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. G

    What is the regularity of weak solutions of the Liouville equation?

    Could someone give me some hint (or some reference) about the study of regularity of weak solutions of (*)\quad \begin{cases} -\Delta u = e^u & \text{in }\Omega\\ u = 0 & \text{on }\partial \Omega \end{cases} where \Omega \subset \mathbb{R}^2 is a bounded domain with smooth boundary (here...
  2. K

    Is every one playing a joke on me? PDE solution question

    Ok I have read/browse 3 books so far about PDEs. They separate variables. Then they suddenly go. solution is X= some thing sin some thing cos Y= some thing -ek some thing ek Where do sine and cos comes from ? How do they know?? where is it explained?
  3. K

    Solving a Partial Differential Equation (PDE)

    Hi, Can somebody help me solve the following PDE? ∂p(x,t)/∂t = -p(x,t) + ∫λ(x-x')p(x',t)dx' with p(x,0)=δ(x) Thanks a lot
  4. T

    MATLAB Solition problems with Matlab (12 PDE with 8 variables))

    Hi, I am a master student comes from USM in Malaysia. I don't know my problem should placed on differential forum or high energy physics forum. Anywhere, My current study is high energy physics subject and my main study is focus on monopole instanton solution in static form which did not...
  5. M

    What can we say about the solution of this PDE?

    Hello! I would like to find some functions F(x,y) which satisfy the following equation \frac{F(x,y)}{\partial x}=\frac{F(y,x)}{\partial y} For example this is obviously satisfied for the function F= exp(x+y) I would like however to find the most general closed form solution...
  6. J

    Numerical PDE boundary problem methodology

    Hi, I'm currently working on a thesis in Economics. I have stumbled upon a system of differential equations that needs to be solved. I am stuck, and have trouble getting the right help from my advisor who is also not very acquainted with numerical methods. For the past couple of days I have...
  7. S

    How can I solve a 4D PDE using numerical methods in MATLAB or Python?

    Hello, I want to solve a 4-dimensional PDE problem using some numerical code. Possibly MATLAB or Python. I have a solved a simple version of the PDE in 2D using MATLAB PDETool. Also I solved a simplified pde in 3D using FiPy library in Python. However, most MATLAB existing tools...
  8. B

    What does 2nd ord pde tell you? like fxx(x,y)

    pure and mixed, what do they tell you about a function?
  9. M

    General solution to partial differential equation (PDE)

    Hi, I have the following PDE-S\frac{\partial\vartheta}{\partial\tau}+\frac{1}{2}\sigma^2\frac{X^2}{S}\frac{\partial^2\vartheta}{\partial\xi^{2}} + [\frac{S}{T} + (r-D)X]\frac{\partial\vartheta}{\partial\xi}I am asked to seek a solution of the form \vartheta=\alpha_1(\tau)\xi + \alpha_0(\tau)...
  10. M

    PDE and finding a general solution

    Hi everyone, I am doing a sheet on Asian Options and The Black Scholes equation. I have the PDE, \frac{∂v}{∂τ}=\frac{1}{2}σ^{2}\frac{X^{2}}{S^{2}}\frac{∂^{2}v}{∂ε^{2}} + (\frac{1}{T} + (r-D)X)\frac{∂v}{∂ε} I have to seek a solultion of the form v=α_{1}(τ)ε + α_{0}(τ) and determine...
  11. N

    Derivation of normal surface vector of a quasilinear PDE

    Hi group, In order to understand the methods of characteristics, I've been reading its wiki entry plus other sources. However, one of the first step of finding the normal surface vector given the PDE remains baffling to me in terms of how it's derived. In short, when provided with a(x...
  12. M

    Black-Scholes PDE and finding the general solution

    Hello, I have the PDE \frac{-∂v}{∂τ}+\frac{1}{2}σ^{2}ε^{2}\frac{∂^{2}v}{∂ε^{2}}+(\frac{1}{T}+(r-D)ε)\frac{∂v}{∂ε}=0 and firstly I need to seek a solution of the form v=α_{1}(τ)ε + α_{0}(τ) and then determine the general solution for α_{1}(τ) and α_{0}(τ). I am given that ε=\frac{I}{TS}...
  13. A

    Help with a 2nd order PDE involving mixed derivatives

    I have a PDE in two variables, u and v, which takes the form \frac{\partial\psi}{\partial u\hspace{1pt}\partial v} + \frac{1}{r}\left(\frac{\partial r}{\partial u} \frac{\partial \psi}{\partial v} + \frac{\partial r}{\partial v}\frac{\partial\psi}{\partial u}\right) for an auxiliary...
  14. I

    How Can Visual Learners Master PDEs and Fourier Transforms?

    Hi, I'm an undergrad in EE who wants to learn the basics of solving PDEs (and Fourier series/transforms), but who has some learning disabilities (developmental, most notably). Before I get criticism for what I'm about to say (which will be asking for an alternative to the obligatory "read...
  15. M

    Inverse Trig Functions as a (unique?) solution to a PDE

    Hi, I know from basic math courses that inverse trig functions are multi valued (e.g. arctan(c)=θ+n*2∏). Now, if I solve a partial differential equation and I get an inverse trig function as part of my solution, does that mean solutions to the pde are non-unique? For example, if...
  16. P

    PDE question: Eigenvalues

    Homework Statement Let λ_n denote the nth eigenvalue for the problem: -Δu = λu in A, u=0 on ∂A (*) which is obtained by minimizing the Rayleigh quotient over all non-zero functions that vanish on ∂A and are orthogonal to the first n-1 eigenfunctions. (i) Show that (*) has no...
  17. J

    Solving PDE: Help with Advection-Diffusion Equation

    Homework Statement The Cauchy problem for the advection-diffusion equation is given by: u.sub.t + c u.sub.x = K u.sub.xx (−∞< x < ∞) u(x, 0) = Phi(x) where c and K are positive constants. The advection-diffusion equation essentially combines the effects of the transport...
  18. M

    MHB Solving First-Order PDE: $u_x+2u_y+2u=0$

    Solve $u_x+2u_y+2u=0,$ $x,y\in\mathbb R$ where $u(x,y)=F(x,y)$ in the curve $y=x.$ I don't know what does mean with the $y=x.$ Well I set up the following $\dfrac{dx}{1}=\dfrac{dy}{2}=\dfrac{du}{-2} ,$ is that correct? but I don't know what's next. Thanks for the help!
  19. B

    MHB Help with PDE: $$yu_x+2xyu_y=y^2$$

    Hi, need some help here so thanks to any replies. PDE: $$yu_x+2xyu_y=y^2$$ edit: Forgot to mention the condition $$u(0,y)=y^2$$ a) characteristic equations: $$dx/ds=y$$ $$dy/ds=2xy$$ $$du/ds=y^2$$ b) find dy/dx and solve $$dy/dx=dy/ds * ds/dx = x/y$$ $$ydy=xdx$$ $$y^2/2=x^2/2 +c$$ $$y=\pm...
  20. M

    MHB Solving PDE by using Laplace Transform

    Given $\begin{aligned} & {{u}_{t}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\ & u(x,0)={{u}_{0}}, \\ & {{u}_{x}}(0,t)=u(0,t). \end{aligned} $ I need to apply the Laplace transform to solve it. I'll denote $u(x,s)=\mathcal L(u(x,\cdot))(s),$ so for the first line I have $s\cdot...
  21. B

    MHB Solution of PDE: General Solution & Modifications

    Dear MHB members, I have the following equation $xy(z_{xx}-z_{yy})+(x^{2}-y^{2})z_{xy}=yz_{x}-xz_{y}-2(x^{2}-y^{2})$. When I transform this into the canonical form via $\xi=2xy$ and $\eta=x^{2}-y^{2}$, I obtain...
  22. M

    Transfer function of a PDE system

    Hi everyone! I want to design a robust controller for a system which is driven by a PDE. I need to acquire its transfer function in 's' parameter which means it should be transferred by Laplace transformation. I know that the result transfer function will be an infinite series of transfer...
  23. Z

    Understanding PDEs: Evaluating a Solution to the 1-Dimensional Wave Equation

    Homework Statement I'm just trying to get an understanding of answering PDEs, so wanted to ask what you thought of my answer to this question. The one-dimensional wave equation is given by the first equation shown in this link; http://mathworld.wolfram.com/WaveEquation1-Dimensional.html...
  24. Z

    PDE for Heat Diffusion Equation

    Homework Statement The one-dimensional heat diffusion equation is given by : ∂t(x,t)/∂t = α[∂^2T(x,t) / ∂x^2] where α is positive. Is the following a possible solution? Assume that the constants a and b can take any positive value. T(x,t) = exp(at)cos(bx) Homework Equations...
  25. M

    MHB PDE and conservation of energy

    Let $u\in\mathcal C^1(\overline R)\cap \mathcal C^2(R)$ where $R=(0,1)\times(0,\infty).$ Suppose that $u(x,t)$ verifies the following wave equation $u_{tt}=K^2 u_{xx}+h(x,t,u)$ where $K>0$ and $h$ is a constant function. a) Determine the total energy of the string. (Well I don't know what does...
  26. M

    MHB PDE and more boundary conditions

    Solve $\begin{aligned} & {{u}_{tt}}={{u}_{xx}}+1+x,\text{ }0<x<1,\text{ }t>0 \\ & u(x,0)=\frac{1}{6}{{x}^{3}}-\frac{1}{2}{{x}^{2}}+\frac{1}{3},\text{ }{{u}_{t}}(x,0)=0,\text{ }0<x<1, \\ & {{u}_{x}}(0,t)=0=u(1,t),\text{ }t>0. \end{aligned} $ Here's something new for me, the boundary...
  27. M

    MHB Solving PDE by using another function

    Solve $\begin{aligned} & {{u}_{tt}}={{c}^{2}}{{u}_{xx}}+A{{e}^{-x}},\text{ }0<x<L,\text{ }t>0, \\ & u(0,t)=B,\text{ }u(L,t)=M,\text{ }t>0, \\ & u(x,0)=0={{u}_{t}}(x,0),\text{ 0}<x<L. \end{aligned} $ What do I need to do first? Homogenize the first boundary conditions? Or first making the...
  28. M

    MHB How to Solve a PDE with an External Function?

    Consider the equation $\begin{aligned} & {{u}_{t}}=K{{u}_{xx}}+g(t),\text{ }0<x<L,\text{ }t>0, \\ & {{u}_{x}}(0,t)={{u}_{x}}(L,t)=0,\text{ }t>0 \\ & u(x,0)=f(x), \\ \end{aligned} $ a) Show that $v=u-G(t)$ satisfies the initial value boundary problem where $G(t)$ is the primitive of...
  29. C

    Object Oriented Programming vs PDE?

    I'm confused on what classes to take next semester. I've talked to my adviser but they're kinda useless as they don't want me to take upper level courses (past calc 3 and ODE). However, I want a dual math and physics degree which would be helpful in gradschool. Right now I have the following...
  30. A. Neumaier

    Classical solution of PDE with mixed boundary conditions

    Nowadays people usually consider PDEs in weak formulations only, so I have a hard time finding statements about the existence of classical solutions of the Poisson equation with mixed Dirichlet-Neumann boundary conditions. Maybe someone here can help me and point to a book or article where I...
  31. K

    Riemann function for a second order hyperbolic PDE

    Homework Statement Find the Riemann function for uxy + xyux = 0, in x + y > 0 u = x, uy = 0, on x+y = 0 Homework Equations The Attempt at a Solution I think the Riemann function, R(x,y;s,n), must satisfy: 0 = Rxy - (xyR)x Rx = 0 on y =n Ry = xyR on x = s R = 1 at (x,y) = (s,n) But I...
  32. I

    How do one solve this PDE

    I have a battle with the following direct partial integration and separation of variables toffee: I have to solve, u(x,y)=\sum_{n=1}^{∞}A_n sin\lambda x sinh \lambda (b-y) If there were no boundary or initial conditions given, do I assume that λ is \frac{n\pi}{L} and do I then solve A_n...
  33. L

    Using change of variables to change PDE to form with no second order derivatives

    Homework Statement Classify the equation and use the change of variables to change the equation to the form with no mixed second order derivative. u_{xx}+6u_{xy}+5u{yy}-4u{x}+2u=0 Homework Equations I know that it's of the hyperbollic form by equation a_{12}^2 - a_{11}*a_{22}, which...
  34. C

    Suggestions for scheme to use to solve PDE numerically

    Hello everyone, I am trying to model the process of laser ablation on a material using MATLAB. The governing equation is of the form: ∂T(x,t)/∂t = ∂/∂x(A*∂T/∂x) + B*exp(-C*t2)*exp(-D*x) with one Initial condition and two boundary conditions. Using the built-in 'pdepe' function in Matlab...
  35. T

    PDE, 2D Laplace Equation, Sep. of Variables, Finding Potential

    Homework Statement A square rectangular pipe (sides of length a) runs parallel to the z-axis (from -\infty\rightarrow\infty). The 4 sides are maintained with boundary conditions (i) V=0 at y=0 (bottom) (ii) V=0 at y=a (top) (iii) V=constant at x=a (right side) (iv) \frac{\partial...
  36. fluidistic

    PDE, inhomogeneous diffusion equation

    Homework Statement Mathews and Walker problem 8-2 (page 253): Assume that the neutron density n inside U_{235} obeys the differential equation \nabla ^2 n+\lambda n =\frac{1}{\kappa } \frac{\partial n }{\partial t} (n=0 on surface). a)Find the critical radius R_0 such that the neutron density...
  37. M

    MHB Another PDE and boundary conditions

    1) Solve $\begin{aligned} {{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\ {{u}_{x}}(0,t)&=0,\text{ }{{u}_{x}}(L,t)=0,\text{ for }t>0, \\ u(x,0)&=6+\sin \frac{3\pi x}{L} \end{aligned}$ 2) Transform the problem so that the boundary conditions get homogeneous: $\begin{aligned}...
  38. V

    Question about PDE solution

    I am trying to solve the following equation in spherical coordinates: \left( \nabla f \right) \cdot \vec{B} = g where g is a known scalar function, \vec{B} is a known vector field, and f is the unknown function. I think the best way to approach this is to expand everything into a...
  39. W

    Is Linearity of PDE Operator Lu = du/dx + u * du/dy Verifiable?

    Homework Statement Is the operator Lu = du/dx + u * du/dy linear? Homework Equations Linearity occurs for L[u+cv] = L[u] + cL[v] The Attempt at a Solution I know this isn't linear because of the second term, but I don't understand why I can't write the operator as L =...
  40. K

    Chain Rule & PDES: Solving ∂z/∂u

    Im new on the forum, so I hope you guys will have some patience with me :-) I have a question about the chain rule and partial differential equations that I can't solve, it's: Write the appropriate version of the chain rule for the derivative: ∂z/∂u if z=g(x,y), where y=f(x) and...
  41. M

    Solving a PDE and finding the jump condition (method of characteristics)

    Here I have my PDE: http://desmond.imageshack.us/Himg718/scaled.php?server=718&filename=pde.png&res=medium I have found the solution by using the method of characteristics two times, one for x<0 and the other for x>0. I have: U(x,y) = o for x<0 and U(x,y) = Uo(x-1)/(1+Uo*y) for x>0...
  42. M

    MHB Solve 1st Order PDE: $u_y+f(u)u_x=0$

    Solve $u_y+f(u)u_x=0,$ $x\in\mathbb R,$ $y>0,$ $u(x,0)=\phi(x).$ What's the easy way to solve this? Fourier Transform? Laplace Transform?
  43. M

    MHB Solution given by sum of functions on a PDE

    Consider $u_t+u_x=g(x),\,x\in\mathbb R,\,t>0$ and $u(x,0)=f(x).$ Given $f,g\in C^1,$ then show that $u(x,t)$ has the form $u(x,t)=f(x-t)+\sqrt{2\pi}(g*h)(x)$ where $h(x)=\chi_{[0,t]}(x).$ So we just apply the Fourier transform to get $\dfrac{{\partial U}}{{\partial t}} + iwU = U(g)$ and...
  44. V

    PDE equation in spherical coordinates

    I am looking for ideas on how to solve this equation: \nabla \cdot \left( \vec{A} + F \hat{b} \right) = 0 where \vec{A} and \hat{b} are known vectors of (r,\theta,\phi) and F is the unknown scalar function to be determined. Also, \nabla \cdot \hat{b} = 0. So the equation can also be expressed...
  45. J

    PDE Characteristic Curve Method

    Homework Statement Solve u_x^2+u_y^2=1 subject to u(x, ax)=1 Homework Equations The Attempt at a Solution I let u_x=p andu_y=q and F=p^2 +q^2 -1=0 Then x'=2p, y'=2q, u'=p.2p+q.2q=2, and p'=0=q'. So p=p_0, q=q_0 are constants. I got x'=2p_0, y'=2q_0 and integrating the...
  46. S

    Really easy PDE, confused about how to put in side conditions

    Hey, before you read this over I'll mention that I've checked the general solution and it works. So if you don't feel like following my steps to get the general solution just jump down to the end of my attempt, because the real problem for me is figuring out what to do with the side conditions...
  47. K

    MHB Need help with transforming one PDE to another

    For quick reference if you have the text, the question is from "Applied Partial Differential Equations" by J. David Logan. Section 1.9 #4 Show that the equation $$u_{tt} - c^2 u_{xx} + au_t + bu_x + du = f(x,t)$$ can be transformed into an equation of the form $$w_{\xi\tau} + kw = g(\xi,\tau)...
  48. A

    PDE Series Problem: Solving for Numerator

    Homework Statement The problem I am having has to do with part (d) in the picture which I have attached. I have managed to get as far as to determine that the coefficients in the series expansion have the recurrence relation shown below in part (2). From this I think that I have been able to...
  49. V

    Finding Solutions to a PDE System with Known Scalar Function

    Hi all, I am looking for ways to solve the following system of equations for \vec{B}: \vec{B} \cdot \nabla f = 0 \left( \nabla \times \vec{B} \right) \cdot \nabla f = 0 \nabla \cdot \vec{B} = 0 and f is a known scalar function. I think we can assume there is a solution since we...
  50. A

    How to Solve the Mixed Partial Differential Equation Given Boundary Conditions?

    (∂^2)(z) / (∂x)(∂y) = (x^2)(y) how do find the general solution of this equation? and how do i find a particular solution for which z(x,0) = x^2 z(1,y) = cosy I have no idea!
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