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. N

    Splitting a second order PDE into a system of first order PDEs/ODEs

    In my research, I'm using a modified version of the wave equation: \[ c^2 \left( {\frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }}} \right) = - \tau c^2 \left( {\frac{{\partial ^3 p}}{{\partial t\partial x^2 }} + \frac{{\partial ^3 p}}{{\partial...
  2. C

    Solving Laplace's Equation with Convode and Initial Conditions

    Could someone tell me how to enter the following PDE into convode (or some other pde engine - feel free to solve it yourself if you like!). Its LaPlace's equation U_xx + U_yy = 0 given U=0 when x=0 U=0 when x=1 U=0 when y=0 U=x when y=1 I've used Convode...
  3. C

    Non-Homogeneous Boundary Conditions: How to Solve PDEs with Green's Function?

    Hey Guys; I'm solving PDE's with the use of Green's function where all the boundary conditions are homogeneous. However, how do you solve ones in which we have non-homogeneous b.c's. In case it helps, the particular PDE I'm looking at is: y'' = -x^2 y(0) + y'(0) = 4, y'(1)= 2...
  4. S

    Is u=0 the only solution for the PDE on the unit disc?

    Homework Statement We the domain be the unit disc D: D=\left \{(x,y):x^{2}+y^{2}<1 \right \} let u(x,y) solve: -\triangle u+(u_{x}+2u_{y})u^{4}=0 on D boundary: u=0 on \partial D One solution is u=0. Is it the only solution?Homework Equations Divergence Theorem "Energy Method"The Attempt at...
  5. N

    A PDE I can't solve by seperation of variables

    Homework Statement Homework Equations After simplification, the PDE is (b^2/a^2)(d^2 v/ d x^2) + (d^2 v/ d y^2) = -1 The Attempt at a Solution Obviously, it can't be solved by separation of variables. And I also failed in similarity solution.
  6. S

    Existence and Uniqueness of Solution for PDE with Boundary Conditions

    Homework Statement Assume we are in the open first quadrant in the (x,y) plane Say we have u(x,y) a C1 function in the closed first quadrant that satisfies the PDE: u_{y}=3u_{x} in the open first quadrant Boundary Conditions: u(0,y)=0 for t greater than or equal to 0 u(x,0)= g(x) for x...
  7. S

    How to find uniqueness in first order pde

    Hi guys, I have a general problem that I'm not quite sure how to solve. Suppose you have a first order pde, like Ut=Ux together with some boundary conditions. You'd do the appropriate transformations that lead to a solution plus an arbitrary function defined implicitly. How would you know...
  8. W

    Mathematica PDE Plot Mathematica(Multivalue)

    I am trying to plot: u(x,y)=sin(x-t u(x,y)) An implicit solution to a PDE. I have no clue how to do this; I've plotted an equation like this before.
  9. W

    PDE Cylindrical and Spherical Symmetry

    Homework Statement Show that the solution u(r,theta) of Laplace's equation (nabla^2)*u=0 in the semi-circular region r<a, 0<theta<pi, which vanishes on theta=0 and takes the constant value A on theta=pi and on the curved boundary r=a, is u(r,theta)=(A/pi)[theta + 2*summation ((r/a)^n*((sin...
  10. U

    Heat Transfer PDE SOV with piecewise BC

    Homework Statement Heat transfer problem with 3 insulated sides and heat flux in and out on one boundary. given values: q & k Homework Equations Governing Equation: \frac{\partial^{2}{T}}{\partial{x}^{2}} + \frac{\partial^{2}{T}}{\partial{y}^{2}} = 0 Boundary Conditions: @ x =...
  11. W

    Finding uniqueness of PDE via. energy method

    Homework Statement consider a solution such that: -\triangle u + b\triangledown u + cu = f in domain Ω and \delta u/\delta n=g in domain δΩ where b is a constant vector and c is a constant scalar. Show that if c is large enough compared to |b|, there is uniqueness Homework Equations Energy...
  12. B

    PDE: a traveling wave solution to the diffusion equation

    Homework Statement Consider a traveling wave u(x,t) =f(x - at) where f is a given function of one variable. (a) If it is a solution of the wave equation, show that the speed must be a = \pm c (unless f is a linear function). (b) If it is a solution of the diffusion equation, find f and show...
  13. W

    I feel really lost in my PDE class. Can somebody explain some things to me?

    I'm not going to blame anyone except for the fact that I'm probably a slow learner. Can somebody explain some of the things I'm learning in layman terms? That way I can have some context when I'm reading about them. Right now, the things I'm reading have no meaning, so it's really hard to...
  14. C

    Question related to PDE y(z_x)+x(z_y)+z=y

    To solve the PDE: y(z_x)+x(z_y)+z=y Use Method of characteristics a=y b=x d-cz=y-z Thus dx/y=dy/b=dz/(y-z) Taking first and second term xdx=ydy x^2-y^2=A x=sqrt(y^2+A) My question is, at this stage of the calculation, must we account for a negative constant A such that...
  15. R

    Any other recommendations for rigorous DE and PDE books?

    My plan is to work thru Rudin's Real and Complex Analysis, and then functional analysis, and then move on to DEs/PDEs. Right now its looking like Arnold for DEs, and Evans for PDEs. Any other recommendations? thanks
  16. P

    What is the role of Fourier transform in solving PDEs?

    Homework Statement (a) Solve \frac{\partial u}{\partial t}=k\frac{\partial ^{2} u}{\partial x^{2}} - Gu where -inf < x < inf and u(x,0) = f(x) (b) Does your solution suggest a simplifying transformation? Homework Equations I used the Fourier transform as: F[f(x)] = F(w) =...
  17. W

    Solving PDE w/Fourier: Obtain All Solutions

    Homework Statement Obtain all solutions of the equation partial ^2 u/partial x^2 - partial u/partial y = u of the form u(x,y)=(A cos alpha x + B sin alphax)f(y) where A, B and alpha are constants. Find a solution of the equation for which u=0 when x=0; u=0 when x = pi, u=x when y=1...
  18. K

    Can a Change of Variables Simplify This PDE to Normal Form?

    Hi I have this problem Reduce to normal form the following equation x2d2f/dx2-y2d2f/dy2=xy. First I guess by normal form , they mean a more standard equation, for example Laplacian. I usually know what to do when the coefficients are constants, but in this case I don't know.I think it is...
  19. J

    Verifying if this PDE is a solution

    PROBLEM: Verify that the functions [x+1]e^(-t) ; e^(-2)sint ; and xt are respectively solutions of the nonhomogeneous equations Hu = -e^(-t)[x+1] ; Hu = e^(-2x)[4sint+cost] ; and Hu = x where H is the 1D heat operator H = \frac{\partial}{\partial t} -...
  20. M

    How Do You Solve a First-Order Inhomogeneous PDE?

    Homework Statement Assume ut+cux = xt, u(x,0) = f(x) for t>0. Find a formula for u(x,t) in terms of f, x, t, and c. The Attempt at a Solution I don't really follow what the professor is doing in class, and his office hours and the textbook weren't much more help, so the only thing I know...
  21. F

    Heat equation PDE for spherical case

    Hello, I believe this is my first post. I would like to solve the heat equation PDE with some special (but not complicated) initial conditions, my scenario is as follows: A perfectly spherical mass of water, where the outer surface is at some particular temperature at t=0 (but not held at...
  22. J

    Help in hard model of RABR - solitons- solving PDE

    hello all, im doing a research on model called RABR which supports dark and bright solitons. as first step i need to find numeric solution for the following equation: F'' = c1*F - c2*F/sqrt(c3+F^2) where c1, c2 , c3 are const, and F is function of x, i.e: F= F(x) i try to make...
  23. M

    Analyzing Nonlinear PDE Systems with Polar Coordinates

    Homework Statement Hi, i have the following system of equation. In the task is that system have periodic solution and have to be used polar coordinates. Homework Equations x'=1+y-x^2-y^2 y'=1-x-x^2-y^2 The Attempt at a Solution After transfer to polar system i tried to use the method...
  24. J

    Reduction of PDE to an ODE by means of linear change of variables

    Homework Statement So it's been a really long time since I've done any ode/linear algebra and would like some help with this problem. Derive the general solution of the given equation by using an appropriate change of variables 2\deltau/\deltat + 3\deltau/\deltax = 0 The thing that...
  25. G

    Finding finite element soluton for a PDE

    Find the Finite element solution for a equation: (∂^2 u)/〖∂x〗^(2 ) +(∂^2 u)/〖∂y〗^2 +λu-c=0 using linear triangular finite elememts. In the above equation u is scalar,λ is a constant and is a body force term(constant). The boundary conditons are in terms of prescribed values of the function...
  26. P

    Solving PDE Problem: du/dt+du/dx=0 with Initial Condition u(x,0)=xe-x2

    Homework Statement which solutions of du/dt+du/dx=0 is equal to xe-x2 Homework Equations The Attempt at a Solution u(x,0) = xe-x2 u(x,t)= (x-t)e(-x-t)2 what else do i need to do?
  27. B

    How should I deal with the expression \frac{d}{dx} (\frac{dx}{dy}) ?

    Hi I have a question regarding a PDE and change of variable. I can follow through the algebra but I have a problem deciding what route to take after I use the chain rule at a later point. I have an expression: - \frac{\partial^2 f}{\partial y^2} and would like to make the variable...
  28. S

    MATLAB Solving PDE without BC with MATLAB

    I am trying to solve numerically the following PDE: dF(x,t) / dt = some function of x and F(x,t) ONLY where 0<x<5. This equation does NOT need boundary conditions at x=0 and x=5 because each point in x evolves independently from the others (the equation doesn't contain spatial derivatives)...
  29. T

    Mastering PDEs: Solving the Non-Constant Coefficient d^2G/dxdy Equation

    d^2G/dxdy+(a-1)*dG/dx*dG/dy=0 where G is a function of x and y. Moreover, what if a is not a constant, but instead a function of x and y?
  30. K

    Solve PDE: dG/dt=(n*s-u)(s-1)dG/ds

    Hi, could anyone tell me what kind of technique I should use to solve the following PDE? dG/dt=(n*s-u)(s-1)dG/ds Many thanks and happy new year to everyone:)
  31. M

    MATLAB How to Solve a Nonlinear PDE System with Non-Diagonal 'c' Matrix in Matlab?

    I'm solving a nonlinear pde system in one space. It looks that the pdepe function won't work, because it only accepts coupled term in 's', not 'c' and 'f'. My equations are like: \partial u1\partial t + c(u2)*\partial u2\partial t = f1(u2)*D^2 u1Dx^2 + s1(u1,u2)...
  32. M

    How Can I Solve a Transport PDE with Numerical Methods and Boundary Conditions?

    Here's my question, friends I have to define initial and boundary condition for a transport PDE: u_t+x(1-x)u_x=0 with x and t is between [0,1], to solve this equation, what kind of numerical method and boundary condition do you recommend and why? What kind of numerical error do you...
  33. M

    About solving Transport PDE

    Here's my question, friends I have to define initial and boundary condition for a transport PDE: u_t+x(1-x)u_x=0 with x and t is between [0,1], to solve this equation, what kind of numerical method and boundary condition do you recommend and why? What kind of numerical error do you...
  34. J

    Fick's Second Law: Laplace Transform to solve PDE in Spherical Coords

    Fick's second law in general form: \frac{\partial C}{\partial t} = D\nabla^2 C In spherical form: \frac{\partial C}{\partial t} = D\frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2\frac{\partial C}{\partial r} \right) (Assume all changes in phi and theta to be zero, so we are only concerned...
  35. L

    Is Advanced Calculus Necessary for Success in PDE?

    Is Advanced calculus absolutely necessary in order to succeed in PDE ? The problem is that my school does not require me to take Adv Calculus since i am an applied math major , i am not even required to take a proof based course here's the link for the major (...
  36. Battlemage!

    Solve a PDE with Separation of Variables

    Homework Statement Solve the 2-D time-independent Schrödinger equation with V (x,y) = 0: Homework Equations -ћ2/2m ( ∂2Ψ(x,y)/∂x2 + ∂2Ψ(x,y)/∂y2 ) = EΨ(x,y) The Attempt at a Solution I started by getting -ћ2/2m to one side: ( ∂2Ψ(x,y)/∂x2 + ∂2Ψ(x,y)/∂y2...
  37. T

    Solving PDE with Laplace Transform

    Homework Statement \frac{\partial^{2}u}{\partial t^{2}} = a^{2} \frac{\partial^{2}u}{\partial x^{2}} (x>0, t>0) with u(0,t) = t, u(x,0) = 0, ut(x,0) = A. Solve the PDE using laplace transform. The Attempt at a Solution I have managed to get the transform: \frac{\partial^{2}U(x,s)}{\partial...
  38. Somefantastik

    Solving 2nd Order PDE: u_{xx} - u_{tt} - au_{t} - bu = 0

    the book gives u_{xx} - u_{tt} - au_{t} - bu = 0; 0<x<L, t>0 says if you multiply it by 2u_{t} you can get \left( 2u_{t}u_{x}\right)_{x} - \left( u^{2}_{x} + u^{2}_{t} + bu^{2}\right)_{t} -2au^{2}_{t} = 0 or \frac{\partial}{\partial x} \left( 2 \frac{\partial...
  39. L

    Implicitly differentiating PDE (multivariable calculus)

    The problem: Find the value of dz/dx at the point (1,1,1) if the equation xy+z3x-2yz=0 defines z as a function of the two independent variables x and y and the partial derivative exists. I don't know how to approach the z3x part. I thought you would use the product rule and get 3(dz/dx)2x +...
  40. O

    Can Laplace's Equation Be Solved in Cylindrical Coordinates on a 3-Sphere?

    I wish to find exact solutions of Laplace's equation in cylindrical coordinates on (a subset of) the 3-sphere. This pde is linear but not separable. The potential {\Phi}(x,z) must fulfil the following pde: (1-{\frac{x^2}{a^2}}){\frac{{\partial}^2}{{\partial}x^2}}{\Phi}(x,z)+...
  41. D

    Linear transformation of a 2nd order pde

    First off I am NOT asking you to solve this for me. I'm just trying to understand the concept behind this problem. Let L be a linear transformation defined by L[p]=(x^2+2)p"+ (x-1)p' -4p I have not seen linear transformations in this format. Usually I see something like L(x)=x1b1+ x2b2...
  42. J

    Basic PDE Help: Simplifying the Confusing Concepts | MathBin

    http://mathbin.net/906 cant figure this one out
  43. J

    PDE in two dependent variables

    That's right, I said dependent. Does anyone have any experience dealing with such beasts. I haven't been able to find a single mention of them in any textbook on PDEs. The thing I'm really curious to know is whether the method of separation of variables works as usual, e.g. if the dep vars...
  44. Somefantastik

    Equation of Diffusion, trouble simplifying, PDE

    isotropic equation, so k, ρ, and c are constant, where k is thermal conductivity, c is specific heat, and ρ is the density of the body. the equation boils down to \left( \frac{c\rho}{k}\right) \left(\frac{\partial u}{\partial t}\right) - \left(\frac{\partial^{2} u}{\partial...
  45. H

    Understanding Fourier Equations in PDE for Beginners

    So I suppose my Fourier knowledge is a little bit rusty. Any help would be greatly appreciated. http://pmgz.net/3259.jpg How do they get from the original DKS equation to the Fourier space DKS equation (from eq 1 to eq 2)? Thanks greatly for any help.
  46. I

    Master Mathematical Techniques for Physics: PHYS 508 vs MATH 442

    Basic background info (which may not be useful):I will be a junior in physics this fall. I am done with all undergraduate level classical mechanics, E&M and quantum mechanics courses. I think I want to do experimental physics. I have been working under an AMO physics professor whose research is...
  47. P

    Another PDE question Where do I begin?

    Homework Statement Consider an electrical cable running along the x-axis which is not well insulated from ground, so that leakage occurs along its entire length. Let V(x,t) and I(x,t) denote the voltage and current at point x in the wire at time t. These functions are related to each other...
  48. S

    PDE with functions for coefficients: f_v g_u + f_u g_v = 0

    I know the solution of f_v g_u - f_u g_v = 0 where f and g are functions of (u,v) and the subscripts _u and _v denote partial derivatives. The equation can be viewed as a PDE for the unknown g with coefficients given by the partial derivatives of the known f. The equation sets the...
  49. P

    First order pde cauchy problem by method of characteristics

    Ok, so I can get through most of this but I can't seem to get the last part... Here is the problem xU_x + (y^2+1)U_y = U-1; U(x,x) = e^x Characteristic equations are: \frac{dx}{x} = \frac{dy}{y^2+1} = \frac{dU}{U-1} Solving the first and third gives: \frac{U-1}{x} = c_1 The...
  50. P

    First Order PDE Cauchy problem Using Method of Characteristics

    Homework Statement Ok, so I can get through most of this but I can't seem to get the last part... Here is the problem xU_x + (y^2+1)U_y = U-1; U(x,x) = e^x Homework Equations The Attempt at a Solution Characteristic equations are: \frac{dx}{x} = \frac{dy}{y^2+1} =...
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