intuitive interpretation of $\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$

I'm trying to visualize what the following equation is saying: $$\frac{\partial S(a,t)}{\partial t} = -\frac{\partial S(a,t)}{\partial a}$$ where $S$ is a probability-density, but I think you can ...
0answers
51 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 ...
1answer
85 views

linearize a nonlinear ode

Could anyone suggest me how to linearize the following system of nonlinear odes (special attention to (2) \begin{align} -cU'&=-U''+UV\tag{1}\\ -cV'&=-k(k+1)V^{k-1}(V')^2+(k+1)V^k ...
1answer
139 views

Existence and uniqueness for a system of first-order PDE

Let $y$ be a scalar and ${\bf t}=(t_1,\ldots,t_K)$ and \begin{align*} \frac{\partial y({\bf t})}{\partial t_k} &=f_k(y({\bf t}),{\bf t}) \qquad k=1,\ldots,K\\ y(t_{10},\ldots,t_{K0}) &=y_0 ...
2answers
44 views

Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y)$$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
2answers
83 views

What do mathematicians mean by “analytical solution of an equation”?

Given a PDE equations of the form: $\dfrac{\partial}{\partial t} u(t,x) = \left(\hat{L}+\hat{N_u}\right)u(t,x) \;\;\;\;\;\;\hspace{10mm}(**)$ where $\hat{L}$ is a linear operator and ...
1answer
28 views

What is the difference between bounded boundary and a bounded domain?

This I found while studying sobolev spaces. In some results they have assumed that $\Omega\subseteq \mathbb{R}^n$ is with bounded boundary. Somebody told me the difference between the set with bounded ...
1answer
56 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 ...
0answers
37 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear SchrÃ¶dinger equation, $$(LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases}$$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
0answers
81 views