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. I

    PDE: How to use Fourier Series to express a real function?

    actually have two questions: here we have a Fourier series.. $$f(t) = \sum c_k e^{2\pi ikt}$$ (c is complex) if we're trying to express a real function via Fourier series, and we do it the following way.. Impose condition: $$\overline{c_k} = c_{-k}$$ $$f(t) = \sum\limits_{k= -n}^n c_k e^{2\pi...
  2. A

    Dirichlet and Nuemann condition on the same boundary

    Hi, My final goal is to solve numerically Schrodinger's equation in 3D with some potential for the unbounded states, meaning that far away from the potential (at infinity) we may find a free wave and not something that goes to zero. The basic idea is that I have a particle in (0,0,0) that...
  3. M

    Linear PDEs: A Simple Explanation

    This isn't a homework problem so hopefully this section is fine. I came across something that's bothering me while reviewing PDEs. Take something like: u_{x}(x,t) = 1. which has the general solution: u(x,t) = c_{1}(t) + x. Wolfram says this is linear but if I take a different solution: v(x,t) =...
  4. A

    Converge pointwise with full Fourier series

    I am working on a simple PDE problem on full Fourier series like this: Given this piecewise function, ##f(x) = \begin{cases} e^x, &-1 \leq x \leq 0 \\ mx + b, &0 \leq x \leq 1.\\ \end{cases}## Without computing any Fourier coefficients, find any values of ##m## and ##b##, if there is any...
  5. N

    Struggling in Diff Eq Class: Advice for a Ditzy Freshman

    I'm a 2nd-semester freshman taking my first upper-level class (partial diff eq) and I'm really struggling. People always ask me what I'm doing in that class as a freshman and I answer by telling them I'm an idiot and a masochist, which is true. I've spent most of my time and energy on that class...
  6. M

    MHB PDE or differentiable manifolds?

    Hello! :o I am doing my master in the field Mathematics in Computer Science. I am having a dilemma whether to take the subject Partial differential equations- Theory of weak solutions or the subject differentiable manifolds. Could you give me some information about these subjects...
  7. S

    MHB PDE Solving Continuity Equation

    Hi, I am trying to find the exact solution of the Continuity Equation. Any Idea how can i start solving it, i need it for some calculation in Image Processing. $$\pd{C}{t}+\pd{UC}{x}+\pd{VC}{y}=0$$ Where $U$ and $V$ is velocity in $X$ and $Y$ direction. The initial condition is as...
  8. J

    Fourier COSINE Transform (solving PDE - Laplace Equation)

    I'm trying to solve Laplace equation using Fourier COSINE Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't think so). NOTE: U(..) is the Fourier Transform of u(..) This are the equations (Laplace...
  9. M

    PDE and differentiating through the sum

    Hi PF! I'm reading my math text and am looking at the heat eq ##u_t = u_{xx}##, where we are are given non-homogenous boundary conditions. We are solving using the method of eigenfunction expansion. Evidently we begin by finding the eigenfunction ##\phi (x)## related to the homogenous...
  10. DrPapper

    Applied PDE for Scientists and Engineers Farlow

    Hello Fellow Physics People, I am just now taking a math methods course for Physicists and we're using Mary Boas book. I wanted to supplement it for better understanding as saw Partial Differential Equations for Scientists and Engineers by Stanley J. Farlow. Reading reviews for this book on...
  11. binbagsss

    Method of characteristics. pde. limits of integral question

    I'm using the method of characteristics to solve a pde of the from ## au_{x}+bu_{y}=c## where ## a=\frac{dx}{d \tau} , b= a=\frac{dy}{d \tau}, c=a=\frac{du}{d \tau}## where initial data is parameterised by ##s## and initial curve given by ##x( \tau)=x_{0}(s)##, ##y( \tau)=y_{0}(s)## and ##u(...
  12. M

    Good texts for solving PDE's by integral transforms

    I look for good books on solving partial diffrential equations (PDE's) using integral transforms specially Fourier and laplace transforms. Do you have any recommendations for such books? I don't look for a book concerned with the theory, rather, with the methods itself (a suitable book for a...
  13. N

    Transforming Partial Differential Equations into Constant Coefficient Form

    Homework Statement The problem statement can be expressed in one of these forms listed in order of preference. [/B] Every character with exception of x, y, t, and C are constants. Homework Equations I require a change of variable or series of subsequent change of variables that can convert...
  14. binbagsss

    Method of Characteristics, PDE, Jacobian condition Q

    Hi, I'm looking at the Jacobian condition which is ## J= a \frac{dy_{0}}{ds}-b\frac{dx_0}{ds}## where the pde takes the form ##c= a\frac{\partial u}{\partial x} + b \frac{\partial u}{\partial y} ##, where ##a=\frac{\partial x}{\partial \tau } ##, ##b=\frac{\partial y}{\partial \tau }##...
  15. J

    Proof of the linearity principle for a 2nd order PDE?

    Homework Statement My textbook (Advanced Engineering Mathematics, seventh edition, Kreyszig) indicates that if u1 and u2 are solutions to a second-order homogeneous partial differential equation, and c1 and c2 are constants, then u where u = c1u1 + c2u2 is also a solution, this is the...
  16. DivergentSpectrum

    What are the Boundary Conditions for Solving Poisson's Equation?

    As i understand, the purpose of laplaces/poissons equation is to recast the question from a geometrical one to a differential equation. im trying to figure out what are the appropriate boundary conditions for poissons equation: http://www.sciweavers.org/upload/Tex2Img_1418842096/render.png...
  17. M

    Inverse Fourier Transform of ##1/k^2## in ##\mathbb{R}^N ##

    Homework Statement This comes up in the context of Poisson's equation Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Homework Equations $$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta }$$...
  18. H

    Multidimensional first order linear PDE

    Hello, I have the following PDE : v_y \frac{\partial f}{\partial y} + \Omega_x(y)\left(v_z\frac{\partial f}{\partial v_y} - v_y\frac{\partial f}{\partial v_z}\right) + \Omega_z(y)\left(v_y\frac{\partial f}{\partial v_x} - v_x\frac{\partial f}{\partial v_y}\right) = 0 (which is the steady...
  19. M

    Find the Best PDE & Fourier Books - Your Input Appreciated!

    Hello everyone, I'm in need for the best books that you know out there for PDE (Partial Differential Equations) and everything related to Fourier (series, transform, etc.). Any help would be much appreciated. Thank you and happy holidays!
  20. 4

    Why is a set of functions v(t) dense in L^2

    Hello, I was going through the following paper: http://www.emis.de/journals/HOA/AAA/Volume2011/142128.pdf In page 6, immediately after equation (3.15), its written that "functions of the form v(t) are dense in L^2". I have been looking for proofs online which verifies the above statement but...
  21. J

    Solving a PDE in spherical with source term

    Homework Statement I have a PDE and I need to solve it in spherical domain: $$\frac{dF(r,t)}{dt}=\alpha \frac{1}{r^2} \frac{d}{dr} r^2 \frac{dF(r,t)}{dr} +g(r,t) $$ I have BC's: $$ \frac{dF}{dr} = 0, r =0$$ $$ \frac{dF}{dr} = 0, r =R$$ Homework Equations So, in spherical coord. First...
  22. P

    Solving 1st Order PDE: Finding L with Ax^2 Form

    How to find L if the form is: $$ (\frac{\partial L}{\partial x})^2 - (\frac{\partial L}{\partial y})^2 = -1$$ The author wrote, $$L = y + ax^2 + ..$$ but I didn't get how?
  23. T

    Solving PDE with smoothing - Time step Query

    Hi, I'm solving an Euler CFD code using the Lax-Wendroff method. It contains a dissipative smoothing term which I'm looking to minimise to optimise the accuracy. The timestep and smoothing terms are uncoupled, however different stable time steps result in different accuracy once the calculation...
  24. binbagsss

    PDE Charactersitic equations colliding and shocks concepts

    I'm new to shocks and trying to get the hang of it. I have 3 sets of characteristic equations,( by a set I mean defined by taking a different fixed value u along the characteristic.) From what I understand,in general talk, we use a shock whenever two sets of characteristics collide as...
  25. 2

    First order, linear PDE with unknown inhomogeneity function

    Homework Statement Solve the following IVP: ##\frac{\partial v(x,t)}{\partial x} + \frac{\partial v(x,t)}{\partial t} + v(x,t) = g(x,t)## Homework Equations The initial values: v(0,t) = a(t) and v(x,0) = b(x)The Attempt at a Solution I applied the Laplace transform x -> s to get...
  26. M

    Solving PDE Heat Equation with Non-Homogeneous Boundary Conditions

    Homework Statement $$u_t = ku_{xx} + \sin(2 \pi x / L)$$ $$u_x(0,t) = u_x(L,t) = 0$$ $$u(x,0) = f(x)$$ Homework Equations none (other than the obvious) The Attempt at a Solution So i started by taking letting ##ku_E''(x) =- \sin(2 \pi x / L)## (notice from the boundary conditions above I...
  27. A

    MATLAB MATLAB code for Computational Fluid Mechanics

    Hello guys, I'm writing to get some help on an exercise I've been thinking but I can't get to solve. I have to write the code for the Example 8.5 of the book White, Fluid Mechanics. Here is the problem and the solution I have to obtain. It is about one duct that has three sections in which I...
  28. M

    Re-writing the PDE Homework Statement | Two-Term Equation Solution

    Homework Statement Provide the missing steps to re-write the equation into one with just two terms $$u_{tt} - c^2(u_{rr}+\frac{2}{r}u_r) = 0$$ Homework Equations Nothing, other than this looks similar to the wave equation hybrid. (I'm just speculating) Also, I'm a little uncertain what is...
  29. Hercuflea

    Math: Job prospects after a PhD in PDE research

    What would the career prospects be for someone who does a Ph. D. in Mathematics with a research focus in partial differential equations? Assuming you got some computer skills along the way like parallel computing, programming, etc? Sure, you could become a professor but most people don't make...
  30. M

    Understanding PDE Solutions: N=0 vs N≥1 Cases

    Hey PF! I have a quick question. When I was solving a PDE via separation of variables, I was able to come up with a same format solution for ##n \geq 1## but when ##n=0## I had a different "type" of solution. This doesn't really bother me since I am dealing with a linear PDE. However, I matched...
  31. O

    Proving a a pitchfork bifurcation: modified swift-hohenberg

    I'm trying to solve question 4.12 from Cross and Greenside "pattern formation and dynamics in nonequilibrium systems". the question is about the equation \partial_t u = r u - (\partial_x ^2 +1)^2 u - g_2 u - u^3 Part A: with the ansatz u=\sum_{n=0}^\infty a_n cos(nx) show that the...
  32. C

    PDE for temperature distribution in rectangle

    Homework Statement A rectangular chip of dimensions a by b is insulated on all sides and at t=o temperature u=0. The chip produces heat at a constant rate h. Find an expression for u(x,y,t) Homework Equations δu/δt = h + D(δ2u/δx2 + δ2u/δy2) x∈(0,a), y∈(0,b) The Attempt at a Solution I'm...
  33. B

    One Physics MS semester left - what courses to take?

    Hello all, I've got one more semester before I earn my physics MS, and I have space for one or two extra courses. I am going into oceanography, and I would like to have a strong foundation in math in order to understand the theory I'll encounter as well as possible. Lots of physical...
  34. binbagsss

    Basic concept Q ,non-linear PDE , kdv

    Ut+6UUx+Uxxx=0 [kdv eq] Why to solve this do you need U(x,t=0)? Why is it a initial value problem? This should probably be really obvious. I think I've forgotten some basic background stuff, just starting my course in solitons... Thanks for your help.
  35. H

    Setting up diffusion PDE in matlab

    I am going to do a numerical simulation of diffusion in matlab. The diffusion coefficient is concentration dependant, and i use an array operation to calculate D(x), so it is known. Based on Fick's second equation: $$ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x} D \frac{\partial...
  36. P

    Deriving a Bound for a System of Coupled PDEs Using the Energy Method

    Homework Statement Hi! Not sure if I'm posting in the right section, this problem is from a course in scientific computing. Anyway, we're considering a set of PDEs: u_t + Au_x = 0 \quad 0<x<1, \ t>0 \\ u(x,0) = f(x) \quad 0 \leq x \leq1 \\ u_1(0,t) = 0 \quad t \geq 0 \\ u_2(1,t) = 0 \quad t...
  37. I

    Parabolic pde with additional term

    Homework Statement Solve \frac{\partial v}{\partial t} = k\frac{\partial^2 v}{\partial x^2} - v 0\leq x \leq L t > 0 Homework Equations v(x,0) = f(x), v(0,t)=0, \frac{\partial v}{\partial x}(L,t) = -v(L,t) The Attempt at a Solution I've already attempted to solve...
  38. M

    MHB Is Wave Phenomena Related to PDE or Just Physics?

    Hey! :o Do you know if "Wave Phenomena" are related to PDE or only to physics?? (Wondering)
  39. DivergentSpectrum

    How do I apply rk4 to a second order pde?

    Im writing a program that calculates the trajectory of a particle in an arbitrary force field. the force field is a vector function of position (x, y, z) AND velocity (x', y', z'). Rk4= runge kutta forth order method Please help. Thanks!
  40. DivergentSpectrum

    Numerical second order pde solver

    Edit:whoops wrong forum mods please move 2nd edit: I just had dinner then got back on the computer, input some points and saw a beautiful elipse.(complete with a fascinating flower petal design due to inaccuracies) Weird lol! No idea why it wasnt working before Now to implement RK4 bwahahaha...
  41. MathematicalPhysicist

    Ultra-hyperbolic pde and F theory

    Is there a conncetion between Fritz john's ultra-hyperbolic pde, which is the equation: u_{tt}+u_{\tau \tau} = u_{xx}+u_{yy} I mean F theory has another dimension of time, and the above pde has also another time variable with regards to the simple wave pde. Any literature on this...
  42. C

    Rotation of coordinates (context of solving simple PDE)

    If you rotate your rectangular coordinate system (x,y) so that the rotated x'-axis is parallel to a vector (a,b), in terms of the (x,y) why is it given by x'=ax+by y'=bx-ay I got x'=ay-bx, y'=by+ax from y=(b/a)x. By the way this is from solving the PDE aux+buy=0 by making one of the...
  43. MathematicalPhysicist

    Maple Solving a Nonlinear PDE in Maple 18 using Numeric Methods

    I wrote the next code: restart; pde := diff(u(x, t), t)+diff(u(x, t), x)-(diff(diff(u(x, t), x), t))^2 = u(x, t); tmax := 0.5e-1; xmin := 0; xmax := 1; N := 10; bc1 := du(xmin, t) = 0; bc2 := u(xmax, t) = 0; ic1 := u(x, 0) = 1; ic2 := du(x, 0) = 2; bcs := {u(x, 0) =...
  44. M

    Finding Advanced PDE Resources for Meteorologists with a Math Background

    I am a meteorologist with Bachelor's Degrees in both Meteorology and Mathematics. I took an intro PDE course in college, but want to learn more. Can anyone suggest a book that would be a good book after only having an intro course? Thanks.
  45. A

    Solving PDE for F and F' in 2D Space: Relation between Variables x, t and t

    If I have a function "F" in a two-dimensional space F(x,t) and its analog F' in another co-ordinate system F'(x',t') and the relation between the two is given by : ∂F/∂t −c(∂F/∂x) =∂F ′/ ∂t ′ How do I find a relation between F and F ′ and between the variables x,t and t ′ ?
  46. R

    Initial condition effect in Nonlinear PDE of a wave

    Hey there, I have modeled a propagating wave in a 1D dispersive media, in which square and cubic nonlinear terms are present. u′′=au3+bu2+cu the propagating pulse starts to steepen with time which is the effect of nonlinearity, but there is an effect which I can't understand. so...
  47. L

    PDE (wave equation) used to find acoustic pressure in a a pipe

    Homework Statement Assume that the wavelength of acoustic waves in an organ pipe is long relative to the width of the pipe so that the acoustic waves are one-dimensional (they travel only lengthwise in the pipe). Therefore, the equation governing the pressure in the wave is: ∂2p/∂t2-c2*∂2p/∂x2...
  48. O

    How to numerically solve a PDE with delta function boundary condition?

    I have a PDE of the following form: f_t(t,x,y) = k f + g(x,y) f_x(t,x,y) + h(x,y) f_y(t,x,y) + c f_{yy}(t,x,y) \\ \lim_{t\to s^+} f(t,x,y) = \delta (x-y) Here k and c are real numbers and g, h are (infinitely) smooth real-valued functions. I have been trying to learn how to do this...
  49. T

    Fourier-Laplace transform of mixed PDE?

    I have a third order derivative of a variable, say U, which is a function of both space and time. du/dx * du/dx * du/dt or (d^3(U)/(dt*dx^2)) The Fourier transform of du/dx is simply ik*F(u) where F(u) is the Fourier transform of u. The Fourier transform of d^2(u)/(dx^2) is simply...
  50. 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...
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