1. ### 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.

7. ### 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 (...
8. ### 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...
9. ### 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...
10. ### 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...
11. ### 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...
12. ### 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...
13. ### 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...
14. ### 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 Equations The Attempt at a Solution First , i associate spectral variables to the x and t variables: ## k ## is the spectral variable...
15. ### 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
16. ### 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!
17. ### 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 ...
18. ### 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...

28. ### I Simple PDE

Hello everybody. I'm about to take a final exam and i've just encountered with this exercise. I know it's simple, but i tried solving it by Separation of variables, but i couldn't reach the result Mathematica gave me. This is the equation: ∂u/∂x = ∂u/∂t Plus i have a condition...
29. ### I Classification of differential equation

Hi, I have an equation that takes the form: ax''-by' + c = 0 where x'' is second order with respect to time and y' is first order with respect to time. Would this be classed as a partial differential equation? Thanks very much for your help :)
30. ### A Buckling PDE

Hi I am trying to verify my manual solution for this problem by any way, so I tried NDSolve, and DSolve, in mathematica with no success. I dont need it in mathematica I just need any way poosible, even matlab, or any other numeric way/soltuion. Can some one help, or even give me the final...