Pde Definition and 743 Threads

Hellow everybody! A simple question: exist a general formulation, a solution general, for a PDE of order 2 like: ## au_{xx}(x,y)+2bu_{xy}(x,y)+cu_{yy}(x,y)+du_x(x,y)+eu_y(x,y)+fu(x,y)=g(x,y) ## ? The maple is able to calculate the solution, however, is a *monstrous* solution!

Given a PDE of order 1 and another of order 2, you could show me what is, or which are, all possible initial conditions? For an ODE of order 2, for example, the initial condition is simple, is (t₀, y₀, y'₀). However, for a PDE, I think that there is various way to specify the initial condition...

Hi guys, I need to simulate wave propagation for a nonlinear dispersive wave PDE and since I can't find proper resources for handling nonlinear PDEs numerically, I would appreciate any help and clues. the PDE is in the form of utt-(au+bu2+cu3+duxx)xx=0 Romik Ps: BC: Clamped at both ends IC...

Hi, Homework Statement I have solved ux + (x/y)uy = 0 using characteristics, to obtain u(x,y)=C (for y=+-x) and f(x2-y2) Homework Equations The Attempt at a Solution I was then given two boundary conditions: (a) u(x=0,y)=cos(y), which I used to obtain u(x,y) = cos(√(y2-x2))...

I want to find some application of the laplace equation on semi-infinite plate on physics where the PDE is looke like $$u_{xx}+u_{yy}=f , for a<x<\infty , c<y<d$$ $$u(a,y)=g(y), u(x,c)=f_{1}(x), u(x,d)=f_{2}(x)$$ $$\lim_{x->\infty} f(x)=\lim_{x->\infty} f_{1}(x)=\lim_{x->\infty}...

Homework Statement now I have a PDE $$u_{xx}+u_{yy}=0,for 0<x,y<1$$ $$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$ Then I want to know whether there are some method to make the PDE become homogeneous boundary condition. $$i.e. u|_{\partialΩ}=0$$

Homework Statement Mod note: Pasted the OP's correction into the original problem.[/color] Solve xe^z\frac{\partial u}{\partial x} - 2ye^z\frac{\partial u}{\partial y} + \left(2y-x \right)\frac{\partial u}{\partial z} = 0 given that for x > 0, u = -x^{-3}e^z when y=-x Homework Equations...

I have a tutorial question for maths involving the heat equation and Fourier transform. {\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}} you are given the initial condition: u(x,0) = 70e^{-{\frac{1}{2}}{x^2}} the answer is: u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}} In this course...

So I have a question in terms of interpreting the boundary conditions for a PDE. It is question 4 in the attached picture. My question is that usually when I have encountered BC problems I have been given that my boundary conditions equal a given value, in terms of the diffusion equation...

Hi everybody, I need to solve a 1st order PDE for my thesis and I'm not a specialist in this field. I've read some texts about this and know one method of solving a 1st order PDE is the method of characteristics. since my equation is nonlinear and a bit complicated, I'm going to solve it...

"Critiquing" separation of variables method for PDE. I am currently taking a course in PDE's and it has been very "applied" and not so much theory based. I can say its been separate this separate that separate this separate that… Enough! We are always "separating variables" and it always...

I'm trying to decide between taking an ODE class or a PDE class next. I have already done Calculus 1,2,3 so I already know some ODEs and PDEs and linear algebra. I'm a 3rd year mathematics major with a minor in Statistics and I'm interested in applied mathematics.ODE course coverage: Ordinary...

Homework Statement solve the heat equation over the interval [0,1] with the following initial data and mixed boundary conditions.Homework Equations \partial _{t}u=2\partial _{x}^{2}u u(0,t)=0, \frac{\partial u}{\partial x}(1,t)=0 with B.C u(x,0)=f(x) where f is piecewise with values: 0...

I've got this far on a pde (second last step) but have no idea how they got this equality(I'm a noob), could someone please explain? I was going to put this under homework but it is not homework and it doesn't really fit the template. Thanks in advance.

Homework Statement Find the general solution of the equation (\zeta - \eta)^2 \frac{\partial^2 u(\zeta,\eta)}{\partial\zeta \, \partial\eta}=0, where ##\zeta## and ##\eta## are independent variables. Homework Equations The Attempt at a Solution I set ##X = \partial u/\partial\eta## so that...

Good day. I was wondering if you could help me solve this first order linear partial differential equation: [∂δ]/[/∂t] = [ρg]/[/μ] δ^2 [∂δ]/[/∂z]. The solution for this is: δ(z, t) = √[μ z]/[/ρg t]. I don't really understand how the PDE became like this. If you could show the...

Is anyone familiar with plotting an infinite domain PDE where the solution is an integral. Take the solution \[ T(x,t) = \frac{100}{\pi}\int_0^{\infty}\int_{-\infty}^{\infty} \frac{\sinh(u(10-y)}{\sinh(10u)} \cos(u(\xi-x))d\xi du \] How could I plot this in Matlab, Mathematica, or Python? As a...

Ok this qusestion has to do with completing the square for a diffusion equation. Initial Cond: u(x,0) = e-x Now they say plug this into the general formula: u(x,t) = 1/(4\pikt)1/2 ∫ e-(x-y)1/2/4kte-y dy where k is a constant now the first step they say is completing the...

Homework Statement By changing variables from (S,t,V) to (x,\tau,u) where \tau = T - t, x = \ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T-t), u=e^{r\tau}V, where r, \sigma, \tau, K are constants, show that the Black-Scholes equation \frac{\partial V}{\partial t} +...

This was something I noticed as I was trying to practice solving PDEs using the method of characteristics. The text has the following example: $$\frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y} = 0$$ This should be easy enough. I let p(x,y) = x and solve for \frac{\partial...

Homework Statement I have 3 equations: \frac{\partial^2 u}{\partial t^2}+\frac{\partial^2 u}{\partial x \partial t}+\frac{\partial^2 u}{\partial x^2} \frac{\partial^2 u}{\partial t^2}+4\frac{\partial^2 u}{\partial x \partial t}+4\frac{\partial^2 u}{\partial x^2} \frac{\partial^2...

Hi all, I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2} subject to the initial condition u(x,0)=g(x)...

Homework Statement If utt - uxx= 1-x for 0<x<1, t>0 u(x,0) = x2(1-x) for 0≤x≤1 ut(x,)=0 for 0≤x≤1 ux(x,)=0 u(1,t)=0 find u(1/4,2) Homework Equations The Attempt at a Solution I was thinking to make a judicious change of variables that not only converts the PDE to a homogenous PDE, but also...

Homework Statement Find the solution of: utt-uxx = sin(∏x) for 0<x<1 u(x,0)=0 for 0<=x<=1 ut(x,0)=0 for 0<=x<=1 u(0,t)=0 u(1,t)=0Homework Equations utt-uxx = sin(∏x) for 0<x<1 u(x,0)=0 for 0≤x≤1...

I'm new here, hope it is the right place to ask the question. The PDE question is ∂2u/∂x2+∂2u/∂y2=0 and u(x,0)=f(x), u(x,1)=0, u(0,y)=0, u(1,y)=0. I use the method of separate the variables, with is let u(x,y)=X(x)Y(y) and get X''/X+Y''/Y=0. Then let X''/X=-Y''/Y=-λ, i.e. X''+λX=0...

Hi, hope this is a right place to ask this question. I work in the soil physics field and this problem has taken lots of my energy for a while! let's state it: Unsaturated horizontal water flow in 2 layer soil: we have, M(for Moisture), K (for hydraulic conductivity), h (for hydraulic...

Solving a "simple" second order PDE, do I need the Fourier? Homework Statement The problem as given: y'' + 2y' + 5y = 10\cos t We want to find the general solution and the steady-state solution. We're using \mu y'' + c y' + k y = F(t) as our general form. OK, so I first want the general...

So up to this point we have only learned 2 forms of PDE's to solve: Constant Coefficient Equations and Variable Coefficient Equations. Questions: Solve: 1) aUx+bUy + cU = 0 2) Ux+ UY = 1 where U = U(x,y) Attempt: Well for 2) I'm thinking that it doesn't necessarily matter...

Homework Statement OK, a PDE: $$a\frac{\partial u}{\partial t} + b \frac{\partial u}{\partial x} = u$$ we want the general solution. 2. The attempt at a solution So, I'll set up a couple of equations thus: r = m11x + m21t s = m12x + m22t (We have a nice matrix of m here if we...

I'm having trouble understanding the boundary conditions and when you would need to use Bessel vs Modified Bessel to solve simple cylindrical problems (I.e. Heat conduction or heat flow with only two independent variables). When do you use Bessel vs Modified Bessel to solve Strum-Louville...

\begin{align*} \psi_t + \psi_{xxx} + f(\psi)\psi_x &= 0 \end{align*} This equation leads to the nonlinear Shrodinger equation but does this equation have a name?

This is a quantum mechanics problem, but the problem itself is reduced (naturally) to a differential equations problem. I have to solve the following equation: \frac{\partial}{\partial t}\psi (x,t) = i\sigma \psi (x,t) where \sigma > 0 The initial condition is: \psi (x,0) =...

Homework Statement Here is a photo of a page in Laser Physics by Hooker: https://www.evernote.com/shard/s245/sh/2172a4e7-63c7-41a0-a0e7-b1d68ac739fc/7ba12c241f76a317a6dc3f2d6220027a/res/642710b5-9610-4b5b-aef4-c7958297e34d/Snapshot_1.jpg?resizeSmall&width=832 I have 3 questions (I'm a bit...

Homework Statement Verify that, for any continuously differentiable function g and any constant c, the function u(x, t) = 1/(2c)∫(x + ct)(x - ct) g(z) dz ( the upper limit (x + ct) and lower limit (x - ct)) is a solution to the PDE utt = c2uxx. Do not use the...

Hi all, Say I am solving a PDE as \frac{\partial y^2}{\partial^2 x}+\frac{\partial y}{\partial x}=f, with the boundary condition y(\pm L)=A. I can understand for the second order differential term, there two boundary conditions are well suited. But what about the first order differential term...

I've been studying Walter A. Strauss' Partial Differential Equations, 2nd edition in an attempt to prepare for my upcoming class on Partial Differential Equations but this problem has me stumped. I feel like it should be fairly simple, but I just can't get it. 10. Solve ##u_{x} + u_{y} + u =...

I took a CFD class last semester (had to leave school though due to personal garbage). I am making a come back this fall and as some extra credit I am trying to numerically solve the unsteady laminar flow equation in a pipe. The equation is \dot{U} + U'' + K = 0 where dots denote the time...

Homework Statement Solve using separation of variables utt = uxx+aux u(0,t)=u(1,t)=0 u(x,0)=f(x) ut=g(x) The Attempt at a Solution if not for the ux I'd set U=XT such that X''T=TX'' and using initial conditions get a solution. In my case I get T''X=T(aX'+X'') which is...

Hey everyone I am going to be a freshman this fall (in college). I am currently having a dilemma in choosing my math class. In high school I took classes all the way up to Honors Differential Equations (ODE). In June I went to the university and signed up for Ordinary Differential Equation...

Homework Statement Hi guys, I'm having trouble with a homework problem: I will have to solve for the IVP of a transport equation on R: the equations are: Ut-4Ux=t^2 for t>0, XER u=cosx for t=0, XER Homework Equations transport equation The Attempt at a Solution...

Homework problem: For the wave equation: Utt-Uxx=0, t>0, xER u(x,0)= 1, |x|<1 0, |x|>1 sketch the solution u as a function of x at t= 1/2, 1, 2, and 3 I am able to use d'Alemberts and solve for u however the boundaries and the odd/even reflections are throwing me off and...

Hi guys, I'm having trouble with a homework problem: I will have to solve for the IVP of a transport equation on R: the equations are: Ut-4Ux=t^2 for t>0, XER u=cosx for t=0, XER I've actually never seen a transportation problem like this and any help would be...

I'm having troubles with PDE. Apply separation of variables, if possible, to found product solutions to the following differential equations. a) x\frac{\partial u}{\partial x}=y\frac{\partial u}{\partial y} I suppose that: u=X(x) \cdot Y(y) Then: xX'Y=yXY' xX'/X=yY'/Y So xX'/X=yY'/Y=c because...

Hello folks, If we have the expression, say \frac{∂f}{∂r}+\frac{∂f}{∂θ}, am I allowed to change it to \frac{df}{dr}+\frac{df}{dr}\frac{dr}{dθ}, if "f" is constrained to the curve r=r(θ). My reasoning is that since the curve equation is known, then f does not really depend on the...

I am trying to solve an ODE and PDE and I am having problems coming up with a method for doing so. The PDE is: k1*(dC/dt) = k2*(dC/dx) But I have an ODE that is an expression for dC/dt: dC/dt = k3*C Where k1,k2 and k3 are constants. I planned to use the method of lines to get...

Hi I'd appreciate any help on identifying the type of PDE the following equation is... *This is NOT homework, it is part of research and thus the lack my explanation of what this represents and boundary conditions. I have a numerical simulation of the solution but I'm looking to have a math...

Homework Statement 1) What is the Equilibrium temperature distributions if α > 0? 2) Assume α > 0, k=1, and L=1, solve the PDE with initial condition u(x,0) = x(1-x) Homework Equations du/dt = k(d^2u/dx^2) - (α*u) The Attempt at a Solution I got u(x) = [(α*u*x)/2k]*[x-L] for...

∂u/∂x=∂u/∂y, can we ensure that u is a constant not dependent on x and y?

Homework Statement Consider the nonlinnear diffusion problem u_t - (u_x)^2 + uu_{xx} = 0, x \in \mathbb{R} , t >0 with the constraint and boundary conditions \int_{\mathbb{R}} u(x,t)=1, u(\pm \inf, t)=0 Investigate the existence of scaling invariant solutions for the equation...

My questions concerns the information in the document. I highlighted the portion that is confusing me and a sample problem at the bottom. Question: Look at the equation 2.2.4 in the document. When I set the function u equal to zero the equation becomes 0 = 0 + 0 + f(x,t) or f(x,t) = 0. Now...