PDE surfaces are used in geometric modelling and computer graphics for creating smooth surfaces conforming to a given boundary configuration. PDE surfaces use partial differential equations to generate a surface which usually satisfy a mathematical boundary value problem.
PDE surfaces were first introduced into the area of geometric modelling and computer graphics by two British mathematicians, Malcolm Bloor and Michael Wilson.
I've got a PDE that I derived from a physical problem, so I suppose it has a solution and that it is unique. I am solving for streamlines in the region having a quarter of an annulus shape, so ##\theta## ranges between ##0## and ##\pi/2## and ##r## ranges between ##r_i## and ##r_o##. The...
I've been studying a few books on PDE's, specifically the heat equation. I have one book that covers this topic in cylindrical coordinates. All the examples are applied to a solid cylinder and result in a general Fourier Bessel series for 3 common cases that can be found easily with an online...
There are some known solutions for 3D Navier-Stokes such as Beltrami flow.
In the literature these Beltrami flow solutions are said to not take into account viscosity, however when I read the information on Beltrami flow, they do seem to involve (kinematic) viscosity:
From incompressible...
Now in my understanding from text ...just to clarify with you guys; the pde is of dimension 2 as ##t## and ##x## are the indepedent variables or it may also be considered to be of dimension 1, that is if there is a clear distinction between time and space variables.
Your insight on this is...
I am writing a 2D hydrocode in Lagrangian co-ordinates. I have never done this before, so I am completely clueless as to what I'm doing. I have a route as to what I want to do, but I don't know if this makes sense or not. I've gone from Eulerian to Lagrangian co-ordinates using the Piola...
Imagine you have a plane wall with constant thermal conductivity, that is the intermediate between two thermal reservoirs:
The reservoir on the left is being kept at temp ##T_s##, and it is a fluid that has very high convective coefficient ##h##. As a result, the boundary condition at the...
Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.
using the equation ##u(x,y)=f(x)g(y)##, first, I substitute the value of ##u_{xx}## and ##u_{yy}## in the given PDE. after that solve the ODEs but I can't understand about the ##u_{t}##.In my solution, I put ##u_{t}=0## because u is only the function of x and y. Is it the right approach, to me...
My PDE:
F,x,t + A(x)*F(x,t)*[(x+t)^(-3/2)] = 0
A(x) is a known function of x.
Trying to separate F(x,t) like
F(x,t) = F1(x)*F2(t)*F3(x+t).
I’m getting desperate to solve,
any suggestions??
Would method of separation of variables lead to a solution to the following PDE?
$$ \frac{1}{r} \frac{ \partial}{\partial r} \left( kr \frac{ \partial T}{ \partial r}\right) = \rho c_p \frac{\partial T }{ \partial t }$$
This would be for the transient conduction of a hollow cylinder, of wall...
My take;
##ξ=-4x+6y## and ##η=6x+4y##
it follows that,
##52u_ξ +10u=e^{x+2y}##
for the homogenous part; we shall have the general solution;
$$u_h=e^{\frac{-5}{26} ξ} f{η }$$
now we note that
$$e^{x+2y}=e^{\frac{8ξ+η}{26}}$$
that is from solving the simultaneous equation;
##ξ=-4x+6y##...
My interest is on the highlighted part only...my understanding is that one should use simultaneous equation... unless there is another way hence my post query.
In my working i have;
##y=\dfrac{2ξ+η}{10}## and ##x=\dfrac{2η-ξ}{10}## giving us...
Hi, I am solving heat equation with internal heat sources both numerically and analytically. My graphs are nearly identical but! analytical one have problem at the beginning and at the end for my domain. Many people have used the same technique to solve it analytically and they got good answers...
I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...
By PDE. The book written by Walter Alexander Strauss perfect described a typical undergraduate PDE course I have in my mind.
It should at least include:
Laplace equations, waves and diffusions
reflection, boundary problems, Fourier series
The content of the book I mentioned can also be found...
We have this type of very famous nicely symmetric pde in our area. However, no one knows how to handle it properly since it is a nonlinear pde.
Suggestions on how it is called in general would help us further googling. I already tried keywords like "bilinear", "dual", "double", but by far could...
Hello,
I wrote a code to solve a non-linear PDE using Canrk nicolson method, but I'm still not able to get a correct final results. can anyone tell me what wrong with it?
I am studying mathematics as bachelor in my second year. At the moment I am taking abstract algebra, analysis (measure and integration theory) and probability course. I don't know exactly what I want to do with maths but the applications in physics always have fascinated me. The next term I have...
Now i learned how to use discriminant i.e ##b^2-4ac## and in using this we have;
##b^2-4ac##=##0-(4×3×2)##=##-24<0,## therefore elliptic.
The textbook has a slight different approach, which i am not familiar with as i was trained to use the discriminant at my undergraduate studies...
see...
Hi all,
I am hoping someone can help me understand a PDE. I am reading a paper and am trying to follow the math. My experience with PDEs is limited though and I am not sure I am understanding it all correctly. I have 3 coupled PDEs, for $n$, $f$ and $c$, that are written in general form, and I...
I have a 2D space-time PDE and I want to solve it numerically over the time axis. The time initial field is already known with respect to space, i.e., the spatial distribution is already known at time `t = 0`. I solved the same PDF in Mathematica and got a solution. I tried to solve it...
I just want to make sure I am on the right track here (hence have not given the other information in the question). In taking the Fourier transform of the PDE above, I get:
F{uxx} = iω^2*F{u},
F{uxt} = d/dt F{ux} = iω d/dt F{u}
F{utt} = d^2/dt^2 F{u}
Together the transformed PDE gives a second...
I feel that my reasoning becomes shaky near the conclusion. So, someone should tell me why it is weak, and suggest how to make it stronger. Thanks.
For ##\delta>0## we define the Appell transform of ##u## by $$u_\delta=(1+\delta t)^{-\frac{n}{2}}exp\Big(-\frac{\delta|x|^2}{4(1+\delta...
dx/dt =1, x(0,s)=0, dy/dt=x, y(0,s) = s, du/dt=(y-1/2x^2)^2, u(0,s)=e^s
I did well at the beginning to get x(t,s) =t and y(t,s)=1/2t^2 + s, but got stuck with the du/dt part.
You can sub in x=t and y=1/2t^2 +s for x and y to get du/dt = s^2. But that's still three variables, and I can't see...
Hi all, I was hoping someone could check whether I computed part (4) correctly, where i find the solution u(t,x) using dAlembert's formula:
$$\boxed{\tilde{u}(t,x)=\frac{1}{2}\Big[\tilde{g}(x+t)+\tilde{g}(x-t)\Big]+\frac{1}{2}\int^{x+t}_{x-t}\tilde{h}(y)dy}$$
Does the graph of the solution look...
Sorry the problem is a bit long to read. thank you to anyone who comments.
We consider the initial value problem for the Burger's equation with viscosity given by
$$\begin{cases} \partial_t u-\partial^2_xu+u\partial_xu=0 & \text{in}\quad (1,T)\times R\\\quad \quad \quad \quad \quad...
This isn't homework, but I was just wondering whether the following PDE has an analytic solution.
$$\partial_x u(t,x)=u(t,x)$$
where ##x\in R^n## and ##\partial_x## implies a derivative with respect to the spatial variables.
(1) From "Radial solutions to Laplace's equation", we know that
$$
\Delta u(x) = v(r)''+\frac{n-1}{r}v(r)'
$$
we re-write the PDE
$$
- \Delta u+m^2u=0
$$
in terms of ##v(r)##
\begin{equation}
- v(r)''-\frac{n-1}{r}v(r)'+m^2v(r)=0
\end{equation}
to give a linear second order ODE with...
Summary:: partial differential equation (PDE) to describe the potential distribution φ in the system
Hey, I need some help with the following question:
We have a stationary electrolyte, a magnetic field "B" and a Current density "j" (2D).
Derive the partial differential equation (PDE) to...
Hi all, I
Fix $$(t,x) ∈ (0,\infty) \times R^n$$and consider auxillary function
$$w(s)=u(t+s,x+sb)$$
Then, $$\partial_s w(s)=(\partial_tu)(t+s,x+sb)\frac{d}{ds}(t+s)+<Du(t+s,x+sb)\frac{d}{ds}(x+sb)>$$
$$=(\partial_tu)(t+s,x+sb)+<b,Du(t+s,x+sb)>$$
$$=-cu(t+s,x+sb)$$...
Hello, please lend give me your wisdom.
I suspect this problem is about the wave equation ##\partial_t^2-\partial_x^2=0## commonly encountered in physics. I tried a search for information but I could not find help.
Attempt at arriving at solution:
So I took the partial derivatives of...
Solution attempt:
We first write ##u(x)=\frac{1}{2}||x||^2## as ##u(x)=\frac{1}{2}(x_1^2+x_2^2+...+x_n^2)##
Operating on ##u(x)## with ##\Delta##, we have ##u(x)=\frac{1}{2}(2+2+...+2)## adding 2 to itself ##n## times.
So ##\Delta u(x)=n## and the function satisfies the first condition...
Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it.
I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...
From some principles in nature we are using in physics the calculus of variations. Let me call it a generator for PDE's. My question: Are there levels above? What I mean is: Is there mathematics where you have principles where the solutions are generators for the generators for PDEs ?
What about...
Hi,
Question: If we have an initial condition, valid for -L \leq x \leq L :
f(x) = \frac{40x}{L} how can I utilise a know Fourier series to get to the solution without doing the integration (I know the integral isn't tricky, but still this method might help out in other situations)?
We are...
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.
Hi everyone!
I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example,
$$
u_t=k u_{xx},
$$
$$
u(t,-1)=\alpha,
$$
$$
u(t,1)=\beta,
$$
$$
u(0,x)=f(x),
$$
$$...
Hello,
Poisson equation and Laplace equation (which is the homogeneous version of Poisson PDE) are important equations in electrostatics where both the electric field ##E## and scalar potential ##\phi## don't depend on time. Poisson's equation is $$\nabla^2 \phi(x,y,z) = - \frac{\rho(x,y,z)}...
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
$$\frac{\partial\rho}{\partial t} + (1-2\rho)\frac{\partial \rho}{\partial x} = \begin{cases} \beta, \hspace{3mm} x < 0 \\...
as you can see, z(x,y) is a function of x, y; and y is a function of x, therefore y'(x) is the total derivative of "y" respect to "x", and y"(x) is the 2nd derivative. y'(x)^2 is just the square of the derivative of y respect to x
I don't have boundary or initial conditions, so you can make up...
I'm trying to compute a 2D Heat diffusion parabolic PDE:
$$
\frac{\partial u}{\partial t} = \alpha \{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \}
$$
by the ADI method. I am actually trying to go over the example in this youtube video. The video is in another...
Problem: Consider the equation $$\frac{\partial v}{\partial t} = \frac{\partial^{2} v}{\partial x^2} + \frac{2v}{t+1}$$ where ##v(x,t)## is defined on ##0 \leq x \leq \pi## and is subject to the boundary conditions ##v(0,t) = 0##, ##v(\pi, t) = f(t)##, ##v(x,0) = h(x)## for some functions...
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...
I am a junior physics major trying to decide if I should squeeze in a (extra) PDE class in my semester which is not required for my degree but obviously can be useful. Though I am only taking 3 (all technical) classes otherwise, I should be quite occupied with GREs and research. Is it worth...
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...