Pde Definition and 856 Threads

Let $$f:\Omega\to\mathbb{R}$$, where $$\Omega\subset\mathbb{R}^d$$, and $$\Omega$$ is convex and bounded. Let $$\{x_i\}_{i=1,2,..N}$$ be a set of points in the interior of $$\Omega$$. $$d_i\in\mathbb{R}$,$i = 1,2,..N$$ I want to solve this weakly formulated pde: $$ 0=\frac{A}{N^{d+1}} \sum_i...

So I'm asked to use separation of variables to find a product solution to the given PDE: (5y + 7)du/dx + (4x+3)du/dy = 0 Since it says to find a product solution, I used the form u(x,y) = XY and plugged that into the PDE. However, I am getting stuck because I'm not sure how exactly I should...

Hello, I have an equation of the form: ##\partial_t f(x,t)+a\partial_x^2 f(x,t)+g(x)\partial_xf(x,t)=0 ## (In my particular case ##g(x)=kx## with ##k>0## and ##a=2k=2g'(x)##) I'd like to know if there is some general technique that i can use to solve my problem (for example: in the first...

Hi. I have this PDE that governs an L x L plate (similar to the Poisson equation, it seems) with boundary conditions F = 0 and F" = 0 along the edges. I have successfully solved the problem by setting up an equality W = ∇2F then I solved the two PDEs simultaneously: W = ∇2F (boundary...

To all who are interested in a source for the treatment of partial differential equations: Victor Ivrii, Toronto, Course notes, 310 pages https://www.ams.org/open-math-notes/omn-view-listing?listingId=110703&utm_content=buffer2458a&utm_medium=social&utm_source=facebook.com&utm_campaign=buffer

Hello, I am working with numerical relativity and spectral methods. Recently I finished a general elliptic PDE solver using spectral methods, so now I want to do Physics with it. I am interested in solving the lapse equation, which fits into this category of PDEs $$ \nabla^2 \alpha = \alpha...

Homework Statement Find the steady state (equilibrium) solution for the following boundary value problem: ∂u/∂t = (1/2)∂2u/∂x2 Boundary condition: u(0,t) = 0 and u(1,t) = -1 Initial condition: u(x,0) = 0 Homework Equations u(x,t) = Φ(x)G(t) The Attempt at a Solution I have found the solution...

Hi All, Does anybody know how to solve the following PDE? I tried a similarity solution method where eta = y/f(x) (which I can do successfully without the C * U term) but was unsuccessful. Thank you very much in advance!

Hello I would like to check my reasoning about solutions of first order PDE. I've spell out (almost) all details. I'll consider the following problem: find ##u=u(t,x)## s.t. : $$ \partial_t u(t,x) + a(x) \cdot \nabla u(x) =0 \qquad \qquad u(0,x) = u_0(x)$$ say with smooth coefficient and...

I have learned that diffusion/heat equation can be used to model groundwater flow in confined conditions. Recently I read a paper where they used linear Boussinesq equation (equation 1 in linked paper) to model groundwater flow in unconfined aquifer. Then in another paper, the auther said, he is...

Additionally, what topics from that same course are relevant to probability? I ask because I'm afraid I might forget some of the topics from my calculus series after one semester of disuse. I mean, I know I should probably brush up on my calculus skills in preparation for any math class that...

Hi. Been a while since I logged in here, I missed this place. Anyway, I have a question (title). Is that even possible? Say for example I have the standard heat equation (PDE) subject to the boundary conditions: T(0,t) = To T(∞,t) = Ti And the initial condition: T(0,t) = Ti I am aware of how...

hi, i know a little bit of ODE but not much about PDE,Some math programs give me the solution but I would like to know what methods they use. The problem is the following: $$I(a,b) = \int_{0}^{\infty} e^{-ax^{2}-\frac{b}{x^2}}$$ through differentiation under the integral sign, substitution...

hi, I do not know much about PDEs and programs like wolfram alpha and maple don't give me a solution. it is possible to calculate the function through PDE?. I would appreciate any help $$\frac{\partial }{\partial a}f(a,b,n)+\frac{\partial }{\partial b}f(a,b,n)=-n f(a,b,n+1)$$...

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...

Hi everyone, I am a student in Mechanical Engineering and I am currently working on an assignment where I am exploiting the possibility of self-similarity for a PDE of a given problem. The PDE in my assignment consists of two independent variables (x for space and t for time), and one dependent...

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 :)

Homework Statement Solve 2D wave eq. ##u_tt=c^2 \nabla^2u## in a circle of radius ##r=a## subject to $$u(t=0)=0\\ u_t(t=0)=\beta(r,\theta)\\u_r(r=a)=0\\$$and then symmetry for ##u_\theta(\theta=\pi)=u_\theta(\theta=-\pi)## and ##u(\theta=\pi)u(\theta=-\pi)##. Homework Equations Lot's I'm sure...

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 don't 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...

Hello, I am currently a High School Senior who has completed Multivariable Calc (up to stokes theorem), basic Linear Algebra ( up to eigenvalues/vectors) and non-theory based ODE (up to Laplace transforms) at my local University. (All with A's) I am hell bent on taking either one of the courses...

Q: Suppose ##u(x,t)## satisfies the heat equation for ##0<x<a## with the usual initial condition ##u(x,0)=f(x)##, and the temperature given to be a non-zero constant C on the surfaces ##x=0## and ##x=a##. We have BCs ##u(0,t) = u(a,t) = C.## Our standard method for finding u doesn't work here...

Homework Statement So I don't really understand what the professor means by "show why the displacements y(x,t) should satisfy this boundary value problem" in problem 1. Doesn't that basically boil down to deriving the wave equation? At least in problem 2 he says what he wants us to show...

Hi PF! I'm wondering if my solution is correct. The PDE is ##h_t = h_{zz}## subject to ##h_z(0,t)=0##, ##h(1,t)=-1##, and let's not worry about the initial condition now. To solve I want homogenous boundary conditions, so let's set ##v = h+1##. Then we have the following: ##v_t = v_{zz}##...

Homework Statement Which algebraic expressions must be solved when you use finite difference approximation to solve the following Possion equation inside of the square : $$U_{xx} + U_{yy}=F(x,y)$$[/B] $$0<x<1$$ $$0<y<1$$ Boundary condition $$U(x,y)=G(x,y)$$ Homework Equations Central...

Hello folks, 1.- In geometry we study for example the conic sections, their exentricity and properties. I was wondering what part of the mathematical science studies the different properties of complex valued distributions. One example are the spherical armonics. I guess mathematicians have...

So I have been studying solving separation of variable, heat equation and came across 2 set of lambda equation. and lambda = 0 have the same equation. Is it different?

I am trying to solve with Laplace Transforms in an attempt to prove duhamels principle but can't find the Laplace transform inverse at the end. The book I am reading just says "from tables"... The problem : $$ U_t = U_{xx}\\\\ U(0,t)=0 \quad 0<t< \infty\\\\ U(1,t)=1\\\\ U(x,0)=0 \quad...

Homework Statement $$\frac{\partial U}{\partial t}=\nu \frac{\partial^{2} U}{\partial y^{2}}$$ $$U(0,t)=U_0 \quad for \quad t>0$$ $$U(y,0)=0 \quad for \quad y>0$$ $$U(y,t) \rightarrow {0} \quad \forall t \quad and \quad y \rightarrow \infty$$ Homework Equations This is a diffusion problem on...

<< Mentor Note -- thread moved from the technical math forums >> I am getting stuck on this partial differential equation. Ut = Uxx - U + x ; 0<x<1 U(0,t) = 0 U(1,t) = 1 U(x,0) = 0 Here is my work so far : U = e-tw + x gives the new eq wt=wxx to get rid of boundary conditions : w=x+W Wt=Wxx...

I am looking to numerically solve the (complex) Time Domain Ginzburg Landau Equation. I wish to write a python simulator to observe the nucleation of fluxons over a square 2D superconductor domain (eventually 3D, cubic domain). I am using a fourth order Runge Kutta solver for this which I made...

Hello physicists, Pretty new to Mathematica here. I'm looking to verify that $$P(s,\tilde{t}|_{s_0}) = 2\tilde{b}_{\rho} \frac{s^{\alpha+1}}{\check{s_0}^{\alpha}}I_{\alpha}(2\tilde{b}_{\rho}s\check{s}_0)exp[-\tilde{b}_{\rho}(s^2+\check{s_0}^2)]$$ Is a solution to...

Does anyone know if it is possible to solve an equation of the type u_x=(sin(x))*(u) on a periodic domain using the fft. I have tried methods using convolutions but have had no success thanks in advance

I've been trying to solve a set of coupled, nonlinear PDEs with the general form: ## \frac {\partial A}{\partial t} = aAC + bBD - cE ## ## \frac {\partial B}{\partial t} = - c(E+F) ## ## \frac {\partial C}{\partial t} = dAD - eE ## ## \frac {\partial D}{\partial t} = dBC + - eF ## ## \frac...

I think I am missing something painfully obvious, but what exactly is the difference in algorithms used to solve PDEs vs ODEs? For example, I've been looking at finite difference methods and the general steps (from what I've seen, although particular approaches may vary) used to numerically...

Homework Statement Homework EquationsThe Attempt at a Solution First write ##\phi(x,t)## as its transform ##\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \! e^{ipx} \widetilde{\phi}(p,t) \, \mathrm{d}p## which I then plug into the PDE in the question to get...

Homework Statement Graph snapshots of the solution in the x-u plane for various times t if \begin{align*} f(x) = \begin{cases} & 3, \text{if } -4 \leq x \leq 0 \\ & 2, \text{if } 4 \leq x \leq 8 \\ & 0, \text{otherwise} \end{cases} \end{align*} Homework Equations Assuming that c=1 and g(x)...

Homework Statement Homework EquationsThe Attempt at a Solution integral[du]= Integral[xt ds] xt=18s2+3sT so, u=Integral[18s2+3sT] u=6s3+(3/2)s2T+C C=eT2 This is what I did and the solution is below. I'm unsure where the missing power on the (3/2)sT went in the u(s,t) equation.[/B]

Is there anyone who has read some of the mentioned texts and can say a few words about how they differ?

Suppose a PDE for a function of that depends on position, ##\mathbf{x}## and time, ##t##, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier...

Hi PF! So my book has boiled the problem down to $$\psi(r,\theta) = c_1 \ln \frac{r}{a}+\sum_{n=1}^\infty A_n\left(r^n-\frac{a^{2n}}{r^n}\right)\sin n\theta$$ subject to ##\psi \approx Ur\sin\theta## as ##r## get big. The book then writes $$\psi(r,\theta) = c_1 \ln...

I'm trying to solve the following PDE: $$u_t+yu_x=-y-\mathbb{H}(y_x)$$ where y satisfies the inviscid Burgers equation $$y_t+yy_x=0$$ and the Hilbert Transform is defined as $$\mathbb{H}(f) = PV \int_{-\infty}^{\infty} \frac{f(x')}{x-x'} \ dx',$$ where PV means principal value. The solutions...

How could you that y(x,t)=ƒ(x/a + t/b), where a and b are constants is a solution to the wave equation for all functions ,f ? many thanks.

The equation is Uxx + Uyy = 0 And domain of solution is 0 < x < a, 0 < y < b Boundary conditions: Ux(0,y) = Ux(a,y) = 0 U(x,0) = 1 U(x,b) = 2 What I've done is that I did separation of variables: U(x,y)=X(x)Y(y) Plugging into the equation gives: X''Y + XY'' = 0 Rearranging: X''/X = -Y''/Y = k...

Hi, I am looking for good books with somewhat of an intuitive explanation on waves physics (acoustic waves), elastic waves, on ODEs, PDEs, and calculus? Also some good ones on DSP Thanks in advance Chirag

Homework Statement Suppose the inner side of the annulus {(r,Φ): r_0 ≤ r ≤ 1} is insulated and the outer side is held at temperature u(1,0) = f(Φ). a) Find the steady-state temperature b) What is the solution if f(Φ) = 1+2sinΦ ? Homework EquationsThe Attempt at a Solution a) A =...

Homework Statement Solve ∇^2u=0 in D subject to the boundary conditions u(x,0) = u(0,y) = u(l,y) = 0, u(x,l) = x(l-x) where D = {(x,y): 0≤x≤l, 0≤y≤l} Homework EquationsThe Attempt at a Solution So, I've looked at the notes and the book and have a gameplan to attack this problem. However...

Homework Statement Show that if v(x,t) and w(y,t) are solutions of the 1-dimensional heat equation (v_t = k*v_xx and w_t = k*w_yy), then u(x,y,t) = v(x,t)w(y,t) satisfies the 2-dimensional heat equation. Can you generalize to 3 dimensions? Is the same result true for solutions of the wave...

Hi everyone. For people who already saw me in this forum, I know I may seem boring with all these questions about PDE, but I promise this will be the last :D Anyway, as the title says, which are the main trends of differential equations research, especially nonlinear differential equations(which...

Homework Statement Part (a) below: Homework EquationsThe Attempt at a Solution There's more to this question but I'm only stuck on this first part so far. I have no idea what specific PDE the equation, θ(x,t) = T(x,t) - T0x/L , obeys In this module (mathematics) we've covered the wave...

I have a PDE which is the following: $$\frac {\partial n}{\partial t} = -G\cdot\frac {\partial n}{\partial L}$$ with boundary condition: $$n(t,0,p) = \frac {B}{G}$$ , where G is a constant, L is length and t is time. G and B depend on a set of parameters, something like $$B = k_1\cdot C^a$$...