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.
If z = f(x,y) and x = r \cos{v}, y = r\sin{v} the object is to show that d = \partial since it's easier to do on computer
Show that:
\frac{d^2 z}{dr^2} + \frac{1}{r} \frac{dz}{dr} + \frac{1}{r^2} \frac{d^2 z}{dv^2} = \frac{d^2 z}{dx^2} + \frac{d^2 z}{dy^2}
It's from Adams calculus, will...
Hi I am having a lot of trouble trying to solve this equation. Any help is appreciated
x^2 \[partial]u/\[partial]x + 2 x \[partial]u/\[partial]t = g (t)
well,
i have a partial differentiation equation that look like this:
c_{1}\frac{\partial^{4}u}{\partial x^{4}}+c_{2}\frac{\partial^{4}u}{\partial x^{3}\partial y}+c_{3}\frac{\partial^{4}u}{\partial x^{2}\partial y^{2}}+c_{4}\frac{\partial^{4}u}{\partial x\partial...
2Uxxy+3Uxyy-Uxy=0 where U=U(x,y)
I made the substitution W=Uxy and then used a change of coordinates (n= 2x+3y, and r=3x-2y) which reduced the problem to solving Uxy=f(3x-2y)exp((2x+3y)/3) because W=f(r)exp(n/3). Now I have no idea where to go from there. Any help would be much appreciated...
Homework Statement
x*u_{x} + y*u_{y}= 1 + y^2
u(x,1) = 1+ x; -infinity < x < +infinity
Solve this parametrically and in terms of x and yHomework Equations
We are supposed to solve this using the method of characteristics
The Attempt at a Solution
My problem is that solving the equation...
Homework Statement
Use Fourier transforms to get solution in terms of f(t) adn g(t)Homework Equations
d4u + K2*d2u =0
dx4 (space) dt2
u(0,t)=f(t)
u'(0,t)=g(t)
u''(L,t)=0
u'''(L,t)=0
The Attempt at a Solution
I been working no it for hours the best I got is
k4U...
Homework Statement
Help, I don't know how to do the following question:
Using Laplace to solve
x' -y =1
2x' +x +y' = (t2-2t+1)e-(t-1)
Homework Equations
x(1)=0
y(3)=0
The Attempt at a Solution
The problem I'm having is the initial conditions aren't at zero, and I'm not sure...
Hi all,
I am Santosh, a grad student at FSU. I am trying to solve a system of 1-D pde's using finite difference scheme.
Here's a brief description of my boundary conditions:
Let my variables be S, T, V, & W, and k1, k2, k3, k4, k5, a, b, and c are constants. At x = 1,dA/dX = Ra,
where...
Homework Statement
x ut +ux =0
intial condition
u(x,0)=f(x)
1. Find the characteristics curves
2. What area of the xt-plane do u expect a solution
3. Find solution when f(x)=cos x
4.Now u(x,0)=f(x) (again), Find the level curves of u i.e for each c find the set Lc={(x,t):u(x,t)=f(c)}...
I am confused by the following example about solving quasilinear first order PDEs.
For the part I circled, the solution is just x^2 + y^2 = k where k is an arbitrary constant. To parametrize it in terms of t, can't we just put x = a cos(t), y = a sin(t) ? Here we only have one arbitrary...
Homework Statement
Draw a picture to illustrate the two-dimensional drumhead in the x-y plane. Label the coordinates of the sides of the drumhead. Use this picture to illustrate the "modes" of vibration.Homework Equations
\frac{\partial^{2}Z}{\partial x^{2}} + \frac{\partial^{2}Z}{\partial...
Homework Statement
u_t = -{{u_{x}}_{x}}
u(x,0) = e^{-x^2}
Homework Equations
The Attempt at a Solution
The initial state is a bell curve centred at x=0. The second partial derivative of u at t=0 is {4x^2}{e^{-x^2}}, which is a Gaussian function, which means nothing to me other than its...
I have to solve:
x(y^2 - z^2) \frac{\partial z}{\partial x} + y(z^2 - x^2) \frac{\partial z}{\partial y} + z(x^2 - y^2)
So, I write out the characteristic system of ODEs:
\frac{dx}{x(y^2 - z^2)} = \frac{dy}{y(z^2 - x^2)} = \frac{dz}{z(x^2 - y^2)}...
the question is , can a delta function /distribution \delta (x-a)
solve a NOnlinear problem of the form F(y,y',y'',x)
the question is that in many cases you can NOT multiply a distribution by itself so you could not deal with Nonlinear terms such as (y)^{3} or yy'
I've run across a PDE that (since I've failed to take a PDE class!) I'm finding some difficulty in solving. Does anyone have any suggestions? It's on a function R(r,t), with functions a(r,t) and b(r,t) and a constant k. If it's easier to solve with a and b not having t-dependence (just being...
Homework Statement
Solve the problem.
utt = uxx 0 < x < 1, t > 0
u(x,0) = x, ut(x,0) = x(1-x), u(0,t) = 0, u(1,t) = 1
Homework Equations
The Attempt at a Solution
Here is what I have so far but I'm not sure if I am on the right path or not.
u(x,t) = X(x)T(t)...
Homework Statement
Find the general solution to the PDE and solve the initial value problem:
y2 (ux) + x2 (uy) = 2 y2, initial condition u(x, y) = -2y on y3 = x3 - 2
2. Homework Equations /concepts
First order linear non-homogeneous PDEs
The Attempt at a Solution
I know that the...
Suppose we have a first order linear PDE of the form:
a(x,y) ux + b(x,y) uy = 0
Then dy/dx = b(x,y) / a(x,y) [assumption: a(x,y) is not zero]
The characteristic equation for the PDE is
b(x,y) dx - a(x,y) dy=0
d[F(x,y)]=0
"F(x,y)=constant" are characteristic curves
Therefore, the...
This is a question from a book in which I can't figure out, but it has no solutions at the back.
Find the general solution to the PDE:
xy ux + y2 (uy) - y u = y - x
I've learned methods such as change of variables and characteristic curves, but I'm not sure how I can apply them in this...
Homework Statement
Solve
(Z+e^x)Z_x + (Z+e^y)Z_y = Z^2 - e^{x+y}
Where Z = Z(x,y)Homework Equations
Equations of the form
PZ_x + QZ_y = R
Where P = P(x,y,z) , Q=Q(x,y,z) , R=R(x,y,z)
Are solved with the Lagrange method.
It is possible to write this in the form:
\frac{dx}{P} =...
Homework Statement
Quote:
" PDE: ∂u/∂x + ∂u/∂y = 0
The general solution is u(x,y) = f(x-y) where f is an arbitrary function.
Alternatively, we can also say that the general solution is u(x,y) = g(y-x) where g is an arbitrary function. The two answers are equivalent since u(x,y) = g(y-x) =...
Homework Statement
Claim:
The solution space of a linear homogeneous PDE Lu=0 (where L is a linear operator) forms a "vector space".
Proof:
Assume Lu=0 and Lv=0 (i.e. have two solutions)
(i) By linearity, L(u+v)=Lu+Lv=0
(ii) By linearity, L(au)=a(Lu)=(a)(0)=0
=> any linear...
Homework Statement
I need to visualize the wave equation with the following initial conditions:
u(x,0) = -4 + x 4<= x <= 5
6 - x 5 <= x <= 6
0 elsewhere
du/dt(x,0) = 0
subject to the following boundary conditions:
u|x=0 = 0
Homework Equations
I'm not sure I understand the...
let be the PDE eigenvalue problem \partial_{t} f =Hf
then if we define its Heat Kernel Z(u)= \sum_{n=0}^{\infty}e^{-uE_{n}} valid only for positive 'u'
then my question is how could get an asymptotic expansion of the Heat Kernel as u approaches to 0
Z(u) \sim...
Homework Statement Derive the differential equation governing the longitudinal vibration of a thin cone which has uniform density p, show that it is
1/x/SUP] d/dx(x du/dx) = (1/c) d u/d[SUP]t
Hint: The tensile force sigma = E du/dx where E is the Young's modulus (a constant), u is the...
I need to solve the following PDE:
\frac{1}{2}F_{\eta \eta }\sigma _{\eta }^{2}\eta ^{2}+\frac{1}{2}F_{pp}\sigma _{p}^{2}+F_{p}k(m-p)+F_{\eta }a\eta -rF=0 \label{6} where p goes from minus to plus infinity and eta goes from zero to plus infinity.
Here p and eta are state variables and all...
I have flipped through the first few pages of Evan's PDE book lately, and I am considering taking a graduate PDE course in the fall. However I don't really understand the purpose of PDE research. Not that I really understand the purpose of ODE research or even analysis research for that matter...
I am trying to decide between two different graduate math classes in the fall. These are the only classes that sound really interesting to me, as well as fitting in with the rest of my schedule. My other courses will consist of several physics type courses, notably, Jackson's electrodynamics...
I am trying to decide between two different graduate math classes in the fall. These are the only classes that sound really interesting to me, as well as fitting in with the rest of my schedule. My other courses will consist of several physics type courses, notably, Jackson's electrodynamics...
1. This problem concerns a nonhomogeneous second order linear ODE L[y] = g(t).
Suppose that:
y1(t) satisfies the ODE with the initial conditions y(0)=1, y'(0) = 0,
y2(t) satisfies the ODE with the initial conditions y(0)=0, y'(0) = 1, and
y3(t) satisfies the ODE with the initial conditions...
Hello,
This is my first post and hopefully my question has not been answered elsewhere already as I realize it is annoying to answer the same type of posts over and over again.
I am working on a system of PDEs with one ODE, coupled. It is an SEIR model with one extra class for the group of...
I make a pricing model as the attachment and write the code as the following,but there are something wrong in the result.Who could help me to check it?!
Thx so much!
the code:
>> dr = 0.005;
Nr = 20;
dt = 0.01;
Nt = 1/dt;
a=0.2339*0.0189;
b=0.2339;
delta=sqrt(0.0073);
T =...
Hi.
When solving a PDE by separation of variables, we obtain a collection of so-called normal modes. My book then tells me to make an "infinite linear combination" of these normal modes, and that this will be a solution to the PDE. But how do we know that this is in fact a solution? I have...
Hi everyone,
Can anyone explain how to use the PDE mode in COMSOL Multiphysics? I used the heat transfer package to model a piece of copper undergoing a change in temperature from 6 Kelvin to 300 Kelvin. Now I want to check to see that I can get the same results with my own equations. I don't...
b]1. Homework Statement [/b]
Find the characteristics, and then the solution, of the partial differential equation
x\frac{\partial u}{\partial x}+xy\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=0
given that u(1, y, z)=yz
Homework Equations
The Attempt at a...
Homework Statement
Hi, so the initial problem was:
given \left.\frac{d^{2}u}{dt^{2}} = \frac{d^{2}u}{dx^{2}}}
\left.-\infty \leq x \leq \infty
\left.u(x,0)=\frac{x}{1+x^{3}} , \frac{du}{dt}(x,0) = 0
Solve the PDE(did this part already) and plot the solution for -20 < x <20 and t =...
Homework Statement
Hi everyone, I just wanted to double check if I've solved this correctly?
Given:
\left.\frac{du}{dx} + sin(x)\frac{du}{dy} = 0
\left.-\infty < x < \infty
y > 0
\left.u(\frac{\pi}{2} , y ) = y^{2}
Solve the PDE
Homework Equations
Method of characteristics
The Attempt...
Hi All,
I am dealing with the heat equation these days and in an attack of originality I thought I would find a new solution to it, namely
(dT/dt)=d^2T/dx^2
has a solution of the type
T(x,t) = ax^2+2t
Now, I do not know much about the existence and uniqueness of PDE solutions, but...
Hi all,
For my thesis I would like to solve the following second order nonlinear PDE for V(x,\sigma,t):
\frac{1}{2}\sigma^2\frac{\partial^2 V}{\partial x^2}+\frac{1}{2}B^2\frac{\partial^2 V}{\partial \sigma^2}+a\frac{\partial V}{\partial \sigma}=0,
subject to the following boundary...
Hi i Would like to solve the the following eqn using Matlab
Since I am new to Matlab, I would request u to help me in this regard
(∂^2 T)/〖∂r〗^2 + 1/r (∂T )/∂r+(∂^2 T)/〖∂z〗^2 = 1/α (∂T )/∂t+τ/α (∂^2 T)/〖∂t〗^2 - { (1+δ(t) )-(1+δ(t-tp) }*IoKa/k 〖exp( - 2r/σ^2 〗_^2)exp(-zka)
where δ(t) =76ns...
Homework Statement
Use the Laplace Transform to solve the PDE for u(x,t) with x>0 and t>0:
x(du/dx) + du/dt = xt
with IC: u(x,0) = 0 and BC: u(0,t) = 0
Homework Equations
The Attempt at a Solution
After taking LT of the PDE wrt t, the PDE becomes
x(dU/dx) + sU = x/(s2)...
So my book says that to solve a PDE by separation of variables, we check the three cases where λ, the separation constant, is equal to 0, -a^2, and a^2. But in this particular problem, instead of substituting λ=0, λ = a^2, λ= -a^2, they substitute the entire coefficient of X, (λ-1)/k =0, (λ-1)/k...
Hello all!
So, I'll be taking first-semester quantum mechanics and partial differential equations this fall, and would like to get a little bit of a head start by reading/working some problems on my own this summer. After some initial browsing, I've heard mixed-to-poor reviews concerning...
Hi
The equation is:
\frac{dP}{dt}-A*\frac{{d}^2P}{dx^2}-B*\frac{dC}{dt}=0
dP/dt=A*d2P/dx^2 was solved using a finite difference method. If the function C(x,t) is known, is it possible to solve the whole equation by using the finite difference solution as a supplement to the complete solution...