# 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...

28. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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.

48. ### 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. ### 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. ### 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...