What is Pde: Definition and 852 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. chwala

    Show that ##f(x,y)=u(x+cy)+v(x-cy)## is a solution of the given PDE

    Looks pretty straightforward, i approached it as follows, ##f_x = u(x+cy) + v(x-cy)## ##f_{xx}=u(x+cy) + v(x-cy)## ##f_y= cu(x+cy) -cv(x-cy)## ##f_{yy}=c^2u(x+cy)+c^2v(x-cy)## Therefore, ##f_{xx} -\dfrac{1}{c^2} f_{yy} = u(x+cy) + v(x-cy) - \dfrac{1}{c^2}⋅ c^2 \left[u(x+cy)+v(x-cy)...
  2. H

    A Discretisation of a PDE in Lagrangian coordinates

    I am writing a 2D hydrocode in Lagrangian co-ordinates. I have never done this before, so I am completely clueless as to what I'm doing. I have a route as to what I want to do, but I don't know if this makes sense or not. I've gone from Eulerian to Lagrangian co-ordinates using the Piola...
  3. Euge

    POTW Neumann Boundary Value Problem in a Half Plane

    Find all bounded solutions to the PDE ##\Delta u(x,y) = 0## for ##x\in \mathbb{R}## and ##y > 0## with Neumann boundary condition ##u_y(x,0) = g(x)##.
  4. E

    I Heat Equation: Solve with Non-Homogeneous Boundary Conditions

    Imagine you have a plane wall with constant thermal conductivity, that is the intermediate between two thermal reservoirs: The reservoir on the left is being kept at temp ##T_s##, and it is a fluid that has very high convective coefficient ##h##. As a result, the boundary condition at the...
  5. Euge

    POTW A Nonlinear Elliptic PDE on a Bounded Domain

    Let ##D## be a smooth, bounded domain in ##\mathbb{R}^n## and ##f : D \to (0, \infty)## a continuous function. Prove that there exists no ##C^2##-solution ##u## of the nonlinear elliptic problem ##\Delta u^2 = f## in ##D##, ##u = 0## on ##\partial D##.
  6. Magnetons

    PDE and the separation of variables

    using the equation ##u(x,y)=f(x)g(y)##, first, I substitute the value of ##u_{xx}## and ##u_{yy}## in the given PDE. after that solve the ODEs but I can't understand about the ##u_{t}##.In my solution, I put ##u_{t}=0## because u is only the function of x and y. Is it the right approach, to me...
  7. B

    A How to separate variables in this PDE?

    My PDE: F,x,t + A(x)*F(x,t)*[(x+t)^(-3/2)] = 0 A(x) is a known function of x. Trying to separate F(x,t) like F(x,t) = F1(x)*F2(t)*F3(x+t). I’m getting desperate to solve, any suggestions??
  8. E

    I PDE - Heat Equation - Cylindrical Coordinates.

    Would method of separation of variables lead to a solution to the following PDE? $$ \frac{1}{r} \frac{ \partial}{\partial r} \left( kr \frac{ \partial T}{ \partial r}\right) = \rho c_p \frac{\partial T }{ \partial t }$$ This would be for the transient conduction of a hollow cylinder, of wall...
  9. chwala

    Find the general solution of the given PDE

    My take; ##ξ=-4x+6y## and ##η=6x+4y## it follows that, ##52u_ξ +10u=e^{x+2y}## for the homogenous part; we shall have the general solution; $$u_h=e^{\frac{-5}{26} ξ} f{η }$$ now we note that $$e^{x+2y}=e^{\frac{8ξ+η}{26}}$$ that is from solving the simultaneous equation; ##ξ=-4x+6y##...
  10. chwala

    A Clarification on the given PDE problem

    My interest is on the highlighted part only...my understanding is that one should use simultaneous equation... unless there is another way hence my post query. In my working i have; ##y=\dfrac{2ξ+η}{10}## and ##x=\dfrac{2η-ξ}{10}## giving us...
  11. A

    MATLAB Boundary conditions in the resolution of a PDE with the FFT method

    How to impose boundary conditions when solving a PDE with fft? For example here: If I copy this code I get periodic boundary conditions. Thank you
  12. U

    Why heat PDE solution does not fully satisfy initial conditions?

    Hi, I am solving heat equation with internal heat sources both numerically and analytically. My graphs are nearly identical but! analytical one have problem at the beginning and at the end for my domain. Many people have used the same technique to solve it analytically and they got good answers...
  13. BiGyElLoWhAt

    I Equations of motion for the Schwarzschild metric (nonlinear PDE)

    I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...
  14. Jamestein Newton

    Courses Is it necessary for theoretical physics students to take a course in PDE?

    By PDE. The book written by Walter Alexander Strauss perfect described a typical undergraduate PDE course I have in my mind. It should at least include: Laplace equations, waves and diffusions reflection, boundary problems, Fourier series The content of the book I mentioned can also be found...
  15. masaakim

    I Looking for what this type of PDE is generally called

    We have this type of very famous nicely symmetric pde in our area. However, no one knows how to handle it properly since it is a nonlinear pde. Suggestions on how it is called in general would help us further googling. I already tried keywords like "bilinear", "dual", "double", but by far could...
  16. H

    I Crank Nicolson method to solve a PDE

    Hello, I wrote a code to solve a non-linear PDE using Canrk nicolson method, but I'm still not able to get a correct final results. can anyone tell me what wrong with it?
  17. F

    Courses Mathematics Bachelor's Degree: Choices ahead

    I am studying mathematics as bachelor in my second year. At the moment I am taking abstract algebra, analysis (measure and integration theory) and probability course. I don't know exactly what I want to do with maths but the applications in physics always have fascinated me. The next term I have...
  18. chwala

    Classify the given second-order linear PDE

    Now i learned how to use discriminant i.e ##b^2-4ac## and in using this we have; ##b^2-4ac##=##0-(4×3×2)##=##-24<0,## therefore elliptic. The textbook has a slight different approach, which i am not familiar with as i was trained to use the discriminant at my undergraduate studies... see...
  19. C

    MHB Writing PDE in terms of x and y

    Hi all, I am hoping someone can help me understand a PDE. I am reading a paper and am trying to follow the math. My experience with PDEs is limited though and I am not sure I am understanding it all correctly. I have 3 coupled PDEs, for $n$, $f$ and $c$, that are written in general form, and I...
  20. Z

    I Can FDM solve any type of PDE same as FEM?

    hello aside from some constraints such as an irregular integration domain, can FDM solve any type of PDE same as FEM ?
  21. tworitdash

    A How to solve simple 2D space-time PDE numerically

    I have a 2D space-time PDE and I want to solve it numerically over the time axis. The time initial field is already known with respect to space, i.e., the spatial distribution is already known at time `t = 0`. I solved the same PDF in Mathematica and got a solution. I tried to solve it...
  22. L

    Transforming to a Normal Form (PDE)

    I don't know how to solve for u(x,y) from where I left of after 5.
  23. H

    Fourier transform to solve PDE (2nd order)

    I just want to make sure I am on the right track here (hence have not given the other information in the question). In taking the Fourier transform of the PDE above, I get: F{uxx} = iω^2*F{u}, F{uxt} = d/dt F{ux} = iω d/dt F{u} F{utt} = d^2/dt^2 F{u} Together the transformed PDE gives a second...
  24. docnet

    Verify the Appell transformed solution also solves the PDE

    I feel that my reasoning becomes shaky near the conclusion. So, someone should tell me why it is weak, and suggest how to make it stronger. Thanks. For ##\delta>0## we define the Appell transform of ##u## by $$u_\delta=(1+\delta t)^{-\frac{n}{2}}exp\Big(-\frac{\delta|x|^2}{4(1+\delta...
  25. T

    PDE Characteristic Method?

    dx/dt =1, x(0,s)=0, dy/dt=x, y(0,s) = s, du/dt=(y-1/2x^2)^2, u(0,s)=e^s I did well at the beginning to get x(t,s) =t and y(t,s)=1/2t^2 + s, but got stuck with the du/dt part. You can sub in x=t and y=1/2t^2 +s for x and y to get du/dt = s^2. But that's still three variables, and I can't see...
  26. docnet

    Solve this PDE with Neumann boundary conditions

    Hi all, I was hoping someone could check whether I computed part (4) correctly, where i find the solution u(t,x) using dAlembert's formula: $$\boxed{\tilde{u}(t,x)=\frac{1}{2}\Big[\tilde{g}(x+t)+\tilde{g}(x-t)\Big]+\frac{1}{2}\int^{x+t}_{x-t}\tilde{h}(y)dy}$$ Does the graph of the solution look...
  27. docnet

    Converting a PDE to an ODE

    Sorry the problem is a bit long to read. thank you to anyone who comments. We consider the initial value problem for the Burger's equation with viscosity given by $$\begin{cases} \partial_t u-\partial^2_xu+u\partial_xu=0 & \text{in}\quad (1,T)\times R\\\quad \quad \quad \quad \quad...
  28. docnet

    Is there a solution to this simple 1st order PDE?

    This isn't homework, but I was just wondering whether the following PDE has an analytic solution. $$\partial_x u(t,x)=u(t,x)$$ where ##x\in R^n## and ##\partial_x## implies a derivative with respect to the spatial variables.
  29. docnet

    The Yukawa PDE equation

    (1) From "Radial solutions to Laplace's equation", we know that $$ \Delta u(x) = v(r)''+\frac{n-1}{r}v(r)' $$ we re-write the PDE $$ - \Delta u+m^2u=0 $$ in terms of ##v(r)## \begin{equation} - v(r)''-\frac{n-1}{r}v(r)'+m^2v(r)=0 \end{equation} to give a linear second order ODE with...
  30. N

    Magnetohydrodynamics - Derivation of PDE

    Summary:: partial differential equation (PDE) to describe the potential distribution φ in the system Hey, I need some help with the following question: We have a stationary electrolyte, a magnetic field "B" and a Current density "j" (2D). Derive the partial differential equation (PDE) to...
  31. docnet

    Modified transport equation (PDE)

    Hi all, I Fix $$(t,x) ∈ (0,\infty) \times R^n$$and consider auxillary function $$w(s)=u(t+s,x+sb)$$ Then, $$\partial_s w(s)=(\partial_tu)(t+s,x+sb)\frac{d}{ds}(t+s)+<Du(t+s,x+sb)\frac{d}{ds}(x+sb)>$$ $$=(\partial_tu)(t+s,x+sb)+<b,Du(t+s,x+sb)>$$ $$=-cu(t+s,x+sb)$$...
  32. docnet

    Verify the PDE has the following solution

    Hello, please lend give me your wisdom. I suspect this problem is about the wave equation ##\partial_t^2-\partial_x^2=0## commonly encountered in physics. I tried a search for information but I could not find help. Attempt at arriving at solution: So I took the partial derivatives of...
  33. docnet

    Verify or refute the function is a solution to a PDE

    Solution attempt: We first write ##u(x)=\frac{1}{2}||x||^2## as ##u(x)=\frac{1}{2}(x_1^2+x_2^2+...+x_n^2)## Operating on ##u(x)## with ##\Delta##, we have ##u(x)=\frac{1}{2}(2+2+...+2)## adding 2 to itself ##n## times. So ##\Delta u(x)=n## and the function satisfies the first condition...
  34. derya

    A Generic Solution of a Coupled System of 2nd Order PDEs

    Hi! I am looking into a mechanical problem which reduces to the set of PDE's below. I would be very happy if you could help me with it. I have the following set of second order PDE's that I want to solve. I want to solve for the generic solutions of the functions u(x,y) and v(x,y). A, B and C...
  35. Omega0

    What comes on top of a generator of a PDE?

    From some principles in nature we are using in physics the calculus of variations. Let me call it a generator for PDE's. My question: Are there levels above? What I mean is: Is there mathematics where you have principles where the solutions are generators for the generators for PDEs ? What about...
  36. M

    How to 'shift' Fourier series to match the initial condition of this PDE?

    Hi, Question: If we have an initial condition, valid for -L \leq x \leq L : f(x) = \frac{40x}{L} how can I utilise a know Fourier series to get to the solution without doing the integration (I know the integral isn't tricky, but still this method might help out in other situations)? We are...
  37. DuckAmuck

    A Separation of variables possible in this problem?

    Is it possible to use separation of variables on this equation? au_{xx} + bu_{yy} + c u_{xy} = u + k Where u is a function of x and y, abck are constant. I tried the u(x,y) = X(x)Y(y) type of separation but I think something more clever is needed. Thank you.
  38. Leonardo Machado

    A Boundary conditions in the time evolution of Spectral Method in PDE

    Hi everyone! I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example, $$ u_t=k u_{xx}, $$ $$ u(t,-1)=\alpha, $$ $$ u(t,1)=\beta, $$ $$ u(0,x)=f(x), $$ $$...
  39. F

    I Poisson's inhomogeneous PDE and its solutions

    Hello, Poisson equation and Laplace equation (which is the homogeneous version of Poisson PDE) are important equations in electrostatics where both the electric field ##E## and scalar potential ##\phi## don't depend on time. Poisson's equation is $$\nabla^2 \phi(x,y,z) = - \frac{\rho(x,y,z)}...
  40. G

    Method of characteristics: Discontinuous source

    Hello all, this question really has me and some friends stomped so advice would be appreciated. Ok so, the relevant (dimensionless) continuity equation I have found to be $$\frac{\partial\rho}{\partial t} + (1-2\rho)\frac{\partial \rho}{\partial x} = \begin{cases} \beta, \hspace{3mm} x < 0 \\...
  41. C

    I Looking for help with this PDE

    as you can see, z(x,y) is a function of x, y; and y is a function of x, therefore y'(x) is the total derivative of "y" respect to "x", and y"(x) is the 2nd derivative. y'(x)^2 is just the square of the derivative of y respect to x I don't have boundary or initial conditions, so you can make up...
  42. Vick

    I Alternating Direction Implicit method for solving 2D Heat diffusion

    I'm trying to compute a 2D Heat diffusion parabolic PDE: $$ \frac{\partial u}{\partial t} = \alpha \{ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \} $$ by the ADI method. I am actually trying to go over the example in this youtube video. The video is in another...
  43. E

    Applying a substitution to a PDE

    Problem: Consider the equation $$\frac{\partial v}{\partial t} = \frac{\partial^{2} v}{\partial x^2} + \frac{2v}{t+1}$$ where ##v(x,t)## is defined on ##0 \leq x \leq \pi## and is subject to the boundary conditions ##v(0,t) = 0##, ##v(\pi, t) = f(t)##, ##v(x,0) = h(x)## for some functions...
  44. F

    Deriving the Adjoint / Tangent Linear Model for Nonlinear PDE

    I am trying to derive the adjoint / tangent linear model matrix for this partial differential equation, but cannot follow the book's steps as I do not know the math. This technique will be used to solve another homework question. Rather than posting the homework question, I would like to...
  45. G

    Courses To take a PDE class or not

    I am a junior physics major trying to decide if I should squeeze in a (extra) PDE class in my semester which is not required for my degree but obviously can be useful. Though I am only taking 3 (all technical) classes otherwise, I should be quite occupied with GREs and research. Is it worth...
  46. JD_PM

    Understanding how to apply the method of images to the wave equation

    Exercise statement Find the general solution for the wave equation ftt=v2fzzftt=v2fzz in the straight open magnetic field tube. Assume that the bottom boundary condition is fixed: there is no perturbation of the magnetic field at or below the photosphere. Solve means deriving the d’Alembert...
  47. T

    A Numerically Solving Scalar Propagation in Curved Spacetime

    Hey everybody, Background: I'm currently working on a toy model for my master thesis, the massless Klein-Gordon equation in a rotating static Kerr-Schild metric. The partial differential equations are (see http://arxiv.org/abs/1705.01071, equation 27, with V'=0): $$ \partial_t\phi =...
  48. D

    I Best method to solve this discretized PDE

    I am attempting to solve the following PDE for Ψ representing a stream function on a 2D annulus grid: (1/s)⋅(∂/∂s)[(s/ρ)(∂ψ/∂s)] + (1/s2)⋅(∂/∂Φ)[(1/ρ)(∂ψ/∂Φ)] - 2Ω + ρ(c0 + c1ψ) = 0 I have made a vertex centered discretization: (1/sj)⋅(1/Δs2)⋅[(sj+1/2/ρj+1/2,l){ψj+1,l - ψj,l} -...
  49. 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 $x$...
  50. 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...