What is Pde: Definition and 855 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. Q

    Simple PDE: Finding the General Solution for u_{xx} + u = 6y

    Homework Statement Find the general solution of u_{xx} + u = 6y, in terms of arbitrary functions.Homework Equations The PDE has the homogeneous solution, u(x,y)=Acos(x)+Bsin(x) . u_{xx} + u = 6y has the particular solution, u(x,y)=6y The Attempt at a Solution Taking a superposition...
  2. M

    Solving a PDE: Analytical Solution for $$\alpha \frac{\partial f}{\partial r}$$

    hey pf! i was wondering if you could help me out with a pde, namely $$\alpha ( \frac{z}{r} \frac{\partial f}{\partial r} + \frac{\partial f}{\partial z} ) = \frac{2}{r} \frac{\partial f}{\partial r} + \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial z^2} + 2 \frac{z}{r}...
  3. M

    Second order PDE (w.r.t 2 variables)

    Homework Statement find the solution to: \frac{\partial^{2}u}{\partial x \partial y} = 0 \frac{\partial^{2}u}{\partial x^{2}} = 0 \frac{\partial^{2}u}{\partial y^{2}} = 0 Homework Equations theorem of integration The Attempt at a Solution now from a previous question I...
  4. N

    Solving 2nd order PDE of single variable

    I've been getting pretty rusty in terms of derivation in recent years. Encountered this problem which I can't derive the steps despite knowing the solution. \frac{\partial^2 u}{\partial r^2} + \frac{\partial u}{\partial r}\left(\beta + \frac{1}{r}\right)+\frac{\beta}{r}u=0 Known...
  5. xortdsc

    How to compute an energy function of PDE ?

    Hi, I have a PDE of the form f(x,y,z)'' = Δf(x,y,z) + f(x,y,z) * (1 - f(x,y,z)^2) where f(x,y,z) is a 3 dimensional vector-field. Now I want to compute an energy function for it such that for any state (f(x,y,z) and its first derivative f(x,y,z)') I can compute its corresponding energy...
  6. T

    PDE Existence/Uniqueness Question

    My equation is: \left(\mathbf{\nabla}\sigma\right)\cdot\left(\mathbf{\nabla}V\right) + \sigma\nabla^2V = 0 If I'm given V(r) on the boundary of some volume, and I know σ(r) inside the volume, is there a unique solution V(r) inside that volume for any arbitrary (well-behaved) function...
  7. C

    General solution for a PDE with new variables

    Homework Statement Find the general solution f = f(x,y) of class C2 to the partial differential equation \frac{\partial^2 f}{\partial x^2}+4\frac{\partial^2 f}{\partial x \partial y}+\frac{\partial f}{\partial x}=0 by introducing the new variables u = 4x - y, v = y. Homework Equations...
  8. J

    Exploring the Shape of Characteristic Curves for a PDE

    Homework Statement suppose u(x,y) satisfies the partial differential equation: -4y\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0 Find the characteristic curves for this equation and name the shape they form The Attempt at a Solution \frac{dy}{dt}=1 \Longrightarrow y=t+y_0...
  9. N

    Assembling mass matrix for transient heat conduction pde

    I have derived the weak form of the transient heat conduction equation (for FEM) and I am having trouble trying to assemble the mass matrix This is the PDE: \frac{\partial U}{\partial t} = \alpha \nabla^2U This is the equation for the mass matrix for an element: M^e = \int \Psi...
  10. J

    Genaral PDE for scalar and vectors

    I realized that a PDE of 2nd order can written like: A:Hf+\vec{b}\cdot\vec{\nabla}f+cf=0 \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}:\begin{bmatrix} \partial_{xx} & \partial_{xy}\\ \partial_{yx} & \partial_{yy}\\ \end{bmatrix}f+\begin{bmatrix} b_1\\ b_2\\...
  11. E

    Solving bioheat PDE numerically

    Hello! My goal here is to plot the solution to the bioheat equation for a tumor as a function of time. I'm plotting this for a fixed radius at r = 0 (the very center of the tumor). The equation to solve is this: \rho_1c_1 \frac{\partial T}{\partial t} = 3(\frac{\partial }{\partial...
  12. I

    1st order PDE, seems easy but still confusing

    Hello, I am doing some physics and I end up with this PDE: \frac{\partial q(x,y,t)}{\partial t} = -(x^2 + y^2)q(x,y,t) + ax\frac{\partial q(x,y,t)}{\partial y} where q(x,y,t) is the scalar field to determine and a is a parameter. I need to consider two types of initial conditions...
  13. V

    Boundary conditions for inhomogeneous non-sepearable 3D PDE

    Hello, I am looking to solve the 3D equation in spherical coordinates \nabla \cdot \vec{J} = 0 using the Ohm's law \vec{J} = \sigma \cdot (\vec{E} + \vec{U} \times \vec{B}) where \sigma is a given 3x3 nonsymmetric conductivity matrix and U,B are given vector fields. I desire the...
  14. P

    Method of characteristics and second order PDE.

    This isn't a homework question per se. Am merely seeking an explanation how the method of characteristics may be applied to a second order PDE. For instance, how is it used to solve utt=uxx-2ut?
  15. maistral

    Poisson PDE discretisation help

    Okay, I'm trying to play around again :D A little overview; I know that the Poisson equation is supposed to be: uxx + uyy = f(x,y) I can manage to discretise the partial derivative terms by Taylor. I don't know how to deal with the f(x,y) though. Say for example, uxx + uyy = -exp(x). what...
  16. J

    General Solution for 2nd Order PDE: Is it Possible?

    Hellow everybody! A simple question: exist a general formulation, a solution general, for a PDE of order 2 like: ## au_{xx}(x,y)+2bu_{xy}(x,y)+cu_{yy}(x,y)+du_x(x,y)+eu_y(x,y)+fu(x,y)=g(x,y) ## ? The maple is able to calculate the solution, however, is a *monstrous* solution!
  17. J

    How is the initial conditions for PDE?

    Given a PDE of order 1 and another of order 2, you could show me what is, or which are, all possible initial conditions? For an ODE of order 2, for example, the initial condition is simple, is (t₀, y₀, y'₀). However, for a PDE, I think that there is various way to specify the initial condition...
  18. E

    Proving convexity for linear parabolic PDE

    I have a parabolic PDE of the form a\frac{\partial^2 f}{\partial x^2} - b\frac{\partial f}{\partial x} + \frac{\partial f}{\partial t} = 0, where (x,t) \in (-\infty, \infty) \times [0, T]. In addition, \lim_{x \to \infty} f(x,t) = 0, \lim_{x\to -\infty} = k (a known positive constant), and...
  19. R

    NUMERICAL approach to NONLINEAR PDE

    Hi guys, I need to simulate wave propagation for a nonlinear dispersive wave PDE and since I can't find proper resources for handling nonlinear PDEs numerically, I would appreciate any help and clues. the PDE is in the form of utt-(au+bu2+cu3+duxx)xx=0 Romik Ps: BC: Clamped at both ends IC...
  20. A

    Multivariable Derivatives/Elliptic PDE

    From http://en.wikipedia.org/wiki/Elliptic_operator: "A nonlinear operator L(u) = F(x, u, (\partial^\alpha u)_{|\alpha| \le 2k}) is elliptic if its first-order Taylor expansion with respect to u and its derivatives about any point is a linear elliptic operator." I'm a bit confused by what we...
  21. P

    Solving ux + (x/y)uy = 0 Using Characteristics

    Hi, Homework Statement I have solved ux + (x/y)uy = 0 using characteristics, to obtain u(x,y)=C (for y=+-x) and f(x2-y2) Homework Equations The Attempt at a Solution I was then given two boundary conditions: (a) u(x=0,y)=cos(y), which I used to obtain u(x,y) = cos(√(y2-x2))...
  22. S

    Application of PDE on physicis

    I want to find some application of the laplace equation on semi-infinite plate on physics where the PDE is looke like $$u_{xx}+u_{yy}=f , for a<x<\infty , c<y<d$$ $$u(a,y)=g(y), u(x,c)=f_{1}(x), u(x,d)=f_{2}(x)$$ $$\lim_{x->\infty} f(x)=\lim_{x->\infty} f_{1}(x)=\lim_{x->\infty}...
  23. S

    Transforming Non-Homogeneous Boundary Conditions in 2D PDEs

    Homework Statement now I have a PDE $$u_{xx}+u_{yy}=0,for 0<x,y<1$$ $$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$ Then I want to know whether there are some method to make the PDE become homogeneous boundary condition. $$i.e. u|_{\partialΩ}=0$$
  24. P

    Solving two coupled PDE from a quantum problem

    Hello Guys. I have to solve two coupled PDE coming from a quantum physical problem, which possesses a cylindrical symmetry. They look like this \left\{-\frac{i\hbar}{2M}\left( D^2_\rho + \frac{1}{\rho}D_\rho + D^2_z - \frac{L_z^2}{\hbar^2 \rho^2} \right) + \frac{M\omega^2_\rho \rho^2}{2} +...
  25. M

    Solving Linear First Order PDE with Initial Conditions

    Homework Statement Mod note: Pasted the OP's correction into the original problem. Solve xe^z\frac{\partial u}{\partial x} - 2ye^z\frac{\partial u}{\partial y} + \left(2y-x \right)\frac{\partial u}{\partial z} = 0 given that for x > 0, u = -x^{-3}e^z when y=-x Homework Equations...
  26. S

    Applying the fourier transform to a PDE

    I have a tutorial question for maths involving the heat equation and Fourier transform. {\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}} you are given the initial condition: u(x,0) = 70e^{-{\frac{1}{2}}{x^2}} the answer is: u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}} In this course...
  27. T

    Solving a separable PDE with BC

    So I have a question in terms of interpreting the boundary conditions for a PDE. It is question 4 in the attached picture. My question is that usually when I have encountered BC problems I have been given that my boundary conditions equal a given value, in terms of the diffusion equation...
  28. R

    Need help solving 1st order pde numerically

    Hi everybody, I need to solve a 1st order PDE for my thesis and I'm not a specialist in this field. I've read some texts about this and know one method of solving a 1st order PDE is the method of characteristics. since my equation is nonlinear and a bit complicated, I'm going to solve it...
  29. S

    Critiquing separation of variables method for PDE.

    "Critiquing" separation of variables method for PDE. I am currently taking a course in PDE's and it has been very "applied" and not so much theory based. I can say its been separate this separate that separate this separate that… Enough! We are always "separating variables" and it always...
  30. E

    Using finite difference method for solving an elliptic PDE with MATLAB

    Homework Statement Given that we the following elliptic problem on a rectangular region: \nabla^2 T=0, \ (x,y)\in \Omega T(0,y)=300, \ T(4,y)=600, \ 0 \leq y \leq 2 \frac{\partial T}{\partial y}(x,0)=0, \frac{\partial T}{\partial y}(x,2) = 0, \ 0\leq x \leq 4 We want to solve this problem...
  31. A

    Upper Division ODE vs PDE class

    I'm trying to decide between taking an ODE class or a PDE class next. I have already done Calculus 1,2,3 so I already know some ODEs and PDEs and linear algebra. I'm a 3rd year mathematics major with a minor in Statistics and I'm interested in applied mathematics.ODE course coverage: Ordinary...
  32. H

    PDE, heat equation with mixed boundary conditions

    Homework Statement solve the heat equation over the interval [0,1] with the following initial data and mixed boundary conditions.Homework Equations \partial _{t}u=2\partial _{x}^{2}u u(0,t)=0, \frac{\partial u}{\partial x}(1,t)=0 with B.C u(x,0)=f(x) where f is piecewise with values: 0...
  33. Superposed_Cat

    How is the equality in this PDE achieved?

    I've got this far on a pde (second last step) but have no idea how they got this equality(I'm a noob), could someone please explain? I was going to put this under homework but it is not homework and it doesn't really fit the template. Thanks in advance.
  34. P

    2nd order PDE using integration by parts

    Homework Statement Find the general solution of the equation (\zeta - \eta)^2 \frac{\partial^2 u(\zeta,\eta)}{\partial\zeta \, \partial\eta}=0, where ##\zeta## and ##\eta## are independent variables. Homework Equations The Attempt at a Solution I set ##X = \partial u/\partial\eta## so that...
  35. S

    Solving a First Order Linear PDE

    Good day. I was wondering if you could help me solve this first order linear partial differential equation: [∂δ]/[/∂t] = [ρg]/[/μ] δ^2 [∂δ]/[/∂z]. The solution for this is: δ(z, t) = √[μ z]/[/ρg t]. I don't really understand how the PDE became like this. If you could show the...
  36. D

    MHB Plotting an infinite domain PDE

    Is anyone familiar with plotting an infinite domain PDE where the solution is an integral. Take the solution \[ T(x,t) = \frac{100}{\pi}\int_0^{\infty}\int_{-\infty}^{\infty} \frac{\sinh(u(10-y)}{\sinh(10u)} \cos(u(\xi-x))d\xi du \] How could I plot this in Matlab, Mathematica, or Python? As a...
  37. T

    How to Complete the Square for a Diffusion Equation?

    Ok this qusestion has to do with completing the square for a diffusion equation. Initial Cond: u(x,0) = e-x Now they say plug this into the general formula: u(x,t) = 1/(4\pikt)1/2 ∫ e-(x-y)1/2/4kte-y dy where k is a constant now the first step they say is completing the...
  38. P

    PDE change of variables Black-Scholes equation

    Homework Statement By changing variables from (S,t,V) to (x,\tau,u) where \tau = T - t, x = \ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T-t), u=e^{r\tau}V, where r, \sigma, \tau, K are constants, show that the Black-Scholes equation \frac{\partial V}{\partial t} +...
  39. A

    [PDE] 2D Vibrating Plate (Unique Solution)

    We have a region Ω in ℝ^2 with a smooth boundary. There is a plate of shape Ω and clamped edges which is approximated by the following equation: $$\frac{∂^2u}{∂t^2}=-Δ^2u$$ $$u(x,t)=0\hspace{4ex} x\in ∂Ω$$ $$Du(x,t)\cdot\hat{n}=0\hspace{4ex} x\in ∂Ω$$ \hat{n} is the outward pointing unit vector...
  40. E

    The sign of terms in a characteristic equation of a PDE

    This was something I noticed as I was trying to practice solving PDEs using the method of characteristics. The text has the following example: $$\frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y} = 0$$ This should be easy enough. I let p(x,y) = x and solve for \frac{\partial...
  41. K

    Classification of Second-Order PDE with Constant Coefficients

    Homework Statement I have 3 equations: \frac{\partial^2 u}{\partial t^2}+\frac{\partial^2 u}{\partial x \partial t}+\frac{\partial^2 u}{\partial x^2} \frac{\partial^2 u}{\partial t^2}+4\frac{\partial^2 u}{\partial x \partial t}+4\frac{\partial^2 u}{\partial x^2} \frac{\partial^2...
  42. J

    Specification of the boundary condition in high order PDE

    Hi all, I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2} subject to the initial condition u(x,0)=g(x)...
  43. K

    Solving a PDE with Non-homogenous Boundary Conditions

    Homework Statement If utt - uxx= 1-x for 0<x<1, t>0 u(x,0) = x2(1-x) for 0≤x≤1 ut(x,)=0 for 0≤x≤1 ux(x,)=0 u(1,t)=0 find u(1/4,2) Homework Equations The Attempt at a Solution I was thinking to make a judicious change of variables that not only converts the PDE to a homogenous PDE, but also...
  44. K

    How Can Variable Transformation Solve a Non-Homogeneous PDE?

    Homework Statement Find the solution of: utt-uxx = sin(∏x) for 0<x<1 u(x,0)=0 for 0<=x<=1 ut(x,0)=0 for 0<=x<=1 u(0,t)=0 u(1,t)=0Homework Equations utt-uxx = sin(∏x) for 0<x<1 u(x,0)=0 for 0≤x≤1...
  45. H

    Solving PDE Using Method of Separation of Variables: Complete Guide

    I'm new here, hope it is the right place to ask the question. The PDE question is ∂2u/∂x2+∂2u/∂y2=0 and u(x,0)=f(x), u(x,1)=0, u(0,y)=0, u(1,y)=0. I use the method of separate the variables, with is let u(x,y)=X(x)Y(y) and get X''/X+Y''/Y=0. Then let X''/X=-Y''/Y=-λ, i.e. X''+λX=0...
  46. R

    Need help solving PDE, numerically which contains some transformations

    Hi, hope this is a right place to ask this question. I work in the soil physics field and this problem has taken lots of my energy for a while! let's state it: Unsaturated horizontal water flow in 2 layer soil: we have, M(for Moisture), K (for hydraulic conductivity), h (for hydraulic...
  47. E

    Solving a simple second order PDE, do I need the Fourier?

    Solving a "simple" second order PDE, do I need the Fourier? Homework Statement The problem as given: y'' + 2y' + 5y = 10\cos t We want to find the general solution and the steady-state solution. We're using \mu y'' + c y' + k y = F(t) as our general form. OK, so I first want the general...
  48. T

    Solving PDE Questions: Constant & Variable Coefficient Equations

    So up to this point we have only learned 2 forms of PDE's to solve: Constant Coefficient Equations and Variable Coefficient Equations. Questions: Solve: 1) aUx+bUy + cU = 0 2) Ux+ UY = 1 where U = U(x,y) Attempt: Well for 2) I'm thinking that it doesn't necessarily matter...
  49. T

    Corner signularity for elliptic PDE

    I'm using spectral element methods to numerical solve a non-linear pde D \psi = f\left(x,\psi \right) in a rectangular domain, with \psi = 0 Here D is a second order elliptic operator. I've found that the rate of convergence of my method depends on my choice of the functional form of...
  50. E

    Solving a PDE with elementary methods

    Homework Statement OK, a PDE: $$a\frac{\partial u}{\partial t} + b \frac{\partial u}{\partial x} = u$$ we want the general solution. 2. The attempt at a solution So, I'll set up a couple of equations thus: r = m11x + m21t s = m12x + m22t (We have a nice matrix of m here if we...
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