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. T

    MHB Solving PDE using laplace transforms

    [Solved] Solving PDE using laplace transforms Hey, I'm stuck on this problem and I don't seem to be making any headway. I took the Laplace transform with respect to t, and ended up with the following ODE: $\frac{\partial^2 W}{\partial x^2}-W(s^2+2s+1)=0$ and the boundry conditions for $x$...
  2. D

    I Help discretizing this PDE (stream function)

    I have a PDE that I want to solve for a stream function ψ(j,l) by discretizing it on a 2D annulus grid in cylindrical coordinates, then solving with guas-seidel methods (or maybe a different method, not the point): (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] Where s and Φ are...
  3. akin-iii

    I How do I derive a PDE for the volume flow rate of a tilting vessel?

    So the other day, I was pouring beer from a can to a mug and I obviously know the flow rate depends on the height of the beer from the bottom of the can (fluid level in the vessel), angle of tilt and I think time as well. I was wondering how to best model the PDE to describe such a phenomenon (...
  4. S

    Mathematica Solving 2-D partial integro-differential equation

    While reproducing a research paper, I came across the following equation, ∂f/∂t−(H(f)(∂f/∂x)=0 where [H(f)] is hilbert transform of 'f.' and f=f(x,t) and initial condition is f(x,0)=cos(x) and also has periodic boundary conditions given by F{H{f(x′,t)}}=i⋅sgn(k)F{f(x,t)}, where F(f(x,t) is...
  5. F

    I Separation of Variables (PDE) for the Laplace Equation

  6. Phys pilot

    Getting the coefficients of inhomogeneous PDE using Fourier method

    Hello, I posted the same in the partial differential equations section but I'm not getting responses and maybe this section is better for help with homework. I have to solve this problem: $$u_t=ku_{xx}+h \; \;\; \; \; 0<x<1 \; \; \,\; t>0$$ $$u(x,0)=u_0(1-\cos{\pi x}) \; \;\; \; \; 0\leq x \leq...
  7. Phys pilot

    I Problem getting the coefficients of a non-homogeneous PDE using the Fourier method

    Hello, I have to solve this problem: $$u_t=ku_{xx}+h \; \;\; \; \; 0<x<1 \; \; \,\; t>0$$ $$u(x,0)=u_0(1-\cos{\pi x}) \; \;\; \; \; 0\leq x \leq 1$$ $$u(0,t)=0 \; \;\; \; \; u(1,t)=2u_0 \; \;\; \; \; t\geq0$$ So I know that I can split the solution in two (I don't know the reason. I would...
  8. E

    MHB How to get a converging solution to a second-order PDE?

    I have been struggling with a problem for a long time. I need to solve the second order partial differential equation $$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$ where $G_{zy}$, $G_{zx}$, $\theta$, $a$, and...
  9. E

    A How to get a converging solution for a second order PDE?

    I have been struggling with a problem for a long time. I need to solve the second order partial differential equation $$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$ where ##G_{zy}##, ##G_{zx}##, ##\theta##...
  10. Safder Aree

    How to apply the Fourier transform to this problem?

    I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a Fourier transform, where I can take the Fourier transform of both sides then solve the general solution in Fourier terms then inverse transform. However, since this...
  11. K

    Solving a PDE with boundary problem

    Homework Statement I want to find the solution to the following problem: $$\begin{cases} \nabla^2 B=c^2 B &\text{ on the half plane } x>0 \\ B=B_0 \hat{z} & \text{ for } x<0 \end{cases}$$ in the ##xz## plane. ##c, B_0 \in \mathbb{R}## Homework Equations I am not really sure what would be...
  12. J

    I Solution for 1st order, homogenous PDE

    ##u_t + t \cdot u_x = 0## The equation can be written as ##<1, t, 0> \cdot <d_t, d_x, -1>## where the second vector represents the perpendicular vector to the surface and since the dot product is zero, the first vector must necessarily represent the tangent to the surface. We parameterize this...
  13. T

    A Determine PDE Boundary Condition via Analytical solution

    I am trying to determine an outer boundary condition for the following PDE at ##r=r_m##: $$ \frac{\sigma_I}{r} \frac{\partial}{\partial r} \left(r \frac{\partial z(r,t)}{\partial r} \right)=\rho_D gz(r,t)-p(r,t)-4 \mu_T \frac{\partial^2z(r,t)}{\partial r^2} \frac{\partial z(r,t)}{\partial t} $$...
  14. N

    Partial Differential Equation with variable coefficients

    Homework Statement This question relates to a very large project I have been assigned in a course on mathematical methods in structural engineering. I have to solve the following equation, in a specific way: (17) Now we have to assume the following solution: (18) It wants me insert...
  15. Auto-Didact

    A PDE: Between Physics and Mathematics

    This is perhaps the single most important mathematical physics papers I have ever read; I think everyone - especially (theoretical) physicists - interested in theoretical physics should read it. In fact, read it now before reading the rest of the thread: Klainerman 2010, PDE as a Unified Subject...
  16. M

    Mathematica 2D PDE Solution with Neumann BC: Plot Comparison

    Hi PF! I'm solving a 2D PDE using NDSolveValueMesh. On one boundary I apply a Neumann BC. Attached are two plots: the top is the boundary condition and the functional value. Notice both are exactly the same. However, the lower plot shows a directional derivative evaluated on the same boundary...
  17. C

    Solving a 2D PDE using the Fourier Transform

    Homework Statement Solve the following partial differential equation , using Fourier Transform: Given the following: And a initial condition: Homework EquationsThe Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...
  18. Jozefina Gramatikova

    Separate the following PDE as much as possible

    Homework Statement [/B]Homework Equations [/B]The Attempt at a Solution [/B] Could you tell me is there something specific that I need to the with sin(xy)? Thanks
  19. M

    MATLAB Solving Chromatography PDE with MOL and ode15s

    Hello all I am using the method of lines to solve the following PDE: ## \frac {\partial C} {\partial t} + F\frac {\partial q} {\partial t} + u \frac {dC} {dz} = D_{ax} \frac{\partial^2 C} {\partial z^2} ## ## \frac {\partial q} {\partial t} = k (q^{*}-q) ## With these initial conditions: ##...
  20. M

    I How can you know if a numerical solution is correct?

    Hi PF, Suppose I numerically solve a nonlinear system of differential equations. How can I know if my solution is correct (if there is no known analytic solution)? What are the standard practices people do? I have a couple of ideas, but I want to know what people are already doing. Danke!
  21. F

    I Solving PDE's with chebychev FFT

    I have seen one lecturer solve a PDE with just using Fast Fourier Transform (##FFT##) of a function ##v## on a chebychev grid. ##v_t=\mu v_x## This lecturer uses ##FFT## on ##V##, then solves the ODE using an ODE solver in Matlab, then inverse ##FFT## to get the real solution ...
  22. T

    What is the characteristic frequency in a PDE modified wave equation?

    Homework Statement I am having a issue understanding this question I have solve the PDE below, but I can't understand where or how you the characteristic frequency, what more confusing is that I don’t know if that lambda is just a constant or a wavelength. Homework EquationsThe Attempt at a...
  23. evinda

    MHB Verify that the formula provides a solution of the pde

    Hello! (Wave) I want to check by direct differentation that the formula $u(x,t)=\phi(z)$, where $z$ is given implicitly by $x-z=t a(\phi(z))$, does indeed provide a solution of the pde $u_t+a(u) u_x=0$.I have tried the following, but we do not get the desired result. Have I done something...
  24. H

    MATLAB Solving 2nd Order PDE System with Crank-Nicholson

    I have the following system of PDEs: \hat{\rho}\hat{c}_{th}\frac{\partial\hat{T}}{\partial\hat{x}}-\alpha_{1}\frac{\partial}{\partial\hat{x}}\left(\hat{k}(\hat{x})\frac{\partial\hat{T}}{\partial\hat{x}}\right)=\alpha_{1}\hat{\sigma}(\hat{x})\hat{E}...
  25. M

    MHB Can we Use Partial Derivatives to Verify the Solution of PDE with Derivatives?

    Hey! :o I want to verify that $$w(x,t)=\frac{1}{2c}\int_0^t\int_{c(t-\tau)-x}^{x+c(t-\tau)}f(y,\tau)dyd\tau$$ is the solution of the problem $$w_{tt}=c^2w_{xx}+f(x,t) , \ \ x>0, t>0 \\ w(x,0)=w_t(x,0)=0, \ \ x>0 \\ w(0,t)=0 , \ \ t\geq 0$$ For that we have to take the partial derivatives of...
  26. mertcan

    A Spectral Theorem to Convert PDE into ODE

    Hi, in the link https://math.stackexchange.com/questions/1465629/numerically-solving-a-non-linear-pde-by-an-ode-on-the-fourier-coefficients there is a nice example related to spectral theorem using Fourier series. Also in the link...
  27. binbagsss

    Solving a Second Order Non-Linear PDE with Undetermined Coefficients

    Homework Statement ##\frac{d^2y}{dx^2}=2xy\frac{dy}{dx}##Homework Equations This is second order non-linear pde of the 'form' ## f(y'',y',y,x) ## . I have read that there are 2 simplified versions of a second order non-linear pde that can be solved easily and these are 1) when there is no y...
  28. D

    I PDE in ℝx(0,∞): Solving the Unknown

    The problem is a PDE uxx+uyy= 0 in ℝx(0,∞) what does this mean ℝx(0,∞) ? Came across it in my math book, and I have not idea what to google to find this.
  29. JTC

    A Example of how a rotation matrix preserves symmetry of PDE

    Good Day I have been having a hellish time connection Lie Algebra, Lie Groups, Differential Geometry, etc. But I am making a lot of progress. There is, however, one issue that continues to elude me. I often read how Lie developed Lie Groups to study symmetries of PDE's May I ask if someone...
  30. SemM

    A How does one "design" a PDE from a physical phenomenon?

    Hi, I have read some on the PDEs for fluids, and particularly for rogue waves, where for instance the extended Dysthe equation and the NLSE look rather intimidating: Take for instance the Non-linear Schrödinger eqn: \begin{equation} \frac{\partial^2 u}{dx^2}-i\frac{\partial d...
  31. Mzzed

    I Boundary Conditions for System of PDEs

    I am unsure how to choose the boundary conditions for a system of PDEs or for a single PDE for that matter. The situation I am stuck with involves a system of 4 PDEs describing plasma in a cylinder. The dependent variables being used are Vr, Vt, Vz, ni, and the independent variables are Rr...
  32. Dor

    A Numerical solution for a two dimensional 3rd order nonlinear diff. eq.

    I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction. I want to solve the following boundary conditioned differential equation: $$a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2...
  33. arpon

    Green's Function for a Partial Differential Equation

    Homework Statement Find out the Green's function, ##G(\vec{r}, \vec{r}')##, for the following partial differential equation: $$\left(-2\frac{\partial ^2}{\partial t \partial x} + \frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2} \right) F(\vec{r}) = g(\vec{r})$$ Here ##\vec{r}...
  34. A

    I Classification of First Order Linear Partial Differential Eq

    How can I classify a given first order partial differential equations? Are all first order linear PDEs hyperbolic? Quora Link:https://www.quora.com/How-do-I-classify-first-order-PDE-elliptic-hyperbolic-or-parabolic-using-method-of-characteristics
  35. H

    A Solving a PDE in four variables without separation of variables

    Within a cylinder with length ##\tau \in [0,2\pi]##, radius ##\rho \in [0,1]## and angular range ##\phi \in [0,2\pi]##, we have the following equation for the dynamics of a variable ##K##: $$\left( - \frac{1}{\cosh^{2} \rho}\frac{\partial^{2}}{\partial\tau^{2}} + (\tanh\rho +...
  36. T

    Boundary condition for PDE heat eqaution

    Homework Statement I am having an issue, not with the maths but with the boundary conditions for this question. A bar 10 cm long with insulated sides, is initially at ##100 ^\circ##. Starting at ##t=0## Find the temperature distribution in the bar at time t. The heat flow equation is...
  37. M

    I Physics? Setting up PDE for air resistance at high velocity

    Not sure if this is more appropriate for physics or for differential equations, but this problem centers around an airplane traveling with air resistance. I am not looking to get the most realistic-possible model, as it is just for a video game I am making. Although this is technically a PDE...
  38. M

    A Unraveling the Confusion: Mistakes in Solving PDEs in Spherical Coordinates?

    Given the PDE $$f_t=\frac{1}{r^2}\partial_r(r^2 f_r),\\ f(t=0)=0\\ f_r(r=0)=0\\ f(r=1)=1.$$ We let ##R(r)## be the basis function, and is determined by separation of variables: ##f = R(r)T(t)##, which reduces the PDE in ##R## to satisfy $$\frac{1}{r^2 R}d_r(r^2R'(r)) = -\lambda^2:\lambda^2 \in...
  39. A

    MHB Problem 13 from section 16.1 of Taylor's PDE textbook.

    I was given as a task to solve this question by my teacher (heck if I had the time I would have solved every problem in both Taylor's and Evans's books on PDE); but didn't succeed to the teacher's satisfaction. In the following link there's a presentation of the problem, and in the attachment...
  40. A

    A Understanding dummy variable in solution of 1D heat equation

    The solution of 1D diffusion equation on a half line (semi infinite) can be found with the help of Fourier Cosine Transform. Equation 3 is the https://ibb.co/ctF8Fw figure is the solution of 1D diffusion equation (eq:1). I want to write a code for this equation in MATLAB/Python but I don't...
  41. P

    Periodic BC's of heat equation

    Homework Statement I have the heat equation $$u_t=u_{xx}$$ $$u(0,t)=0$$ $$u(1,t) = \cos(\omega t)$$ $$u(x,0)=f(x)$$ Find the stable state solution. The Attempt at a Solution I used a transformation to complex to solve this problem, and then I can just take the real part to the complex...
  42. P

    Solving a wave equation with seperation of variables.

    Homework Statement I am trying to solve the given wave equation using separation of variables, u_{tt} - 4u_{xx} = 4 for 0 < x < 2 and t > 0 (BC) u(0,t) = 0 , u(2,t) = -2, for t>0 (IC) u(x,0)=x-x^2 , u_t(x,0)=0 for 0\leq x \leq2 Homework Equations We are told we will need to use, x =...
  43. Telemachus

    I Resolution of a PDE with second order Runge-Kutta

    Hi, I want to solve the p.d.e.: ##\frac{\partial u(x,t)}{\partial t} - \frac{\partial^2 u(x,t)}{\partial x^2}=f(x,t)##, with periodic boundary conditions ##u(x,t)=u(L,t)##. using a second order Runge-Kutta method in time. However, I am not having the proper results when I apply this method to...
  44. A

    MHB Well-posedness of a complex PDE.

    I asked my question at math.stackexchange with no reply as of yet, here's my question: https://math.stackexchange.com/questions/2448845/well-posedness-of-a-complex-pde Hope I could have some assistance here. [EDIT by moderator: Added copy of question here.] I have the following PDE: $$u_t=...
  45. i_hate_math

    Heat Kernel at t=0: Dirac Delta Intuition

    Homework Statement Show that k(x,0)=δ(x). Where k(x,t) is the heat kernel and δ(x) is the Dirac Delta at x=0. Homework Equations k(x,t) = (1/Sqrt[4*π*D*t])*Exp[-x^2/(4*D*t)] The Attempt at a Solution I am just clueless from the beginning. I am guessing this is got to do with convolution...
  46. S

    A Transforming a PDE with Laplace method

    Hello, I have the following PDE equation: a*b/U(u)*V(v) = 0 where a and b are arbitrary constants, and U an V are two unknown functions. To me it appears this has no solution, however I would like to ask if anyone has some suggestions, such as transforming it to another type using Fourier or...
  47. A

    Solving PDE heat problem with FFCT

    Homework Statement solve the following heat problem using FFCT: A metal bar of length L, is at constant temperature of ## U_0 ## , at ##t=0## the end ##x=L## is suddenly given the constant temperature of ##U_1## and the end x=0 is insulated. Assuming that the surface of the bar is insulated...
  48. A

    Solving partial differential equation with Laplace

    Homework Statement am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it? appreciated, Homework EquationsThe Attempt at a Solution my attempt is the same as in the attached picture...
  49. R

    A Solution of a weakly formulated pde involving p-Laplacian

    Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$ I want to solve this weakly formulated pde: $$ 0=\frac{A}{N^{d+1}} \sum_i...
  50. U

    MHB Find a product solution to the following PDE

    So I'm asked to use separation of variables to find a product solution to the given PDE: (5y + 7)du/dx + (4x+3)du/dy = 0 Since it says to find a product solution, I used the form u(x,y) = XY and plugged that into the PDE. However, I am getting stuck because I'm not sure how exactly I should...
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