Pde Definition and 743 Threads

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, 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...

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 $$B = k_1\cdot C^a$$...

Homework Statement Solve ∇2T(x, y) = 0 with boundary conditions T(0, y) = T(L, y) = T0 T(x, L/2) = T(x, -L/2) = T0 + T1sin(πx/L) Homework EquationsThe Attempt at a Solution Set T(x, y) = X(x)Y(y) Then ∇2T(x,y) = (∂2X/∂x2) Y + (∂2Y/∂y2) X = 0 Rearrange to find two separate ODEs: d2X/dx2 =...

I've recently been making some posts around the web and on this forum attempting to figure out how to use a PDE that models traffic flow in concrete examples. I realize that I have to solve this PDE in order to use it, but I'm sort of lost on how exactly one solves it. The PDE is as follows...

Homework Statement Consider the wave equation: u_{tt} - c^2u_{xx} = f(x,t), \hspace{1cm} for \hspace{1cm} 0 < x < l \\ u(0,t) = 0 = u(l,t) \\ u(x,0) = g(x), u_t(x,0) = f(x) \\ Find a nontrivial solution. Homework EquationsThe Attempt at a Solution Here's what I did, but I have little...

Homework Statement Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π). Homework EquationsThe Attempt at a Solution Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π) (ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...

Homework Statement Solve y*∂Ψ/∂x-(x/3)∂Ψ/∂y Homework EquationsThe Attempt at a Solution My teacher told me to try separation of variables but and I tried to set Ψ=X(x)Y(y) where X is a function of just X and Y is a function of just y but when I got the solution and put it into the original pde...

Hi everyone. In electrical engineering, when you study control theory, you're taught that electrical circuits can be used to simulate the behaviour of complex systems. What I don't understand is, what are the limitation of this sistem, and why it can't be obviouslly used in a general way to...

Sorry to keep the title too broad and general. I'm starting learning pde by myself , using "linear partial differential equations for scientists and engineers" I'm having some problems with the basics "I took ODE". The following differentiation is totally new to me, can some one explain to me...

Homework Statement Consider the nonlinear (ordinary) differential equation u' = u(1-u). a) Show that u_1 (x) = e^x/(1+e^x) and u_2(x) = 1 are solutions. b) Show that u_1+u_2 is not a solution. c) For which values of c is cu_1 a solution? How about cu_2 ? Homework Equations N/a The Attempt at...

Homework Statement In my PDE course we have a homework question stating the following: Let ϑ(x) = x in the interval [-pi, pi ]. Find its Fourier Coefficients. Homework Equations From my notes on this type of question: a_o = 2c_o = 1/pi * integral from -pi to pi [f(x) dx] a_n = c_n + c_(-n)...