Pde Definition and 743 Threads

I look for good books on solving partial diffrential equations (PDE's) using integral transforms specially Fourier and laplace transforms. Do you have any recommendations for such books? I don't look for a book concerned with the theory, rather, with the methods itself (a suitable book for a...

Homework Statement The problem statement can be expressed in one of these forms listed in order of preference. [/B] Every character with exception of x, y, t, and C are constants. Homework Equations I require a change of variable or series of subsequent change of variables that can convert...

Hi, I'm looking at the Jacobian condition which is ## J= a \frac{dy_{0}}{ds}-b\frac{dx_0}{ds}## where the pde takes the form ##c= a\frac{\partial u}{\partial x} + b \frac{\partial u}{\partial y} ##, where ##a=\frac{\partial x}{\partial \tau } ##, ##b=\frac{\partial y}{\partial \tau }##...

Homework Statement My textbook (Advanced Engineering Mathematics, seventh edition, Kreyszig) indicates that if u1 and u2 are solutions to a second-order homogeneous partial differential equation, and c1 and c2 are constants, then u where u = c1u1 + c2u2 is also a solution, this is the...

As i understand, the purpose of laplaces/poissons equation is to recast the question from a geometrical one to a differential equation. im trying to figure out what are the appropriate boundary conditions for poissons equation: http://www.sciweavers.org/upload/Tex2Img_1418842096/render.png...

Homework Statement This comes up in the context of Poisson's equation Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Homework Equations $$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta }$$...

Hello everyone, I'm in need for the best books that you know out there for PDE (Partial Differential Equations) and everything related to Fourier (series, transform, etc.). Any help would be much appreciated. Thank you and happy holidays!

Hello, I was going through the following paper: http://www.emis.de/journals/HOA/AAA/Volume2011/142128.pdf In page 6, immediately after equation (3.15), its written that "functions of the form v(t) are dense in L^2". I have been looking for proofs online which verifies the above statement but...

Homework Statement I have a PDE and I need to solve it in spherical domain: $$\frac{dF(r,t)}{dt}=\alpha \frac{1}{r^2} \frac{d}{dr} r^2 \frac{dF(r,t)}{dr} +g(r,t) $$ I have BC's: $$ \frac{dF}{dr} = 0, r =0$$ $$ \frac{dF}{dr} = 0, r =R$$ Homework Equations So, in spherical coord. First...

How to find L if the form is: $$ (\frac{\partial L}{\partial x})^2 - (\frac{\partial L}{\partial y})^2 = -1$$ The author wrote, $$L = y + ax^2 + ..$$ but I didn't get how?

I'm new to shocks and trying to get the hang of it. I have 3 sets of characteristic equations,( by a set I mean defined by taking a different fixed value u along the characteristic.) From what I understand,in general talk, we use a shock whenever two sets of characteristics collide as...

Homework Statement Solve the following IVP: ##\frac{\partial v(x,t)}{\partial x} + \frac{\partial v(x,t)}{\partial t} + v(x,t) = g(x,t)## Homework Equations The initial values: v(0,t) = a(t) and v(x,0) = b(x)The Attempt at a Solution I applied the Laplace transform x -> s to get...

Homework Statement $$u_t = ku_{xx} + \sin(2 \pi x / L)$$ $$u_x(0,t) = u_x(L,t) = 0$$ $$u(x,0) = f(x)$$ Homework Equations none (other than the obvious) The Attempt at a Solution So i started by taking letting ##ku_E''(x) =- \sin(2 \pi x / L)## (notice from the boundary conditions above I...

Hello guys, I'm writing to get some help on an exercise I've been thinking but I can't get to solve. I have to write the code for the Example 8.5 of the book White, Fluid Mechanics. Here is the problem and the solution I have to obtain. It is about one duct that has three sections in which I...

Homework Statement Provide the missing steps to re-write the equation into one with just two terms $$u_{tt} - c^2(u_{rr}+\frac{2}{r}u_r) = 0$$ Homework Equations Nothing, other than this looks similar to the wave equation hybrid. (I'm just speculating) Also, I'm a little uncertain what is...

What would the career prospects be for someone who does a Ph. D. in Mathematics with a research focus in partial differential equations? Assuming you got some computer skills along the way like parallel computing, programming, etc? Sure, you could become a professor but most people don't make...

Hey PF! I have a quick question. When I was solving a PDE via separation of variables, I was able to come up with a same format solution for ##n \geq 1## but when ##n=0## I had a different "type" of solution. This doesn't really bother me since I am dealing with a linear PDE. However, I matched...

I'm trying to solve question 4.12 from Cross and Greenside "pattern formation and dynamics in nonequilibrium systems". the question is about the equation \partial_t u = r u - (\partial_x ^2 +1)^2 u - g_2 u - u^3 Part A: with the ansatz u=\sum_{n=0}^\infty a_n cos(nx) show that the...

Homework Statement A rectangular chip of dimensions a by b is insulated on all sides and at t=o temperature u=0. The chip produces heat at a constant rate h. Find an expression for u(x,y,t) Homework Equations δu/δt = h + D(δ2u/δx2 + δ2u/δy2) x∈(0,a), y∈(0,b) The Attempt at a Solution I'm...

Hello all, I've got one more semester before I earn my physics MS, and I have space for one or two extra courses. I am going into oceanography, and I would like to have a strong foundation in math in order to understand the theory I'll encounter as well as possible. Lots of physical...

Ut+6UUx+Uxxx=0 [kdv eq] Why to solve this do you need U(x,t=0)? Why is it a initial value problem? This should probably be really obvious. I think I've forgotten some basic background stuff, just starting my course in solitons... Thanks for your help.

I am going to do a numerical simulation of diffusion in matlab. The diffusion coefficient is concentration dependant, and i use an array operation to calculate D(x), so it is known. Based on Fick's second equation: $$ \frac{\partial C}{\partial t} = \frac{\partial}{\partial x} D \frac{\partial...

Homework Statement Hi! Not sure if I'm posting in the right section, this problem is from a course in scientific computing. Anyway, we're considering a set of PDEs: u_t + Au_x = 0 \quad 0<x<1, \ t>0 \\ u(x,0) = f(x) \quad 0 \leq x \leq1 \\ u_1(0,t) = 0 \quad t \geq 0 \\ u_2(1,t) = 0 \quad t...

Homework Statement Solve \frac{\partial v}{\partial t} = k\frac{\partial^2 v}{\partial x^2} - v 0\leq x \leq L t > 0 Homework Equations v(x,0) = f(x), v(0,t)=0, \frac{\partial v}{\partial x}(L,t) = -v(L,t) The Attempt at a Solution I've already attempted to solve...

Hey! :o Do you know if "Wave Phenomena" are related to PDE or only to physics?? (Wondering)

Im writing a program that calculates the trajectory of a particle in an arbitrary force field. the force field is a vector function of position (x, y, z) AND velocity (x', y', z'). Rk4= runge kutta forth order method Please help. Thanks!

Edit:whoops wrong forum mods please move 2nd edit: I just had dinner then got back on the computer, input some points and saw a beautiful elipse.(complete with a fascinating flower petal design due to inaccuracies) Weird lol! No idea why it wasnt working before Now to implement RK4 bwahahaha...

Is there a conncetion between Fritz john's ultra-hyperbolic pde, which is the equation: u_{tt}+u_{\tau \tau} = u_{xx}+u_{yy} I mean F theory has another dimension of time, and the above pde has also another time variable with regards to the simple wave pde. Any literature on this...

If you rotate your rectangular coordinate system (x,y) so that the rotated x'-axis is parallel to a vector (a,b), in terms of the (x,y) why is it given by x'=ax+by y'=bx-ay I got x'=ay-bx, y'=by+ax from y=(b/a)x. By the way this is from solving the PDE aux+buy=0 by making one of the...

I wrote the next code: restart; pde := diff(u(x, t), t)+diff(u(x, t), x)-(diff(diff(u(x, t), x), t))^2 = u(x, t); tmax := 0.5e-1; xmin := 0; xmax := 1; N := 10; bc1 := du(xmin, t) = 0; bc2 := u(xmax, t) = 0; ic1 := u(x, 0) = 1; ic2 := du(x, 0) = 2; bcs := {u(x, 0) =...

I am a meteorologist with Bachelor's Degrees in both Meteorology and Mathematics. I took an intro PDE course in college, but want to learn more. Can anyone suggest a book that would be a good book after only having an intro course? Thanks.

If I have a function "F" in a two-dimensional space F(x,t) and its analog F' in another co-ordinate system F'(x',t') and the relation between the two is given by : ∂F/∂t −c(∂F/∂x) =∂F ′/ ∂t ′ How do I find a relation between F and F ′ and between the variables x,t and t ′ ?

Hey there, I have modeled a propagating wave in a 1D dispersive media, in which square and cubic nonlinear terms are present. u′′=au3+bu2+cu the propagating pulse starts to steepen with time which is the effect of nonlinearity, but there is an effect which I can't understand. so...

Homework Statement Assume that the wavelength of acoustic waves in an organ pipe is long relative to the width of the pipe so that the acoustic waves are one-dimensional (they travel only lengthwise in the pipe). Therefore, the equation governing the pressure in the wave is: ∂2p/∂t2-c2*∂2p/∂x2...

I have a PDE of the following form: f_t(t,x,y) = k f + g(x,y) f_x(t,x,y) + h(x,y) f_y(t,x,y) + c f_{yy}(t,x,y) \\ \lim_{t\to s^+} f(t,x,y) = \delta (x-y) Here k and c are real numbers and g, h are (infinitely) smooth real-valued functions. I have been trying to learn how to do this...

I have a third order derivative of a variable, say U, which is a function of both space and time. du/dx * du/dx * du/dt or (d^3(U)/(dt*dx^2)) The Fourier transform of du/dx is simply ik*F(u) where F(u) is the Fourier transform of u. The Fourier transform of d^2(u)/(dx^2) is simply...

Homework Statement Find the general solution of u_{xx} + u = 6y, in terms of arbitrary functions.Homework Equations The PDE has the homogeneous solution, u(x,y)=Acos(x)+Bsin(x) . u_{xx} + u = 6y has the particular solution, u(x,y)=6y The Attempt at a Solution Taking a superposition...

hey pf! i was wondering if you could help me out with a pde, namely $$\alpha ( \frac{z}{r} \frac{\partial f}{\partial r} + \frac{\partial f}{\partial z} ) = \frac{2}{r} \frac{\partial f}{\partial r} + \frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\partial z^2} + 2 \frac{z}{r}...

Homework Statement find the solution to: \frac{\partial^{2}u}{\partial x \partial y} = 0 \frac{\partial^{2}u}{\partial x^{2}} = 0 \frac{\partial^{2}u}{\partial y^{2}} = 0 Homework Equations theorem of integration The Attempt at a Solution now from a previous question I...

I've been getting pretty rusty in terms of derivation in recent years. Encountered this problem which I can't derive the steps despite knowing the solution. \frac{\partial^2 u}{\partial r^2} + \frac{\partial u}{\partial r}\left(\beta + \frac{1}{r}\right)+\frac{\beta}{r}u=0 Known...

Hi, I have a PDE of the form f(x,y,z)'' = Δf(x,y,z) + f(x,y,z) * (1 - f(x,y,z)^2) where f(x,y,z) is a 3 dimensional vector-field. Now I want to compute an energy function for it such that for any state (f(x,y,z) and its first derivative f(x,y,z)') I can compute its corresponding energy...

My equation is: \left(\mathbf{\nabla}\sigma\right)\cdot\left(\mathbf{\nabla}V\right) + \sigma\nabla^2V = 0 If I'm given V(r) on the boundary of some volume, and I know σ(r) inside the volume, is there a unique solution V(r) inside that volume for any arbitrary (well-behaved) function...

Homework Statement Find the general solution f = f(x,y) of class C2 to the partial differential equation \frac{\partial^2 f}{\partial x^2}+4\frac{\partial^2 f}{\partial x \partial y}+\frac{\partial f}{\partial x}=0 by introducing the new variables u = 4x - y, v = y. Homework Equations...

Homework Statement suppose u(x,y) satisfies the partial differential equation: -4y\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}=0 Find the characteristic curves for this equation and name the shape they form The Attempt at a Solution \frac{dy}{dt}=1 \Longrightarrow y=t+y_0...

I have derived the weak form of the transient heat conduction equation (for FEM) and I am having trouble trying to assemble the mass matrix This is the PDE: \frac{\partial U}{\partial t} = \alpha \nabla^2U This is the equation for the mass matrix for an element: M^e = \int \Psi...

Hello! My goal here is to plot the solution to the bioheat equation for a tumor as a function of time. I'm plotting this for a fixed radius at r = 0 (the very center of the tumor). The equation to solve is this: \rho_1c_1 \frac{\partial T}{\partial t} = 3(\frac{\partial }{\partial...

Hello, I am doing some physics and I end up with this PDE: \frac{\partial q(x,y,t)}{\partial t} = -(x^2 + y^2)q(x,y,t) + ax\frac{\partial q(x,y,t)}{\partial y} where q(x,y,t) is the scalar field to determine and a is a parameter. I need to consider two types of initial conditions...

Hello, I am looking to solve the 3D equation in spherical coordinates \nabla \cdot \vec{J} = 0 using the Ohm's law \vec{J} = \sigma \cdot (\vec{E} + \vec{U} \times \vec{B}) where \sigma is a given 3x3 nonsymmetric conductivity matrix and U,B are given vector fields. I desire the...

This isn't a homework question per se. Am merely seeking an explanation how the method of characteristics may be applied to a second order PDE. For instance, how is it used to solve utt=uxx-2ut?

Okay, I'm trying to play around again :D A little overview; I know that the Poisson equation is supposed to be: uxx + uyy = f(x,y) I can manage to discretise the partial derivative terms by Taylor. I don't know how to deal with the f(x,y) though. Say for example, uxx + uyy = -exp(x). what...