Is it possible to use separation of variables on this equation?
au_{xx} + bu_{yy} + c u_{xy} = u + k
Where u is a function of x and y, abck are constant.
I tried the u(x,y) = X(x)Y(y) type of separation but I think something more clever is needed.
Thank you.
Hello all, this question really has me and some friends stomped so advice would be appreciated.
Ok so, the relevant (dimensionless) continuity equation I have found to be
$$\frac{\partial\rho}{\partial t} + (1-2\rho)\frac{\partial \rho}{\partial x} = \begin{cases} \beta, \hspace{3mm} x < 0 \\...
I'm trying to compute a 2D Heat diffusion parabolic PDE:
$$
\frac{\partial u}{\partial t} = \alpha \{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \}
$$
by the ADI method. I am actually trying to go over the example in this youtube video. The video is in another...
I am trying to derive the adjoint / tangent linear model matrix for this partial differential equation, but cannot follow the book's steps as I do not know the math. This technique will be used to solve another homework question. Rather than posting the homework question, I would like to...
Exercise statement
Find the general solution for the wave equation ftt=v2fzzftt=v2fzz in the straight open magnetic field tube. Assume that the bottom boundary condition is fixed: there is no perturbation of the magnetic field at or below the photosphere. Solve means deriving the d’Alembert...
Hey everybody,
Background:
I'm currently working on a toy model for my master thesis, the massless Klein-Gordon equation in a rotating static Kerr-Schild metric.
The partial differential equations are (see http://arxiv.org/abs/1705.01071, equation 27, with V'=0):
$$ \partial_t\phi =...
So the other day, I was pouring beer from a can to a mug and I obviously know the flow rate depends on the height of the beer from the bottom of the can (fluid level in the vessel), angle of tilt and I think time as well.
I was wondering how to best model the PDE to describe such a phenomenon (...
While reproducing a research paper, I came across the following equation,
∂f/∂t−(H(f)(∂f/∂x)=0
where [H(f)] is hilbert transform of 'f.'
and f=f(x,t) and initial condition is f(x,0)=cos(x) and also has periodic boundary conditions given by
F{H{f(x′,t)}}=i⋅sgn(k)F{f(x,t)},
where F(f(x,t) is...
I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a fourier transform, where I can take the fourier transform of both sides then solve the general solution in fourier terms then inverse transform. However, since this...
Homework Statement
I want to find the solution to the following problem:
$$\begin{cases} \nabla^2 B=c^2 B &\text{ on the half plane } x>0 \\ B=B_0 \hat{z} & \text{ for } x<0 \end{cases}$$
in the ##xz## plane. ##c, B_0 \in \mathbb{R}##
Homework Equations
I am not really sure what would be...
##u_t + t \cdot u_x = 0##
The equation can be written as ##<1, t, 0> \cdot <d_t, d_x, -1>## where the second vector represents the perpendicular vector to the surface and since the dot product is zero, the first vector must necessarily represent the tangent to the surface. We parameterize this...
Homework Statement
This question relates to a very large project I have been assigned in a course on mathematical methods in structural engineering. I have to solve the following equation, in a specific way:
(17)
Now we have to assume the following solution:
(18)
It wants me insert...
This is perhaps the single most important mathematical physics papers I have ever read; I think everyone - especially (theoretical) physicists - interested in theoretical physics should read it. In fact, read it now before reading the rest of the thread:
Klainerman 2010, PDE as a Unified Subject...
Homework Statement
Solve the following partial differential equation , using Fourier Transform:
Given the following:
And a initial condition:
Homework Equations
The Attempt at a Solution
First , i associate spectral variables to the x and t variables:
## k ## is the spectral variable...
Homework Statement
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Homework Equations
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The Attempt at a Solution
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Could you tell me is there something specific that I need to the with sin(xy)? Thanks
Hi PF,
Suppose I numerically solve a nonlinear system of differential equations. How can I know if my solution is correct (if there is no known analytic solution)? What are the standard practices people do? I have a couple of ideas, but I want to know what people are already doing.
Danke!
I have seen one lecturer solve a PDE with just using Fast Fourier Transform (##FFT##) of a function ##v## on a chebychev grid. ##v_t=\mu v_x##
This lecturer uses ##FFT## on ##V##, then solves the ODE using an ODE solver in Matlab, then inverse ##FFT## to get the real solution ...
I am unsure how to choose the boundary conditions for a system of PDEs or for a single PDE for that matter.
The situation I am stuck with involves a system of 4 PDEs describing plasma in a cylinder. The dependent variables being used are Vr, Vt, Vz, ni, and the independent variables are Rr...
I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction.
I want to solve the following boundary conditioned differential equation:
$$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2...
How can I classify a given first order partial differential equations?
Are all first order linear PDEs hyperbolic?
Quora Link:https://www.quora.com/How-do-I-classify-first-order-PDE-elliptic-hyperbolic-or-parabolic-using-method-of-characteristics
The solution of 1D diffusion equation on a half line (semi infinite) can be found with the help of Fourier Cosine Transform. Equation 3 is the attached figure is the solution of 1D diffusion equation (eq:1). I want to write a code for this equation in MATLAB/Python but I don't understand what...
Homework Statement
I have the heat equation
$$u_t=u_{xx}$$
$$u(0,t)=0$$
$$u(1,t) = \cos(\omega t)$$
$$u(x,0)=f(x)$$
Find the stable state solution.
The Attempt at a Solution
I used a transformation to complex to solve this problem, and then I can just take the real part to the complex...
Homework Statement
I am trying to solve the given wave equation using separation of variables,
u_{tt} - 4u_{xx} = 4 for 0 < x < 2 and t > 0
(BC) u(0,t) = 0 , u(2,t) = -2, for t>0
(IC) u(x,0)=x-x^2 , u_t(x,0)=0 for 0\leq x \leq2
Homework Equations
We are told we will need to use...
Homework Statement
Show that k(x,0)=δ(x).
Where k(x,t) is the heat kernel and δ(x) is the Dirac Delta at x=0.
Homework Equations
k(x,t) = (1/Sqrt[4*π*D*t])*Exp[-x^2/(4*D*t)]
The Attempt at a Solution
I am just clueless from the beginning. I am guessing this is got to do with convolution...
Hello, I have the following PDE equation:
a*b/U(u)*V(v) = 0
where a and b are arbitrary constants, and U an V are two unknown functions. To me it appears this has no solution, however I would like to ask if anyone has some suggestions, such as transforming it to another type using Fourier or...
Homework Statement
am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it?
appreciated,
Homework Equations
The Attempt at a Solution
my attempt is the same as in the attached...
Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$
I want to solve this weakly formulated pde:
$$
0=\frac{A}{N^{d+1}} \sum_i...
Hello everybody. I'm about to take a final exam and i've just encountered with this exercise. I know it's simple, but i tried solving it by Separation of variables, but i couldn't reach the result Mathematica gave me. This is the equation:
∂u/∂x = ∂u/∂t
Plus i have a condition...
Hi,
I have an equation that takes the form: ax''-by' + c = 0 where x'' is second order with respect to time and y' is first order with respect to time. Would this be classed as a partial differential equation?
Thanks very much for your help :)
Hi I am trying to verify my manual solution for this problem by any way, so I tried NDSolve, and DSolve, in mathematica with no success. I dont need it in mathematica I just need any way poosible, even matlab, or any other numeric way/soltuion. Can some one help, or even give me the final...