Pde Definition and 856 Threads

Hi, I have a test tomorrow and I'd like you to guys help me please. Solve the following: $\begin{align*} & {{u}_{tt}}={{u}_{xx}}+1+x,\text{ }0<x<1,\text{ }t>0. \\ & u(x,0)=\frac{1}{6}{{x}^{3}}-\frac{1}{2}{{x}^{2}}+\frac{1}{3},\text{ }{{u}_{t}}(x,0)=0,\text{ }0<x<1. \\ &...

I haven't worked with partial derivatives since high school 25 years ago so I'm quite a bit rusty and need a little guidance. I'm reading through a paper and would like to write a program to simulate it. The equations are: Eq 1. Eq 13. M=f*Po+f*(Pb-Po)*(1-ln(e)^(-0.693*time/Halftime) Eq 10...

Could someone give me some hint (or some reference) about the study of regularity of weak solutions of (*)\quad \begin{cases} -\Delta u = e^u & \text{in }\Omega\\ u = 0 & \text{on }\partial \Omega \end{cases} where \Omega \subset \mathbb{R}^2 is a bounded domain with smooth boundary (here...

Ok I have read/browse 3 books so far about PDEs. They separate variables. Then they suddenly go. solution is X= some thing sin some thing cos Y= some thing -ek some thing ek Where do sine and cos comes from ? How do they know?? where is it explained?

Hi, Can somebody help me solve the following PDE? ∂p(x,t)/∂t = -p(x,t) + ∫λ(x-x')p(x',t)dx' with p(x,0)=δ(x) Thanks a lot

Hi, I am a master student comes from USM in Malaysia. I don't know my problem should placed on differential forum or high energy physics forum. Anywhere, My current study is high energy physics subject and my main study is focus on monopole instanton solution in static form which did not...

Hello! I would like to find some functions F(x,y) which satisfy the following equation \frac{F(x,y)}{\partial x}=\frac{F(y,x)}{\partial y} For example this is obviously satisfied for the function F= exp(x+y) I would like however to find the most general closed form solution...

Hi, I'm currently working on a thesis in Economics. I have stumbled upon a system of differential equations that needs to be solved. I am stuck, and have trouble getting the right help from my advisor who is also not very acquainted with numerical methods. For the past couple of days I have...

Hello, I want to solve a 4-dimensional PDE problem using some numerical code. Possibly MATLAB or Python. I have a solved a simple version of the PDE in 2D using MATLAB PDETool. Also I solved a simplified pde in 3D using FiPy library in Python. However, most MATLAB existing tools...

pure and mixed, what do they tell you about a function?

Hi, I have the following PDE-S\frac{\partial\vartheta}{\partial\tau}+\frac{1}{2}\sigma^2\frac{X^2}{S}\frac{\partial^2\vartheta}{\partial\xi^{2}} + [\frac{S}{T} + (r-D)X]\frac{\partial\vartheta}{\partial\xi}I am asked to seek a solution of the form \vartheta=\alpha_1(\tau)\xi + \alpha_0(\tau)...

Hi everyone, I am doing a sheet on Asian Options and The Black Scholes equation. I have the PDE, \frac{∂v}{∂τ}=\frac{1}{2}σ^{2}\frac{X^{2}}{S^{2}}\frac{∂^{2}v}{∂ε^{2}} + (\frac{1}{T} + (r-D)X)\frac{∂v}{∂ε} I have to seek a solultion of the form v=α_{1}(τ)ε + α_{0}(τ) and determine...

Hi group, In order to understand the methods of characteristics, I've been reading its wiki entry plus other sources. However, one of the first step of finding the normal surface vector given the PDE remains baffling to me in terms of how it's derived. In short, when provided with a(x...

Hello, I have the PDE \frac{-∂v}{∂τ}+\frac{1}{2}σ^{2}ε^{2}\frac{∂^{2}v}{∂ε^{2}}+(\frac{1}{T}+(r-D)ε)\frac{∂v}{∂ε}=0 and firstly I need to seek a solution of the form v=α_{1}(τ)ε + α_{0}(τ) and then determine the general solution for α_{1}(τ) and α_{0}(τ). I am given that ε=\frac{I}{TS}...

I have a PDE in two variables, u and v, which takes the form \frac{\partial\psi}{\partial u\hspace{1pt}\partial v} + \frac{1}{r}\left(\frac{\partial r}{\partial u} \frac{\partial \psi}{\partial v} + \frac{\partial r}{\partial v}\frac{\partial\psi}{\partial u}\right) for an auxiliary...

Hi, I'm an undergrad in EE who wants to learn the basics of solving PDEs (and Fourier series/transforms), but who has some learning disabilities (developmental, most notably). Before I get criticism for what I'm about to say (which will be asking for an alternative to the obligatory "read...

Hi, I know from basic math courses that inverse trig functions are multi valued (e.g. arctan(c)=θ+n*2∏). Now, if I solve a partial differential equation and I get an inverse trig function as part of my solution, does that mean solutions to the pde are non-unique? For example, if...

Homework Statement Let λ_n denote the nth eigenvalue for the problem: -Δu = λu in A, u=0 on ∂A (*) which is obtained by minimizing the Rayleigh quotient over all non-zero functions that vanish on ∂A and are orthogonal to the first n-1 eigenfunctions. (i) Show that (*) has no...

Homework Statement The Cauchy problem for the advection-diffusion equation is given by: u.sub.t + c u.sub.x = K u.sub.xx (−∞< x < ∞) u(x, 0) = Phi(x) where c and K are positive constants. The advection-diffusion equation essentially combines the effects of the transport...

Solve $u_x+2u_y+2u=0,$ $x,y\in\mathbb R$ where $u(x,y)=F(x,y)$ in the curve $y=x.$ I don't know what does mean with the $y=x.$ Well I set up the following $\dfrac{dx}{1}=\dfrac{dy}{2}=\dfrac{du}{-2} ,$ is that correct? but I don't know what's next. Thanks for the help!

Hi, need some help here so thanks to any replies. PDE: $$yu_x+2xyu_y=y^2$$ edit: Forgot to mention the condition $$u(0,y)=y^2$$ a) characteristic equations: $$dx/ds=y$$ $$dy/ds=2xy$$ $$du/ds=y^2$$ b) find dy/dx and solve $$dy/dx=dy/ds * ds/dx = x/y$$ $$ydy=xdx$$ $$y^2/2=x^2/2 +c$$ $$y=\pm...

Given $\begin{aligned} & {{u}_{t}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\ & u(x,0)={{u}_{0}}, \\ & {{u}_{x}}(0,t)=u(0,t). \end{aligned} $ I need to apply the Laplace transform to solve it. I'll denote $u(x,s)=\mathcal L(u(x,\cdot))(s),$ so for the first line I have $s\cdot...

Dear MHB members, I have the following equation $xy(z_{xx}-z_{yy})+(x^{2}-y^{2})z_{xy}=yz_{x}-xz_{y}-2(x^{2}-y^{2})$. When I transform this into the canonical form via $\xi=2xy$ and $\eta=x^{2}-y^{2}$, I obtain...

Hi everyone! I want to design a robust controller for a system which is driven by a PDE. I need to acquire its transfer function in 's' parameter which means it should be transferred by Laplace transformation. I know that the result transfer function will be an infinite series of transfer...

Homework Statement I'm just trying to get an understanding of answering PDEs, so wanted to ask what you thought of my answer to this question. The one-dimensional wave equation is given by the first equation shown in this link; http://mathworld.wolfram.com/WaveEquation1-Dimensional.html...

Homework Statement The one-dimensional heat diffusion equation is given by : ∂t(x,t)/∂t = α[∂^2T(x,t) / ∂x^2] where α is positive. Is the following a possible solution? Assume that the constants a and b can take any positive value. T(x,t) = exp(at)cos(bx) Homework Equations...

Let $u\in\mathcal C^1(\overline R)\cap \mathcal C^2(R)$ where $R=(0,1)\times(0,\infty).$ Suppose that $u(x,t)$ verifies the following wave equation $u_{tt}=K^2 u_{xx}+h(x,t,u)$ where $K>0$ and $h$ is a constant function. a) Determine the total energy of the string. (Well I don't know what does...

Solve $\begin{aligned} & {{u}_{tt}}={{u}_{xx}}+1+x,\text{ }0<x<1,\text{ }t>0 \\ & u(x,0)=\frac{1}{6}{{x}^{3}}-\frac{1}{2}{{x}^{2}}+\frac{1}{3},\text{ }{{u}_{t}}(x,0)=0,\text{ }0<x<1, \\ & {{u}_{x}}(0,t)=0=u(1,t),\text{ }t>0. \end{aligned} $ Here's something new for me, the boundary...

Solve $\begin{aligned} & {{u}_{tt}}={{c}^{2}}{{u}_{xx}}+A{{e}^{-x}},\text{ }0<x<L,\text{ }t>0, \\ & u(0,t)=B,\text{ }u(L,t)=M,\text{ }t>0, \\ & u(x,0)=0={{u}_{t}}(x,0),\text{ 0}<x<L. \end{aligned} $ What do I need to do first? Homogenize the first boundary conditions? Or first making the...

Consider the equation $\begin{aligned} & {{u}_{t}}=K{{u}_{xx}}+g(t),\text{ }0<x<L,\text{ }t>0, \\ & {{u}_{x}}(0,t)={{u}_{x}}(L,t)=0,\text{ }t>0 \\ & u(x,0)=f(x), \\ \end{aligned} $ a) Show that $v=u-G(t)$ satisfies the initial value boundary problem where $G(t)$ is the primitive of...

I'm confused on what classes to take next semester. I've talked to my adviser but they're kinda useless as they don't want me to take upper level courses (past calc 3 and ODE). However, I want a dual math and physics degree which would be helpful in gradschool. Right now I have the following...

Nowadays people usually consider PDEs in weak formulations only, so I have a hard time finding statements about the existence of classical solutions of the Poisson equation with mixed Dirichlet-Neumann boundary conditions. Maybe someone here can help me and point to a book or article where I...

Homework Statement Find the Riemann function for uxy + xyux = 0, in x + y > 0 u = x, uy = 0, on x+y = 0 Homework Equations The Attempt at a Solution I think the Riemann function, R(x,y;s,n), must satisfy: 0 = Rxy - (xyR)x Rx = 0 on y =n Ry = xyR on x = s R = 1 at (x,y) = (s,n) But I...

I have a battle with the following direct partial integration and separation of variables toffee: I have to solve, u(x,y)=\sum_{n=1}^{∞}A_n sin\lambda x sinh \lambda (b-y) If there were no boundary or initial conditions given, do I assume that λ is \frac{n\pi}{L} and do I then solve A_n...

Homework Statement Classify the equation and use the change of variables to change the equation to the form with no mixed second order derivative. u_{xx}+6u_{xy}+5u{yy}-4u{x}+2u=0 Homework Equations I know that it's of the hyperbollic form by equation a_{12}^2 - a_{11}*a_{22}, which...

Hello everyone, I am trying to model the process of laser ablation on a material using MATLAB. The governing equation is of the form: ∂T(x,t)/∂t = ∂/∂x(A*∂T/∂x) + B*exp(-C*t2)*exp(-D*x) with one Initial condition and two boundary conditions. Using the built-in 'pdepe' function in Matlab...

Homework Statement A square rectangular pipe (sides of length a) runs parallel to the z-axis (from -\infty\rightarrow\infty). The 4 sides are maintained with boundary conditions (i) V=0 at y=0 (bottom) (ii) V=0 at y=a (top) (iii) V=constant at x=a (right side) (iv) \frac{\partial...

Homework Statement Mathews and Walker problem 8-2 (page 253): Assume that the neutron density n inside U_{235} obeys the differential equation \nabla ^2 n+\lambda n =\frac{1}{\kappa } \frac{\partial n }{\partial t} (n=0 on surface). a)Find the critical radius R_0 such that the neutron density...

1) Solve $\begin{aligned} {{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\ {{u}_{x}}(0,t)&=0,\text{ }{{u}_{x}}(L,t)=0,\text{ for }t>0, \\ u(x,0)&=6+\sin \frac{3\pi x}{L} \end{aligned}$ 2) Transform the problem so that the boundary conditions get homogeneous: $\begin{aligned}...

I am trying to solve the following equation in spherical coordinates: \left( \nabla f \right) \cdot \vec{B} = g where g is a known scalar function, \vec{B} is a known vector field, and f is the unknown function. I think the best way to approach this is to expand everything into a...

Homework Statement Is the operator Lu = du/dx + u * du/dy linear? Homework Equations Linearity occurs for L[u+cv] = L[u] + cL[v] The Attempt at a Solution I know this isn't linear because of the second term, but I don't understand why I can't write the operator as L =...

Im new on the forum, so I hope you guys will have some patience with me :-) I have a question about the chain rule and partial differential equations that I can't solve, it's: Write the appropriate version of the chain rule for the derivative: ∂z/∂u if z=g(x,y), where y=f(x) and...

Here I have my PDE: http://desmond.imageshack.us/Himg718/scaled.php?server=718&filename=pde.png&res=medium I have found the solution by using the method of characteristics two times, one for x<0 and the other for x>0. I have: U(x,y) = o for x<0 and U(x,y) = Uo(x-1)/(1+Uo*y) for x>0...

Solve $u_y+f(u)u_x=0,$ $x\in\mathbb R,$ $y>0,$ $u(x,0)=\phi(x).$ What's the easy way to solve this? Fourier Transform? Laplace Transform?

Consider $u_t+u_x=g(x),\,x\in\mathbb R,\,t>0$ and $u(x,0)=f(x).$ Given $f,g\in C^1,$ then show that $u(x,t)$ has the form $u(x,t)=f(x-t)+\sqrt{2\pi}(g*h)(x)$ where $h(x)=\chi_{[0,t]}(x).$ So we just apply the Fourier transform to get $\dfrac{{\partial U}}{{\partial t}} + iwU = U(g)$ and...

I am looking for ideas on how to solve this equation: \nabla \cdot \left( \vec{A} + F \hat{b} \right) = 0 where \vec{A} and \hat{b} are known vectors of (r,\theta,\phi) and F is the unknown scalar function to be determined. Also, \nabla \cdot \hat{b} = 0. So the equation can also be expressed...

Homework Statement Solve u_x^2+u_y^2=1 subject to u(x, ax)=1 Homework Equations The Attempt at a Solution I let u_x=p andu_y=q and F=p^2 +q^2 -1=0 Then x'=2p, y'=2q, u'=p.2p+q.2q=2, and p'=0=q'. So p=p_0, q=q_0 are constants. I got x'=2p_0, y'=2q_0 and integrating the...

Hey, before you read this over I'll mention that I've checked the general solution and it works. So if you don't feel like following my steps to get the general solution just jump down to the end of my attempt, because the real problem for me is figuring out what to do with the side conditions...

For quick reference if you have the text, the question is from "Applied Partial Differential Equations" by J. David Logan. Section 1.9 #4 Show that the equation $$u_{tt} - c^2 u_{xx} + au_t + bu_x + du = f(x,t)$$ can be transformed into an equation of the form $$w_{\xi\tau} + kw = g(\xi,\tau)...

Homework Statement The problem I am having has to do with part (d) in the picture which I have attached. I have managed to get as far as to determine that the coefficients in the series expansion have the recurrence relation shown below in part (2). From this I think that I have been able to...