All Questions
Tagged with perturbation-theory partial-differential-equations
44
questions
0
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0
answers
24
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Solving a new system of PDEs using solutions of an old system
I got stuck in my research. Briefly speaking, the following is a system of 6 variables ($u,v,p,h_{11},h_{12},h_{22}$) I need to analyze:
\begin{equation}
g^2\frac{\partial u}{\partial X}+\frac{\...
- 382
2
votes
0
answers
52
views
Leading order perturbation to the solution of a dynamical system
I was reading the paper 'A Proposal on Machine Learning via Dynamical Systems', where I came across the following steps:
Consider a system-
$$\frac{dz}{dt} = f(A(t),z),$$ with $z(0) = x.$
So, the ...
- 21
1
vote
1
answer
96
views
Two Timing (Multiple Time Scales) with Coupled IVPs
Question: Find the leading-order approximation for times of order $\epsilon^{-1}$ to $$\ddot{x} + x = y,~~\dot{y} = \epsilon(xy - 2y^2),~~x(0) = 1,~~\dot{x}(0) = 0,~~y(0)=1.$$
My approach: Let the ...
- 3,965
0
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0
answers
38
views
Bound for $\left\|\nabla^2 u\right\|_{L^2(\Omega)}$
Consider the elliptic equation
$$
\begin{aligned}
-\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\
u=0, & \text { on } \partial \Omega.
\end{aligned}
$$
where $\Omega$ is a bounded domain....
- 445
2
votes
0
answers
101
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Poisson Equation for a perturbed sphere - both exterior and interior solutions
I am trying to solve a Heat transfer problem in a slightly ellipsoidal geometry using the Poission Equation, and Cauchy matching conditions on the boundary between the interior (which is a finite ...
- 31
0
votes
1
answer
91
views
Asymptotics of a nonlinear PDE
Consider the partial differential equation with boundary conditions
\begin{equation}
\frac{\partial u}{\partial t} = \varepsilon \bigg( (u+1)\frac{\partial u}{\partial x}\bigg), \qquad \frac{\partial ...
- 635
1
vote
0
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67
views
Graph of the function which involves Dirac Delta function
I am working on Modified Homotopy perturbation transformation method to solve a PDE.
Referred paper is given below.
DOI 10.1007/s10598-015-9278-x
In the above paper, they got $w(r,t)$ as
\begin{...
- 445
3
votes
0
answers
75
views
The linear heat eqaution with potential $V\in L_{t,x}^{\frac{n+2}2}$ is well-posed for initial data in $L^p$ for any $p\in(1,\infty)$
Consider the linear heat equation with potential
$$u_t-\Delta u-V(x,t)u=0\qquad \text{in }\ \ \mathbb R^n\times[0,\infty),$$
where $V\in L_{t,x}^{\frac{n+2}2}$. Show that this equation is well-posed ...
- 13.7k
1
vote
0
answers
21
views
Large $t$ solution to $\partial_t S(x,v,t) = v \partial_x S + (a+v) \partial_v S + \partial_v^2 S$
I have the partial differential equation
$$\partial_t S(x,v,t) = v \partial_x S - (v + a ) \partial_v S + \partial_v^2 S$$
subject to the following boundary conditions:
$$S(x,v,0)=1,$$
$$S(0+,|v|,t)=...
- 1,454
1
vote
0
answers
20
views
Converting an equation with "small displacements" retains a zeroth-order term -- can it become a PDE?
I have come across an equation that looks like
$$ f\!\left(t + \tau, x\right) = f\!\left(t, x - \xi\right) - f\!\left(t, x + \xi\right) $$
where $\tau$ and $\xi$ are both small. Every bone in my body ...
- 179
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0
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30
views
Can I solve $\partial_t(\partial_t +a ) W(x,t) = (\partial_t + b)\hat{L} W(x,t)$ using the solution $\partial_t W_0 = \hat{L}W_0$?
I have a challenging partial differential equation which is second order in time:
$$\partial_t(\partial_t +a ) W(x,t) = (\partial_t + b)\hat{L} W(x,t)$$
Here $L$ is some linear operator which involves ...
- 1,454
1
vote
0
answers
53
views
Spotting distinguished limits from Robin boundary conditions
I am currently working on a problem involving solving PDEs and using boundary layers. In this problem, $f(x,y)$ is the main dependent variable, $f(x,y)=f_R (X_R ,y)$ and $f(x,y)=f_B(x,Y_B)$ where $X_R$...
- 173
1
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0
answers
25
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Deriving solution of a second order ode from a closely related one with a different scalar coefficient for the term in X(x)
I solved the wrong differential equation!
I was looking at a parabolic pde in the style of a heat equations which upon separation of variables I wrote as
\begin{equation}
a x^2\left(\frac{d^2 X}{d x^...
- 53
1
vote
2
answers
291
views
Find the first two terms of the perturbation series solution to the IVP
Here is the equation for the initial value problem, $y'=1+(1+\epsilon)y^2$ with initial condition $y(0)=1$ with $t>0$ where $0<\epsilon\ll1.$. Find exact solution and compare approximation.
My ...
- 403
1
vote
1
answer
277
views
Expanding a PDE in powers of a small parameter?
I'm working on an assignment for my quantum mechanics class and I've arrived at a nonlinear inhomogeneous partial differential equation for a complex function $S:\mathbb{R}^2\to\mathbb{C}~;~S:(x,t)\...
- 12.5k