What is Pde: Definition and 854 Discussions

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.

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  1. D

    MHB Is This Equation Known as the Korteweg-de Vries Equation?

    \begin{align*} \psi_t + \psi_{xxx} + f(\psi)\psi_x &= 0 \end{align*} This equation leads to the nonlinear Shrodinger equation but does this equation have a name?
  2. K

    PDE - Boundary value problem found in QM

    This is a quantum mechanics problem, but the problem itself is reduced (naturally) to a differential equations problem. I have to solve the following equation: \frac{\partial}{\partial t}\psi (x,t) = i\sigma \psi (x,t) where \sigma > 0 The initial condition is: \psi (x,0) =...
  3. G

    Separable PDE for electric field in a cavity

    Homework Statement Here is a photo of a page in Laser Physics by Hooker: https://www.evernote.com/shard/s245/sh/2172a4e7-63c7-41a0-a0e7-b1d68ac739fc/7ba12c241f76a317a6dc3f2d6220027a/res/642710b5-9610-4b5b-aef4-c7958297e34d/Snapshot_1.jpg?resizeSmall&width=832 I have 3 questions (I'm a bit...
  4. P

    Verifying Solution of PDE utt = c2uxx with FTC

    Homework Statement Verify that, for any continuously differentiable function g and any constant c, the function u(x, t) = 1/(2c)∫(x + ct)(x - ct) g(z) dz ( the upper limit (x + ct) and lower limit (x - ct)) is a solution to the PDE utt = c2uxx. Do not use the...
  5. J

    Doubts on the boundary conditions of PDE

    Hi all, Say I am solving a PDE as \frac{\partial y^2}{\partial^2 x}+\frac{\partial y}{\partial x}=f, with the boundary condition y(\pm L)=A. I can understand for the second order differential term, there two boundary conditions are well suited. But what about the first order differential term...
  6. T

    Reducing a PDE to an ODE Using a Change of Coordinates

    I've been studying Walter A. Strauss' Partial Differential Equations, 2nd edition in an attempt to prepare for my upcoming class on Partial Differential Equations but this problem has me stumped. I feel like it should be fairly simple, but I just can't get it. 10. Solve ##u_{x} + u_{y} + u =...
  7. A

    How Can Numerical Stability Be Achieved in Unsteady Laminar Flow Equations?

    I took a CFD class last semester (had to leave school though due to personal garbage). I am making a come back this fall and as some extra credit I am trying to numerically solve the unsteady laminar flow equation in a pipe. The equation is \dot{U} + U'' + K = 0 where dots denote the time...
  8. T

    PDE: Wave equation with first order derivative

    Homework Statement Solve using separation of variables utt = uxx+aux u(0,t)=u(1,t)=0 u(x,0)=f(x) ut=g(x) The Attempt at a Solution if not for the ux I'd set U=XT such that X''T=TX'' and using initial conditions get a solution. In my case I get T''X=T(aX'+X'') which is...
  9. P

    Can't decide between PDE or ODE or both

    Hey everyone I am going to be a freshman this fall (in college). I am currently having a dilemma in choosing my math class. In high school I took classes all the way up to Honors Differential Equations (ODE). In June I went to the university and signed up for Ordinary Differential Equation...
  10. R

    Transport Equation IVP Solution

    Homework Statement Hi guys, I'm having trouble with a homework problem: I will have to solve for the IVP of a transport equation on R: the equations are: Ut-4Ux=t^2 for t>0, XER u=cosx for t=0, XER Homework Equations transport equation The Attempt at a Solution...
  11. R

    PDE Wave equation with phi(x) as initial boundaries

    Homework problem: For the wave equation: Utt-Uxx=0, t>0, xER u(x,0)= 1, |x|<1 0, |x|>1 sketch the solution u as a function of x at t= 1/2, 1, 2, and 3 I am able to use d'Alemberts and solve for u however the boundaries and the odd/even reflections are throwing me off and...
  12. R

    PDE for IVP on R for a transport equation

    Hi guys, I'm having trouble with a homework problem: I will have to solve for the IVP of a transport equation on R: the equations are: Ut-4Ux=t^2 for t>0, XER u=cosx for t=0, XER I've actually never seen a transportation problem like this and any help would be...
  13. S

    Solving Partial Differential Equations Using Separation of Variables

    I'm having troubles with PDE. Apply separation of variables, if possible, to found product solutions to the following differential equations. a) x\frac{\partial u}{\partial x}=y\frac{\partial u}{\partial y} I suppose that: u=X(x) \cdot Y(y) Then: xX'Y=yXY' xX'/X=yY'/Y So xX'/X=yY'/Y=c because...
  14. S

    Can I Modify a PDE Expression If It's Constrained to a Curve?

    Hello folks, If we have the expression, say \frac{∂f}{∂r}+\frac{∂f}{∂θ}, am I allowed to change it to \frac{df}{dr}+\frac{df}{dr}\frac{dr}{dθ}, if "f" is constrained to the curve r=r(θ). My reasoning is that since the curve equation is known, then f does not really depend on the...
  15. T

    How can I solve a coupled PDE and ODE using the method of lines?

    I am trying to solve an ODE and PDE and I am having problems coming up with a method for doing so. The PDE is: k1*(dC/dt) = k2*(dC/dx) But I have an ODE that is an expression for dC/dt: dC/dt = k3*C Where k1,k2 and k3 are constants. I planned to use the method of lines to get...
  16. C

    Modified diffusion equation PDE

    Hi I'd appreciate any help on identifying the type of PDE the following equation is... *This is NOT homework, it is part of research and thus the lack my explanation of what this represents and boundary conditions. I have a numerical simulation of the solution but I'm looking to have a math...
  17. A

    What Are the Conditions for Uniqueness in Solving This PDE?

    Consider the PDE $$ U_{xy}+\frac{2}{x+y}\left(U_{x}-U_{y}\right)=0 $$ with the boundary conditions $$ U(x_{0},y)=k(x_{0}-y)^{3}\\ U(x,y_{0})=k(x-y_{0})^{3} $$ where $k$ is a constant given by $k=U_{0}(x_{0}-y_{0})^{3}$. $x_{0}$, $y_{0}$ and $U(x_{0},y_{0})=U_{0}$ are known. The solution...
  18. E

    Find the Equilibrium temperature distribution of a PDE

    Homework Statement 1) What is the Equilibrium temperature distributions if α > 0? 2) Assume α > 0, k=1, and L=1, solve the PDE with initial condition u(x,0) = x(1-x) Homework Equations du/dt = k(d^2u/dx^2) - (α*u) The Attempt at a Solution I got u(x) = [(α*u*x)/2k]*[x-L] for...
  19. A

    Solving the Constant PDE ∂u/∂x=∂u/∂y

    ∂u/∂x=∂u/∂y, can we ensure that u is a constant not dependent on x and y?
  20. B

    Scaling Invariant, Non-Linear PDE

    Homework Statement Consider the nonlinnear diffusion problem u_t - (u_x)^2 + uu_{xx} = 0, x \in \mathbb{R} , t >0 with the constraint and boundary conditions \int_{\mathbb{R}} u(x,t)=1, u(\pm \inf, t)=0 Investigate the existence of scaling invariant solutions for the equation...
  21. M

    PDE Linear Equation Q: Homogeneous vs Nonhomogeneous

    My questions concerns the information in the document. I highlighted the portion that is confusing me and a sample problem at the bottom. Question: Look at the equation 2.2.4 in the document. When I set the function u equal to zero the equation becomes 0 = 0 + 0 + f(x,t) or f(x,t) = 0. Now...
  22. G

    Predicting the form of solution of PDE

    Predicting the functional form of solution of PDE How do you conclude that the solution of the PDE u(x,y)\frac{∂u(x,y)}{∂x}+\upsilon(x,y)\frac{∂u(x,y)}{∂y}=-\frac{dp(x)}{dx}+\frac{1}{a}\frac{∂^{2}u(x,y)}{dy^{2}} is of the functional form u=f(x,y,\frac{dp(x)}{dx},a) ? I know this...
  23. A

    On solving PDE using separating the variable.

    hi.. with refrenence to http://www.math.uah.edu/howell/MAPH/Archives/Old_Notes/PDEs/PDE1.pdf page 7, “Observe” that the only way we can have formula of t only= formula of x only is for both sides to be equal to a single constant. here I do understand that for these to being equal...
  24. H

    Upwinding method for convection terms 2nd order PDE

    I'm trying to solve the equation $$ \frac{\partial u}{\partial t} + \frac{\partial}{\partial x}\left(Cu\right) - \frac{\partial}{\partial x}\left(D\frac{\partial u}{\partial x}\right) = f(x,t) $$ where C and D allow for linearity. I'm using a discontinuous Galerkin method in space and...
  25. G

    Method of Characteristics for Hyperbolic PDE

    I am trying to build a program in Matlab to solve the following hyperbolic PDE by the method of characteristics ∂n/∂t + G(t)∂n/∂L = 0 with the inital and boundary conditions n(t,0)=B(t)/G(t) and n(0,L)=ns Here ns is an intial distribution (bell curve) but I don't have a function to...
  26. N

    Problem with solution of a PDE, Neumann functions

    hello everyone i'm in my sixth semester of undergraduate physics and currently taking a math methods of physics class. So far we've been working with boundary value problems using PDE's. In the textbook we're using and from which I've been reading mostly (mathematical physics by eugene...
  27. T

    Weak solutions to PDE with different ICs

    Homework Statement Let ##U\subset\mathbb{R}^n## be a bounded open set with smooth boundary ##\partial U##. Consider the boundary value problem $$\begin{cases}\bigtriangleup^2u=f&\text{on }U\\u=\frac{\partial u}{\partial n}=0&\text{on }\partial U\end{cases}$$where ##n## is the outward pointing...
  28. T

    Hyperbolic PDE, Cauchy-type problem

    Homework Statement Consider the equation 4y^2u_{xx} + 2(1-y^2)u_{xy} - u_{yy} - \frac{2y}{1+y^2} (2u_x - u_y) = 0 Find the solution u(x,y) which satisfies u(x,0) = g(x), and u_y(x,0) = f(x), where f, g \in \mathcal{C}^2(\mathbb{R}) are arbitrary functions. Homework Equations I used...
  29. M

    How Do You Solve a PDE Involving Heat Transfer in a Circular Heat Source?

    I have a circular heat source of inner radius r1 and r2=r1+Δr on top of a pcb board. This heat source is transferring heat along the radius and the length of the beaker which is say L. I have to find temperature distribution along the length of the beaker so T(r.z). The beaker is filled with...
  30. M

    PDE Wave Equation and Energy Conservation

    Homework Statement Just looking back through my notes and it looks like I'm missing some. Just a few questions. For one example in the notes I have the wave utt-c2uxx + u3 = 0 and that the energy density 1/2u2t + c2/2u2x + 1/4u4 I have that the differential form of energy conservation...
  31. E

    Is this PDE linear or non-linear?

    hello, guys Below is the equation I am concerned with: Is the above equation non-linear because of (delta P/delta x)^2 term assuming other variables are constant and don't change with pressure , P?
  32. L

    Learning Intro PDE: Farlow vs Hillen vs Pinsky

    So I am currently a math undergraduate (senior though) taking an introduction partial differential equations. We are using the PDE book by Farlow (Dover reprint). It seems to be a solid book though my professor does diverge from the methods used in it fairly regularly (like not making...
  33. D

    How to Solve This Coupled PDE System Involving Complex Variables?

    (r^2 \nabla^2 - 1) X(r,\theta,z) + 2 \frac{\partial}{\partial \theta} Y(r,\theta,z) = 0 (r^2 \nabla^2 - 1) Y(r,\theta,z) - 2 \frac{\partial}{\partial \theta} X(r,\theta,z) = 0 any suggestions are greatly appreciated :)
  34. kai_sikorski

    Moving average filter for MC solution of PDE

    I have a PDE that can be interpreted as basically an exit time problem for a certain stochastic process. I would like to use this to verify an analytical solution I've found. If I start the stochastic process at (x,y), then the average exit time from a certain region will be equal to the value...
  35. kai_sikorski

    Probobalistic interpretation of a PDE

    Consider the following PDE. A lot of this is from "Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation" by J. Franklin and E. Rodemich. \frac{1}{2} \frac{\partial^2 T}{\partial y^2} + y \frac{\partial T}{\partial x} = -1 With |x|<1, |y| < \infty and we require...
  36. A

    [PDE] Transforming Hard Equations into Easier Ones

    I have a PDE and I have to transform it into an easier one using a substitution: u_t=u_{xx}-\beta u I am supposed to use the following substitution: u(x,t)=e^{-\beta t}w(x,t) I am supposed to get something that looks like this: w_t=w_{xx} Can someone show me the steps?
  37. AntSC

    Solutions to PDE: Understanding and Simplifying the Process

    I have seen a couple of solutions to this PDE - \frac{\partial x}{\partial u}=\frac{x}{\sqrt{1+y^{2}}} One is - u=\ln \left | y+\sqrt{1+y^{2}} \right |+f\left ( x \right ) I have no idea how this is arrived at or if it's correct. This is what i want to know. The solution I've...
  38. A

    [PDE] Transforming Nonhomogeneous BCs into Homogeneous Ones

    So there's this problem in my text that's pretty challenging. I can't seem to work out the answer that is given in the back of the book, and then I found a solution manual online that contains yet another solution. The problem is a the heat equation as follows: PDE: u_{t} = α^2u_{xx} BCs...
  39. K

    Solving PDEs: Deriving Wave Equation from u(x ± ct)

    It's been a little too long since I've has to do this. Can someone please remind me, how do you get from: ∂u/∂t = C(∂u/∂g) to ∂^2u/∂t^2 = (C^2)(∂^2u/∂t^2) The notation here is a little clumsy, but I'm just taking the second PDE of each side. How does the C^2 get there...
  40. A

    PDE: Initial Conditions Contradicting Boundary Conditions

    Suppose we have the following IBVP: PDE: u_{t}=α^{2}u_{xx} 0<x<1 0<t<∞ BCs: u(0,t)=0, u_{x}(1,t)=1 0<t<∞ IC: u(x,0)=sin(πx) 0≤x≤1 It appears as though the BCs and the IC do not match. The derivative of temperature with respect to x at position x=1 is a constant 1...
  41. H

    DG method for nonlinear elliptic PDE

    Preface: just want to start by saying that I'm 99% sure I'm having a stability issue here in the way I'm implementing the time step since if I set \Delta t \ge 1 then for any stopping time > 1, the algorithm works as it should. For time steps smaller than 1, as the time step gets smaller and...
  42. D

    Local solutions to semilinear parabolic PDE with a singular nonlinearity

    Hi everyone, I met a specific semilinear second-order PDE given by \frac{\partial u(x,t)}{\partial t} = {\bf A}(x,t) u(x,t) + U(x,t) u(x,t)^{-p} + A(x,t),...(1) u(x,0) = b>0, where p>0,\ \ (x,t) = (x_1,x_2,t)\in{\bf R}_+^2\times [0,T] , and {\bf A}(x,t)u =...
  43. P

    Solve Nonhomogenous PDE: Equilibrium Temp Distribution, B Value

    Homework Statement du/dt=(d^2 u)/dx^2+1 u(x,0)=f(x) du/dx (0,t)=1 du/dx (L,t)=B du/dt=0 Determine an equilibrium temperature distribution. For what value of B is there a solution? Homework Equations Not really sure what to put here. The Attempt at a Solution I started by trying to separate...
  44. S

    Second order pde - on invariant?

    second order pde -- on invariant? What the meaning for a second order pde is rotation invariant? Is all second order pde are rotation invariant? or only laplacian?
  45. S

    First Order PDE Solution Method Issues

    I'd really appreciate help with two little questions relating to first order partial differential equations. Just to quickly let you know what I'm asking, the first is about solution methods t first order PDE's & pretty much requires you to have familiarity, by name, with Lagrange's method...
  46. M

    Understanding the Heat Equation and its Practical Applications

    given the heat equation \frac{\partial u}{\partial x}=\frac{\partial^2 u}{\partial x^2} what does \frac{\partial^2 u}{\partial x^2} represent on a practical, physical level? I am confused because this is not time-space acceleration, but rather a temperature-spacial derivative. thanks all!
  47. J

    PDE with variable boundary condition

    Homework Statement I am trying to solve this PDE with variable boundary condition, and I want to use combination method. But I have problem with the second boundary condition, which is not transformed to the new variable. Can you please give me some advise? Homework Equations (∂^2 T)/(∂x^2...
  48. S

    Mathematica How to Solve a Second Order PDE in Mathematica?

    Hello! I am trying to solve the following second order PDE (copy that into mathematica): \!\( \*SubscriptBox[\(\[PartialD]\), \(x, t\)]\(\[Delta][x, t]\)\) + b \!\( \*SubscriptBox[\(\[PartialD]\), \(t\)]\(\[Delta][x, t]\)\) + a \!\( \*SubscriptBox[\(\[PartialD]\), \(\(x\)\(\...
  49. L

    Non Linear PDE in 2 dimensions

    Hi all. I'm trying to solve this PDE but I really can't figure how. The equation is f(x,y) + \partial_x f(x,y) - 4 \partial_x f(x,y) \partial_y f(x,y) = 0 As a first approximation I think it would be possible to consider \partial_y f a function of only y and \partial_x f a function of only...
  50. B

    Natural Vibration of Beam - PDE

    I am just wondering the author is doing in this calculation step. Given ##\displaystyle \rho A \frac {\partial^2 w}{\partial x^2} - \rho I \frac{\partial^4 w}{\partial t^2 \partial x^2} +\frac {\partial^2 }{\partial x^2}EI \frac {\partial^2 w}{\partial x^2}=q(x,t)## where ##w(x,t)=W(x)e^{-i...
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